# American Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options

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1 American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010

2 Early Exercise Since American style options give the holder the same rights as European style options plus the possibility of early exercise we know that C e C a and P e P a.

3 Early Exercise An American option (a call, for instance) may have a positive payoff even when the corresponding European call has zero payoff. S t K T t

4 Trade-offs of Early Exercise Consider an American-style call option on a dividend-paying stock. If the option is exercised early you, own the stock and are entitled to receive any dividends paid after exercise,

5 Trade-offs of Early Exercise Consider an American-style call option on a dividend-paying stock. If the option is exercised early you, own the stock and are entitled to receive any dividends paid after exercise, pay the strike price K early foregoing the interest ( ) K e r(t t) 1 you could have earned on it, and

6 Trade-offs of Early Exercise Consider an American-style call option on a dividend-paying stock. If the option is exercised early you, own the stock and are entitled to receive any dividends paid after exercise, pay the strike price K early foregoing the interest ( ) K e r(t t) 1 you could have earned on it, and lose the insurance provided by the call in case S(T) < K.

7 Parity Recall: European options obey the Put-Call Parity Formula: P e + S = C e + Ke rt

8 Parity Recall: European options obey the Put-Call Parity Formula: P e + S = C e + Ke rt American options do not satisfy a parity formula, but some inequalities must be satisfied.

9 Parity Recall: European options obey the Put-Call Parity Formula: P e + S = C e + Ke rt American options do not satisfy a parity formula, but some inequalities must be satisfied. Theorem Suppose the current value of a security is S, the risk-free interest rate is r, and C a and P a are the values of an American call and put respectively on the security with strike price K and expiry T > 0. Then C a + K S + P a

10 Proof Assume to the contrary that C a + K < S + P a. Sell the security, sell the put, and buy the call. This produces a cash flow of S + P a C a. Invest this amount at the risk-free rate, If the owner of the American put chooses to exercise it at time 0 t T, the call option can be exercised to purchase the security for K. The net balance of the investment is (S + P a C a )e rt K > Ke rt K 0. If the American put expires out of the money, exercise the call to close the short position in the security at time T. The net balance of the investment is (S + P a C a )e rt K > Ke rt K > 0. Thus the investor receives a non-negative profit in either case, violating the principle of no arbitrage.

11 Another Inequality Theorem Suppose the current value of a security is S, the risk-free interest rate is r, and C a and P a are the values of an American call and put respectively on the security with strike price K and expiry T > 0. Then S + P a C a + Ke rt

12 Proof Suppose S + P a < C a + Ke rt. Sell an American call and buy the security and the American put. Thus C a S P a is borrowed at t = 0. If the owner of the call decides to exercise it at any time 0 t T, sell the security for the strike price K by exercising the put. The amount of loan to be repaid is (C a S P a )e rt and (C a S P a )e rt + K = (C a + Ke rt S P a )e rt (C a + Ke rt S P a )e rt since r > 0. By assumption S + P a < C a + Ke rt, so the last expression above is positive.

13 Combination of Inequalities Combining the results of the last two theorems we have the following inequality. S K C a P a S Ke rt

14 Example Example The price is a security is currently \$36, the risk-free interest rate is 5.5% compounded continuously, and the strike price of a six-month American call option worth \$2.03 is \$37. The range of no arbitrage values of a six-month American put on the same security with the same strike price is S K C a P a S Ke rt

15 Example Example The price is a security is currently \$36, the risk-free interest rate is 5.5% compounded continuously, and the strike price of a six-month American call option worth \$2.03 is \$37. The range of no arbitrage values of a six-month American put on the same security with the same strike price is S K C a P a S Ke rt P a 36 37e 0.055(6/12)

16 Example Example The price is a security is currently \$36, the risk-free interest rate is 5.5% compounded continuously, and the strike price of a six-month American call option worth \$2.03 is \$37. The range of no arbitrage values of a six-month American put on the same security with the same strike price is S K C a P a S Ke rt P a 36 37e 0.055(6/12) 2.03 P a 3.03

17 A Surprising Equality We know C a C e, but in fact:

18 A Surprising Equality We know C a C e, but in fact: Theorem If C a and C e are the values of American and European call options respectively on the same underlying non-dividend-paying security with identical strike prices and expiry times, then C a = C e.

19 A Surprising Equality We know C a C e, but in fact: Theorem If C a and C e are the values of American and European call options respectively on the same underlying non-dividend-paying security with identical strike prices and expiry times, then C a = C e. Remark: American calls on non-dividend-paying stocks are not exercised early.

20 Proof Suppose that C a > C e. Sell the American call and buy a European call with the same strike price K, expiry date T, and underlying security. The net cash flow C a C e > 0 would be invested at the risk-free rate r. If the owner of the American call chooses to exercise the option at some time t T, sell short a share of the security for amount K and add the proceeds to the amount invested at the risk-free rate. At time T close out the short position in the security by exercising the European option. The amount due is (C a C e )e rt + K(e r(t t) 1) > 0. If the American option is not exercised, the European option can be allowed to expire and the amount due is (C a C e )e rt > 0.

21 American Puts Theorem For a non-dividend-paying stock whose current price is S and for which the American put with a strike price of K and expiry T has a value of P a, satisfies the inequality (K S) + P a < K.

22 Proof Suppose P a < K S. Buy the put and the stock (cost P a + S). Immediately exercise the put and sell the stock for K. Net transaction K P a S > 0 (arbitrage).

23 Proof Suppose P a < K S. Buy the put and the stock (cost P a + S). Immediately exercise the put and sell the stock for K. Net transaction K P a S > 0 (arbitrage). Suppose P a > K. Sell the put and invest proceeds at risk-free rate r. Amount due at time t is P a e rt. If the owner of the put chooses to exercise it, buy the stock for K and sell it for S(t). Net transaction S(t) K + P a e rt > S(t) + K(e rt 1) > 0. If the put expires unused, the profit is P a e rt > 0.

24 Example Remark: in contrast to American calls, American puts on non-dividend-paying stocks will sometimes be exercised early. Example Consider a 12-month American put on a non-dividend paying stock currently worth \$15. If the risk-free interest rate is 3.25% per year and the strike price of the put is \$470, should the option be exercised early?

25 Solution We were not told the price of the put, but we know P a < K = 470. If we exercise the put immediately, we gain = 455 and invest at the risk-free rate. In one year the amount due is 455e = > K > P a. Thus the option should be exercised early.

26 American Calls Theorem For a non-dividend-paying stock whose current price is S and for which the American call with a strike price of K and expiry T has a value of C a, satisfies the inequality (S Ke rt ) + C a < S.

27 Variables Determining Values of American Options (1 of 3) Theorem Suppose T 1 < T 2 and then let C a (T i ) be the value of an American call with expiry T i, and let P a (T i ) be the value of an American put with expiry T i, C a (T 1 ) C a (T 2 ) P a (T 1 ) P a (T 2 ).

28 Proof Suppose C a (T 1 ) > C a (T 2 ). Buy the option C a (T 2 ) and sell the option C a (T 1 ). Initial transaction, C a (T 1 ) > C a (T 2 ) > 0 If the owner of C a (T 1 ) chooses to exercise the option, we can exercise the option C a (T 2 ). Transaction cost, (S(t) K) (S(t) K) = 0. Since we keep the initial transaction profit, arbitrage is present.

29 Variables Determining Values of American Options (2 of 3) Theorem Suppose K 1 < K 2 and then let C a (K i ) be the value of an American call with strike price K i, and let P a (K i ) be the value of an American put with strike price K i, C a (K 2 ) C a (K 1 ) P a (K 1 ) P a (K 2 ) C a (K 1 ) C a (K 2 ) K 2 K 1 P a (K 2 ) P a (K 1 ) K 2 K 1.

30 Variables Determining Values of American Options (3 of 3) Theorem Suppose S 1 < S 2 and then let C a (S i ) be the value of an American call written on a stock whose value is S i, and let P a (S i ) be the value of an American put written on a stock whose value is S i, C a (S 1 ) C a (S 2 ) P a (S 2 ) P a (S 1 ) C a (S 2 ) C a (S 1 ) S 2 S 1 P a (S 1 ) P a (S 2 ) S 2 S 1.

31 Proof (1 of 3) Suppose C a (S 1 ) > C a (S 2 ).

32 Proof (1 of 3) Suppose C a (S 1 ) > C a (S 2 ). Ordinarily we would argue to purchase the call C a (S 2 ) and sell the call C a (S 1 ); however,

33 Proof (1 of 3) Suppose C a (S 1 ) > C a (S 2 ). Ordinarily we would argue to purchase the call C a (S 2 ) and sell the call C a (S 1 ); however, the stock has only one price for all buyers and sellers, initially S(0).

34 Proof (1 of 3) Suppose C a (S 1 ) > C a (S 2 ). Ordinarily we would argue to purchase the call C a (S 2 ) and sell the call C a (S 1 ); however, the stock has only one price for all buyers and sellers, initially S(0). Define x 1 = S 1 S(0) and x 2 = S 2 S(0).

35 Proof (2 of 3) Sell x 1 options C a (S(0)) where x 1 C a (S(0)) = C a (S 1 ) and buy x 2 options C a (S(0)) where Initial transaction is x 2 C a (S(0)) = C a (S 2 ). C a (S 1 ) C a (S 2 ) > 0. If the owner of option C a (S 1 ) chooses to exercise it, option C a (S 2 ) is exercised as well. Transaction profit is x 2 (S(t) K) x 1 (S(t) K) = (x 2 x 1 )(S(t) K) > 0. Arbitrage is present.

36 Proof (3 of 3) Suppose P a (S 1 ) P a (S 2 ) > S 2 S 1, this is equivalent to the inequality P a (S 1 ) + S 1 > P a (S 2 ) + S 2. Buy x 2 put options P a (S(0)), sell x 1 put options P a (S(0)), and buy x 2 x 1 shares of stock. Initial transaction, x 1 P a (S(0)) x 2 P a (S(0)) (x 2 x 1 )S(0) = P a (S 1 ) P a (S 2 ) (S 2 S 1 ) > 0. If the owner of put P a (S 1 ) chooses to exercise the option, we exercise put P a (S 2 ) and sell our x 2 x 1 shares of stock. (x 2 x 1 )S(t)+x 2 (K S(t)) x 1 (K S(t)) = (x 2 x 1 )K > 0 Arbitrage is present.

37 Binomial Pricing of American Puts Assumptions: Strike price of the American put is K, Expiry date of the American put is T > 0, Price of the security at time t with 0 t T is S(t), Continuously compounded risk-free interest rate is r, and Price of the security follows a geometric Brownian motion with variance σ 2.

38 Definitions u: factor by which the stock price may increase during a time step. u = e σ t > 1 d: factor by which the stock price may decrease during a time step. 0 < d = e σ t < 1 p: probability of an increase in stock price during a time step. 0 < p = 1 2 ( ( r 1 + σ σ ) t < 1 2)

39 Illustration S 0 p 1 p S T u S 0 S T d S 0

40 Intrinsic Value Observation: an American put is always worth at least as much as the payoff generated by immediate exercise. Definition The intrinsic value at time t of an American put is the quantity (K S(t)) +.

41 Intrinsic Value Observation: an American put is always worth at least as much as the payoff generated by immediate exercise. Definition The intrinsic value at time t of an American put is the quantity (K S(t)) +. The value of an American put is the greater of its intrinsic value and the present value of its expected intrinsic value at the next time step.

42 One-Step Illustration (1 of 2) K u S 0 K S 0 K d S 0

43 One-Step Illustration (2 of 2) At expiry the American put is worth { P a (K us(0)) (T) = + with probability p, (K ds(0)) + with probability 1 p.

44 One-Step Illustration (2 of 2) At expiry the American put is worth { P a (K us(0)) (T) = + with probability p, (K ds(0)) + with probability 1 p. At t = 0 the American put is worth P a (0) = max { (K S(0)) +, e rt [ p(k us(0)) + + (1 p)(k ds(0)) +]} = max {(K S(0)) +, e rt E [ (K S(T)) +]} { } = max (K S(0)) +, e rt E [P a (T)].

45 Two-Step Ilustration (1 of 2) K u 2 S 0 K u S 0 K S 0 K u d S 0 K d S 0 K d 2 S 0

46 Two-Step Ilustration (2 of 2) At t = T/2, if the put has not been exercised already, an investor will exercise it, if the option is worth more than the present value of the expected value at t = T. { } P a (T/2) = max (K S(T/2)) +, e rt/2 E [P a (T)]

47 Two-Step Ilustration (2 of 2) At t = T/2, if the put has not been exercised already, an investor will exercise it, if the option is worth more than the present value of the expected value at t = T. { } P a (T/2) = max (K S(T/2)) +, e rt/2 E [P a (T)] Using the same logic, the value of the put at t = 0 is the larger of the intrisic value at t = 0 and the present value of the expected value at t = T/2. { } P a (0) = max (K S(0)) +, e rt/2 E [P a (T/2)]

48 Example Example Suppose the current price of a security is \$32, the risk-free interest rate is 10% compounded continuously, and the volatility of Brownian motion for the security is 20%. Find the price of a two-month American put with a strike price of \$34 on the security.

49 Example Example Suppose the current price of a security is \$32, the risk-free interest rate is 10% compounded continuously, and the volatility of Brownian motion for the security is 20%. Find the price of a two-month American put with a strike price of \$34 on the security. We will set t = 1/12, then u d p

50 Stock Price Lattice

51 Intrinsic Value Lattice

52 Pricing the Put at t = 1/12 If S(1/12) = then P a (1/12) = max { ( ) +, e 0.10/12 (0.5574( ) + + ( )(34 32) + ) } = If S(1/12) = then P a (1/12) = max { ( ) +, e 0.10/12 (0.5574(34 32) + + ( )( ) + ) } =

53 Pricing the Put at t = 0 P a (0) = max { (34 32) +, } e 0.10/12 [(0.5574)(0.8779) + (0.4426)(3.7952)] =

54 American Put Lattice

55 Summary: P a (t) (K S(t)) + S(t)

56 General Pricing Framework Using a recursive procedure, the value of an American option, for example a put, is given by P a (T) = (K S(T)) + { } P a ((n 1) t) = max (K S((n 1) t)) +, e r t E [P a (T)] P a ((n 2) t) = max { (K S((n 2) t)) +, } e r t E [P a ((n 1) T)] P a (0) = max. { } (K S(0)) +, e r t E [P a ( T)].

57 Early Exercise for American Calls If a stock pays a dividend during the life of an American call option, it may be advantageous to exercise the call early so as to collect the dividend. Example Suppose a stock is currently worth \$150 and has a volatility of 25% per year. The stock will pay a dividend of \$15 in two months. The risk-free interest rate is 3.25%. Find the prices of two-month European and American call options on the stock with strikes prices of \$150.

58 Solution (1 of 3) If t = 1/12, then p = 1 2 u = e σ t d = e σ t ( ( r 1 + σ σ ) t 2)

59 Solution (2 of 3) Stock Prices

60 Solution (3 of 3) Call Prices

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