# FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing

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1 FUNDING INVESTMENTS FINANCE 238/738, Spring 2008, Prof. Musto Class 5 Review of Option Pricing I. Put-Call Parity II. One-Period Binomial Option Pricing III. Adding Periods to the Binomial Model IV. Black-Scholes I. Put-Call Parity Suppose you have a stock, selling for S, and two options on that stock Option to buy that stock for X on a given day in the future This is a call option, and it costs C X is the strike price The given day in the future is the expiration date Because the option applies only to the expiration date, this is called a European option. If it were usable before the expiration date, it would be called an American option. Option to sell that stock for X on a given day in the future This is a put option, and it costs P Note: same strike price and expiration date as the call You can also buy a STRIP maturing on the expiration date with face value X We will denote its current price as PV(X) (i.e. the Present Value of getting X on the expiration date) On the expiration date, the stock will be some price, call it S *. Note that On that date, the option to buy the stock for X will be worthless if S * <X (better just to buy it at the market price) worth S * -X if S * >X (buy the stock for X, sell it for S * ) The option to sell the stock for X will be worth X-S * if S * <X (buy the stock for S *, sell it for X) worthless if S * >X (better just to sell it at the market price) 1

2 Put-Call Parity refers to a necessary relation between the prices C, P, S, and PV(X): it must be true that C + PV(X) = S + P or else there s an easy and foolproof arbitrage. Compare the expiration-date values of two portfolios: Portfolio #1: Buy the call and the STRIP Portfolio #2: Buy the stock and the put Portfolio #1: On the expiration day, S * will be either higher or lower than X. S * <X S * >X Value of call: 0 S * -X Value of STRIP: X X TOTAL: X S * Portfolio #2: S * <X S * >X Value of put: X-S * 0 Value of stock: S * S * TOTAL: X S * So the two portfolios have exactly the same value on the expiration date And neither pays anything before the expiration date So their payoffs are identical, so their current prices should be identical If they aren t, buy the cheap one and sell the rich one Portfolio #1 costs C+PV(X) Portfolio #2 costs P+S So there it is. One implication of put-call parity is that (in the context of European options) you only have to know how to calculate a call price, because you just plug that price into the put-call parity relation to get the put price. 2

3 II. One-Period Binomial Option Pricing The basis of all option pricing everywhere is the binomial option pricing model, sometimes abbreviated BOPM. What to notice about this approach is that it is a simple arbitrage argument, repeated over and over. Suppose there are two dates which we ll call now and a year from now, and there are two securities you could buy A risk-free bond paying the risk-free rate r F Pay 1/(1+r F ) today, get 1 in a year, or Get 1/(1+r F ) today, pay 1 in a year, i.e. you can borrow or lend at r F For example, r F =4.04%, so pay 1/1.0404= today, get 1 in a year, or get today, pay 1 in a year A risky stock that costs S today, and will either Go up to S up, or Go down to S down This is the big simplification of this initial model, that there are only two values the risky asset can have in a year For example, a stock that costs 50 today, and will either go up to 80 or down to 30 Notice, I m not saying anything about the probability of going up vs. going down. We want to know the value C of a call option with strike price of X and expiration date of next year In our example, a call with strike price 55 Step 1: Figure out what the call would be worth if the stock went up, and if the stock went down. Call these numbers C up and C down. Stock goes up: Call is worth max{0,s up -X} = C up If the stock goes to 80, the option to buy for 55 is worth 80-55=25 Stock goes down: Call is worth max{0,s down -X} = C down If the stock goes to 30, the option to buy for 55 is worth 0 So we don t yet know what the call is worth today, but we do know what it will be worth in each scenario next year. 3

4 Step 2: Calculate the portfolio of the stock and the bond that will be worth C up if the stock goes up, and C down if the stock goes down. Simple Math Problem: Suppose you Buy n shares of the stock, which costs ns, and will be worth ns up if the stock goes up ns down if the stock goes down Lend B at the risk-free rate, which will be worth B(1+r F ), no matter what This stock will be worth C up if the stock goes up and C down if the stock goes down, if ns up + ns down + B(1+r F ) = C up B(1+r F ) = C down So we have two equations, and two n, and B unknowns. The solution: n = (C up -C down )/(S up -S down ) B = (1/(1+r F ))[(C down S up -C up S down )/(S up -S down )] So the price of the call is the cost of buying n shares and lending B dollars, i.e. C = ns+b Going back to our example numbers, we get n= (25-0)/(80-30) = 0.5 B = (1/1.0404)[((0)(80)-(25)(30))/(80-30)]=(1/1.0404)[(-750)/50] = So we would buy half a share, and lend , or in other words borrow so that we would owe 15 next year. The total effect is Stock goes to 80: ½ share is worth 40, owe 15, so 25 net, same as C up Stock goes to 30: ½ share is worth 15, owe 15, so 0, same as C down Exactly the same payoff as the Call, no matter what happens! Step 3: Price of the call option = price of the portfolio, ns+b In our example, (0.5)(50) + ( ) =

5 III. Adding Periods to the Binomial Model The one-period model gives the exact, arbitrage-free call price, but it is not directly useful for pricing a real-world option In the real world, there are many possible future prices, not just two We can address this shortcoming, while retaining the arbitrage logic, by subdividing the period In each subperiod, the stock goes up to one price or down to another Let s go back to the example from section II, and divide the year in half. The stock still starts at 50, but in half a year it will either Go up to 65, or Go down to 40 If it goes to 65 in half a year, then over the next half-year it will either Go up to 80, or Go down to 50 If it goes to 40 in half a year, then over the next half-year it will either Go up to 50, or Go down to 30 We can represent the potential outcomes with a binomial tree: 5

6 If you invest 1 in the bond, you get 1.02 in half a year Compounded for a year, that s = , so same r F as before As before, we want the price today of a call option with strike price 55, expiring in one year. We ll get it by applying the arbitrage logic, working backwards through the tree. Step 1: Determine the option s value at expiration for each possible outcome. Easy at expiration, the option is either exercised or worthless In this case, the only possible outcome where the call is exercised is where the stock is worth 80, in which case the call is worth 25 Step 2: Move back from the last period to the previous period, and solve for the portfolio of the stock and bond that replicates the option over the final subperiod Once you ve calculated the values at expiration, from the perspective of the period just before expiration, you re looking at the exact same oneperiod problem we just solved The stock goes either up or down You know what the call is worth in both cases You know the bond s payoff Solve two equations with two unknowns In our example, the previous period is half a year, when the stock is either 65 or 40 Stock is 65: Over the last half-year, Stock goes to 80 and call goes to 25, or Stock goes to 50 and call goes to 0 Bond pays 1.02 So we plug in S up =80, S down =50, C up =25, C down =0 and get n = (25-0)/(80-50) = 5/6 B = 1/(1.02)[((0)(80)-(25)(50))/(80-50)] = /1.02 =

7 So if the stock goes to 65 in the first half-year, then the portfolio that replicates the call over the second half-year is Buy 5/6 shares, which costs Borrow 40.85, which brings in Net cost is = If the stock goes to 40 in the first half year, the call will be worthless no matter what the stock does next, so it is worth 0 Step 3: Go back another subperiod, and do the same thing. Repeat until you re done The key insight of multiperiod option pricing: Once we know what the option is worth at every node of period t, we can solve for the period t-1 values with the one-period model. And once we ve done that, we can move back to t-2, then to t-3, etc. We repeat this folding-back until we re done. In our example, we have established that, after half a year the call is worth if the stock goes to 65 The portfolio that pays the same as the call, given that the stock went to 65, costs if the stock goes to 40 No hope of a positive future value So, at the beginning, the call is worth the same as a portfolio of the stock and bond which is worth if the stock goes to 65 0 if it goes to 40 So we plug S up =65, S down =40, C up =13.32, C down =0 and get n = ( )/(65-40) = B = 1/(1.02)[((0)(65)-(13.32)(40))/(65-40)] = Cost is (0.5328)(50) =

8 The call is worth 5.75 because for 5.75 we can implement a dynamic trading strategy We don t just buy and hold, we rebalance every subperiod And this strategy is self-financing Amount of cash we need at each point is exactly the current value of the portfolio and will pay off exactly what the call pays off on the expiration date Convince yourself that this is an arbitrage argument: in the simple world of this model, the dynamic trading strategy can not fail to have the same payoff as the call option. 8

9 IV. Black-Scholes Formula The famous Black-Scholes formula takes the binomial model to its logical extreme. As the number of subperiods goes to infinity, or in other words the length of each subperiod goes to zero, the value of the option converges to the number produced by the B-S formula To put it another way, if you could rebalance your portfolio of the stock and the bond continuously day and night with no transactions costs and also if the stock price moves smoothly, then the B-S formula gives you the arbitrage-free value of the call. Of course these modeling assumptions are not literally true; the operative question is whether they are close enough. Here s the formula: where: and and: C S X r T σ 2 N() C = S N(d 1 )-PV(X) N(d 2 ) 2 log( S / X ) + ( r + σ / 2) T d1 = σ T d 2 =d 1 -σt 1/2 = Call Price = Stock Price = Strike Price = Annual interest rate = Time to maturity = Variance of the stock return = cdf of the standard normal distribution Notice that you need an estimate of the variance of the stock to price the option. Or in other words, an option price imputes a variance estimate, known as the implied volatility. You are not responsible for this formula for this class, but I will use it for some illustrations 9

10 AN EXAMPLE: AMAZON This shows calls expiring July 19, 2008 (FWIW, the standard option expiration dates are the Saturdays after the third Fridays of the indicated months) Look at row 8, marked GP 80. If you buy this option at the Ask of 13.85, you have the right to buy a share of Amazon for \$80 at any time up to 7/19/08. Amazon shares currently trading at days to expiration Interest rate assumed to be 3.09% Volatility implied by the ask of is 56.36% o Above the recent historical volatility of 46.09% This would be considered the at-the-money July call, since the strike of 80 is closest to the share price of Dividends can complicate the relation between American options (such as this) and European options (what B-S actually prices) but Amazon doesn t pay dividends. 10

11 Main point to bear in mind about B-S: The formula gives the arbitrage-free price in the case where it is feasible to replicate the option with dynamic trading. If you can t replicate the option with dynamic trading, then the Black-Scholes price is just a guide for thinking about the option s value. Is there a liquid market for the underlying security? Is there a liquid market for something correlated with the underlying security? In dotcom days, many people owned options on companies that hadn t gone public yet, and were not highly correlated with stocks they could trade. The B-S formula does not really apply in these situations That is, the option holder can t lay off his risk by undertaking the dynamic hedging strategy, so he must bear some risk to hold the options, so their value to him depends on his attitude toward risk But people use the formula anyhow, as a rough estimate 11

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