BUSM 411: Derivatives and Fixed Income

Transcription

1 BUSM 411: Derivatives and Fixed Income 12. Binomial Option Pricing Binomial option pricing enables us to determine the price of an option, given the characteristics of the stock other underlying asset The binomial option pricing model assumes that the price of the underlying asset follows a binomial distribution that is, the asset price in each period can move only up down by a specified amount The binomial model is often referred to as the Cox-Ross-Rubinstein pricing model A one-period binomial tree We begin with a simple example: A stock price is currently $20 At the end of 3 months it will be either $22 $18. The risk-free interest rate is 12% per annum, continuously compounded We want to value a European call option to buy the stock f a strike price of $21 in 3 months. t = 0 t = 3 months Stock price = $22 Option price = $1 Stock price = $20 Option price =? Stock price = $18 Option price = $0 Figure 1: Numerical example of a one-period binomial tree 1

2 We can use a relatively simple argument to price the option in this example. The only assumption needed is no arbitrage. 1. Set up a ptfolio of the stock and the option such that there is no uncertainty about the value of the ptfolio at the end of the 3 months. Note that because there are to securities and only two possible outcomes (up down), it is always possible to set up a riskless ptfolio. 2. Then, because the ptfolio has no risk, the return must equal the risk-free rate 3. Since we know the certain value of the ptfolio at the end of three months, and that its return must equal the risk free rate, we can back out the cost of the ptfolio today. 4. Given the cost of the ptfolio today and the price of the stock, we can then solve f the price of the option. To price the option in this example: Consider a ptfolio that is long shares of the stock and a sht one call option. If the stock price moves up from $20 to $22, the value of the shares is 22 and the value of the option is 1, so that the total value of the ptfolio is If the stock price moves down from $20 to $18, the value of the shares is 18 and the value of the option is 0, so that the total value of the ptfolio is 18. The ptfolio is riskless if is chosen so that the final value of the ptfolio is the same f both outcomes. That is, 22 1 = 18 = 0.25 A ptfolio that is long 0.25 shares of the stock and sht 1 call option is therefe riskless. [Note: is actually the ratio of shares to options needed to fm a riskless ptfolio. Thus, a ptfolio that is long 1 share and sht 4 options would also be riskless, as would a ptfolio that is long 25 shares and sht 100 options.] To confirm, note that if the stock price goes to $22, the value of the ptfolio is = 4.5. If the stock price falls to $18, the value of the ptfolio is =

3 Because the ptfolio is riskless, absence of arbitrage implies that it must earn the risk-free rate of return. Thus, the value of this ptfolio today must equal the PV of 4.5, 4.5e /12 = We know that the value of the stock price today is $20, and the cost of fming the ptfolio is equal to c 0, where c 0 is the price of the call option today. We thus have c 0 = which yields c 0 = This shows that, in the absence of arbitrage opptunities, the current value of the option must be A generalized one-period binomial tree We can generalize the no-arbitrage argument above: Consider a stock whose price is S 0, and an option on the stock ( any derivative dependent on the stock) whose price is c 0, and which lasts f time T Suppose that during the life of the option, the stock price can either increase from S 0 to us 0, where u > 1, decrease from S 0 to ds 0 where d < 1. If the price moves up to us 0, the value of the option will be c u ; if the value of the stock falls to ds 0, then the value of the option will be c d. t = 0 t = T us 0 c u S 0 c 0 ds 0 c d Figure 2: Stock and option prices in a general one-period binomial tree 3

4 As befe, we construct a ptfolio that is long shares of the stock and sht one option, and we will choose to make the ptfolio riskless. If the stock price moves up, the value of the ptfolio is us 0 c u ; if the stock price moves down, the value of the ptfolio becomes ds 0 c d. Setting these values equal gives us 0 c u = ds 0 c d which yields = c u c d us 0 ds 0 (1) Denoting the risk-free interest rate by r, the PV of the ptfolio is (us 0 c u )e rt and the cost of setting up the ptfolio is S 0 c 0 Setting these equal gives S 0 c 0 = (us 0 c u )e rt c 0 = S 0 (1 ue rt ) + c u e rt Substituting f, we obtain c 0 = S 0 ( cu c d us 0 ds 0 ) (1 ue rt ) + c u e rt c 0 = c u(1 de rt ) + c d (ue rt 1) u d c 0 = e rt [p c u + (1 p )c d ] (2) where p = ert d u d (3) 4

5 In the numerical example above, u = 1.1, d = 0.9, r = 0.12, T = 0.25, c u = 1, and c d = 0. Substituting these values into the equations (3) and (2) gives p = e.12 3/ = and c 0 = e /12 [ ( ) 0] = which agrees with our answer from befe Irrelevance of the stock s expected return Note that the option pricing fmula in equation (2) does not involve the probabilities of the stock price moving up down! Perhaps surprisingly, we would get the same option price whether the probability of an upward movement is The reason f this is that we are not pricing the option in absolute terms we calculate its value in terms of the price of the underlying stock. Since the probabilities of up down movements are already incpated into the stock price, we do not need to take them into account again Risk-neutral valuation We can now introduce a very imptant principle in derivatives pricing known as riskneutral valuation. This means that, when valuing a derivative we can make the assumption that invests are risk-neutral, so that the value of a security to invests is simply equal to the present value of its expected payoff, regardless of its risk. A risk-neutral wld has two features that greatly simplify the pricing of derivatives: 1. The expected return on a stock ( any other investment) is the risk-free rate 2. The discount rate used f the expected payoff of an option ( any other instrument) is the risk-free rate In reality, invests are definitely not risk-neutral, but we can assume that they are and find the probabilities of up and down movements that would make the return on the stock equal to the risk-free rate 5

6 These risk-neutral probabilities are hypothetical probabilities they do not crespond to the actual probabilities of up and down price movements in the real wld. But, it turns out that computing the expected payoff of an option using these riskneutral probabilities and discounting the expected value to the present using the riskfree rate gives us the crect option price in the real wld! Returning to equation (3), we can interpret the parameter p as the probability of an up movement in a risk-neutral wld, so that 1 p is the risk-neutral probability of a down movement. Thus, the expression p c u + (1 p )c d is the expected payoff of the option in a risk-neutral wld, and c 0 = e rt [p c u + (1 p )c d ] is the value of the option today. To prove the validity of this interpretation of p, note that if p is the probability of an up movement, the expected stock price E(S T ) at time T is E(S T ) = p us 0 + (1 p )ds 0 Substituting from equation (3) f p gives E(S T ) = p S 0 (u d) + ds 0 E(S T ) = S 0 e rt Thus, the expected return on the stock is the risk-free rate when p is the probability of an up movement. Returning to our earlier example, we can compute the probability of an up movement that would make the return on stock equal to the risk-free rate: 22p + (1 p )18 = 20e /12 6

7 Solving f p yields p = The expected value of the option payoff under the risk-neutral probability is ( ) 0 = Discounting this expected payoff to the present using the risk-free rate gives e /12 = Two-period binomial trees We can extend the previous analysis to a two-step binomial tree Stock price starts at $20, and in each period may go up by 10% down by 10% Each time step is 3 months long Risk-free interest rate is 12% per annum, continuously compounded Consider a 6-month option with a strike price of $21 t = 0 t = 3 months t = 6 months B D A E C F Figure 3: Stock and option prices in a general two-period binomial tree We wk backwards to find the value of the option today 7

8 At the end of 6 months, there are now 3 possible outcomes. If the stock price is 24.2, the value of the option will be = 3.2. If the stock price is , the option expires out of the money and its value is zero. At node C where the stock price is 18, the value of the option is 0 because both of the possible outcomes from that point yield a payoff of 0. We can solve f the option value at node B using the previous method, based on the stock price at node B and the stock and option values at nodes D and E. Given that u, d, r, and T remain the same as in the previous example, the risk-neutral probability remains p = Hence the value of the option at node B is e ( ( ) 0) = Similarly, the value of the option at node A is e ( ( ) 0) = A generalized two-period binomial tree Stock price is initially S 0 During each time step it either moves up to u times its previous value, down to d times its previous value The risk-free rate is r and the length of the time step is t years Because the length of the time step is now t rather than T, equations (2) and (3) become c 0 = e r t [p c u + (1 p )c d ] (4) and p = er t d u d (5) Repeated application of equation (4) gives c u = e r t [p c uu + (1 p )c ud ] (6) c d = e r t [p c ud + (1 p )c dd ] (7) 8

9 Substituting equations (6) and (7) into (4), we get c 0 = e 2r t [p 2 c uu + 2p (1 p )c ud + (1 p ) 2 c dd ] (8) This is consistent with risk-neutral valuation, where p 2, 2p (1 p ), and (1 p ) 2 are the probabilities of reaching the upper, middle, and lower final nodes, respectively We can easily add me steps to the binomial tree, and the risk-neutral valuation principle continues to hold 9