Example 1. Consider the following two portfolios: 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r).

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1 Chapter 4 Put-Call Parity 1 Bull and Bear Financial analysts use words such as bull and bear to describe the trend in stock markets. Generally speaking, a bull market is characterized by rising prices. Indeed, an investor is sometimes called a bull when he buys commodities or securities in anticipation of a rise in prices, or tries by speculative purchases to effect such a rise. These bulls are potential writers of put options and buyers of call options. The word bear is used just in the opposite way to bull. In short, a bearish market is usually characterized by falling stock-market prices. The bears are also potential customers for put options and writers of call options on the underlying, as these investors expect (or speculate) the asset price to fall. In the sequel, we say that we are bullish about the market if we think that the price of the market will rise; and we say that we are bearish about the market if we think that the price of the market will fall. A portfolio is said to be good for a bullish market if it will bring a profit when the market rises; and we say a portfolio is good for a bearish market if the portfolio will bring a profit when the market falls. By combining calls and puts with various exercise prices one can construct portfolios that suit one s market view. Example 1. Consider the following two portfolios: 1. Buy one c(s(t), 10, τ, r) and sell one c(s(t), 20, τ, r); 2. Buy one c(s(t), 20, τ, r) and sell one c(s(t), 10, τ, r). Which one is good for a bullish market? Consider Portfolio 1 first. Clearly, Π(t) = c(s(t), 10, τ, r) c(s(t), 20, τ, r), which is positive because of Proposition 2.2, and hence one needs money to set up this portfolio. At expiry, Π(T ) = max(s(t ) 10, 0) max(s(t ) 20, 0) 0, if S(T ) 10, = S(T ) 10, if 10 < S(T ) 20, 20 10, if 20 < S(T ). If the underlying asset price S(T ) is very large (i.e., bullish), the profit of this portfolio is then given by Π(T ) Π(t) = (c(s(t), 10, τ, r) c(s(t), 20, τ, r))

2 38 MAT4210 Notes by R. Chan which is nonnegative according to Proposition 2.2. The portfolio can be regarded as good for bullish markets. This portfolio is called a bullish vertical spread. We will explain these terminologies in Chapter 5. Next consider Portfolio 2. Indeed, Π(t) = c(s(t), 20, τ, r) c(s(t), 10, τ, r), and Π(T ) = max (S(T ) 20, 0) max (S(T ) 10, 0) 0, if S(T ) 10, = 10 S(T ), if 10 < S(T ) 20, 10, if 20 < S(T ). When S(T ) is very large, the profit is Π(T ) Π(t) =(S(T ) 20) (S(T ) 10) (c(s(t), 20, τ, r) c(s(t), 10, τ, r)) =(10 20) (c(s(t), 20, τ, r) c(s(t), 10, τ, r)) 0. Obviously, this portfolio is not good for bullish markets. The appeal of these strategies is in their ability to redirect risk. In exchange for the premium which is the maximum possible loss and is known from the start one can construct portfolios to benefit from virtually any move in the underlying asset. 2 Properties of Stock Options In this section, we prove some simple facts about options. Intuitively, one should exercise an American call option if it is deep in the money. But we prove in the following that this is not the case. Proposition 1. Suppose a share pays no dividend between t and T. Then C(S(t), E, τ, r) = c(s(t), E, τ, r), for all τ = T t 0. (1) Proof. It has been shown in Proposition 3.2 that C(t) c(t). We need to show that the strict inequality C(t) > c(t) is not valid. Suppose, however, C(S(t), E, τ, r) > c(s(t), E, τ, r). Then we can form the portfolio: write one American call C(S(t), E, τ, r) and buy a European call c(s(t), E, τ, r). Obviously, the value of this portfolio is Π(t) = c(s(t), E, τ, r) C(S(t), E, τ, r) < 0. Observe that, there is an initial cash inflow in forming such a portfolio. As we are the writer of an American call, we have to monitor the action taken by the buyer of this option. There are two possibilities: (i) The buyer of this American call does not exercise the option early. In other words, the American call option survives till the end of the contract. The terminal value of this portfolio is then given by Π(T ) = max (S(T ) E, 0) max (S(T ) E, 0) = 0. Note the profit Π(T ) Π(t) > 0, and thus, an arbitrage opportunity exists, which is a contradiction.

3 Put-Call Parity 39 (ii) The buyer exercises the option early, say, at time t 1 < T. Put τ 1 = T t 1 τ. At this time, S(t 1 ) E > 0, otherwise the buyer would not exercise it (why?). Now that the buyer exercises it at t 1, we, as the writer, have to sell him the underlying asset at the price E. What can we do? We can short sell the asset S(t 1 ) to the buyer, from whom we receive E. (Alternatively, we can short sell the asset S(t 1 ) in the market, and give the buyer S(t 1 ) E from S(t 1 ). Note that at this stage, we still use no money out of our own fund.) Next, let us deposit E in a bank account so that it earns the risk-free interest rate r. After having done that, the value of the adjusted portfolio at time t 1 is given by Π(t 1 ) = c(s(t 1 ), E, τ 1, r) S(t 1 ) + E. At time T, the terminal payoff of the above portfolio is Π(T ) = max (S(T ) E, 0) S(T ) + Ee r τ 1 { S(T ) E S(T ) + E e rτ 1 = E[e rτ 1 1] > 0, if S(T ) E, = S(T ) + Ee rτ1 > Ee rτ1 E > 0, if S(T ) < E. In either case, Π(T ) > 0. As the original premium Π(t) < 0, we have made a profit, and hence found an arbitrage opportunity. A contradiction! Proposition 1 implies that American call options are the same as European call options. Hence one shouldn t exercise American call options before expiry. This is proved formally below. Proposition 2. Given interest rate r > 0, it is never optimal to exercise an American call option on a non-dividend-paying stock before the expiry date. Proof. By Proposition 3.1 and Proposition 1, we have S(t) Ee rτ c(s(t), E, τ, r) = C(S(t), E, τ, r). (2) In other words, at any time t 1 [t, T ), S(t 1 ) E < C(t 1 ). Thus for any time t 1 < T, the American call option always worth more than what it gets when it exercises. Therefore, we deduce that it can never be optimal to exercise early. Thus there are no advantages to exercise an American call early if the holder plans to keep the stock for the remaining life of the option. If you exercise the option early, you forfeit the interest you could have earned on the strike price E by investing it until maturity. There is no offsetting benefit to early exercise, so it should not be done. What if the holder thinks the stock is currently overpriced and is wondering whether to exercise the option and sell the stock? In this case, the holder is better off selling the option than exercising it. Note that we impose in this proposition the assumption that no dividend payment is made over the lifetime of options. A natural question one would like to ask is that: What would happen if dividend payments are made? We shall return to the question later in the chapter. Nevertheless, it can be optimal to exercise an American put option on a nondividend-paying stock early. Indeed, at any given time during its life, a put option should always be exercised early if it is sufficiently deep in the money. To illustrate, we consider an extreme situation.

4 40 MAT4210 Notes by R. Chan Example 2. Suppose that the strike price is $10 and the stock price is virtually zero. By exercising immediately, a trader makes an immediate gain of $10. If the trader waits, the gain from exercising might be less than $10 but cannot be more than $10, because negative stock prices are impossible. Furthermore, receiving $10 now is preferable to receiving $10 in the future. It follows that in this case, the put option should be exercised immediately. Consider an investor who holds the stock plus an in-the-money put option. The advantage of exercising immediately is that the strike price is received early and can be invested to earn additional interest. The disadvantage is that, in the event the stock price rises above the strike price, the investor will be worse off. The decision to exercise early is in essence a trade-off of these two considerations. In general, the early exercise of a put option becomes more attractive as S(t) decreases, as r increases, and as the volatility decreases. Because there are some circumstances when it is desirable to exercise an American put option early, it follows that an American put option is always worth more than the corresponding European put option, i.e., p(t) P (t). Further, because an American put is sometimes worth its intrinsic value, it follows that a European put option must sometimes be worth less than its intrinsic value. Because of the no arbitrage assumptions, the price of the options are limited within certain intervals. We have seen some of the bounds (e.g. Proposition 2.3). Let us summarize some here. Proposition 3. For options on non-dividend-paying stocks, we have max(s(t) Ee rτ, 0) c(s(t), E, τ, r) S(t), (3) max(ee rτ S(t), 0) p(s(t), E, τ, r) Ee rτ, (4) max(s(t) Ee rτ, 0) C(S(t), E, τ, r) S(t), (5) max(e S(t), 0) P (S(t), E, τ, r) E. (6) Proof. The bounds for the European calls are proven in Propositions 2.1 and 3.3. The bounds for the European puts can be proven likewise. The bounds for the American calls is easy because of (1) and (3). Let us only prove the inequalities for the American puts. The left inequality is intuitively trivial as max(e S(t), 0) is the intrinsic value of the American put option. To formally prove it, we assume that E S(t) > P (t). Then we borrow E to buy a S(t) and a P (t), i.e. Π(t) = P (t) + S(t) E < 0. Then we immediately exercise the put option and note that Π(t + ) = max(e S(t), 0) + S(t) E 0. Hence we have an arbitrage opportunity, a contradiction. The proof of the right inequality is left as an exercise. Finally, we prove that the time value of an option increases as time to expiry increases. Proposition 4. For European calls c(s(t), E, τ, r) on non-dividend-paying stocks, they become more valuable as the time to expiration increases. That is, for T 1 < T 2, where τ i = T i t for i = 1, 2. c(s(t), E, τ 1, r) c(s(t), E, τ 2, r), (7)

5 Put-Call Parity 41 Proof. We just prove the case for the European calls (the same for American calls as they have the same price). The proof is almost the same as that in Proposition 1. Suppose that c(s(t), E, τ 1, r) > c(s(t), E, τ 2, r). Let us form the following portfolio: long one c(s(t), E, τ 2, r) and short one c(s(t), E, τ 1, r). The value of the portfolio at t is Π(t) = c(s(t), E, τ 2, r) c(s(t), E, τ 1, r) < 0. At T 1, the value of the portfolio is then given by Π(T 1 ) = c(s(t 1 ), E, T 2 T 1, r) max (S(T 1 ) E, 0). If S(T 1 ) E, the buyer will not exercise. Then Π(T 1 ) = c(s(t 1 ), E, T 2 T 1, r), i.e. we are left with the call option we are holding. Obviously, at T 2, Π(T 2 ) = (S(T 2 E) + 0. If S(T 1 ) > E, then the buyer will exercise the option. In this case, we should do exactly what we did in Proposition 1, namely short a stock to the buyer to obtain E from him, and put E in the bank. Then our portfolio at T 1 becomes Π(T 1 ) = c(s(t 1 ), E, T T 1, r) S(T 1 )+E. We can show that at T 2, Π(T 2 ) = E(e r(t 2 T 1 ) 1) > 0 if S(T 2 ) E, and Π(T 2 ) = Ee r(t2 T1) S(T 2 ) > 0 if S(T 2 ) < E. Thus, in either case, Π(T 1 ) Π(t) > 0. This shows the existence of an arbitrage opportunity, which is a contradiction. We are done. For European puts, we need the assumption that r = 0. The proof is left as an exercise. Proposition 5. For European puts p(s(t), E, τ, r) on non-dividend-paying stocks, they become more valuable as the time to expiration increases if the interest rate r = 0. That is, for T 1 < T 2, p(s(t), E, τ 1, r) p(s(t), E, τ 2, r), (8) where τ i = T i t for i = 1, 2. Note that for American options, (7) and (8) are always true no matter what the assumptions are. It is because the T 2 -expiry option can always be exercised at T 1, so it has more right than the T 1 -expiry option. One can verify that mathematically by no arbitrage arguments. 3 Put-Call Parity Relation Although call and put options are superficially different, in fact they can be combined in such a way that they are perfectly correlated. This is demonstrated by the following identity. Proposition 6. Assume that no dividend payment is made over the lifetime of options. The following identity holds: S(t) + p(s(t), E, τ, r) c(s(t), E, τ, r) = E e rτ. (9) This relationship between the underlying asset and its options is called the put-call parity.

6 42 MAT4210 Notes by R. Chan Proof. Suppose that S(t)+p(S(t), E, τ, r) c(s(t), E, τ, r) < E e rτ. Form the following portfolio: long one share S(t), long one put p(t), short one call c(t), and borrow E e rτ from a bank. Note that both the call and the put have the same expiry date T and the same exercise price E. The value of this portfolio is At T, the terminal payoff is given by Π(t) = S(t) + p(t) c(t) E e rτ < 0. Π(T ) = S(T ) + max (E S(T ), 0) max (S(T ) E, 0) E = (S(T ) E) + max (E S(T ), 0) max (S(T ) E, 0) = max (0, S(T ) E) max (S(T ) E, 0) = 0. The profit is Π(T ) Π(t) > 0, which implies that an arbitrage opportunity is found. On the other hand, suppose S(t) + p(s(t), E, τ, r) c(s(t), E, τ, r) > E e rτ. We can also present a corresponding portfolio to show that an arbitrage opportunity exists and hence a contradiction. The parity identity shows that there is a close relationship between the different financial instruments. For example, a long position in a stock combined with a short option in a call is equivalent to a short put option plus a certain amount of cash. The put-call parity (9) can be yet derived by considering the following two portfolios: (A) long one European call option plus an amount of cash equal to Ee τr ; (B) long one European put option plus one share. Both the call and put options have the same strike price and the expiration date. One can check easily that both portfolios are worth max (S(T ), E) at expiration of the options. Because the options are European, they cannot be exercised prior to the expiration date. By Proposition 3.3, the portfolios must, therefore, have identical values today. Thus, (9) is established. We illustrate in the next two examples that if the values of portfolios (A) and (B) are not the same, then there will be an arbitrage opportunity open to a trader. Example 3. Suppose European call and put options on a non-dividend-paying stock each have a strike price $30 and an expiration date in three months. Say, the stock price is $31 today, the risk-free interest rate is 10% per annum, the price of a threemonth European call option is $3, and the price of a three-month European put option is $2.25. Identify the arbitrage opportunity open to a trader. Then the value of Portfolio (A) is c(t) + E e rτ = $3 + $30 e $32.26, while the value of Portfolio (B) is given by p(t) + S(t) = $ $31 = $ In such a case, Portfolio (B) is overpriced compared to Portfolio (A). An arbitrage strategy is to buy the securities in Portfolio (A) and short the securities in Portfolio (B). This involves buying the call and shorting both the put and the stock. The strategy generates a positive cash flow of M = $3 + $ $31 = $30.25

7 Put-Call Parity 43 upfront. We put this money M in the bank. Our portfolio in essence is: Π(t) = c(t) p(t) S(t) + M = 0. When invested at the risk-free interest rate, M grows to $30.25 e $31.02 in three months. Three months later, if the stock price at expiration of the option is greater than $30, the call will be exercised. If it is less than $30, the put will be exercised. In either case, the investor ends up buying one share for $30. This share can be used to close out the short position. Hence, Π(T ) = E + Me r/4, and the net profit is Π(T ) Π(t) =$31.02 $30.00 = $1.02. Example 4. As in Example 3, but assume that the call price is $3 and the put price is $1. In such case, one can easily check that Portfolio (A) is now overpriced to Portfolio (B). Indeed, c(t) + E e rτ = $3 + $30 e $32.26, p(t) + S(t) = $ $31.00 = $ An arbitrageur can short the securities in Portfolio (A) and buy the securities in Portfolio (B) to lock in a profit. Take note that this strategy involves an initial investment of $31 + $1 $3 = $29. When financed at the risk-free interest rate, a repayment of $29 e $29.73 is required at the end of the three months. As in Example 3, either the call or the put will be exercised at expiration, and the net profit is $30.00 $29.73 = $0.27. To write it mathematically, we set up a portfolio Π(t) = p(t)+s(t) c(t) M, where M = 29 is the money we get by selling a put and a stock and buying a call. We put M in the bank. Thus initially Π(t) = 0. At T, Π(T ) = E Me rt = = Suppose we have special information that the stock price will increase significantly over the next month. How do we take advantage of this information? Do we buy calls, sell puts, buy the stock, or buy the stock with borrowing? The put-call parity gives us the answer. If you believe the stock price cannot fall, you should buy stock with borrowing. According to (9), S(t) Ee r(t t) < c(t), i.e. you need less money than buying a call. Do not just buy a call because by put-call parity this involves purchasing an insurance policy (the put option) that you do not need. On the other hand, you could simultaneously buy calls and write puts, but again by put-call parity this is the same as a levered stock position. Note that the put-call parity in Proposition 6 holds only for European options. However, it is possible to derive some relationships for American option prices. We will here mention only one of such relationships. Proposition 7. When the stock pays no dividends, S(t) E C(S(t), E, τ, r) P (S(t), E, τ, r) S(t) Ee rτ. (10) Proof. Observe that C(t) = c(t) for non-dividend-paying stocks, and P (t) p(t) always. Thus, it is trivial to see C(t) P (t) c(t) p(t) = S(t) Ee rτ by making use of the put-call parity for European options. It remains to derive the lower bound for C(t) P (t). For this it suffices to show that c(t) + E P (t) + S(t) as C(t) = c(t). If by contradiction, c(t) + E < P (t) + S(t), then we consider the portfolio Π(t) = c(t) + E P (t) S(t), where E is put in a bank account earning an interest rate r.

8 44 MAT4210 Notes by R. Chan The value of the portfolio is negative (positive cash flow) at the initial time t. First consider the case where the put option is never exercise in [t, T ], then Π(T ) = max(s(t ) E, 0) + Ee r(t t) S > 0. If the put option is exercised at t 0 [t, T ], then at t 0 Π(t 0 ) = c(t 0 ) + Ee r(t 0 t) (E S(t 0 )) S(t 0 ) = c(t 0 ) + E[e r(t 0 t) 1] 0. From t 0 on until T, Π(t) 0. Hence in both cases, we have Π(T ) Π(t) > 0, a contradiction. Example 5. An American call option on a non-dividend-paying stock with exercise price $20.00 and maturity in five months is worth $1.50. This must also be the value of a European call option on the same stock with the same exercise price and maturity. Suppose that the current stock price is $19.00 and the risk-free interest rate is 10% per annum. By the put-call parity relationship for European options, we get the price of a European put with exercise price $20 and maturity in five months From (10), e 0.1 5/12 19 = C P 19 20e 0.1 5/12 or 0.18 P C 1. This shows that P lies between $1.68 and $2.50. In other words, upper and lower bounds for the price of an American put with the same strike price and expiration date as the American call are $2.50 and $ The Effect of Dividends The results produced so far have assumed that we are dealing with options on a nondividend-paying stock. In the following we will examine the impact of dividends. In the US, exchange-traded stock options generally have less than a year to maturity. The dividends D payable during the life of the option can usually be predicted with reasonable accuracy. Recall from (3.1) that if t d is the ex-dividend date, S(t + d ) = S(t d ) D. Note that since the holder of an option does not receive any dividend, the value of the option should be the same just before and after the ex-dividend date: Proposition 8. For any options V (S(t), t) = V (S(t), E, τ, r) on an asset S(t) with dividend payment at t d (t, T ), we have V (S(t d ), t d ) = V (S(t+ d ), t+ d ). (11) Proof. Let us prove this for American put, and leave the proofs for the other options as exercises. Since S drops across t d, we know that P (S(t d ), t d ) P (S(t+ d ), t+ d ). Now if P (S(t d ), t d ) < P (S(t+ d ), t+ d ), then one can buy the option at t d and sell it back at to earn an immediate riskless profit. t + d The proposition seems to be counter-intuitive in that dividends have the effect of reducing the stock price S(t) across t d, and hence it should be bad news for the value of call options and good news for the put options. The fact that the option price is continuous across t d, even though the asset value is not, does not mean that the option

9 Put-Call Parity 45 value is unaffected by the dividend payments. The effect of (11) is felt throughout the life of the option, and is propagated by the underlying equation (the Black-Scholes partial differential equation) that governs its value. More precisely, a call option should be less valuable if one foresees that the underlying stock will pay cash dividends within the lifetime of the option. Next we derive the put-call parity equality when we have dividends. Proposition 9. Suppose the dividend payment D is only made once at t d (t, T ), and the asset does not pay dividend at any other time over the lifetime of options. Then, for European options, the following identity holds : S(t) + p(s(t), E, τ, r) c(s(t), E, τ, r) De r(t d t) = Ee rτ. (12) Proof. We consider two portfolios: (A) one European call option plus an amount of cash equal to De r(t d t) + Ee rτ ; (B) one European put option plus one share. It is easy to show that both give De r(t t d) + max(s, E) at time T. Thus by Proposition 3.3, they must have the same value at t < t d. Notice that if t > t d, then portfolio (B) in the proof above is S(t)+p(t)+De r(t t d) while portfolio (A) is c(t) + De r(td t) + Ee rτ. Hence by equating the two portfolios and canceling De r(td t) on both sides, we get S(t) + p(t) = c(t) + E rτ which is precisely the put-call parity we have in (9). We get this put-call parity because t > t d and the stock will not pay dividend between t and T. By (12), we get the following inequality: c(s(t), E, τ, r) S(t) De r(t d t) Ee rτ, p(s(t), E, τ, r) Ee rτ + De r(t d t) S(t), for t < t d. For American options, we have the following result. Proposition 10. Suppose the dividend payment D is only made once at t d (t, T ), and the asset does not pay dividend at any other time over the lifetime of options. Then, the put-call parity becomes S(t) De r(t d t) E C(t) P (t) S(t) Ee rτ. The proof of this equation is similar to that in Proposition 7. We leave it as an exercise. By Propositions 2.3 and 4.1, we know that for a non-dividend paying stock, C(t) = c(t) S(t) E, for all t T. However the same inequality is true for dividend-paying stocks. Proposition 11. For any stock, whether it is dividend-paying or not, C(S(t), E, T t, r) S(t) E and P (S(t), E, T t, r) E S(t), (13) for all t T. Proof. If say at any time t T, C(t) < S(t) E, then the holder of the option can exercise his call option to get a profit of S(t) E, and then he can use C(t) amount of money (which is straightly less than S(t) E) to buy back the option. By doing so, he reaps an instantaneous riskless profit of S(t) E C(t) > 0, a contradiction. The same can be said about the American puts.

10 46 MAT4210 Notes by R. Chan With no dividend payment to be made during the life of an American call option, it has been explained that it is never optimal to exercise it before the expiry date. When dividends are expected, however, we can no longer assert that an American call option will not be exercised early. Sometime it is optimal to exercise an American call immediately prior to an ex-dividend date. This is because the dividend causes the stock price to jump down, making the option less attractive. It turns out that it is never optimal to exercise a call at other times. Proposition 12. Suppose the dividend payment D is only made once at t d (t, T ), and the asset does not pay dividend at any other time over the lifetime of options. Then, an American call option will only be exercised at t d or T. Proof. If the option is exercised at T, then we are done. Suppose it is not. First consider (t + d, T ). Since there are no dividends in this time interval, the American call behaves like the European call. Hence according to Proposition 2, one would not exercise the call option in this period. One would not exercise at t + d also. For if one exercise, one gets S(t + d ) E = S(t d ) D E. This is clearly less than S(t d ) E which one would get if one exercises at t d. If the option is exercised at t d, then we are done too. It remains to show that the option will not be exercised in any time in the interval [t, t d ). By (13), we have at t d, S(t d ) E C(t d ). (14) From (14), we claim that S(t) E < C(t) for all t [t, t d ), and hence it is never optimal to exercise the option before t d. The proof of this is similar to the proof in Proposition 2, but we repeat it here for clarity. By contradiction, if at some t [t, t d ), S( t) E C( t), then we buy a C( t), short a S( t) and put E in the bank. The portfolio at t is Π( t) = C( t) S( t) + E 0. Since we are the holder of the American call C(t), we can choose to keep it at least until t d. Then we have Π(t d ) = C(t d ) S(t d ) + Eer(t d t) > C(t d ) S(t d ) + E which is nonnegative by (14). Thus Π(t d ) Π( t) > 0, a contradiction. From the proof, we see that the only time that C(t) can be (and not necessarily must be) equal to S(t) E, the exercise price, is either at T or at t d. At all other time, C(t) > S(t) E, and therefore one must not exercise the option. The following proposition for American options, first shown in Proposition 2.2 for European options, holds regardless of whether dividends are paid out or not. Proposition 13. For E 1 and E 2 with 0 E 1 E 2, 0 C(S(t), E 1, τ, r) C(S(t), E 2, τ, r) E 2 E 1. (15) 0 P (S(t), E 2, τ, r) P (S(t), E 1, τ, r) E 2 E 1. (16) Proof. We only prove (15) and leave (16) as an exercise. Let us prove the left inequality in (15) first. If it is not true, we form the portfolio: buy one an American call C(S(t), E 1, τ, r) and write one an American call C(S(t), E 2, τ, r). The value of the portfolio is Π(t) = C(S(t), E 1, τ, r) C(S(t), E 2, τ, r) < 0.

11 Put-Call Parity 47 Note that we are now the writer of an American call C(S(t), E 2, τ, r) and we have to wait for the buyer s decision on whether or not he would like to exercise the call C(S(t), E 2, τ, r) early. Suppose the buyer of this call would not like to exercise the option early, then we can hold the portfolio till the expiry date of the calls. At T, the terminal payoff of the portfolio is Π(T ) = max (S(T ) E 1, 0) max (S(T ) E 2, 0) 0, if S(T ) E 1, = S(T ) E 2, if E 1 < S(T ) E 2, E 2 E 1, if E 2 < S(T ). Obviously, Π(T ) 0 in any case. Thus, an arbitrage opportunity exists, which is a contradiction. Suppose the buyer exercises the call early, say at t 0. We must have S(t 0 ) E 2. Then, as the holder of the call with the strike price E 1, we also exercise our option. The payoff of the portfolio at t 0 is Π(t 0 ) = S(t 0 ) E 1 (S(t 0 ) E 2 ) = E 2 E 1 0. It also has an arbitrage opportunity. Hence we have proved the left inequality in (15). Next we prove the right inequality in (15). Suppose that C(S(t), E 1, τ, r) C(S(t), E 2, τ, r) > E 2 E 1. Then form the portfolio: sell one C(S(t), E 1, τ, r), buy one C(S(t), E 2, τ, r), and then deposit E 2 E 1 amount of cash in a bank. The value of the portfolio is Π(t) = C(S(t), E 1, τ, r) + C(S(t), E 2, τ, r) + (E 2 E 1 ) < 0. Again we divide into two cases. If the buyer of the call with the strike price E 1 does not exercise early, then we also hold the portfolio till T. Thus, Π(T ) = (S(T ) E 1 ) + + (S(T ) E 2 ) + + (E 2 E 1 ) e rτ (E 2 E 1 ) e rτ, if S(T ) E 1, = (E 2 E 1 ) e rτ (S(T ) E 1 ), if E 1 < S(T ) E 2, (E 2 E 1 ) e rτ (E 2 E 1 ), if E 2 < S(T ). It is trivial to see that Π(T ) 0 when either S(T ) E 1 or E 2 < S(T ); and when E 1 < S(T ) E 2, Π(T ) E 2 E 1 (S(T ) E 1 ) = E 2 S(T ) 0. Therefore, the profit of this portfolio is always positive, which means an arbitrage opportunity exists. A contradiction. If the buyer of the call with the strike price E 1 does exercise early, say, at time t 0. Thus S(t 0 ) E 1 0. Then, as the holder of the call (with the strike price E 2 ), we do according to one of the following two scenarios:

12 48 MAT4210 Notes by R. Chan (a) If S(t 0 ) E 2, we exercise our call option at the same time; and hence, the value of the portfolio is given by Π(t 0 ) = (S(t 0 ) E 1 ) + (S(t 0 ) E 2 ) + (E 2 E 1 )e r(t 0 t) = (E 2 E 1 )[e r(t0 t) 1] 0. An arbitrage opportunity exists, which is a contradiction. (b) If S(t 0 ) < E 2, then we just sell our option C(S(t 0 ), E 2, T t 0, r), which is not negative anyway. The value of the portfolio is now given by Π(t 0 ) = (S(t 0 ) E 1 ) + C(S(t 0 ), E 2, T t 0, r) + (E 2 E 1 )e r(t 0 t) (S(t 0 ) E 1 ) + C(S(t 0 ), E 2, T t 0, r) + (E 2 E 1 ) = E 2 S(t 0 ) + C(S(t 0 ), E 2, T t 0, r) > 0, Again we obtain an arbitrage opportunity, and hence a contradiction. Note that we have to close the short position on the stock at t 0 to ensure that Π(t) > 0 for all t t 0. For if we do not close out the short position on the stock, the portfolio may become negative at T if S(T ) is very large. 5 Put-Call Parity for Digital Options The original and still the most common contracts are the vanilla calls and puts. Increasingly important are the binary or digital options. These contracts have a payoff at expiry that is discontinuous in the underlying asset price. Say, for a simple example of a binary call, it pays a fixed amount $Q at expiry time T, if the asset price is greater than or equal to the exercise price E; and it pays nothing at expiry if the asset price ends up below the strike price. This is a kind of cash-or-nothing call. Why would you invest in a binary call? If you think that the asset price will rise by expiry, to finish above the strike price, then you might choose to buy either a vanilla call or a binary call. The vanilla call has the best upside potential, growing linearly with S beyond the strike. The binary call, however, can never pay off more than $Q. If you expect the underlying to rise dramatically, then it may be best to buy the vanilla call. If you believe that the asset price rise will be less dramatic, then buy the binary call. The gearing of the vanilla call is greater than that for a binary call if the move in the underlying is large. A cash-or-nothing binary put can be defined analogously to a cash-or-nothing binary call. The holder of such a put receives $Q if the asset is straightly below E at expiry. The binary put would be bought by someone expecting a modest fall in the asset price. There is a particularly simple binary put-call parity relationship. What do you get at expiry if you hold both a binary call and a binary put with the same strikes and expiries? The answer is that you will always get $Q regardless of the level of the underlying at expiry. Thus, according to Proposition 3.3, Binary call + Binary put = Q e r (T t).