By Neha Arya For HRC Eco Hons Sem 6 FE A binomial tree is one of the popular ways of pricing an options contract ID: 1028316
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1. Binomial TreesOptions, Futures and Other Derivatives By- Neha Arya For HRC, Eco Hons (Sem 6), FE
2. A binomial tree is one of the popular ways of pricing an options contract.This model assumes that the underlying stock’s price follows a random walk.The binomial tree represents the likely (with given probabilities) paths that the stock price may follow over the life of an option.Introduction
3. A stock is currently priced at $20In 3 months it will either be worth $22 or $18Stock price= $20 One-step Binomial ModelStock Price = $22Stock Price = $18
4. We want to find the price of a European call option to buy the stock for $21 in 3 months.If the stock price turns out to be $22 at the end of 3 months, the value of the option will be $1If the stock price at the end of 3 months is $18 the value of the option will be $0.The simple “No Arbitrage opportunities exist” assumption is used to price the option.A Call Option
5. A 3-month call option on the stock has a strike price of $21.Stock price= $20 Option Price= ?Stock Price = $22Option Price= $1Stock Price = $18Option Price= $0Up moveDown move
6. We create a portfolio of the stock and the option in a way that there is no uncertainty about the portfolio’s value at the end of 3 months.Since there are 2 possible outcomes in this case, the number of shares can be so chosen that the final value of this portfolio is same in both cases. This makes the portfolio riskless.In absence of arbitrage opportunities, riskless portfolios must earn risk-free interest rate.Creating a Riskless Portfolio
7. Using these facts, the price of the European call option can be easily found.Let the portfolio comprise of: long x shares + short 1 call optionPortfolio is riskless when $(22x-1) = $18xUp moveDown movePortfolio value= $(22x – 1)Portfolio value= $(18x)
8. Or when x = 0.25Hence, the riskless portfolio is: Long 0.25 shares + Short 1 call optionThe value of the portfolio in 3 months = $4.50If the risk-free rate is 12% p.a, then the portfolio’s value today must = 4.5*e –(0.12*3/12) = $4.3670
9. The portfolio that is long 0.25 shares and short 1 option, is worth $4.367.The value of the shares= 0.25*20 = $5.00Hence the value of the option today (in the absence of arbitrage opportunities) = 5 – 4.367= $0.633Valuing the Option
10. Consider an option lasting for time T and the underlying stock price likely to go up Su or down Sd during the life of the option. S0 The current price of the option is f. S0u represents a move up from current stock price, with u>1.GeneralizationUp moveS0u fuS0d fdDown move
11. While S0d represents a move down from current stock price, with d<1.The price of the option in either case is represented by fu and fd respectively.Value of the portfolio that is long x shares and short 1 option:Up moveDown moveS0u*x - fuS0d*x - fd
12. The portfolio is riskless when S0u*x - fu = S0d*x - fd or x = (fu – fd)/(S0u - S0d)Value of the portfolio at time T = S0u*x - fu Value off the portfolio today is: S0x – f = (S0u*x - fu)e-rT This gives, f = [pfu + (1-p) fd]e-rT where p = (erT- d)/ (u-d)
13. The p here can naturally be interpreted as probability of up and down movements.The value of the option is then its expected payoff discounted at the risk-free rate S0 p as a ProbabilityS0u fuS0d fdp1-p
14. When the probability of an up and down movements are p and (1-p), the expected stock price at time T is S0erT This shows that the stock price earns the risk-free rate.Binomial trees illustrate the general result that to value a derivative we can assume that the expected return on the underlying asset is the risk-free rate and discount at the risk-free rate.This is known as using risk-neutral valuation.Risk-Neutral Valuation
15. Say the current stock price is $20 and in each of the two time steps (each 3 month long) may go up by 10% or down by 10%. The risk-free rate is 12% p.a.We consider a 6-month option with a strike price of $21.The aim is to find the option price at the initial node of the tree.Two-step Binomial Trees
16. Stock prices in a two-step tree: $20By finding the option price at each node, like earlier, the option prices at the final nodes of the tree can be easily calculated. (*Refer to examples in the book)$22$24.2$18$16.2$19.8
17. The End