A short note on American option prices

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1 A short note on American option Filip Lindskog April 27, The set-up An American call option with strike price K written on some stock gives the holder the right to buy a share of the stock (exercise the option) at the strike price K at any time before and including the time of maturity T. The value of exercising the option at time t [0, T ] is max(s t K, 0), where S t is the share price at time t. Similarly for an American put option. American options are not necessarily stock option. Other examples include American futures options where the underlying price process is the process of futures on some asset or good. Write c A t = c A t (S t, K, t, T ) and c E t for the option price at time t of an American and European call option, respectively. Write p A t and p E t for the option price at time t of an American and European put option, respectively. Since the cash flow of an American option held to maturity is the cash flow of the corresponding European option, c A t c E t and p A t p E t must hold in order not to violate the Law of One Price. In what follows we assume that we can take long and short positions of arbitrary sizes, that the bid and ask coincide, and that short-selling does not lead to additional costs (fees, commissions, etc). We assume that zero-coupon bonds of all maturities are available and that the zero rates are strictly positive. We also assume that we can trade in the underlying asset on which the options are written. The stock is said to be a pure investment asset if it does not pay dividends or give other benefits before time T. This short note is intended as a complement to the lecture notes [1] used in the course SF2701 Financial Mathematics at KTH. More details and further properties of American options can be found in [2] on which this note is based. The notation used follows that in [1]. 2 No-arbitrage relations Theorem 1. If T 1 [0, T 1 ]. < T 2, then c A t (S t, K, t, T 1 ) c A t (S t, K, t, T 2 ) for t 1

2 Proof. Suppose that the inequality does not hold and consider the following strategy. At time t buy the (cheaper) call option with maturity T 2 and shortsell the (more expensive) call option with maturity T 1. This gives a strictly positive cash flow at time t. Whenever the short-sold call option is exercised, exercise the call option maturing at T 2. This produces a zero cash flow at the exercise time of the short-sold call option and cancels the call option position. The strategy violates the Law of One Price since it produces a strictly positive cash flow at time t and no other cash flows. Theorem 2. If the stock is a pure investment asset, then c E t Z t,t K, 0) for t [0, T ]. max(s t Proof. Consider the following strategy: at time t buy the call option, buy K zero-coupon bonds maturing at T with face value 1, short-sell the underlying asset and close the short position at time T. The payoff at time T is max(s T K, 0)+K S T = max(s T, K) S T 0. The initial cash flow is c E t Z t,t K + S t. Since the payoff is nonnegative, the initial cash flow must be nonpositive in order not to violate the Law of One Price. Similarly, c E t 0. Therefore, c E t max(s t Z t,t K, 0). Theorem 3. If the stock is a pure investment asset, then an American call option is not exercised prior to maturity and c A t = c E t for t [0, T ]. Proof. The value of exercising the American call option at time t is max(s t K, 0). Moreover, c A t c E t. The call option is only exercised if S t > K and in this case Theorem 2 gives c A t c E t max(s t Z t,t K, 0) > max(s t K, 0), t < T. In particular, at time t < T selling the option is always better for the holder than exercising it. Since the cash flow of an American call option held to maturity is identical to a European call option the Law of One Price implies that the option must coincide. Theorem 4. The American call option price c A t (S t, K, t, T ) is a convex function of K, i.e. if λ [0, 1], K 1 < K 2, and K 3 = λk 1 + (1 λ)k 2, then c A t (S t, K 3, t, T ) λc A t (S t, K 1, t, T ) + (1 λ)c A t (S t, K 2, t, T ). Proof. Suppose that the inequality does not hold and consider the following strategy. At time t buy λ call options with strike price K 1, buy 1 λ call options with strike price K 2, and short-sell one call option with strike price K 3. This gives a strictly positive cash flow at time t. Whenever the short-sold call option is exercised, exercise the other two call options. This produces the cash flow C = λ max(s K 1, 0) + (1 λ) max(s K 2, 0) max(s λk 1 (1 λ)k 2, 0) 2

3 at the exercise time of the short-sold call option and cancels the call option position, where S denotes the share price at the exercise time. Since max(x, 0) is a convex function, max(s λk 1 (1 λ)k 2, 0) = max(λ(s K 1 ) + (1 λ)(s K 2 ), 0) λ max(s K 1, 0) + (1 λ) max(s K 2, 0). Therefore C 0 so the strategy produces a strictly positive cash flow at time t, a nonnegative cash flow at the exercise time of the short-sold call option, and no other cash flows. The strategy thereby violates the Law of One Price. Theorem 5. c A t p A t S t Z t,t K for all t [0, T ]. Proof. Consider the following portfolio. At time t buy a put option with strike K and maturity T, short-sell a call option with the same strike and maturity, buy a share of the stock, and short-sell K zero-coupon bonds maturing at T with face value one. Consider the following strategy. If the call option is held to maturity, then we hold the put option to maturity and at that time sell the share. The net payoff in this case is zero. The value of the portfolio excluding the short position in the call option at time u prior to maturity is p A u + S u Z u,t K = p A u + K(1 Z u,t ) + S u K > S u K which is strictly greater than the exercise value for the call option at that time. If the call option is exercised prior to maturity, then we pay the call option payoff, sell the put option and the share, and close out the short position in the zero-coupon bonds by buying bonds. The net cash flow is in that case positive. The cash flow of the strategy is therefore always nonnegative for the holder of the portfolio. In order not to violate the Law of One Price the cost for buying the portfolio must therefore by nonnegative, i.e. p A t c A t + S t Z t,t K 0. Problems Consider the American IBM stock option (in $) at closing on April 26 in Table 1. The IBM share price at that time was $ A dividend of $0.85 per share is paid on June 9 to stock holders of record May 10. The current zero rates for all maturities less than three months is assumed to be 1% per year. The in Table 1 are last and not necessarily at which you can trade. Assume, however, that you can trade at these and take both long and short positions in the options, the stock, and in zero-coupon bonds with arbitrary maturities. 3

4 Problem 1. Determine any violations of the Law of One Price (or determine that there are no such violations) in the option in Table 1. If you find a violation of the Law of One Price, construct a strategy that capitalizes on the inconsistent. For the computation of discount factors, assume that a year consists of 365 days. Strike Last Vol Open Int Last Vol Open Int Call, exp. at close May 18 Put, exp. at close May Call, exp. at close June 15 Put, exp. at close June Table 1: Prices of American stock options at closing on April 26. Set S 0 = , r = 0.01, T 1 = 22/365, and T 2 = 50/365. We first check whether (K, T ) = c A 0 (K, T ) pa 0 (K, T ) S 0 + Z T K 0 for all T = T 1, T 2 and K = 190, 195, 200, 205, 210, 215. We find that the (K, T )s are T = T 1 : T = T 2 : We notice that the inequality does not hold for (K, T ) = (190, T 1 ). Consider the following portfolio. At time 0 buy a put option with strike 190 and maturity T 1, short-sell a call option with the same strike and maturity, buy a share of the stock, and short-sell 190 zero-coupon bonds maturing at T 1 with face value one. The initial cash flow is here > 0. Consider the following strategy. If the call option is held to maturity, then we hold the put option to maturity and at that time sell the share. The net payoff in this case is max(190 S T1, 0) max(s T1 190, 0)+S T1 190 = 0 (plus a dividend payment later). If the call option is exercised at time u prior to maturity, then we pay 4

5 the call option payoff, sell the put option and the share, and close out the short position in the zero-coupon bonds by buying bonds. The net cash flow is in that case (S u 190) + p A u (190, T 1 ) + S u 190Z u,t1 = p A u (190, T 1 ) + 190(1 Z u,t ) > 0. Next we check whether c A 0 (K, T 1) c A 0 (K, T 2) 0 for all K and p A 0 (K, T 1) p A 0 (K, T 2) 0 for all K. Call: Put: Finally we check whether the option are convex as functions of the strike price by plotting the linearly interpolated. Figure 1 shows that the convexity property is not violated. References [1] Harald Lang, Lectures on Financial Mathematics, Lecture notes, KTH Mathematics, [2] Robert Merton (1973), Theory of Rational Option Pricing, The Bell Journal of Economics and Management Science, 4,

6 Call exp. May 18 Call exp. June Put exp. May 18 Put exp. June Figure 1: Convexity of option as functions of strike price 6

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