Chapter 5 Option Pricing Model - PowerPoint Presentation

1. Chapter 5Option Pricing ModelThe Black-Scholes-Merton Model1

2. The Black-Scholes model for calculating the premium of an option was introduced in 1973 in a paper entitled, "The Pricing of Options and Corporate Liabilities" published in the Journal of Political Economy. The formula, developed by three economists – Fischer Black, Myron Scholes and Robert Merton – is perhaps the world's most well-known options pricing model.2

3. The knowledge required to develop the Black-Scholes-Merton model came from mathematics and physics.Black–Scholes–Merton model is a mathematical model of a financial market containing certain derivative investment instruments. From the model, the Black–Scholes formula, gives a theoretical estimate of the price of European-style options.3

4. The model makes certain assumptions, including: The options are European and can only be exercised at expiration.Stock prices are random and log normally distributedThe risk-free rate and volatility of the underlying are known and constant. There are no taxes, transaction costs, or dividends.4

5. The Black–Scholes–Merton call option pricing formula gives the call price in terms of theStock priceExercise priceTime to expiration Risk-free interest rateStandard deviation.5

6. Stock PriceThe higher the stock price, the more a given call option is worthA call option holder benefits from a rise in the stock price6

7. Exercise PriceThe lower the striking price for a given stock, the more the option should be worthBecause a call option lets you buy at a predetermined striking price7

8. Time to ExpirationThe longer the time until expiration, the more the option is worthThe option premium increases for more distant expirations for puts and calls8

9. Risk Free Interest RateThe higher the risk-free interest rate, the higher the option premium, everything else being equalA higher “discount rate” means that the call premium must rise for the put/call parity equation to hold9

10. Volatility or Standard deviationThe greater the price volatility, the more the option is worth10

11. The Black–Scholes–Merton Formula C = S0N(d1) – Xeˉͬ ͭ N(d2) Where d1 = ln(S0/ X) +( rc + σ²/2) T σ√T d2 = d1 - σ√T N(d1), N(d2) = cumulative normal probabilities σ = annualized volatility (standard deviation) of stock return rc = risk-free interest rate T = time until option expiration S0 = current stock price C = theoretical call premium 11

12. The model is essentially divided into two parts:The first part, SN(d1), multiplies the price by the change in the call premium in relation to a change in the underlying price. This part of the formula shows the expected benefit of purchasing the underlying outright. 12

13. The model is essentially divided into two parts:The second part, Xeˉͬ ͭ N(d2), provides the current value of paying the exercise price upon expiration (remember, the Black-Scholes model applies to European options that are exercisable only on expiration day). The value of the option is calculated by taking the difference between the two parts, as shown in the equation.13

14. Using the normal distributionA standard normal random variable is called z statistic.One can take any normally distributed random variable, convert it to a standard normal or z statistic, and use a table to determine the probability that an observed value of the random variable will be less than or equal to the value of interest.14

15. The table 5.1 gives the cumulative probabilities of the standard normal distribution.Suppose that we want to know the probability of observing a value of z less than or equal to 1.57.We go to the table and look down the first column for 1.5, then move over to the right under the column labeled 0.07 that is the1.5 and 0.07 add up to the z value, 1.57.The entry in the 1.5raw\0.07 column is 0.9418.15

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17. Example;Let us use the Black-scholes-Merton model to price the DCRB June 125 call. The inputs are;1. computed d1; 17

18. 2. computed d2;3. look up N(d1)N(0.17) = 0.5675.4.look up N(d2)N(-0.08) = 1-N(0.08)= 1- 0.5319= 0.4681. 18

19. 5. plug into formula for C C= 125.94(0.5675) - 125 19

20. Known Discrete DividendsThe black-scholes-Merton model can price Euoropean options on stocks with known discrete dividends by substracting the present value of the dividends from the stock price.20

21. Suppose that stocks pays a dividend of Dt at some time during the option‘s life.If we make a small adjustment to the stock price, the Black-Scholes-Merton model will remain applicable to the pricing of the option.21

22. Calculating the Black-Scholes-Merton price when there are known discrete dividendsOne dividend of 2, ex-dividend in 21 days (May 14 to June 4).1.Determine the amount of thee dividend;Dt =$2.2.Determine the time to the ex-dividend date; 21 days, so t= 21/365 = 0.0575. 22

23. 3. Substract the present value of the dividend from the stock price to obtain $125.94 – 24. Compute d1 = 0.1122 23

24. 5.Compute d26. Look up N(d1) = N(0.11) = 0.5438.7. Look up N(d2) =N(-0.14)= 1-N(0.14) = 1- 0.5557 = 0.4443.8.C= 123.9451(0.5438)-125 24

25. Put option pricing modelsThe Black-Schole-Merton European put option pricing model;Where d1 and d2 the same as in the call option pricing model.In the previous example of the DCRB June 125 call, the values of N(d1) and N(d2) were 0.5692 and 0.4670 respectively.P=125 25

26. Problems Using the Black-Scholes ModelDoes not work well with options that are deep-in-the-money or substantially out-of-the-moneyProduces biased values for very low or very high volatility stocksIncreases as the time until expiration increasesMay yield unreasonable values when an option has only a few days of life remaining26