1 PART 3: REGULATED DERIVATIVES - PowerPoint Presentation

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PART 3: REGULATED DERIVATIVESFUTURES & OPTIONS

REGULATED

DERIVATIVES:

FUTURES

AND OPTIONS CONTRACTS

ON STOCKS, STOCK INDEXES, COMMODITIES, CURRENCIES

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Introduction to regulated derivatives

Derivatives are quite complicated instruments

. They are called that way because the behaviour of a derivative contract derives more or less closely (depending from case to case) from that of another asset, named

underlying

.

Trading in derivatives attracts many investors thanks to the

high intrinsic leverage

of these instruments, allowing to reach high gains with small money amounts. And since the leverage works in the same way in both directions, amplifying both profits and losses, these instruments demand for a deep knowledge, both in terms of risk/reward profiles and the ways to manage open positions.

Difficulties in the field of derivatives arise from seven sources (which we’ll see in details):

they have a

maturity

, at which they cease to exist and are replaced by new contracts;

at maturity they can involve

cash settlements or physic settlements

;

the purpose of trades held till maturity (

hedging

of risks of some sort) is totally different from that of short term trades (just

speculation

);

their intrinsic leverage demands for a

strong risk management

, based on a daily regulation of profits and losses, according to rules we’ll see;

every open contract is made of

two active counterparties

, having opposed expectations;

pricing

of derivatives changes according to the nature of the underlying;

prices often follow dynamics not completely explained by market variables

(especially in the field of options).

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Buyers Vs Writers

For any derivative contract active in the market there is always someone long (buyer) and someone short (writer).

The buyer of a futures contract

, for instance, expects that the price will increase, the writer expects it will decrease. Things are different in the field of options, as we’ll see, and yet for each open contract there are still two counterparties active at the same time, with different (not exactly opposed) expectations.

Derivatives always need two active counterparties to exist

.

They can be traded exactly like stocks, on the long, mid, short or very short term. They can be purchased and sold also hundreds times a day, and transfered among traders just with electronic transactions. Short selling of derivatives does not require borrowing or lending. There are no special commissions and costs for short selling.

Profits and losses of buyers and writers are symmetrical:

the derivatives game is a zero-sum one

.

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Transfers of contracts among traders

Derivatives can be trasnfered from a trader to another at any time, symply clicking on the mouse. A buyer and a writer who created a contract in the first place are not bound one to the other: in the moment one of them wants to get out of the contract he just need to transfer his side of the contract to someone else willing to take the same side. Buyers and writers are not bound each other, then.

Let’s consider the case of investors A and B: A thinks the price of an asset is going to increase, so he buys a derivative contract written on that asset. B has opposed feelings, hence he decides to be the writer. Let’s use the symbol + for long positions and – for short positions:

Imagine that after some time B changes his mind and wants to get out: he needs to buy to cover, so he needs someone else willing to sell. Be C this person:

A

+1

B

-1

A

+1

B

-1

+1

=0

C

-1

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Transfers of contracts among traders

The short side of the contract has been transfered from B to C.

Now imagine that A wants to get out: he has to sell, so he needs someone willing to buy and takeover the long side of the contract. Be D this guy:

What happens, then, after a while is this:

The existing contract is still the same created in the beginning by A and B: it just changed hands. Should, in the end, C and D decide to get out at the same time, then they could meet in the market and close their opposed positions, becoming flat both at the same time. In that situation the contract would disappear.

But if E and F showed up, buying the first, writing the second, a new contract would be created out of the blue.

A

+1

-1

=0

C

-1

D

+1

C

-1

D

+1

A

flat

B

flat

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Futures

DEFINITION

a futures contract is an agreement between two parties to buy or sell an asset (that can be financial or physic) at some date in the future, at some specific price. It’s a derivative contract, that is, its behaviour is linked to the one of another asset, named underlying.

This is the classic definition of futures contract, but to be stringent

it applies only in a specific context

: when the two parties keep the contract opened till its maturity, the good is physic, and it can be delivered. This behaviour, indeed, is the answer to a specific need: the need for the underlying at some date in the future at some price decided in advance;

this is a typical need for companies

willing to hedge a risk of some sort (a concept we will see in full details forward).

To the contrary, in most of the cirumstances a futures contract is just a way to speculate on the trend of an asset, on a short or very short term, with a huge leverage.

Apart from some exceptions which we’ll see at the right moment, the futures price

moves generally together

with the spot price (the price of the underlying) but most of the time it is different: it can be higher, equal or lower, depending on the contingent situation and the peculiar features of the underlying asset (read forward for details).

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Futures

Futures are

standardized

contracts, quoted on

regulated

markets.

This means that for any contract everything is set in advance: the

asset to

be traded, the quantity, the quality (in case oh physic goods), delivery dates,

minimum

price change (tick), maximum daily price ranges, and so on.A parte of the regulation is dedicated to the management of the high intrinsic leverage of these assets, as we’ll see in details.

Futures are given of a maturity, at which they are closed, regulated and retired from negotiation.

If the underlying is physic and can be delivered (like cereals, metals, stocks) there happens to be a physic delivery of the underlying; in many cases though there is just a cash settlement.

The physic trading of goods can happen only between institutions and individuals (named commercials) registered in a specific registry. Private investors are not allowed to hold futures contracts till the maturity if there is a physic settlement: in case of need their positions are closed by force by their intermediaries.

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An example: a stock index futures

Futures contract written on the FTSE MIB index:

underlying:

1 FTSE

Mib index

quotes: in index points

value of a point: 5€

minimum price variation (tick): 5 points (25€)

contract size: price of the futures times the value of a point (for example, if the index quotes 20000 points, its value is 100000€)

maturities: every quarter of

the first year, then every six months for two more years, then every year for up to five years

expiration day: third Friday of the expiration month settlement price at maturity: opening price of the expiration day

settlement: cash initial margin (see slide 10): usually between 10 and 15%, self-adaptive to market volatility

The variation field of the initial margin, which, as we’ll see in next slide, is a measure of risk, is the answer to the need for a risk management policy istantaneously adaptable to changes in the risk of the underlying:

the higher the risk, the higher the initial margin, automatically.What does all of this mean?

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An example: a stock futures

Futures contract written on the

stock Eni:

underlying (lot): 500 Eni shares

quotes: in

euros and eurocents

minimum price variation (tick) : 0.0001 euros

contract size: futures price times lot; for instance, being 15€ per share the futures price, the contract size is 7500 euros

maturities: two closest months, four closest quarters, then every six months for several years settlement day: third Friday of expiration month

settlement price at maturity: opening price of expiration day settlement type at maturity: cash or physic (both isted)

initial margin: still related to the risk of the underlying, generally much higher than that of the index

The main difference between stock index futures and stock futures is the lot, equal to the number of units of underlying controlled with the contract.

The lot changes from stock to stock. The full list is available on the internet site of Borsa Italiana, in the derivatives section  contract specifications.

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Initial margin and leverage

The

initial margin

is the money needed to open a position on the FTSE Mib

futures, and as we’ll see

it is not an expense, but just a cash deposit

: a money amount deposited as a trade is opened, for the coverage of its risk. In the moment the trade is closed the initial margin

is restituted

.

If the margin requested by the market is 10%, this

means that to open a position valued 100k€ we just need 10k€.The point is that we use 10k but we are exposed for 100k

, that is profits and losses are referred to 100k, not 10k. It takes, then, just a +10% do double the money, or a -10% to lose all the money.

This is the main peculiar feature of futures, then: they have a huge intrinsic leverage. In this case the need for capital is just a tenth of the value of the position on the market (there are futures with leverages higher than 100…). The leverage is what makes futures very attractive for speculators.

The initial margin is a safe deposit asked for the coverage of a risk. It is then integrated and managed according to some clear rules having the purpose to protect the market from insolvency risk.In order to fully understand this issue we have to introduce the role of the clearing house. But we need to understand the nature of the problem first.

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Risk/reward profile (payoff)

The payoff is a chart that shows the result – in terms of money – of an investment, depending on the price of the asset at some date. In the field of derivatives, the payoff is used to visualize the profit or loss of a trade at the maturity of contracts for several prices of the underlying. It’s a way to see what happens in case of some possible outcomes. On the horizontal axis we place the prices of the underlying on arbitrary intervals, on the vertical axis we place the economic result of the position.

Let’s first consider the case of a stock: suppose

To buy 1000 shares of the stock ABC at 18€

each. The payoff of this position is a straight line,

with a 45° slope, growing from bottom left to top

right: profits and losses are straight proportional

to the movement (up or down) of the stock:

In case of futures things are slightly different, since

the futures’ price is not equal to the underlying’s

price (spot price), for reasons that we’ll see.

18€

19€

+1000€

17€

-1000€

ABC price

Payoff

Profit or loss

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Potential profits and losses

In passing from stocks to futures, to understand the payoff we have to first understand a key difference in terms of potential profits and losses and real profits and losses. Here we need to focus on a paramount concept: on the stock market,

till

a trade

is not

closed, profits and losses are only nominal

. This happens because the profit or loss is the difference between what we paid on purchase and what we get back selling.

If we buy a stock at 10€ and the price rises to 11€, if we do not sell those shares the profit is not real; at the same time, if the price drops to 9€, the loss is not real, if we do not close the trade.

This means that trades that generate nominal profits or losses do not have any impact on the capital of the investor while they are opened:

only when they are closed they generate a real profit or loss

.

Moreover, in stock trading there is no insolvency risk: the value of a position is totally paid in the moment of purchasing, so the only individual exposed to a risk is who holds the stock in his hands. His

counter party – who sold the stock to him – already got paid of the whole amount, so he doesn’t need to care about that stock anymore. In other words, if a stock holder loses money it is just his business: no one else is at risk in that case.

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Buyers Vs writers

From the point of view of the holder of the stock, the maximum risk is given by the loss of the entire capital invested, due to the bankruptcy of the company.

Under normal conditions, the entire value of the stock is paid on purchase with money owned by the buyer. If the trader uses a leverage he has a debt with the bank, and that bank asked for collaterals as an insurance for the money lending. And in any case the entire value of the stock is paid, so there is not a counter party somewhere out there at risk to lose money because someone is not going to pay him.

In futures trading, as we already know, things are different:

who loses money in a trade loses that money in favour of a counterparty set on the other side of the contract, who gains the same money amount

.

Since the initial margin covers just a fraction of the value of the contract,

the risk here is that the loss could be higher than the money deposited

: what if I decided not to pay my dues in case of losses higher than the initial margin?

Here insolvency risk could arise

: one of the two parties in the contract could not be able (or willing) to pay his debts.

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Insolvency risk

A

strong source of risk

in derivatives trading is

given by the leverage

, that puts the market in front of the insolvency risk, originated by the possibility for losses to be higher than the

money

deposited in the first place.

The problem is that if, for instance, the buyer faces a loss higher than the money deposited and has no money to pay for those losses, the writer could not be paid the money he gains. This issue involves the need of some sort of risk management policies.

There is a paramount difference between buying a stock and buying a futures written on that stock: to buy 1000 shares of ABC at 18€ each we have to pay 18000€ cash immediately, and this provides a full coverage of the maximum potential risk of the trade (the stock goes bankrupt and its price falls to zero).Now let’s consider a futures written on ABC, assuming it’s lot is 1000 shares and the initial margin is 10%. The investment is valued 18000€, but to open it we just need 1800€. This means that the risk is covered only up to a 10%. This generates a risk for both parties: the buyer faces the risk to be not paid if the price rises more than a 10%; the writer faces the risk to be not paid if the price drops more than a10%.

Generally speaking: both the buyer and the writer are exposed to insolvency risk.

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Marking to market

The management of insolvency risks requires the use of a specific methodology, called

marking to market

.

Marking to market is applied to all regulated derivatives and is

based on a daily regulation of profits and losses

: as the market closes, every day, nominal profits are immediately credited, nominal losses have to be immediately paid

. To be stringent,

intermediaries today apply this mechanism in real time

, but to make things easier let’s take it as an end of day procedure

Daily profits and losses are managed by the clearing house, that acts as counter-party for any open trade: the clearing house pays who has to be paid and asks money from the banks whose clients are losing. Banks then take money out from the accounts of clients losing money.This way

the clearing house always ensures the good ending of any transaction, always paying who has to be paid.

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How do margins work

The initial margin serves as intial safe deposit and

has to cover a risk calculated on a statistical basis

: it expresses the maximum expected loss from a day to another on the underlying asset. This calculation accounts for the impact that a significant increase or decrease in the value of the underlying would have on the trader’s account.

To explain it in a few words, today the systems account for the worst of the worst case scenarios on a 1, or 2 or 3 days basis, according to the Normal model and the interval included in the mean of past returns plus and minus three standard deviations.

Since both counter-parties face insolvency risk

they are both required to deposit the initial margin

, calculated as a percentage of the value of the contract deriving from some kind of “stress test” on the underlying, as decided by the clearing house.

Doing this, risk is secured till the next day only. Then, depending on how the trend of the underlying evolves and so to always keep under strict control risk, positions are monitored every day. At the end of any day the closing price is compared to that of the previous day, and the

initial margin is adjusted

to a new value (see also next slide).

Then profits are immediately credited, losses debited. These money transfers are named

variation margins (see an example in slide 19).

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Initial margin adjustments

The initial margin is set as a fraction of the contract value; this implies that it changes according to changes in the value of the futures. Every time the closing price of a day is different from that of the previous, then, the amount of initial margin changes, hence the value already deposited

has to be adjusted

.

According to what written in slide 8, initial margins can change both because of a change in the value of the futures and in the percentage automatically set by the IT systems according to the volatility of the market. The two effects could compensate or amplify each other.

In other words, initial margin can change from day to day both due to the trend in the underlying and the risk of the underlying, measured by its volatility: the systems automatically adapt the percentage of the initial margin as a consequence of the latter and the monetary value of that percentage according to the former.

Every night, then, initial margin is adjusted according to the evoultion of the market

: if it reduces, some money amount is released on the account, and viceversa.

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Summing all up

To conclude, initial margin is just a safe deposit, adjusted every day depending on the evolution in price and risk.

When the position is closed all the initial margin (including any adjustment)

is entirely restituted

.

Variation margins are the real profit or loss of the trade

: they are credits or debits collected or paid on a daily basis

so to leave untouched the initial margin, so that it could always keep protecting the two counter-parties from insolvency risks

.

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Example 1 – A long trade

.

Initial margin

adjustm.

Note: for the sake of this example it has been considered a 10.25% fixed initial margin, day after day

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Example 2 – a short trade

.

Initial margin

adjustm.

Note: for the sake of this example it has been considered a 10.25% fixed initial margin, day after day

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In some specific situations it can happen that positions in futures evolve very unfavorably, so that every day huge margin integrations are

required. Think

back, for

instance,

to September 11, 2001: in that day (and in the following ones as well) investors who had long positions on stock futures and stock index futures had to face sizable variation margins.

The deposit of margins is required by the market regulation, hence

it is mandatory

.

If, on a specific day, as the markets close, an investor has not enough money on his account to face variation

margins

the intermediary has to warn him to close his positions the next day on open; if he doesn’t, the intermediary has the power (by law) to close positions by force. Some intermediaries, indeed, do not wait for the market close to intervene: being trading continuously monitored, at 5.15 p.m. they check positions and if things are getting bad they warn clients to close positions before 5.30, otherwise the bank will do it in their behalf.

It has been proven that the meticulous application of marking to maket prevents insolvency risk in almost a 100% of cases, and in any case uncovered losses can be very reduced.

Unfortunately, not all the intermediaries apply correctly these “well oiled” risk management policies.

Sizable losses

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To understand what it means to trade on futures we just need to do some basic calculations on market data.

Let’s look at the italian market. The average daily variation field (distance between minimum and maximum) for the price of the futures FTSE Mib over the 2469 days between June the 1st 2007 and March the 31st 2017 is 430 points.

If we had a trading system able to catch every day average profits equal even just to a 25% of the daily range, we could make 107.5 points of gain every day, that is 537.5 euros, trading only one contract every time.

With which capital? Hard to tell exactly, but we can make a precautionary estimate: being always 15% the initial margin, the average daily margin over the period of time above mentioned would have been about 16100 euros.

Using the mini-sized futures, valued 1 euro per point, the initial margin would be a fifth of that amount, that is about 3220 euros. Profits would be 107.5 euros per day, on average.

Either way, the average daily return would be 3.33%...

Average daily potential profits on futures

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Previously it has been stated that the futures price generally differs from the price of the underlying: it can be equal, higher or lower; generally speaking, the gap between the spot and the futures price depends on the physic characteristics of the underlying

good and

the equilibrium between supply and

demand.

The futures price can be derived with some math reasoning

under a clear purpose: to eliminate any arbitrage opportunity

.

An arbitrage is a trade that restitues a sure profit without any risk; such situations usually happen when the same product is quoted on two different markets with two different prices: if an investor can have access to both markets, and on those markets there are no limits to short selling, then he can buy the good where it’s cheaper and short it where it is more expensive.

Sooner or later the two prices will align; in the moment the two prices go back in line, the arbitrager can close both the positions, gaining for sure without any risk

.

A preliminary note: the futures price is generally different from the spot price during the lifetime of the contract,

but it is exactly equal to the spot price in the moment of expiration

(why, will be clear going on). Keep this in mind in order to understand everything that follows.

Futures pricing: introduction

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Suppose to have growing expectations on a listed stock, on which are written futures; given a specific capital amount, if we buy the stock we need some money amount, and if we buy the futures we can use a capital amount surely lower, thanks to the leverage. This makes the futures advantageous in any

case:

we can use it to save liquidity to be used elsewhere

.

Being, just to say, 10% the initial margin on the futures (and assuming there are no variation margins, to make it easier), the remainder 90% can be invested in risk free assets (a government bond, for example), rising the global result on the whole capital whatever happens.

Be, in fact, one year the timeline of the investment, 5% the interest rate on

1-year maturity government bonds,

10% the initial margin requested to buy a futures written on the stock we want to buy, 10€

the

price of that stock, and 1000 shares the lot of the futures written on

it.

Finally, be 11€ the price of the stock at the end of the year.

Buying 1000 shares of the stock means to invest 10000 euros, and at the end of the year it restitutes 11000 euros, that is, a 10% profit.Futures pricing: stock futures

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Assuming the futures price is equal to the spot price (e.g. the price of the underlying), buying a futures means to deposit 1000 euros in terms of initial margin, and to get a 100% profit at the end of the year on that initial margin; this because in futures trading profits and losses are on the real value of the position, not on the

money used.

And since to get that result we used only a 10% of the available money we can use the remainder 90% to buy a

1-year maturity government bond,

that will restitute a 5% gain at the end of the

period,

with no risk

.

Summing up, on the 10% of the capital we gain a 100%, and on the remainder 90% we gain a 5%. The global return is the

weighted average

of the two returns:total return = 100% (return) * 10% (ratio of capital) + 5% (return) * 90% (ratio) = 14.5%

It is clear, then, that thanks to the leverage it is always possible to enhance profits, no matter the direction of the

price of the underlying (see next slide).Futures pricing: stock futures

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Suppose, in fact, the stok loses a 10% in the current year: in this case the profit on the futures is a 100% loss, but the gain on the BOT is still achieved. The global profit is again the weighted average of the two results:

total profit = -100% * 0.1 + 5% * 0.9 = -5.5%

It should be clear then that the leverage allows investors to gain more (or to lose less) no matter the direction of the price:

it amplifies

profits

if things go well, it reduces

losses

if things go bad

.

The consequence is that

stock futures need to have a cost of some sort, linked to the interest rate and the residual life of the contract: that cost has to erase the interest rate achievable on the money we do not use. In other words, the price of a stock futures (for different underlying assets the matter gets a bit more complicated) has to be equal to the spot price only in the moment the futures expires

; in any other moment it has to be higher than the spot price.This applies to stock futures (just in part for stock index futures as well). In case of other underlying assets, as we’ll see, other variables come into play.

Futures pricing: stock futures

Slide27

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Stock futures price: basic hypothesis

The futures pricing procedure requires the assumption of some specific initial conditions:

there are no

limitations

to short selling and to the use of the money collected from it

money can be lended and borrowed at the same interest rate

it is not possible to make arbitrages

there are no commissions on trades

Under those conditions (whether they are reasonable or not), the futures price can be achieved after the analysis of two limit

situations: that the futures price was way under the spot price or way over it.

We shouldn’t even take into consideration the first, according to what seen in previous slides, but it’s time to give mathematical proof that in that case arbitrages would come into play.

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First scenario: futures < spot

Suppose a stock quotes 10€ and the price of the futures with a one year maturity written on it is lower than the spot price (we know it couldn’t but we still have to prove it); let’s say 9€.

In that case it is possible to make an arbitrage: we can

short sell

the underlying, collecting 10€ a share; a small part of that amount can be used to buy the futures at 9€, and the remainder liquidity can be lended at the risk free interest rate (buying a BOT, for instance).

The final result is a profit on money we don’t even have!

Suppose, indeed, for instance, that at maturity the stock price is 7€; in that case the short position on the stock restitutes a profit of 3€ a share: sold at 10€, bought to cover at 7€, 3€ are in our pocket; the long position on the futures restitutes a loss of 2€; bought at 9€, sold at maturity at 7€, 2€ are gone: summing up the two positions we have a net profit of 1€.

If at maturity the stock price is 13€, just to say, the short position on the stock restitutes a loss of 3€, but the long position on the futures restitutes a profit of 4€; a net gain of 1€ again, then.

In both scenarios, to the 1€ net profit we can add the profit coming out of the money lending.

No matter the direction of the market, then, we gain. And moreover we gain on someone else’s money!

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Second scenario: futures >> spot

Now suppose the stock quotes 10€ and the price of the futures expiring in one year is much higher than the spot price, let’s say 11€.

An arbitrage can be done in this situation too: we buy the underlying at 10€, sell the futures at 11€ and get a 1€ profit for sure after one year.

At the end of the year, indeed, the futures expires and

its price

coincides with the underlying, and it is easy to prove that for whatever price of the underlying we have a 1€ profit, that is, a 10% profit (in this case there is no initial margin, since the position is risk free, hence the capital invested is 10€ per share) without any risk, since losses on one asset are always compensated by the profits on the other, in both directions.

Be, for example, 8€ the price of the underlying at the maturity of the futures: we lose 2€ on the underlying, we gain 3€ on the futures, so the net result is +1€; if the price at maturity is 16, just to say, we gain 6€ on the underlying, we lose 5€ on the futures, so the net result is still +1€.

Conceptually speaking, then, the futures price has

to

be higher than the spot price, so to erase

the advantage of the leverage (in terms of extra gains, like in slides 25-26), but not too much, otherwise it generates arbitrages’ opportunities. The solution is quite simple: the futures price has to capitalize the advantage of the leverage, in order to erase it: you pay the advantage in the futures price.

Slide30

30

The fair valure of a stock futures

It is quite easy to prove, then, that the correct futures price is equal to the capitalization of the spot price at the risk free intrerest rate on a number of days equal to the residual life of the contract; in maths:

Fut = S * exp(r * T)

where

S

is the spot price,

exp

is the exponential function,

r

is the yearly gross risk free interest rate,

T is the residual life on an annual basis, that is in days to maturity (including the present day) divided by 365Note that in the formula there is not any component related to the actions of supply and demand: it is just a mathematical formula, related to the risk free interest rate and the maturity of the contract.There are no expectations in the futures price: it is not the expected price at maturity.The futures price is just the price of the advantage of not holding the underlying today.A question arises, then: is this formula valid for any kind of futures?

Slide31

31

Validity of the formula

The formula on previous slide is the more accurate one for the calculation of

stock

futures’ prices

If, on one hand, it doesn’t take care of the dynamics in supply and demand, on the other hand the price of a stock futures cannot be different from the one that comes out from the formula, otherwise it would leave room for arbitrages.

Once again:

there is

not

any kind of expectation in

the stock

futures price; if it were, arbitragers would immediately take advantage from it.

This because long and short positions can be opened both on stocks and futures, so arbitragers can combine them in any way so to take advantage of disalignments in prices. All of this under the hypothesis seen in slide 27, of course (lthe imitation to the use of money collected from short selling of stock, indeed, is sufficient to nullify all the reasoning seen in previus slides).

More or less the formula on previous slide can be used to price stock index futures too, but here arbitrages are not so easy to take,

since to buy or sell an index in order to make an arbitrage is not as simple as to buy or sell a single stock: indexes are not negotiable, so it often happens that stock index futures do not respect the formula (arbitragers find it much harder to take advantage of prices’ disalignments).

Slide32

32

Example

The stock

abc

quotes 10€ a share; calculate the futures price with a 130 days maturity, given a 1% risk free interest rate.

Fut = 10 * exp(1% * 130 / 365) = 10.0357

To prove that this price is correct is quite easy. Suppose for example to try and make an arbitrage, buying the stock at 10 and selling the futures at the price just calculated.

After 130 days the profit is

0.0357, exactly

equal to the interest that can be collected depositing the value of the position on the stock on an account remunerated at

a 1% annual interest rate continuously compounded.

Looking at the formula we can state that:1. the futures price is a crescent function of the risk free interest rate;2. the futures price is a crescent function of the time to maturity.The higher the interest rate, the longer the maturity, the higher the stock futures’ price.

Slide33

33

Futures Vs spot

Comparing the charts of the spot price of a stock and the futures price, we can see that the futures price moves always atop of the spot price, but the gap between them reduces day after day; at maturity they coincide:

Slide34

34

Futures payoff: cash settlement

We can now finally give an explanation to a sentence previously stated: the payoff of a futures at maturity and the payoff of an equivalent position on the underlying are slightly different

. For the buyer, indeed, the extra-cost due to risk free interest rates and time is eroded day after day, and is zero at maturity. Referring to data in slide 32, then, if at maturity the spot price is still 10€ the buyer lost 0.0357€ per share. If the settlement price is 12€ he gains 2€ minus 0.0357€ per share. If the settlement price is 8€ he loses 2€ plus 0.0357€ per share.

For the short seller it’s the opposite: the extra-cost paid by the buyer is a little profit, no matter the final result of the trade.

Spot price

underlying

Profit / loss

Futures

Long

position

Spot price

underlying

Profit / loss

Futures

Short

position

Slide35

35

Futures payoff: physic delivery

What seen on previous slide is what happens buying or selling a futures at some time and

keeping it till its maturity

, when the underlying cannot be delivered, so there is just a cash settlement. When the underlying is a stock, at the maturity there is a physic delivery of the underlying: the

holder

of the futures has the

obligation

to buy the stock, the writer of the futures has the

obligation

to sell the stock to the buyer.

The delivery happens at the settlement price, the opening of the third Friday of the expiration month (on the italian futures market). So the payoff at maturity is conceptually different, because the trade changes nature: from a position in a futures the two parties pass into a position on shares, with different implications on both sides, depending also on what happened between the moment the futures was opened and its maturity. We then have to distinguish between three possible outcomes: price up, price stable, price down.In the first case the holder of the futures was right about the future trend of the spot price. He got paid the difference between the spot price at maturity and the futures price in time zero, as a sum of daily variation margins; then, the day of the maturity of the futures he buys the stock for the settlement price. This has a clear consequence: no more leverage, because in the moment of purchase of the stock the entire value has to be paid. Since the value of the position is the value of the settlement price, all the gain achieved on the futures plus the money needed to buy the stock in time zero is invested in the new position, that after that moment can move in any direction.

Slide36

36

Futures payoff: physic delivery

The conclusion is that

speculators do never

hold positions

in futures

till

the maturity

, because it’s a contraddiction in terms to use futures for the leverage and keep them till the maturity to lose the leverage!

On the side of the

writer

of the futures, he paid the distance between the futures price in time zero and the spot price at maturity to the holder, and he has the obligation to sell shares to the holder at the settlement price; if he does not possess shares he has the obligation to sell them anyway, and it means he has to open a short selling position.All of this happens also if the price of the stock at maturity is equal to the price in time zero or below that level: in the first case the buyer buys the stock for the same price as in time zero and lost the gap between the futures price in time zero and the spot price at maturity. The seller sells the stock for the same price and collected the money.In the second case the buyer buys for a reduced price but lost money day after day in terms of variation margins; the seller has to sell shares at a reduced price but he collected the difference between the final and the initial price in cash.In any case, the bottom line is that shares are bought and sold practically for the exact initial price! That’s the reason why no matter where the underlying goes a futures intended to be held till maturity allows both parties to set in advance the price for a trade delayed in

time (as from the definition of futures contract).

Slide37

37

Dividends and stock futures price

As from contract specifications the buyer of a futures i

s not entitled to

collect dividends

paid during the life of the contract (if the dividend is paid after the maturity it is a nonsense to consider its potential effects on the futures).

Now, since the dividend causes a correspondent drop

in the

price of a stock in the moment it is paid, its value has to be somehow cut from the futures price, otherwise

this would leave room for arbitrages: knowing that from a day to the next one the price will drop for sure due to the payment of the dividend, some investors could short sell the futures on closing of the day before the payment and take a sure profit the next day on opening.

Here how it

works in words: the value of the dividend the day of the payment has to be discounted to the present day, in order to get to its present value; this value has to be subtracted from the spot price before calculating the futures price.

Slide38

38

ExampleBe 3.75% the risk free interest rate; consider a stock that quotes 10€ and pays a 0.5€ dividend 20 days from today. Calculate the fair value of the futures that expires 30 days from now. The present value (PV) of the dividend that will be paid in time T (D

T

) – that is, the money amount that invested today at the risk free interest rate on the next 20 days would restitute the amount of the dividend that will be paid in T – is equal to:

D = PV(D

T

) = 0.5 * exp (-3.75% * 20/365) = 0.498974

The futures price is then:

Fut = (10 –

0.498974

) * exp (

3.75% * 30 / 365) = 9.53Generally speaking, then, the formula for the pricing of stock futures in case of dividends is the following:Fut = (S – D) * exp(r * T)Where D is the present value of the dividend and all the other parameters are already known.Dividends and stock futures price

Slide39

39

Futures Vs spot

It’s interesting now to see what happens to the gap between spot and futures price when there are dividends paid during the lifetime of the futures:

Slide40

40

A complication arises when we have to calculate the fair value of a stock index futures when there are dividends going to be paid between the current date and the maturity of the futures.

Here the problem is dual: on one hand it happens that the stocks included in the index pay dividends at

multiple dates

between the present day and the maturity of the futures. On the other hand we have to

cumulate all the dividends

according to their dates of payment.

The procedure is quite complicated, but basically we need to follow these steps:

take

the composition file of the index (to know the weights of the stocks in the index)

take

the dividends for each stock and calculate their incidence on the price of the stock weigh that incidence on the weight of the stock on the whole index sum up all the incidences for each payment date calculate the value of the cumulated incidence on each date on the value of the index day after dayIn this way, day after day, ww have the total value of the dividends paid at some date, to discount to the pressent day and then subtract from the spot price.If there are more dividends paid during the lifetime of the futures we have to account for them all, discounting all of them at their respective time to payment.

Dividends and stock index futures price

Slide41

41

Commodity futures

In case of physic goods, like cereals, metals, or oil, the stock exchanges that issue futures written on those goods have warehouses for the storing of goods that are traded on the markets as underlying assets for futures contracts. Storing of those goods causes costs that have to be somehow included in the futures price (transportation is an additional cost that the buyer of the goods have to pay separately). As a matter of fact, storing costs act like negative dividends, hence their present value has to be added to the spot price before capitalizing it to the maturity at the risk free interest rate. Being

X

the present value of storing costs, then, the futures price is the following:

Fut = (S + X) * exp(r * T)

EXAMPLE

Suppose the price of an oil barrel is 90$ and that storing costs are 5$ per barrel per year. Calculate the price of the oil futures expiring 3 months from now, given a 2% annual interest rate.

The storing cost for a quarter of a year is 1.25$, and its present value is the following:

The futures price is then:

In order to discuss the validity of this formula we need to introduce some more issues.

Slide42

42

Contango and backwardation

What is not considered in the formula on previous slide is the dynamics acting on demand and supply, which

could disalign prices among different maturities

, depending on several events.

Indeed, the commodity futures price depends also on expectations in terms of production and consumption. Sometimes, prices are driven by climatic events that occur in specific regions of the world (think of hurricanes and their effects on agricultural production in the eastern american states) or by other factors that can significantly impact on the balance between supply and demand. All of this often leads to commodity futures prices not following math formulas.

In the end it is the market who decides the right price, according to facts and expectations.

An important issue here is that many events impact on prices just for limited time periods: it often happens that some factors affect shorter maturities only, leaving untouched the longer ones (see forward for some examples).

Under normal circumstances – when supply and demand are not affected by peculiar situations – longer maturities quote higher prices due to the capitalization of the same spot price to a longer time and with higher storing costs.

This “natural” situation is called

contango

. The longer the maturity, the higher the contango in respect to shorter maturities.

Slide43

43

Contango and backwardation

The equilibrium between supply and demand can disalign in two opposite directions, with opposite effects. And here

expectations come into play

.

When there is an excess in the immediate supply in respect with the current demand, while the expected future supply and demand are in equilibrium, the

contango amplifies

.

At the end of an unexpected mild winter, for example, oil producers might have a lot of oil barrels ready to be sold; demand decreases because spring is coming, so

current prices decrease

in respect with

the prices that are expected to show up in late summer, waiting for the new cold season. Short term maturities’ futures prices are much lower than longer term maturities’ futures prices.Sometimes the opposite situation, called backwardation, can show up: shorter maturities quote prices higher than longer maturities.

Such situations show up, for example, when in the middle of an unexpected very cold winter the (poor) oil reserves accumulated in fall on the basis of the weather forecasts reduce much faster than expected and an immediate need for more oil shows up; buyers need the oil immediately, so they are willing to pay more just to be sure to have it:

short term maturities have much higher prices than longer ones.

Slide44

44

Currency futures

In case of

exchange rates between currencies

there are two interest rates to take care of in the calculation of the

futures

price: the interest rates of the two countries involved

. The dynamics in pricing here have to neutralize the opportunity to get higher risk free returns just moving money from a country to another, making the two investments absolutely equal.

In maths, the need above mentioned is accomplished capitalizing the spot exchange rate to the difference between the interest rates of the two countries, in a specific way. The formula is the following:

Fut = S * exp [(R

i

– Rf)*T]Where i and f are for internal and foreign. The internal rate of interest is that of the currency at the denominator of the exchange rate, the foreign one is that of the currency on the numerator.Example: be 1.0675 the spot exchange rate EUR Vs USD (euros on the numerator, US dollars on the denominator), 0.5% the american interest rate, 0.1% the european one, one year the maturity. The futures price is then: Fut = 1.0675 * exp [(0.5% - 0.1%) * 1] = 1.07178

Slide45

45

Currency futures

Why is – and it is meant to be – the futures price higher than the spot?

The answer is that if I am a european citizen, I have money in an account in any european country and I buy a european bond, I get a 0.1% rate of return (risk free interest).

Looking at the american market and seeing it returns 0.5% I could change my euros in US dollars and buy an american bond, getting a higher risk free return.

But I am obviously exposed to an exchange rate risk

, which I have to protect myself against.

If I didn’t, the extra-return on the US bond could be erased partially or totally by a loss in terms of exchange rates between currencies.

To hedge that risk I can look at the futures market, so to decide right now the exhange rate for the conversion back from US dollars to euros (see next slide for numbers). Doing this, if the future exchange rate is equal to the current one I can get a higher return with no risks of any sort.

If, on the other hand, the future exchange rate is higher than the current, then the advantage of the extra-return of the american bond is reduced by the cost of the hedging.

Finally,

If this cost is exactly equal to the gap between the two interest rates the advantage is fully neutralized, making the two investments absolutely replaceable

.

Slide46

46

Currency futures: an example

Let’s see it in numbers. Suppose to have 100k Euros in time 0 and to invest them in the american bond. We change euros and we get 106750 US dollars, that we invest for a 0.5% yearly interest rate.

In time T (after 1 year) we get:

Final value US bond = 106750 * 1.005 = 107283.75$

Changing back this money amount into euros for the exchange rate we had in time 0 it would be:

Final value US bond in euros = 107283.75 / 1.0675 = 100500€

In this case I would keep all the benefit of the extra-return. If, instead, the exchange rate in time T is 1.07178, then we have:

Final value US bond in euros = 107283.75 / 1.07178 = 100100€

Given the higher value of the exchange rate, the real return on the american bond is exactly the same of the european bond:

the futures price neutralized the convenience of the american bond, cutting away any arbitrage opportunity

.

Slide47

47

Corporate hedging: what derivatives are for

To do hedging means to apply strategies in order to erase or at least minimize a series of risks companies may be exposed to in their

activity.

Risks can be of different types:

price: supplies to produce goods, energy to move plants, oil to make airplanes fly, cereals to produce pasta, oranges to produce orange juice are only some examples

interest rates on loans and mortgages

exchange rates: if a company exports goods or imports supplies it can happen it is exposed to exchange rates’ risks

unpredictable events (think for example at the effects of bad weather conditions on agriculture!)

Slide48

48

Conditions for the need of hedging

Risks need to be hedged when we have at least one of the following conditions:

volatility: if prices are stable, then we do not need to make any kind of hedging; viceversa, if the volatility of the risk is high, then hedging is necessary

exposition: if the company can charge the volatility of costs on the price of the goods it sells, then there is no need of hedging; utilities companies, for example, can increase prices of their services if their costs increase; but if the company cannot charge higher costs on market prices, then hedging is necessary

incidence of costs on the final price: if the price of a good needed for production incides only a few on the price of the output, then hedging is not necessary; and viceversa if the incidence is high

Slide49

49

The hedging process

The first step

consists

in the costs analysis: we first need to analyse each single entry in the costs’ chain

Then we have to evaluate the exposition

And finally the volatility

Once we have identified the components that need to be neutralized the final step consists in searching assets to build the hedging strategy

Slide50

50

The real purpose of hedging

As previously stated, the main target of hedging is not to erase risks, but just to minimize them

On another point of view this means the objective of hedging is the minimizing of volatility of costs

And it means to minimize the volatility of returns, and of companies’ results as a consequence

Benefits due to

a correct

hedging are then the following:

stability of returns

easier access to credit

higher appeal for investors

higher competitivity protecion from unpredictable events that could even put in danger the survival of the company

Slide51

51

Hedging with futures

Now let’s

consider

the case of an airline, who needs fuel to make aircrafts fly; the cost of the fuel is a variable that incides considerably on the total cost of a flight, since a higher cost of the fuel cannot be completely debited to the clients, otherwise they could decide to use a competitor’s flights; moreover, the price of the fuel is very volatile: the hedging is absolutely necessary, then, since we have volatility, high incidence and exposition.

In order to understand how this kind of risk can be neutralized we first need to draw a chart that shows the risk position of the company.

Be 100$ the cost of a fuel barrel on the physic market; if the price increases, the company has a higher cost, that causes a reduction in earnings, proportional to the growth of the cost of the fuel.

If the cost of the fuel decreases, the firm has a lower cost, that causes higher earnings, proportional to the drop in the cost of the fuel.

These relationships are drawn on the chart on next slide.

Slide52

52

Fuel price

Line of the firm’s risk

Cost / Saving

100$

150$

+ 50$

50$

- 50$

Note that the risk position of the company is like the payoff of a short futures on the fuel: if the price decreases the company gains, and viceversa.

In order to neutralize this risk we need to open a position coherent with the risk to hedge, that is, we need to open a long position on the fuel.

Hedging with futures

Slide53

53

A long position on the futures written on fuel means indeed to have a profit if the risk shows up.

If the market is in equilibrium, the price of the futures will be a consequence of a standard contango, given by the capitalization of interest rates and storage costs to the maturity of the futures; be 105$ the price of a six-months futures.

If, after six months, the spot price of the fuel is still 100$, the futures is valued 100$ too, that is we have a loss of 5$ a barrel; if the spot price is 150$ the futures is valued 150$, so we have a profit of 45$ a barrel; if the spot price is 50$ the futures is valued 50$, so we have a loss of 55$ a barrel.

Fuel price

at maturity

Proft / loss on futures

100$

150$

+ 45$

50$

- 55$

- 5$

Hedging with futures

Slide54

54

Referring to the three scenarios on previous slide it is quite easy to see that the global result of the hedging position (the vertical sum of the two positions) is exactly the same: no matter where the price of the fuel goes the hedging strategy restitutes a 5$ loss per barrel. In facts:

Scenario 1

The spot price at maturity is still 100$: the futures is valued 100$ and we have the obligation to buy a specific number of fuel barrels at 100$ each; the cost of the hedging operation was 5$, and it means that the final cost of each barrel is 105$

Comment: the hedging was not necessary, but nobody coul tell that in advance; the price paid can be seen as a sustainable cost for the protection against a potentially high risk

Scenario 2

The spot price at maturity is 150$; the futures is valued 150$ and we have the obligation to buy fuel barrels at 150$ each; in front of this higher cost we collected 45$ a barrel in terms of daily variation margins; buying at 150$ having 45$ cash in our pockets isa exactly like to buy at 105$ per barrel

Comment: in this case the hedging allows to save about 90% of the higher cost

Hedging with futures

Slide55

55

Scenario 3The price of the fuel at maturity is 50$: the futures is valued 50$ and we have the obligation to buy fuel barrels at 50$ each; apart from this lower cost we have a loss of 55$ a barrel, as a sum of daily varation margins; that loss is already paid entirely and it increases the real cost of the fuel, to 105$.

Comment: the hedging was not necessary; indeed it was counter-productive: if, in fact, no hedging was in place, the company would have saved 50$ a barrel; moreover we paid 5$ to lose 50$ of savings…

In conclusion, the hedging of price risks with futures is not a good strategy, because it neutralizes risks but erases benefits as well.

In the previous example, indeed, the cost of the fuel will always be 105$ no matter where the real cost goes.

Hedging with futures

Slide56

56

A different hedging

In the field of hedging futures can have a different application.

Since, in fact, to buy a futures and to keep it to the maturity means to have the obligation to buy the underlying,

it also means to

be forced to receive that

underlying

.

If, then, a company needs some kind of supply on any future date, for a specific quantity, and of a specific quality, it can use the futures

to be sure to have the good when it will be needed

.

In such situations it is not a matter of the price, but of the availability of the supply, that is, the possibility to programme the production fo the company under the certainty that the goods needed will be available at the right time.Such hedging strategies are very useful when – for any reason – a supply that is fundamental in the production cycle of the company becomes hard to find: in these situations it often shows up the backwardation; buying futures at the right time can have two advantages, then: the availability of the good, and a low price…These are the cases when futures – in the field of hedging – can be very effective.

Slide57

57

Introduction to options

Options are the more versatile assets available in the markets so far.

Their main peculiarity is they allow us to look at the markets with different eyes: it no longer is just a matter of where the price will go within some time. Indeed, with options we can take advantage of much many more situations, such as expected directional movements, and also bi-directional movements (I think the price will move sharply but I cannot foretell the direction), or even non-directional (I think the price will remain more or less stable).

A complication here is given by the fact our expectations need to be bi-dimensional:

it is not just a matter of where the price will go, but also when it will do that

.

And here it can be proven that it is statistically much easier to foretell price thresholds that will hardly be overspassed within some time than to do the opposite, making the short selling of options a strategy much more profitable in the long run,

although very risky

.

Slide58

58

What is a call option

Options are financial contracts that can be of two different types:

call or put

. Both can be bought or short sold, so here we have four different positions that can be taken, with different consequences: long call, long put, short call, short put.

The buyer of a call option (long call) pays a price (premium) to buy the right to buy

a specific amount of an underlying asset, set by the contract,

at or within

a specific date in the future, at a specific price – called

strike price

.

To be stringent, though, this definition applies only in a specific context: when the buyer keeps the contract opened till its maturity, the good is physic, and it can be delivered. This behaviour, indeed, is the answer to a specific need: the need for the underlying at some date in the future at some price decided in advance; this is a typical need for companies willing to hedge a risk of some sort.If, for instance, the cost for a ton of wheat is 100$, i’m afraid that the price could increase in the next months, and i want to be sure to have the chance to buy it for not more than 115$ within the next three months, then i can buy a call option strike 115$ expiring within 90 days, paying a price to get the right to do that, only if it will be convenient for me.

Slide59

59

What is a call option

Should

at any time before maturity or at the maturity

the wheat’s price increase to a level much higher than the strike, let’s say 150$ per ton, I could decide to exercise the right bought, getting assigned of a specific number of tons of wheat at 115$ each.

On the other hand, should the price of wheat

at maturity

be lower than the strike, let’s say 100$ (like the initial price) or 90$, then I could just let the contract expire without doing anything: I would have no reason, in fact, in being assigned of tons of wheat for 115$ each when I could buy them on the markets for 100$ or less. Again, as the buyer of the option, I can let the contract expire without value, losing only the price initially paid to buy the right to buy.

Now, what to do at maturity or before the maturity depends on whether the exercise is permitted only at the maturity or at any time, and also on whether the exercise is physic or there is just a cash settlement.

The right for exercise and the settlement are crucial issues, then (see forward).

Slide60

60

SPECIFICS OF OPTION CONTRACTS

type:

underlying:

strike price, or exercise price, or basis:

style:

price:

value of a point:

premium:

contract size:

maturity: settlement and exercise:AN EXAMPLE: FTSE MIB CALL 20,000 DECEMBER 2017calla half FTSE Mib index20,000european468 points

2.5€ each468 points * 2.5€ = 1,170€strike * value of a pointthird friday of december 2017cash settlement, no exercise

Slide61

61

Maturity, exercise, style and payoff

If at any time before maturity the underlying reaches and crosses the strike price the holder of a call option could find convenient to exercise his right, and buy the underlying for a specific price.

This so called

early exercise

can happen only if the underlying is physic, and can be delivered; in these situations options are said to be

american style

: they can be exercised at

any time

before maturity and at maturity. Stock options are generally american style; and commodity options too.

European

style options can be exercised only at maturity; in the pratice there is no exercise at all: at maturity there is just a cash settlement. Stock index options are always european style, since the underlying can never be delivered.Options cannot be used only for hedging purposes: they can be used for speculation too: they can be seen as a way to speculate on the trend of an asset, on a long, mid, or short term, with a huge leverage. And they can even be used to gain from sideways ranges (as we’ll see).Now it’s interesting to see what happens at maturity for european and american style options (what happens before maturity is much more complicated; we will deal with it forward).Let’s first consider cash settled options.

Slide62

62

Options Vs futures

As from what seen so far there are several differences between options and futures:

options give rights to their holders, not obligations

rights can be exercised at the maturity or before it

if it is not convenient to exercise the

right,

then the holder can leave the option expire, without the need to do anything

the price of exercise is set in the moment of purchase, not at the maturity

The right to do something is peculiar in the field of options, so it’s better to repeat it once more: the main advantage of options is

the right to do something, not the obligation

. The holder of a call option has the opportunity to decide to exercise his right to buy the underlying only if it is convenient for him.This right has a price, of course, but as from contract specifications, in the worst case the buyer can lose the value of the right, nothing more; in other words,at most he can lose the premium paid. An implied stop loss!

Slide63

63

VALUE at maturity of a cash settled call(no matter long or short)

Since the call option represents an advantage for the holder only if the price crosses the strike, then for any price of the underlying at maturity equal or lower than the strike the value of the option is zero

.

For

any price higher than the strike, instead, the option has a positive value, proportional to the gap between the spot price and the strike price: 1 euro over the strike means a 1 euro value for the call option. The chart of the value for a call option

-

no

matter if it’s long or

short

-

is then the following:

strike

Spot price

profit/loss

Call option

Technically speaking, when the spot price is lower than the strike the call option is

defined

out

of the money

: it is worthless.

When the spot price is close to the strike price the option is

defined

near the money, or

at the money; then, for any spot price higher than the strike it is defined in the money.These “labels” are referred to as the moneyness of an option.

Slide64

64

PAYOFF at maturity of a cash settled LONG call

Given the chart of the value of a

cash

settled call option at maturity now we have all the elements to draw the

payoff

for the buyer of that option

:

Conclusions:

maximum prefixed loss, unlimited potential gain. Nice, but… too good to be true??? What is the drawback?

If the option expires out of the money or at the money, that is, the spot price is lower or at most equal to the strike, the option is worthless, so

all the premium

paid is

gone.As the spot price crosses the strike, that is, the option gets in the money, the holder starts recovering the premium; the breakeven point is then set at strike + premium.

For any spot price higher than strike + premium, then, there is a net gain.

strike price

spot price

profit/loss

-premium

Call option

Slide65

65

PAYOFF at maturity of a cash settled SHORT call

The writer of the option collects the premium paid by the buyer, but assumes an obligation: the obligation to pay the holder if the spot price is higher than the strike at maturity.

Indeed, on options there is marking to market exactly like in the futures market; hence, if the spot price moves up, the writer of the call option faces losses that need to be paid daily to the holder. The payoff at maturity is then the algebric sum of daily variation margins

.

Maximum potential gain, unlimited potential loss. Smart buyer, stupid writer?

If the option expires out of the money or at the money the writer keeps all the premium collected from the buyer.

For any spot price at maturity between strike and strike + premium he keeps only a part of the premium.

For any spot price higher than spot + premium the writer loses money

.

strike price

spot price

profit/loss

premium

Short call option

Slide66

66

Buyers Vs writers – AT MATURITY

Generally speaking, the position of the buyer of options is the expression of the expectation of a price movement in a specific direction within a specific date in the

future.

The buyer of a call expects an up movement. But that movement has an expiration date.

Two are then the hypothesis implied in the behaviour of the buyer:

direction of the price, and time sufficient for that to happen

.

Think of stocks: a first strong distinction between stocks and stock options is that when we buy shares the price has not to go immediately on the favourable direction to make the buyer have a profit: if he can wait, it might just be a matter of time to reach a profit.

That’s not a luxury the buyer of an option can afford, because sooner or later there will be a maturity to deal with.

Time is the worst enemy for the option buyer

.Again, the buyer of options has to deal with two key variables: price and time. Time is money, and when events do not go to the right direction in option trading, time is a cost.Indeed to answer the question “is it better to be buyers or writers of options” we need another element: the probabilities on each side (see forward).

Slide67

67

Payoff at maturity of a physic settled call option

When the settlement is physic things are completely different.

Assuming the holder of a call option

holds it

till the maturity beacuse he wants to buy the underlying at some specific price, then what happens at the maturity

basicly depends on the moneyness:

1. the option is at the money or out of the money: there is no exercise, the holder of the option does not buy the underlying and

he lost

all the premium

paid;

2. the option is in the money: the exercise is automatic; the bank asks the counter party of the option to deliver the underlying; this is accounted on the account of the former option’s holder the third working day after; in the meantime the premium paid to buy the option is gone, even if the option is in the money at maturity: basicly, the value of a physic settled option at maturity is always zero; the result, then, is that there is no gain at all: the holder of the option transforms his position into a new one on the underlying, then it all depends on what the spot price does. And once again, the premium paid is entirely lost.

The main consequence in the exercising of a call option is that the holder has to pay the whole amount of the underlying, losing the advantage of the leverage. And moreover, since the option is gone, the only thing in the hands of the buyer is a stock with a potential profit, not a sure one!

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What is a put option

The buyer of a put option pays a price (premium) to buy the right to sell

a specific amount of a specific underlying asset at or within a specific date in the future, at a specific price (strike price) set by the contract

Looking at it in case it is held till the maturity, a put option is typically an insurance contract that can be used, for example, to protect a position on a stock from the risk of a price drop

Suppose for example we want to buy xyz shares at 10€, and we want to set a maximum potential risk of 1€ a share. Buying a put option strike 9€ we buy the right to sell our shares for 9€ each at or within some date in the future.

Doing this we hedge our position from the risk of a strong price drop. If at the maturity of the option the price of xyz shares is 5€ we have the right to sell shares at 9€ each, limiting the loss to 1€ a share (plus the price paid for the option). Call it stop loss!

This happens for the buyer of put options. And if the buyer pays a price to get rights, the writer of options collects the price and assumes obligations, in favour of the buyer, of course.

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VALUE at maturity of a cash settled put(no matter long or short)

Since the put option represents an advantage for the holder only if the price crosses the strike on the downside, then for any price of the underlying at maturity equal or higher than the strike the value of the option is zero.

For any price lower than the strike, instead, the option has a positive value, proportional to the gap between the the strike price and the spot price: 1 euro under the strike means a 1 euro value for the put option. The chart of the value for a put option (no matter if it’s long or short) is then the following:

Technically speaking, when the spot price is higher than the strike the put option is

defined

out

of the money

: it is worthless.

When the spot price is close to the strike price the option is

defined

near the money, or at the money; then, for any spot price lower than the strike it is in

the money.

strike

profit/loss

Put option

Spot price

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PAYOFF at maturity of a cash settled LONG put

Given the chart of the value of a long cash settled put option at maturity now we have all the elements to draw the payoff

:

And again, maximum prefixed loss, very high (not unlimited, this time) potential gain. Too good to be true again?

If the option expires out of the money or at the money, that is, the spot price is higher or at most equal to the strike, the put option is worthless, so all the premium is gone.

As the spot price crosses the strike on the downside, that is, the option gets in the money, the holder starts recovering the premium; the breakeven point is then set at

(strike – premium).

For any spot price lower than

(strike – premium),

then, there is a net gain

.

strike price

spot price

profit/loss

premium

Put option

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PAYOFF at maturity of a cash settled SHORT put

The writer of the option collects the premium paid by the buyer, but assumes an obligation: the obligation to pay the holder if the spot price is lower than the strike at maturity.

Indeed, on options there is marking to market exactly like in the futures market; hence, if the spot price moves down, the writer of the put option faces losses that need to be paid daily to the holder. The payoff at maturity is then the algebric sum of daily variation margins

.

Maximum potential gain, unlimited potential loss. Smart buyer, stupid writer?

If the option expires out of the money or at the money the writer keeps all the premium collected from the buyer.

For any spot price at maturity between strike and

(strike – premium)

he keeps only a part of the premium.

For any spot price lower than

(strike – premium)

the writer loses money

.

strike price

spot price

profit/loss

premium

Short put option

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Payoff at maturity of a physic settled put option

When the settlement is physic things are completely different.

Assuming the holder of a put option keeps it till the maturity beacuse he wants to sell the underlying at some specific price, then what happens at the maturity is basicly this:

1. the option is at the money or out of the money: there is no exercise, the holder of the option does not sell the underlying and lost all the premium paid

2. the option is in the money: the exercise is automatic; the bank takes the stock from the holder’s account and sends it to the writer, who has the obligation to buy that stock from the holder; the premium paid to buy the option is gone; the advantage for the holder of the option is to have the chance to sell a stock for a price higher than the current one; to have that right he paid a premium, and it is not going to be restituted, in any case.

The main consequence in the exercising of a put option is that the holder sells the underlying, so his position in the end is out of the option and out of the stock

.

If the holder of the put doesn’t have the underlying he has to sell it anyway, meaninghe will open a short selling position on the stock.

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Summing up: rights and obligations for buyers

and writers

in case of physic settled options

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Hedging with options

Let’s go back to the example of the air company and imagine we are trying to hedge the risk of the price of the fuel without erasing potential benefits.

Le’ts consider a call strike 100$ written on the fuel, then; purchsing that call we pay a premium that will never be restituted, in any case; if the price is stable or decreases the maximum loss will be equal to the premium paid; if the price increases we will have the right to buy the fuel at 100$; be 15$ the premium of a six-months call strike 100$ on the fuel (higher than the cost fo the futures, because of the capitalization of the volatility aside of all the other parameters); the cost / saving chart is then the following:

Fuel price

at maturity

Cost / saving

100$

150$

+ 35$

50$

- 15$

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Referring to the three scenarios at maturity seen on previous slide:Scenario 1

The price of the fuel at maturity is 100$: the option expires at the money and is not exercised; we buy the fuel for 100$ per barrel, and since we paid 15$ for the hedging strategy is like to spend 115$ per barrel (we have a loss – or a higher cost – of 15$ per barrel)

Scenario 2

The price of the fuel at maturity is 150$ per barrel; the option expires in the money and is exercised automatically; we buy the fuel for 100$ per barrel (the strike), and since we paid 15$ per barrel to have that right, it is like buying at 115$ per barrel (we save 35$ per barrel)

Scenario 3

The price of the fuel at maturity is 50$ per barrel: the option expires out of the money and is not exercised; we buy the fuel for 50$ per barrel, but since we paid 15$ for the hedging strategy it is like buying at 65$ (we save 35$ per barrel, but we could save 50$)

As it comes out of these scenarios, the option allows the firm to save the most part of the benefit of a reduction in the price of the fuel, setting at the same time a maximum cost for the fuel no matter what happens.

If the hedging is necessary, the cost of the option is gone, but the price of the fuel is lower; if the hedging is not necessary the option is a cost that increases the cost of the fuel, but the firm can still keep the most part of the saving.

Hedging with options

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76

It is interesting to overlay the line of the firm’s risk and the payoff of the long call option, to see the resulting total payoff:

What we get is a “synthetic” long put 100$, that is, a put option achieved combining two different products.

The resulting payoff

Price of the fuel

at maturity

Line of the firm’s risk

Cost / saving

100$

150$

50$

- 15$

long call strike 100$

Total payoff

+ 35$

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77

A source of problems in hedging strategies with options comes out from the strike price; in the previous example we purchased an at the money option, but that is not the only way to do the hedging.

Suppose then we want to reduce the cost of the hedging option; in order to do that we have to move towards out of the money strike prices, wose cost will be progressively lower; let’s think of a call strike 110$, still six-months maturity; the cost will be lower, let’s say 10$.

Scenario 1

The price of the fuel at maturity is 100$: the option expires out of the money and is not exerciesd; we buy the fuel for 100$ and we paid 10$ for the hedging, hence our real final cost is 110$ per barrel (we lose 10$ per barrel).

Scenario 2

The price of the fuel at maturity is 150$ per barrel: the option expires in the money and is automatically exercised; we buy the fuel for 110$ per barrel (the strike), and since we paid 10$ to have that right it is like purchasing the fuel for 120$ per barrel (we save 30$ per barrel).

Scenario 3

The price of the fuel at maturity is 50$ per barrel: the options expires out of the money and is not exercised; we buy the fuel for 50$ per barrel, but since we paid 10$ for an unnecessary hedging it is like buying for 60$ (we save 40$ per barrel, but we could save 50$).

The choice of the strike

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78

The more we take out of the money call options, the more the effects seen on previous slide get highlighted: advantages of scenarios 1 and 3, and drawbacks of scenario 2 amplify.

Let’s consider a call strike 120$, always on a six-months maturity; be 7$ its premium.

Scenario 1

The price of the fuel at maturity is 100$: the option expires out of the money and is not exerciesd; we buy the fuel for 100$ and we paid 7$ for the hedging, hence our real final cost is 107$ per barrel (we lose 7$ per barrel).

Scenario 2

The price of the fuel at maturity is 150$ per barrel: the option expires in the money and is automatically exercised; we buy the fuel for 120$ per barrel (the strike), and since we paid 7$ to have that right it is like purchasing the fuel for 127$ per barrel (we save 23$ per barrel).

Scenario 3

The price of the fuel at maturity is 50$ per barrel: the options expires out of the money and is not exercised; we buy the fuel for 50$ per barrel, but since we paid 7$ for an unnecessary hedging it is like buying for 57$ (we save 43$ per barrel, but we could save 50$).

The choice of the strike

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79

The conclusion is quite obvious: since the hedging has to cover the risk of the fuel price, the more that risk is covered, the better it is.

In other words, generally speaking, the best call should be that at the money.

Indeed it all depends on how much the firm can tolerate a little higher cost; if, for example, the firm can sustain a cost of the fuel up to 120$ per barrel without any problem, then there are no reasons not to choose the call strike 110, given its cost of 10$ per barrel: it makes the cost of the fuel at most 120$ and provides a higher saving in case of a drop in the price of the fuel.

Another way could be the use of in the money options; indeed they have advantages and drawbacks too.

Suppose, for example, to purchase a call strike 90$, valued 22$.

Let’s see what happens in the three scenarios.

The choice of the strike

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80

Scenario 1The price of the fuel at maturity is 100$: the option expires in the money and is automatically exercised; we buy the fuel for 90$ and we paid 22$ for the hedging, hence our real final cost is 112$ per barrel (we lose 12$ per barrel).

Scenario 2

The price of the fuel at maturity is 150$ per barrel: the option expires in the money again and is automatically exercised; we buy the fuel for 90$ per barrel (the strike), and since we paid 22$ to have that right it is like purchasing the fuel for 112$ per barrel (we save 38$ per barrel).

Scenario 3

The price of the fuel at maturity is 50$ per barrel: the options expires out of the money and is not exercised; we buy the fuel for 50$ per barrel, but since we paid 22$ for an unnecessary hedging it is like buying for 72$ (we save 28$ per barrel, but we could save 50$).

In the money options provide a higher saving in case the hedging is necessary, then, but they also reduce dramatically the benefits of a drop in the price of the supply, when the hedging is useless.

In the money options

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81

1. Let’s consider a cattle farmer, who needs to buy the cattle feed for his animals; which risk is he exposed to? How can he manage it with options?

2. On the other side of the cattle framer there is the feed’s producer; what about his risk? And the hedging of it (with options again)?

3.

Imagine to be the manager of a european company that buys supplies from the USA, paying them in US$, and sells its final products in Europe, collecting Euros. What kind of risk is your company exposed to? Draw a line of your risk, in terms of EUR vs US$ exchange rate, assuming that in time zero the exchange rate is 1.35 (for 1 EUR you get 1.35 US$). Imagine to know that you will need to buy supplies within 3 months and that your financial analysts forecast a strenghtening of US$ vs Euro in the same period. Use the proper at the money option to hedge your risk, and draw it on the chart of the line of risk. Draw the payoff of the global position too. Suppose the cost of the option is 0.07. What happens at maturity for a spot exchange rate equal to 1.50? And for 1.20?

4. Consider a company that extracts and sells gold in South Africa. The internal management of the company (supplies, human work, and the like) is in South African rands. The gold extracted is sold on the international market, in US dollars. How many and which sources of risk do you see here? Draw the lines of risk (one for each source of risk) and make an hypothesis of hedging strategies with options.

5. A sweet producer needs to buy huge amounts of sugar for his plants. Draw his line of risk and think of an hedging strategy using at the money options, assumin the initial price of sugar is 50$ per tonn and the cost of the option is 2$. Then do the same for the sugar producer, being all data equal.

Exercises on hedging with options

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82

Everything seen so far in the field of option is about what happens at the maturity. Now it’s time to sse what happens between the pressent date and the maturity.

In the next slides we will deal with premiums, intrinsic value and time value, variables affecting both of them.

And also elasticity and leverage

We will then introduce the delta of an option, as a measure of the reactivity of its price to the movements in the underlying.

Dynamics in options’ prices

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83

Price, contract size and premium

The price of an option listed on the italian market can be in

points

(stock index options) or in

euros

and

fractions of euros

(stock options).

In case of stock options the price in the book is

per share

; and on options there are minimum lots too: 100 shares, 500, shares, 1000 shares, and so on, depending on the underlying. It means that with one option the holder controls a specific number of shares.The price times the number of shares leads to the total cost – premium – of the option.The contract size is the dimension that the position would have if it was converted in a position on the underlying, exercising the right included in the option; it is, then, equal to the strike times the lot

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84

Strike price and leverage

Generally speaking, the leverage is the ratio between the gain achievable given some price movement of the underlying and the money needed to get that gain, compared to the same ratio in case the investment is made on the underlying, with no leverage.

In case of futures:

If we buy 1,000 shares for 10€ each with no leverage we invest 10,000€. If the price goes to 11 we gain 1,000€, 10% on the investment. The same trade taken with a stock futures with a 1,000 shares lot, that requires a 10% margin, means to gain 100% on the money invested. Same price movement in the stock, gain 10 times higher.

In case of options:

Given the same trade on the stock and the same 10% gain using no leverage, if we use a call option, strike 10, lot 1,000 shares, that in the beginning costs 1€ and that in front of the same price movement of the underlying raises to 1,6€, means we gain 600€ in front of 1000€ invested, 60% gain, six times higher.

Now a question could arise: why the futures price follows the spot 1:1 and the option’s price doesn’t?

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85

The elasticity of options’ prices to spot prices

The point is that not all the options have the same elasticity in front of changes in the spot price. This reflects also in their leverage.

Generally speaking, in the money options have a higher elasticity than at the money or out of the money options; but at the same time they have a lower leverage.

To prove all of this we need to look at a typical option chain, that is, the matrix that shows all the listed options for a specific underlying, divided in tables, one for each maturity.

These tables are symmetrical: in the central column there are the strike prices, on the left call options, on the right put options (see next slide).

Since it is all referred to the strikes, call options on the upper part of the table (those in blue in the picture on next slide) are in the money, those on the lower part are out of the money (white). The threshold between them is the strike closest to the current price of the underlying, the at (or near) the money option. On the side of put options is the opposite: upper options

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86

Strike price and leverage

The following chain shows Ftse Mib Options captured on October the 16° 2014, 12:30 p.m., maturity december 2014, when the spot price was quoting about 17680 points:

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87

The elasticity of options’ prices to spot prices

Now, in order to understand leverage and elasticity, let’s make a few examples, assuming that the price of the underlying moves up 500 points (+2.83% on the initial value), the distance between two consecutive strike prices.

Being all other things equal, what happens to all the options’ prices? They change, assuming the value of any contiguous one: all the prices shift down one position, since the at the money call becomes the first in the money one, the first out of the money becomes the at the money, and so on.

Look at the ask side of the book, the prices we should hit if we wanted to buy: the call 18000, that in the first place is valued 795 points, now quotes 1050 points. The spot price increases a 2.83% on the initial value; the option’s price increases 255 points, a 32% on the initial value; the leverage is then 11.3 times (32%/2.83%). But the price movement on the option (its elasticity) is only a half of that of the underlying: indeed the option captured only a 51% (255/500) of the spot price movement.

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88

The elasticity of options’ prices to spot prices

Now let’s look at the call 17000, in the money at first, more in the money later: its price moves from 1370 to 1735, 365 points higher, in front of the same movement in the underlying, 500 points. The gain is 26.6%, so the leverage is 9.4 (given by 26.6%/2.83%, the ratio between the two gains, that of the option and that of the spot). At the same time, the elasticity is higher, since the option’s price caught a 73% of the price change in the underlying (365/500 points).

Finally, let’s look at the call 19000, out of the money in the beginning, still out of the money later: its price moves from 422 to 585, 163 points higher. The gain is then 38.6%, the leverage is 13.6% (38.6%/2.83%), while the elasticity is 32.6% (163/500 points).

In conclusion, in passing from in the money to at the money and out of the money options the leverage increases, but the elasticity decreases.

Which one is the better to buy? It is all a matter of expectations! And expectations have to look for probabilities (see forward).

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89

Intrinsic value

A very important question now, looking at the prices in the option chain, could arise: how does the market make the price of each listed option? The answer is not easy, because it depends on many parameters.

The premium of an option is always given by the sum of two elements: intrinsic value and time value. The first (from now on I.V.) is given by a simple formula:

I.V. of a call = MAX(0,spot price - strike price)

I.V. of a put = MAX(strike price - spot price,0)

According to the formula, the intrinsic value cannot fall under zero, that is to say it cannot reach negative values (this is a specific set by the contract); in other words, the maximum risk in options buying is prefixed and it's equal to the premium paid

Examples

: if a stock quotes 23.6€, the call 23 has 0.60€ of I.V., the call 23.5€ has 0.1€. On the other hand the put 23€ has 0€ of I.V., the 23.5€ has 0€ too, while the 24€ has 0.4€. If the spot price gets down to 23€, then the put 24 has 1€ of I.V., the put 23.5 has 0.5€ and the 23€ has still 0€ of I.V.

Attention

: the intrinsic value is

not affected by the time to maturity

: in case of a call 23€ and a spot price of 23.6€ the I.V. is always 0.6€, whether the time to maturity is three months, six months, a year or whatever

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Time value

Time value is the component we have to add to the intrinsic value to get the total premium of an option.

Substantially it means how much one is willing to pay under the expectation the spot price will move in a specific direction, and enough to reach a specific target price.

That is to say time value is the price of the expectation that the option will have a positive intrinsic value at maturity. As time passes, time value decreases, since the lesser is the time to maturity the lesser is the probability the underlying will reach the strike price.

Example

: if a stock quotes 23.6€ and the call 23 expiring within a month quotes 0.75€, then it has 0.6€ of intrinsic value and 0.15€ of time value. If maturity is not within a month, but within three months, intrinsic value is still 0.6€, but time value is higher, let’s say 0.4€.

Time value is affected by many variables; we’ll see them in details forward.

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Intrinsic value Vs time value

We said at any time the total premium of an option is given by the sum of two elements. Intrinsic value cannot get under zero, but it can be null many days before maturity; time value can neither get under zero, but of course it will get null at maturity only

That is the reason why options with strike prices very far from the spot price do never get null before maturity: there is always a (very low) probability for them to get a not null value at expiry, so they quote a price higher than zero (even if it’s very low)

At maturity the price of an option is all given by its intrinsic value only

Example

: suppose a stock quotes 26.3€ and the call 26.5€ expiring within the next three months quotes 0.5€:

it is all time value

, since the spot price is under the strike price

If the price of the underlying at maturity is 27.5€ then the value of the call is 1€ and

it is all intrinsic value

, since now the time value is nullIf the price of the underlying at maturity is 26€ then the value of the call is null, since both intrinsic and time value are null

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Value at maturity Vs value at now

In the field of options nothing is certain but their value at maturity, for each price of the underlying: given some specific spot price, indeed, the option’s value is completely predictable.

The problem is how to manage what happens before the maturity, because the dynamics in options’ prices can be very hard to understand sometimes.

The first concept to get familair with in order to understand how it works is the at now curve. What follows is the at now curve of the value of a long call strike 10 in respect with the curve of the value at maturity:

Basicly, the problem is that the shape of the at now curve is not constant, since it depends on what happens to all the parameters that incide in the option’s price. The only sure thing is that at maturity it will be exactly equal to the at maturity curve.

10

Spot price

profit/loss

Value at now

Value at maturity

The slope of the curve of the at now value is the delta of the option, which, then, is the first derivative of the price curve in respect with the spot price.

Indeed, as we’ll understand from now on, it is the first partial derivative of the price curve, since the variables the price depends on are six; spot price is just one of them

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Variables affecting the premium of options

As we saw the total premium of an option is given by the sum of two components: one is simply mathematical; the other one includes some kind of expectation

That is to say the price of an option depends

both on the actual and the expected

trend of the underlying

There are six quantifiable factors: spot price, strike price, volatility, time to maturity, risk free interest rates, dividends

There are also some not quantifiable factors, such as the expectations about the spot price's trend and its volatility

Now we’ll see in details how each of the quantifiable factors affects the premium

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94

Quantifiable factors

Pay attention: all the following concepts are valid only on the basis of a specific hypothesis: when a parameter moves all the others stay still

underlying

: if it grows call options’ prices increase, put options’ prices decrease; and viceversa if the spot price goes down

strike price

: the higher it is, the lower is the price of call options and the higher is the price of put options; generally speaking, the farther is the strike price the lower is the price both of a call and a put

time to maturity

: as the expiry date gets close, the probability of an option to get in the money decreases, hence its value is lower; at maturity time value is zero

volatility

: the wider are price movements of the underlying, the higher is the probability of an option to be in the money at maturity, that is to say both call and put options’ prices are higher; a complication here is given by the fact that the volatility that acts on options’ prices is not the one that can be observed on the markets (the so called hystorical volatility), but the perspective one (expected, or implied volatility) which cannot be calculated, but only estimated (see forward for more details).

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95

Quantifiable factors

risk-free interest rates

: according to the economics theory, if interest rates increase, the demand for leveraged products increases, because the capital that is not used for investments can be lended at a higher interest; in case of options we need to make a distinction between calls and puts: a higher interest rate means that to invest on an uptrend it is better to buy a call, saving liquidity that can be invested in a government bond, instead of using the whole money to buy a stock; the demand for call options, then, rises, and so their price does. Viceversa, to go short, instead of buying a put option it is better to short sell the stock, collect the money and lend it at a higher interest rate; hence put option prices decrease, together with their demand. Viceversa if interest rates decrease.

Dividends

:

when a dividend is paid, the price of a stock drops proportionally to its value; this causes a loss in value for call options and a gain in value for put options; but since dividends are known in advance by the market (and if they are not available they can be estimated on the basis of the previous year’s value and the expected growth rate) operators tend to discount the decrease of the price due to the dividend, quonting in advance higher prices for put options and lower prices for call options; in other words, dividends are almost always already included in the prices of options; if, as sometimes happens, the expectation in terms of dividends (in case the real value is not public yet) is wrong, then it happens that prices adjust a bit, but it happens very rearly.

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96

Quantifiable factors: an overview

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97

Clearing houses and margin deposits

Options can be both bought or short sold, but the specific risk of this two positions is extremely different

The buyer in fact pays a premium and has the certainty that wherever the underlying will go it will always be his maximum risk

The writer collects the premium from the buyer and his only certainty is that it will be his maximum potential profit, and at the same time he is exposed to unpredictable risks

This means that in each operation in option only one of the two sides is exposed to a not prefixed risk

The clearing house has to collect margins in order to hedge itself from unpredictable risks; hence in case of options trading

the writer only

is required to deposit margins

The mechanism of margins in option trading is exactly the same as that in case of futures trading: as an option is written the writer deposits an initial margin, calculated on a statistic basis and set by the contract; then according to marking to market every day there are variation margins in order to keep the risk always covered

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Strategies

SPECULATIVE STRATEGIES

Theese are the most common option strategies for speculators:

simple buying of a call or a put

short selling of a call or a put

straddles and strangles

strips and straps

vertical, orizontal and diagonal spreads

butterfly spreads (simple and multiple)

short straddles and short strangles

condor spreads

covered call writing, naked put writing

HEDGING STRATEGIES

Theese are the most common option strategies for hedging purposes:

static hedging of a stock or a basket dynamic hedging low cost default protection

OPTION STRATEGIES FOR COMPANIES

fix in advance the price of a good

secure a payment in a foreign currency

fix a zero-cost channel for a payment or a collection in a foreign currency, for the cost of a supply, etc…

OTHER USES OF OPTIONS

Options, stocks, futures and other assets can be combined in order to build structured products, such as certificates; for example we'll see how to replicate the ABN Amro's Double Up certificate

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99

Six ingredients for any recipe: structured products

Let’s look back at the payoff of a long and a short position on a linear instrument and the payoffs of the four basic positions in options:

Spot price

Profit / loss

Long position

Spot price

Profit / loss

Short position

Combining these six instruments two or more at a time we can have a large number of different payoffs at some maturity.

Combining options with different strike prices and maturities there is an even higher number of possible payouts; indeed the number of reasonable strategies is almost infinite!