Options, continued

1. Options, continued March 10, 2004

2. March 10, 2004 2 Put-Call Parity Consider the payoff on a portfolio consisting of a call plus the (present value of the) exercise price: At the exercise date of the call, the payoff equals the exercise price of the call or the price of the stock, whichever is greater. We’re assuming European options here (no early exercise)

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5. March 10, 2004 5 Now consider a portfolio consisting of a holding of the stock, plus a put (with the same exercise price and exercise date as the call).

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8. March 10, 2004 8 The payoffs (orange line) are the same! Since the payoffs on the two portfolios are the same, so must be their current values. (Otherwise there would exist an arbitrage; be sure you understand how you would set up the arbitrage to exploit a discrepancy).

9. March 10, 2004 9 Put-call parity, applied to current values: Call premium + PV of the exercise price = Put premium + current stock price. Implication: if you can value a call, you can value a put via put-call parity.

10. March 10, 2004 10 Option Valuation value (price) = intrinsic value + time value intrinsic value = value if exercised now time value = whatever is left over ... Time value can’t be negative, since otherwise you would throw away an out of the money option, and exercise an in the money option.

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12. March 10, 2004 12 Bounds on option value: obvious: the value of a call is (1) always positive; (2) always less than value of stock (3) always greater than or equal to its intrinsic value Less obvious: call value > stock price - present value of exercise price.

13. March 10, 2004 13 A lower bound on call value

14. March 10, 2004 14 Derivation via Put-Call Parity You can derive this bound via put-call parity. Claim: C > S - PVE, or: C - S + PVE > 0 Put-call parity: C - S + PVE = P, so the claim reduces to P > 0. The current price of a put is always positive.

15. March 10, 2004 15 Direct derivation If C > S - PVE weren’t satisfied, there would be an arbitrage: Suppose C < S - PVE. (1) buy the call, and (2) invest the present value of the exercise price, and (3) short the stock. This generates a positive amount of cash now.

16. March 10, 2004 16 If the call matures in-the-money, exercise it and cover the short position. Zero cash at the exercise date. If the call matures out-of-the-money, throw it away. Use cash to close out the short position. Positive cash at the exercise date. So it’s an arbitrage: profitable no matter what happens to the stock price!

17. March 10, 2004 17 Major determinants of option value exercise price -- the higher the exercise price, the lower (higher) the value of a call (put) exercise date -- the farther in the future is the exercise date, the greater is the value of either a call or a put stock price volatility -- the greater the stock price volatility, the greater the value of either a call or a put

18. March 10, 2004 18 Early exercise of options Is it ever optimal to exercise a call (in the case of an American option) early? No (on non-dividend-paying stock). Conclusion: an American call is effectively the same as a European call. In particular, they have the same price.

19. March 10, 2004 19 However, this isn’t necessarily true if the stock pays a dividend (since the holder of an unexercised call doesn’t get the dividend). Also, it’s not necessarily true of puts. We’ll discuss this later. Now let’s think about calls on non-dividend-paying stock

20. March 10, 2004 20 Here’s why it’s never optimal to exercise a call on a (non-dividend-paying) stock: Suppose the call is in the money (otherwise you certainly don’t want to exercise it). Assume the price of the stock now is S0, and that the price of the stock at the exercise date is ST (not known now, of course).

21. March 10, 2004 21 If you exercise now, the current value of your portfolio is S0 - X. If you hold the stock, the subsequent change is ST - S0. Add these together. The sum equals ST - X By not exercising the call, you get max (ST - X, 0), which is at least as good.

22. March 10, 2004 22 So if you exercise the call now and the stock subsequently drops below the exercise price, you’ll wish you hadn’t exercised. This is the basis for the time value of a call: the call is worth more unexercised because that way you’re protected against the price dropping below the exercise price.

23. March 10, 2004 23 Suppose you exercise and sell the stock You might think the preceding exercise isn’t realistic, since you could lock in a profit by exercising the call now and selling the stock, whereas if you hold the call the stock price might drop, in which case the call might expire out of the money

24. March 10, 2004 24 This makes 2 changes at once: exercising the call now and selling the stock now, vs. exercising the call (or not exercising it, if it’s out of the money) at the exercise date and selling the stock then.

25. March 10, 2004 25 Instead … You want to compare apples to apples. Instead, compare (a) early exercise and sale of stock now with (b) exercise at the exercise date plus a short sale now. Delaying exercise plus an immediate short sale dominates exercise now plus selling the stock now

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27. March 10, 2004 27 However … If the stock pays dividends before the exercise date, you might want to exercise early so as to get the dividend. Most simply, think of a liquidating dividend. With puts, if you exercise early you get paid, so the time value of money suggests early exercise.

28. March 10, 2004 28 Black-Scholes Model example: S0 = 100 X = 95 r = .1 T = .25 s = .5

29. March 10, 2004 29 Black-Scholes formula

30. March 10, 2004 30 N(x) is the value of the cumulative normal distribution evaluated at x. This is available in Excel under the function Normsdist(x) (Hit fx button, choose statistical functions) Black-Scholes value of call = $13.70