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).


 Mary Martin
 2 years ago
 Views:
Transcription
1 Chapter 4 PutCall 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 stockmarket 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 PutCall 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 riskfree 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 nondividendpaying 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 nondividendpaying 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 inthemoney 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 tradeoff 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 nondividendpaying 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 nondividendpaying 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 PutCall 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 nondividendpaying 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 PutCall 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 putcall 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 putcall 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 nondividendpaying stock each have a strike price $30 and an expiration date in three months. Say, the stock price is $31 today, the riskfree interest rate is 10% per annum, the price of a threemonth European call option is $3, and the price of a threemonth 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 PutCall 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 riskfree 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 riskfree 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 putcall 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 putcall 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 putcall parity this is the same as a levered stock position. Note that the putcall 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 nondividendpaying 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 putcall 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 nondividendpaying 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 riskfree interest rate is 10% per annum. By the putcall 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 nondividendpaying stock. In the following we will examine the impact of dividends. In the US, exchangetraded 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 exdividend 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 exdividend 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 counterintuitive 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 PutCall 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 BlackScholes 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 putcall 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 putcall parity we have in (9). We get this putcall 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 putcall 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 nondividend paying stock, C(t) = c(t) S(t) E, for all t T. However the same inequality is true for dividendpaying stocks. Proposition 11. For any stock, whether it is dividendpaying 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 exdividend 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 PutCall 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 PutCall 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 cashornothing 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 cashornothing binary put can be defined analogously to a cashornothing 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 putcall 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).
Lecture 5: Put  Call Parity
Lecture 5: Put  Call Parity Reading: J.C.Hull, Chapter 9 Reminder: basic assumptions 1. There are no arbitrage opportunities, i.e. no party can get a riskless profit. 2. Borrowing and lending are possible
More information9 Basics of options, including trading strategies
ECG590I Asset Pricing. Lecture 9: Basics of options, including trading strategies 1 9 Basics of options, including trading strategies Option: The option of buying (call) or selling (put) an asset. European
More informationOptions Markets: Introduction
Options Markets: Introduction Chapter 20 Option Contracts call option = contract that gives the holder the right to purchase an asset at a specified price, on or before a certain date put option = contract
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the
More informationExpected payoff = 1 2 0 + 1 20 = 10.
Chapter 2 Options 1 European Call Options To consolidate our concept on European call options, let us consider how one can calculate the price of an option under very simple assumptions. Recall that the
More information1 Strategies Involving A Single Option and A Stock
Chapter 5 Trading Strategies 1 Strategies Involving A Single Option and A Stock One of the attractions of options is that they can be used to create a very wide range of payoff patterns. In the following
More informationFinance 400 A. Penati  G. Pennacchi. Option Pricing
Finance 400 A. Penati  G. Pennacchi Option Pricing Earlier we derived general pricing relationships for contingent claims in terms of an equilibrium stochastic discount factor or in terms of elementary
More informationChapter 9 Parity and Other Option Relationships
Chapter 9 Parity and Other Option Relationships Question 9.1. This problem requires the application of putcallparity. We have: Question 9.2. P (35, 0.5) C (35, 0.5) e δt S 0 + e rt 35 P (35, 0.5) $2.27
More informationFinance 350: Problem Set 8 Alternative Solutions
Finance 35: Problem Set 8 Alternative Solutions Note: Where appropriate, the final answer for each problem is given in bold italics for those not interested in the discussion of the solution. All payoff
More informationSession IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics
Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock
More informationCHAPTER 7: PROPERTIES OF STOCK OPTION PRICES
CHAPER 7: PROPERIES OF SOCK OPION PRICES 7.1 Factors Affecting Option Prices able 7.1 Summary of the Effect on the Price of a Stock Option of Increasing One Variable While Keeping All Other Fixed Variable
More informationEconomics 1723: Capital Markets Lecture 20
Economics 1723: Capital Markets Lecture 20 John Y. Campbell Ec1723 November 14, 2013 John Y. Campbell (Ec1723) Lecture 20 November 14, 2013 1 / 43 Key questions What is a CDS? What information do we get
More informationLecture 4: Properties of stock options
Lecture 4: Properties of stock options Reading: J.C.Hull, Chapter 9 An European call option is an agreement between two parties giving the holder the right to buy a certain asset (e.g. one stock unit)
More informationFinance 436 Futures and Options Review Notes for Final Exam. Chapter 9
Finance 436 Futures and Options Review Notes for Final Exam Chapter 9 1. Options: call options vs. put options, American options vs. European options 2. Characteristics: option premium, option type, underlying
More informationLecture 3: Put Options and DistributionFree Results
OPTIONS and FUTURES Lecture 3: Put Options and DistributionFree Results Philip H. Dybvig Washington University in Saint Louis put options binomial valuation what are distributionfree results? option
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION 1. Cost Profit Call option, X = 95 12.20 10 2.20 Put option, X = 95 1.65 0 1.65 Call option, X = 105 4.70 0 4.70 Put option, X = 105 4.40 0 4.40 Call option, X
More informationChapter 2 Introduction to Option Management
Chapter 2 Introduction to Option Management The prize must be worth the toil when one stakes one s life on fortune s dice. Dolon to Hector, Euripides (Rhesus, 182) In this chapter we discuss basic concepts
More informationThis question is a direct application of the PutCallParity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have:
Chapter 3 Insurance, Collars, and Other Strategies Question 3.1 This question is a direct application of the PutCallParity [equation (3.1)] of the textbook. Mimicking Table 3.1., we have: S&R Index S&R
More information2. Exercising the option  buying or selling asset by using option. 3. Strike (or exercise) price  price at which asset may be bought or sold
Chapter 21 : Options1 CHAPTER 21. OPTIONS Contents I. INTRODUCTION BASIC TERMS II. VALUATION OF OPTIONS A. Minimum Values of Options B. Maximum Values of Options C. Determinants of Call Value D. BlackScholes
More informationWeek 1: Futures, Forwards and Options derivative three Hedge: Speculation: Futures Contract: buy or sell
Week 1: Futures, Forwards and Options  A derivative is a financial instrument which has a value which is determined by the price of something else (or an underlying instrument) E.g. energy like coal/electricity
More informationChapter 2 Questions Sample Comparing Options
Chapter 2 Questions Sample Comparing Options Questions 2.16 through 2.21 from Chapter 2 are provided below as a Sample of our Questions, followed by the corresponding full Solutions. At the beginning of
More informationChapter 1: Financial Markets and Financial Derivatives
Chapter 1: Financial Markets and Financial Derivatives 1.1 Financial Markets Financial markets are markets for financial instruments, in which buyers and sellers find each other and create or exchange
More informationChapter 3 Insurance, Collars, and Other Strategies
Chapter 3 Insurance, Collars, and Other Strategies Question 3.1. This question is a direct application of the PutCallParity (equation (3.1)) of the textbook. Mimicking Table 3.1., we have: S&R Index
More informationArbitrage spreads. Arbitrage spreads refer to standard option strategies like vanilla spreads to
Arbitrage spreads Arbitrage spreads refer to standard option strategies like vanilla spreads to lock up some arbitrage in case of mispricing of options. Although arbitrage used to exist in the early days
More informationUnderlying (S) The asset, which the option buyer has the right to buy or sell. Notation: S or S t = S(t)
INTRODUCTION TO OPTIONS Readings: Hull, Chapters 8, 9, and 10 Part I. Options Basics Options Lexicon Options Payoffs (Payoff diagrams) Calls and Puts as two halves of a forward contract: the PutCallForward
More informationOptions. Moty Katzman. September 19, 2014
Options Moty Katzman September 19, 2014 What are options? Options are contracts conferring certain rights regarding the buying or selling of assets. A European call option gives the owner the right to
More informationLecture 4 Options & Option trading strategies
Lecture 4 Options & Option trading strategies * Option strategies can be divided into three main categories: Taking a position in an option and the underlying asset; A spread which involved taking a position
More informationAmerican Options. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan American Options
American Options An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Early Exercise Since American style options give the holder the same rights as European style options plus
More informationFIN40008 FINANCIAL INSTRUMENTS SPRING 2008. Options
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the socalled plain vanilla options. We consider the payoffs to these
More informationFigure S9.1 Profit from long position in Problem 9.9
Problem 9.9 Suppose that a European call option to buy a share for $100.00 costs $5.00 and is held until maturity. Under what circumstances will the holder of the option make a profit? Under what circumstances
More informationChapter 21: Options and Corporate Finance
Chapter 21: Options and Corporate Finance 21.1 a. An option is a contract which gives its owner the right to buy or sell an underlying asset at a fixed price on or before a given date. b. Exercise is the
More informationFactors Affecting Option Prices
Factors Affecting Option Prices 1. The current stock price S 0. 2. The option strike price K. 3. The time to expiration T. 4. The volatility of the stock price σ. 5. The riskfree interest rate r. 6. The
More informationProperties of Stock Options. Chapter 10
Properties of Stock Options Chapter 10 1 Notation c : European call option price C : American Call option price p : European put option price P : American Put option price S 0 : Stock price today K : Strike
More informationTrading Strategies Involving Options. Chapter 11
Trading Strategies Involving Options Chapter 11 1 Strategies to be Considered A riskfree bond and an option to create a principalprotected note A stock and an option Two or more options of the same type
More informationCall and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options
Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder
More information7: The CRR Market Model
Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney MATH3075/3975 Financial Mathematics Semester 2, 2015 Outline We will examine the following issues: 1 The CoxRossRubinstein
More informationOverview. Option Basics. Options and Derivatives. Professor Lasse H. Pedersen. Option basics and option strategies
Options and Derivatives Professor Lasse H. Pedersen Prof. Lasse H. Pedersen 1 Overview Option basics and option strategies Noarbitrage bounds on option prices Binomial option pricing BlackScholesMerton
More informationModelFree Boundaries of Option Time Value and Early Exercise Premium
ModelFree Boundaries of Option Time Value and Early Exercise Premium Tie Su* Department of Finance University of Miami P.O. Box 248094 Coral Gables, FL 331246552 Phone: 3052841885 Fax: 3052844800
More informationDerivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.
OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss
More informationChapter 7: Option pricing foundations Exercises  solutions
Chapter 7: Option pricing foundations Exercises  solutions 1. (a) We use the putcall parity: Share + Put = Call + PV(X) or Share + Put  Call = 97.70 + 4.16 23.20 = 78.66 and P V (X) = 80 e 0.0315 =
More informationEC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals
EC372 Bond and Derivatives Markets Topic #5: Options Markets I: fundamentals R. E. Bailey Department of Economics University of Essex Outline Contents 1 Call options and put options 1 2 Payoffs on options
More informationCHAPTER 20: OPTIONS MARKETS: INTRODUCTION
CHAPTER 20: OPTIONS MARKETS: INTRODUCTION PROBLEM SETS 1. Options provide numerous opportunities to modify the risk profile of a portfolio. The simplest example of an option strategy that increases risk
More informationLecture 12. Options Strategies
Lecture 12. Options Strategies Introduction to Options Strategies Options, Futures, Derivatives 10/15/07 back to start 1 Solutions Problem 6:23: Assume that a bank can borrow or lend money at the same
More informationStochastic Processes and Advanced Mathematical Finance. Options and Derivatives
Steven R. Dunbar Department of Mathematics 203 Avery Hall University of NebraskaLincoln Lincoln, NE 685880130 http://www.math.unl.edu Voice: 4024723731 Fax: 4024728466 Stochastic Processes and Advanced
More informationOther observable variables as arguments besides S.
Valuation of options before expiration Consider European options with time t until expiration. Value now of receiving c T at expiration? (Value now of receiving p T at expiration?) Have candidate model
More informationChapter 11 Properties of Stock Options. Options, Futures, and Other Derivatives, 9th Edition, Copyright John C. Hull
Chapter 11 Properties of Stock Options 1 Notation c: European call option price p: European put option price S 0 : Stock price today K: Strike price T: Life of option σ: Volatility of stock price C: American
More informationOption pricing. Vinod Kothari
Option pricing Vinod Kothari Notation we use this Chapter will be as follows: S o : Price of the share at time 0 S T : Price of the share at time T T : time to maturity of the option r : risk free rate
More informationOption Premium = Intrinsic. Speculative Value. Value
Chapters 4/ Part Options: Basic Concepts Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options An OptionPricing Formula Investment in
More informationOn BlackScholes Equation, Black Scholes Formula and Binary Option Price
On BlackScholes Equation, Black Scholes Formula and Binary Option Price Abstract: Chi Gao 12/15/2013 I. BlackScholes Equation is derived using two methods: (1) riskneutral measure; (2)  hedge. II.
More informationOptions Pricing. This is sometimes referred to as the intrinsic value of the option.
Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the PutCall Parity Relationship. I. Preliminary Material Recall the payoff
More informationLecture 7: Bounds on Options Prices Steven Skiena. http://www.cs.sunysb.edu/ skiena
Lecture 7: Bounds on Options Prices Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Option Price Quotes Reading the
More information1 Introduction to Derivative Instruments
1 Introduction to Derivative Instruments The past few decades have witnessed a revolution in the trading of derivative securities in world financial markets. A financial derivative may be defined as a
More informationOptions. The Option Contract: Puts. The Option Contract: Calls. Options Markets: Introduction. A put option gives its holder the right to sell
Options Options Markets: Introduction Derivatives are securities that get their value from the price of other securities. Derivatives are contingent claims (A claim that can be made when certain specified
More informationCHAPTER 21: OPTION VALUATION
CHAPTER 21: OPTION VALUATION 1. Put values also must increase as the volatility of the underlying stock increases. We see this from the parity relation as follows: P = C + PV(X) S 0 + PV(Dividends). Given
More informationRisk / Reward Maximum Loss: Limited to the premium paid up front for the option. Maximum Gain: Unlimited as the market rallies.
Bullish Strategies Bullish options strategies are employed when the options trader expects the underlying stock price to move upwards. It is necessary to assess how high the stock price can go and the
More informationCHAPTER 20. Financial Options. Chapter Synopsis
CHAPTER 20 Financial Options Chapter Synopsis 20.1 Option Basics A financial option gives its owner the right, but not the obligation, to buy or sell a financial asset at a fixed price on or until a specified
More informationDetermination of Forward and Futures Prices
Determination of Forward and Futures Prices 3.1 Chapter 3 3.2 Consumption vs Investment Assets Investment assets assets held by significant numbers of people purely for investment purposes Examples: gold,
More informationOptions Strategies in a Bear Market
Class: Options Strategies in a Bear Market www.888options.com 1.888.678.4667 This document discusses exchangetraded options issued by The Options Clearing Corporation. No statement in this document is
More informationPutCall Parity. chris bemis
PutCall Parity chris bemis May 22, 2006 Recall that a replicating portfolio of a contingent claim determines the claim s price. This was justified by the no arbitrage principle. Using this idea, we obtain
More informationStrategies for a moderate price decrease
Course #: Title Module 6 Strategies for a moderate price decrease Topic 1: Strategy overview... 3 Construction... 3 Limited profits, low cost... 3 Volatility... 4 Time decay... 4 Bull spread, bear spread...
More informationIntroduction to Mathematical Finance 2015/16. List of Exercises. Master in Matemática e Aplicações
Introduction to Mathematical Finance 2015/16 List of Exercises Master in Matemática e Aplicações 1 Chapter 1 Basic Concepts Exercise 1.1 Let B(t, T ) denote the cost at time t of a riskfree 1 euro bond,
More informationChapter 6 Arbitrage Relationships for Call and Put Options
Chapter 6 Arbitrage Relationships for Call and Put Options Recall that a riskfree arbitrage opportunity arises when an investment is identified that requires no initial outlays yet guarantees nonnegative
More informationOptions and Derivative Pricing. U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University.
Options and Derivative Pricing U. NaikNimbalkar, Department of Statistics, Savitribai Phule Pune University. email: uvnaik@gmail.com The slides are based on the following: References 1. J. Hull. Options,
More informationThis page intentionally left blank
This page intentionally left blank Trading Strategies Involving Options 239 Table 11.1 ST^K^ K\ < ST < K2 S T ^ K 2 a bull spread created using calls. long call option &T ^ 1 &T ^ 1 short call option
More informationLecture 17. Options trading strategies
Lecture 17 Options trading strategies Agenda: I. Basics II. III. IV. Single option and a stock Two options Bull spreads Bear spreads Three options Butterfly spreads V. Calendar Spreads VI. Combinations:
More informationCHAPTER 20 Understanding Options
CHAPTER 20 Understanding Options Answers to Practice Questions 1. a. The put places a floor on value of investment, i.e., less risky than buying stock. The risk reduction comes at the cost of the option
More information11 Option. Payoffs and Option Strategies. Answers to Questions and Problems
11 Option Payoffs and Option Strategies Answers to Questions and Problems 1. Consider a call option with an exercise price of $80 and a cost of $5. Graph the profits and losses at expiration for various
More informationSession X: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics. Department of Economics, City University, London
Session X: Options: Hedging, Insurance and Trading Strategies Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Option
More informationChapter 5 Financial Forwards and Futures
Chapter 5 Financial Forwards and Futures Question 5.1. Four different ways to sell a share of stock that has a price S(0) at time 0. Question 5.2. Description Get Paid at Lose Ownership of Receive Payment
More informationSection 1  Overview and Option Basics
1 of 10 Section 1  Overview and Option Basics Download this in PDF format. Welcome to the world of investing and trading with options. The purpose of this course is to show you what options are, how they
More informationConvenient Conventions
C: call value. P : put value. X: strike price. S: stock price. D: dividend. Convenient Conventions c 2015 Prof. YuhDauh Lyuu, National Taiwan University Page 168 Payoff, Mathematically Speaking The payoff
More information2. How is a fund manager motivated to behave with this type of renumeration package?
MØA 155 PROBLEM SET: Options Exercise 1. Arbitrage [2] In the discussions of some of the models in this course, we relied on the following type of argument: If two investment strategies have the same payoff
More informationHedging. An Undergraduate Introduction to Financial Mathematics. J. Robert Buchanan. J. Robert Buchanan Hedging
Hedging An Undergraduate Introduction to Financial Mathematics J. Robert Buchanan 2010 Introduction Definition Hedging is the practice of making a portfolio of investments less sensitive to changes in
More informationThe BlackScholes Formula
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 The BlackScholes Formula These notes examine the BlackScholes formula for European options. The BlackScholes formula are complex as they are based on the
More informationOption Basics. c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153
Option Basics c 2012 Prof. YuhDauh Lyuu, National Taiwan University Page 153 The shift toward options as the center of gravity of finance [... ] Merton H. Miller (1923 2000) c 2012 Prof. YuhDauh Lyuu,
More informationOne Period Binomial Model
FIN40008 FINANCIAL INSTRUMENTS SPRING 2008 One Period Binomial Model These notes consider the one period binomial model to exactly price an option. We will consider three different methods of pricing
More informationChapter 20 Understanding Options
Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interestrate options (III) Commodity options D) I, II, and III
More informationDerivative strategies using
FP CLASSROOM Derivative strategies using options Derivatives are becoming increasingly important in the world of finance. A Financial Planner can use the strategies to increase profitability, hedge portfolio
More informationCaput Derivatives: October 30, 2003
Caput Derivatives: October 30, 2003 Exam + Answers Total time: 2 hours and 30 minutes. Note 1: You are allowed to use books, course notes, and a calculator. Question 1. [20 points] Consider an investor
More informationChapter 9 Trading Strategies Involving Options
Chapter 9 Trading Strategies Involving Options Introduction Options can be used to create a wide range of different payoff functions Why? /loss lines involving just underlying assets (forward) and money
More informationOption Pricing Basics
Option Pricing Basics Aswath Damodaran Aswath Damodaran 1 What is an option? An option provides the holder with the right to buy or sell a specified quantity of an underlying asset at a fixed price (called
More informationDetermination of Forward and Futures Prices. Chapter 5
Determination of Forward and Futures Prices Chapter 5 Fundamentals of Futures and Options Markets, 8th Ed, Ch 5, Copyright John C. Hull 2013 1 Consumption vs Investment Assets Investment assets are assets
More informationI. Derivatives, call and put options, bounds on option prices, combined strategies
I. Derivatives, call and put options, bounds on option prices, combined strategies Beáta Stehlíková Financial derivatives Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava
More informationTable of Contents. Bullish Strategies
Table of Contents Bullish Strategies 1. Long Call 03 2. Synthetic Long Call 04 3. Short Put 06 4. Covered Call 07 5. Long Combo 08 6. The Collar 09 7. Bull Call Spread 10 8. Bull Put Spread 11 9. Call
More informationDetermination of Forward and Futures Prices
Determination of Forward and Futures Prices Chapter 5 5.1 Consumption vs Investment Assets Investment assets are assets held by significant numbers of people purely for investment purposes (Examples: gold,
More informationOptions. + Concepts and Buzzwords. Readings. PutCall Parity Volatility Effects
+ Options + Concepts and Buzzwords PutCall Parity Volatility Effects Call, put, European, American, underlying asset, strike price, expiration date Readings Tuckman, Chapter 19 Veronesi, Chapter 6 Options
More informationOption Properties. Liuren Wu. Zicklin School of Business, Baruch College. Options Markets. (Hull chapter: 9)
Option Properties Liuren Wu Zicklin School of Business, Baruch College Options Markets (Hull chapter: 9) Liuren Wu (Baruch) Option Properties Options Markets 1 / 17 Notation c: European call option price.
More informationProtective Put Strategy Profits
Chapter Part Options and Corporate Finance: Basic Concepts Combinations of Options Options Call Options Put Options Selling Options Reading The Wall Street Journal Combinations of Options Valuing Options
More informationManual for SOA Exam FM/CAS Exam 2.
Manual for SOA Exam FM/CAS Exam 2. Chapter 7. Derivatives markets. c 2009. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall
More informationNumerical Methods for Option Pricing
Chapter 9 Numerical Methods for Option Pricing Equation (8.26) provides a way to evaluate option prices. For some simple options, such as the European call and put options, one can integrate (8.26) directly
More informationBuying Call or Long Call. Unlimited Profit Potential
Options Basis 1 An Investor can use options to achieve a number of different things depending on the strategy the investor employs. Novice option traders will be allowed to buy calls and puts, to anticipate
More informationChapter 11 Options. Main Issues. Introduction to Options. Use of Options. Properties of Option Prices. Valuation Models of Options.
Chapter 11 Options Road Map Part A Introduction to finance. Part B Valuation of assets, given discount rates. Part C Determination of riskadjusted discount rate. Part D Introduction to derivatives. Forwards
More informationSolutions to practice questions Study Session 11
BPP Professional Education Solutions to practice questions Study Session 11 Solution 11.1 A mandatory convertible bond has a payoff structure that resembles a written collar, so its price can be determined
More informationLikewise, the payoff of the betteroftwo note may be decomposed as follows: Price of gold (US$/oz) 375 400 425 450 475 500 525 550 575 600 Oil price
Exchange Options Consider the Double Index Bull (DIB) note, which is suited to investors who believe that two indices will rally over a given term. The note typically pays no coupons and has a redemption
More informationTHE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING
THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The BlackScholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages
More informationComputational Finance Options
1 Options 1 1 Options Computational Finance Options An option gives the holder of the option the right, but not the obligation to do something. Conversely, if you sell an option, you may be obliged to
More informationIntroduction to Options. Derivatives
Introduction to Options Econ 422: Investment, Capital & Finance University of Washington Summer 2010 August 18, 2010 Derivatives A derivative is a security whose payoff or value depends on (is derived
More informationOPTION TRADING STRATEGIES IN INDIAN STOCK MARKET
OPTION TRADING STRATEGIES IN INDIAN STOCK MARKET Dr. Rashmi Rathi Assistant Professor Onkarmal Somani College of Commerce, Jodhpur ABSTRACT Options are important derivative securities trading all over
More informationBasics of Spreading: Butterflies and Condors
1 of 31 Basics of Spreading: Butterflies and Condors What is a Spread? Review the links below for detailed information. Terms and Characterizations: Part 1 Download What is a Spread? Download: Butterflies
More informationChapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS
Chapter 8 Financial Options and Applications in Corporate Finance ANSWERS TO ENDOFCHAPTER QUESTIONS 81 a. An option is a contract which gives its holder the right to buy or sell an asset at some predetermined
More information