Option Pricing Principles Tutorial: for Chapter 3 of An Introduction to Derivatives and Risk Management, 8th ed. D. Chance and R. Brooks What are the upper.

Option Pricing Principles Tutorial: for Chapter 3 of An Introduction to Derivatives and Risk Management, 8th ed. D. Chance and R. Brooks What are the upper and lower limits on the prices of call and put options? What are the prices at expiration of call and put options? What can these results tell us about what call and put options are worth prior to expiration? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Principles of Option Pricing
Some important points to remember. Today’s stock price is S0. At expiration, the stock price is ST. You can create a risk-free zero coupon bond with any face value by borrowing or lending the present value of the face value. For example, if X is the face value, the present value is X(1 + r)-T. You can borrow or lend this amount and pay back or receive X at maturity. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing For calls remember that
The current price is Ce(S0,T,X) At expiration a call is worth Max(0,ST – X). So, if ST ≥ X, the call is worth ST – X If ST < X, the call is worth 0. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing A call option can never be a liability to its holder; therefore, its value must always be non-negative, that is, C(S0,T,X) ≥ 0. The following figure illustrates this result for European calls. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

European Call Call Price Stock Price (S0)
Stock Price (S0) The European call price is somewhere in the grey area, that is, between 0 and infinity Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing An American call option must always be worth at least its intrinsic value, that is Ca(S0,T,X) ≥ Max(0,S0 – X). Otherwise, investors can buy the call and exercise it, thereby earning a profit at no risk. This result is illustrated in the following figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

American Call Call Price Stock Price (S0)
Max(0,S0 – X) X Stock Price (S0) The American call price is somewhere in the grey area, which is bounded from below by the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Any call, European or American, cannot be worth more than the price of the underlying stock. That is, C(S0,T,X) ≤ S0. This is because no investor would pay more for the option than for the stock, inasmuch as the option is only the right to acquire the stock. This result is illustrated in the following figure for European calls. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

European Call Call Price Stock Price (S0)
Stock Price (S0) The European call price is somewhere in the grey area, which is bounded from above by the stock price. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing For American calls, when we impose the upper limit of the stock price with the lower limit of the intrinsic value, we obtain the following figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

American Call Call Price Stock Price (S0)
Max(0,S0 – X) X Stock Price (S0) The American call price is somewhere in the grey area, which is bounded from above by the stock price and from below by the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing At expiration, both American calls and European calls are worth the intrinsic value. At this point both options will be exercised if in-the-money, or not exercised if not in-the-money. This result is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Value of a Call at Expiration
Call Price C(ST,0,X) Max(0,ST – X) X Stock Price at Expiration (ST) At expiration, both the European and American call values must equal the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing What effect does time to expiration have on the price of a call option? Consider two otherwise identical American calls with different times to expiration, T1 and T2, where T2 is the longer time. Which price is higher, Ca(S0,T1,X) or Ca(S0,T2,X)? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing The expiring (shorter-term) option is worth the intrinsic value. The longer-term option is worth the intrinsic value plus more. T1 (shorter-term option expires) T2 (longer-term option expires) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing The longer-term option can do everything the shorter-term option can do, plus it has additional time for further gains from favorable stock price movements. Thus, the value today must reflect this feature. We cannot yet make such a statement about European calls. Ca(S0,T2,X) ≥ Ca(S0,T1,X) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing The effect of time on an option’s price is called time value. An option price consists of intrinsic value plus time value. Time value will be greatest when the stock price is close to the exercise price. Time value goes away as the option approaches expiration. This is called time value decay. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing We are interested in what an option is worth prior to expiration. A complete understanding of that point requires an option pricing model. For now, just understand that the relationship between the option price and the stock price is non-linear and bounded from below by the intrinsic value for American calls. The call price curve, however, converges to the intrinsic value as expiration approaches. This is the time value decay. This point is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for an American Call
Call Price Ca(S0,T,X) Max(0,S0 – X) X Stock Price (S0) Prior to expiration, the American call is a non-linear function of the stock price and lies above the intrinsic value. As expiration approaches, the price converges to the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Let us examine how the exercise price affects the value of a call option. Consider two European calls with different exercise prices, X1 and X2, where X1 < X2. Their prices are Ce(S0,T,X1) and Ce(S0,T,X2). Which is greater? In other words, does a higher exercise price make a call worth more or less? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Construct a portfolio called Portfolio A as follows: Buy the call with exercise price X1. Sell the call with exercise price X2. Your initial outlay will be Ce(S0,T,X1) - Ce(S0,T,X2). Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Construct a portfolio called Portfolio B as follows: Buy a risk-free zero coupon bond paying X2 - X1 at expiration. This portfolio will cost (X2 - X1) (1 + r)-T Now let’s look at what happens at expiration. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing ( < ) ( ≤ ) ( = ) Value at Expiration ST ≤ X1
X1 ≤ ST > X2 ST ≥ X2 Portfolio A Long X1 Call ST – X1 ST – X1 -(ST – X2) Short X2 Call Total ST – X1 ≥ 0 X2 – X1 > 0 ( < ) ( ≤ ) ( = ) Portfolio B X2 – X1 > 0 X2 – X1 > 0 X2 – X1 > 0 Bond Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Ce(S0,T,X1) ≥ Ce(S0,T,X2)
Note first that Portfolio A always pays off zero or a higher amount. It never has a negative payoff. Consequently, the cost of investing in A must be zero or positive; therefore Ce(S0,T,X1) - Ce(S0,T,X2) ≥ 0 and Thus, the lower exercise price call must be worth no less than the higher exercise price call. Ce(S0,T,X1) ≥ Ce(S0,T,X2) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Now recall that Portfolio B always performs at least as well as Portfolio A. Therefore, the value of B must be no less than the value of A: If that is true, then this must be true: (X2 – X1)(1 + r)-T ≥ Ce(S0,T,X1) - Ce(S0,T,X2) (X2 – X1) ≥ Ce(S0,T,X1) - Ce(S0,T,X2) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing If the calls are American, however, A can outperform B by exercising early for X2 – X1 plus interest until expiration. For American calls, we have to change the statement slightly to the last one on the previous page: (X2 – X1) ≥ Ca(S0,T,X1) - Ca(S0,T,X2) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing For European calls, the minimum value is not the intrinsic value because these options cannot be exercised prior to expiration. Construct a portfolio called Portfolio A, consisting of one share of stock. This will cost S0. Construct a portfolio called Portfolio B, consisting of one call and a risk-free bond paying X at expiration. This portfolio will cost Ce(S0,T,X) + X(1 + r)-T. Let us see what happens at expiration. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Value at Expiration ST ≤ X ST > X Portfolio A
Stock ST ST Portfolio B Call ST - X B is better The same Bond X X Total X ST Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing The value of B can, therefore, never be less than the value of A: Ce(S0,T,X) + X(1 + r)-T ≥ S0. Rearranging If the right-hand side is negative, we know that the call cannot be worth less than zero. Therefore, the lower bound is Ce(S0,T,X) ≥ S0 - X(1 + r)-T Ce(S0,T,X) ≥ Max(0,S0 - X(1 + r)-T) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing To repeat, for European calls, the minimum value is not the intrinsic value because these options cannot be exercised prior to expiration. The minimum value is the lower bound of Max(0,S0 – X(1 + r)-T). The lower bound is above the intrinsic value because X(1 + r)-T < X. This point is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for a European Call
Call Price Ce(S0,T,X) Max(0,S0 – X(1 + r)-T) Max(0,S0 – X) X Stock Price (S0) Prior to expiration, the European call is a non-linear function of the stock price and lies above the minimum value. As expiration approaches, the price converges to the minimum value, which itself converges to the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing We have been assuming no dividends on the stock during the option’s life. Therefore, there is no reason to exercise early. Early exercise does three things Throws away any remaining time value Pays out the exercise price early instead of at expiration Gives up the right to decide later if you want the stock Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Even if the call holder is tempted to exercise the call because he feels the stock is going no higher, he should consider two things: If the stock is going no higher, why does he want it so urgently? Why not sell the call? Its price is at least S0 – X(1 + r)-T, which is more than what he gets for exercising it, S0 – X. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Thus, American calls on non-dividend paying stocks will never be exercised early. Therefore, prior to expiration, the American call price will equal the European call price, that is, Ca(S0,T,X) = Ce(S0,T,X). The lower bound for the American call price is, therefore, Max(0,S0 – X(1 + r)-T) and not Max(0,S0 – X). Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for an American Call Redux
Call Price Ca(S0,T,X) Max(0,S0 – X(1 + r)-T) Max(0,S0 – X) X Stock Price (S0) Prior to expiration, the American call price curve will equal the European call price curve and must lie above the lower bound, which is above the intrinsic value. This establishes the lower bound as the minimum price of an American call. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Dividends affect the price of a call option.
Define the present value of the dividends during the life of the option as The stock price minus the present value of the dividends is Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Then replace S0 with S0′:
For a currency paying interest at the rate ρ: Lower bound: Max(0,S0′ – X(1 + r)-T) Lower bound: Max(0,S0(1 + ρ)-T – X(1 + r)-T) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Dividends, however, can trigger early exercise of American calls. If the size of the dividend is sufficiently large in relation to the time value, exercise just before the ex-dividend instant can be justified. The option holder gives up the time value and captures the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing There is no other time prior to expiration at which calls should be exercised early. Exercise at the instant before the stock goes ex-dividend provides the minimal loss of time value. The stock price drops by the amount of the dividend but the investor earns the dividend, thereby avoiding this loss from the stock price fall. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Call Option Pricing Two other factors affect a call option price:
Volatility makes a call option be worth more by increasing the payoffs in outcomes in which the stock price is high not decreasing the payoffs in outcomes in which the stock price is low Interest rates make a call option be worth more but the effect is not very strong and difficult to explain without an option pricing model. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing For puts remember that
The current price is Pe(S0,T,X) At expiration a put is worth Max(0,X - ST). So, if ST ≤ X, the put is worth X - ST If ST > X, the put is worth 0. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing A put option can never be a liability to its holder; therefore, its value must always be non-negative, that is, P(S0,T,X) ≥ 0. The following figure illustrates this result for European puts. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

European Put Put Price Stock Price (S0)
Stock Price (S0) The European put price is somewhere in the grey area, that is, between 0 and infinity Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing An American put option must always be worth at least its intrinsic value, that is Pa(S0,T,X) ≥ Max(0,X - S0). Otherwise, investors can buy the put and exercise it, thereby earning a profit at no risk. This result is illustrated in the following figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

American Put Put Price Stock Price (S0)
X Max(0,X - S0) X Stock Price (S0) The American put price is somewhere in the grey area, which is bounded from below by the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing A put can never pay off more than the exercise price (which occurs if the stock price is zero at expiration). Because a European put cannot be exercised today its maximum value today is the present value of the exercise price. This point is illustrated in the following figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

European Put Put Price Stock Price (S0)
X X(1 + r)-T X Stock Price (S0) The European put price is somewhere in the grey area, which is bounded from above by the present value of the exercise price. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Likewise, an American put cannot pay off more than the exercise price, but it can do so immediately. Hence, its maximum value is the exercise price. This point is illustrated in the following figure, which combines the upper and lower bounds established for American puts. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

American Put Put Price Stock Price (S0)
X Max(0,X - S0) X Stock Price (S0) The American put price is somewhere in the grey area, which is bounded from above by the exercise price and from below by the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing At expiration, both American puts and European puts are worth the intrinsic value. At this point either option will be exercised if in-the-money, or not exercised if not in-the-money. This result is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Value of a Put at Expiration
Put Price X P(ST,0,X) Max(0,X - ST) X Stock Price at Expiration (ST) At expiration, the put price is equal to the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing What effect does time to expiration play on the price of a put option? Consider two otherwise identical American puts with different times to expiration, T1 and T2, where T2 is the longer time. Which price is higher, Pa(S0,T1,X) or Pa(S0,T2,X)? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing The expiring (shorter-term) option is worth the intrinsic value. The longer-term option is worth the intrinsic value plus more. T1 (shorter-term option expires) T2 (longer-term option expires) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing As with calls, the longer-term option can do everything the shorter-term option can do, plus it has additional time for further gains from favorable stock price movements. Thus, the value today must reflect this feature: Pa(S0,T2,X) ≥ Pa(S0,T1,X) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing As noted previously, the effect of time on an option’s price is called time value. For calls and puts, the option price consists of intrinsic value plus time value. Time value will be greatest when the stock price is close to the exercise price. Time value goes away as the option approaches expiration. This is called time value decay. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Unlike calls and American puts, however, we cannot assert that the shorter-term European put is worth less. Longer time to expiration does give the put more time value, but it means waiting longer to exercise and receive the exercise price. (Note: with calls, longer expiration means more waiting longer to pay the exercise price.) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing As with calls, a complete understanding of the put price in relation to the stock price requires an option pricing model. For now, just understand that the relationship between the put option price and the stock price is non-linear and bounded from below by the intrinsic value for American puts. The put price curve, however, converges to the intrinsic value as expiration approaches. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing American put options might be exercised early.
For example, if the stock is at zero, there is no reason for the holder of an American put to delay exercising. But the stock does not have to go all the way to zero. There is a critical price at which a put would be exercised early. The American put value can never fall below this price. This point is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for an American Put
Put Price X Pa(S0,T,X) Max(0,X - S0) Critical stock price for early exercise X Stock Price (S0) Prior to expiration, the American put is a non-linear function of the stock price and lies no lower than the intrinsic value. As expiration approaches, the price converges to the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Let us examine how the exercise price affects the value of a put option. Consider two European puts with different exercise prices, X1 and X2, where X1 < X2. Their prices are Pe(S0,T,X1) and Pe(S0,T,X2). Which is greater? In other words, does a higher exercise price make a put worth more or less? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Construct a portfolio called Portfolio A as follows: Buy the put with exercise price X2. Sell the put with exercise price X1. Your initial outlay will be Pe(S0,T,X2) - Pe(S0,T,X1). Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Construct a portfolio called Portfolio B as follows: Buy a risk-free zero coupon bond paying X2 - X1 at expiration. This portfolio will cost (X2 - X1) (1 + r)-T Now let’s look at what happens at expiration. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing ( = ) ( ≤ ) ( < ) Value at Expiration ST ≤ X1
X1 ≤ ST > X2 ST ≥ X2 Portfolio A Long X2 Put X2 - ST X2 – ST Short X1 Put -(X1 – ST) Total X2 – X1 > 0 X2 – ST ≥ 0 ( = ) ( ≤ ) ( < ) Portfolio B X2 – X1 > 0 X2 – X1 > 0 X2 – X1 > 0 Bond Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Pe(S0,T,X2) ≥ Pe(S0,T,X1)
Note first that Portfolio A always pays off zero or a higher amount. It never has a negative payoff. Consequently, the cost of investing in A must be zero or positive; therefore Pe(S0,T,X2) - Pe(S0,T,X1) ≥ 0 and Thus, the higher exercise price put must be worth no less than the lower exercise price put. Pe(S0,T,X2) ≥ Pe(S0,T,X1) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Now recall that Portfolio B always performs at least as well as Portfolio A. Therefore, the value of B must be no less than the value of A: If that is true, then this must be true: (X2 – X1)(1 + r)-T ≥ Pe(S0,T,X2) - Pe(S0,T,X1) (X2 – X1) ≥ Pe(S0,T,X2) - Pe(S0,T,X1) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing If the puts are American, however, A can outperform B by exercising early for X2 – X1 plus interest until expiration. For American puts, we have to change the statement slightly to the last one on the previous page: (X2 – X1) ≥ Pa(S0,T,X2) - Pa(S0,T,X1) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing For European puts, the minimum value is not the intrinsic value because these options cannot be exercised prior to expiration. Construct a portfolio called Portfolio A, consisting of one share of stock. This will cost S0. Construct a portfolio called Portfolio B, consisting of one short put and a risk-free bond paying X at expiration. This portfolio will cost -Pe(S0,T,X) + X(1 + r)-T. Let us see what happens at expiration. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Value at Expiration ST < X ST ≥ X Portfolio A
Stock ST ST Portfolio B Short put -(X – ST) The same A is better Bond X X Total ST X Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing The value of B can, therefore, never be more than the value of A: -Pe(S0,T,X) + X(1 + r)-T < S0. Rearranging: If the right-hand side is negative, we know that the put cannot be worth less than zero. Therefore, the lower bound is Pe(S0,T,X) ≥ X(1 + r)-T - S0 Pe(S0,T,X) ≥ Max(0,X(1 + r)-T – S0) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing To repeat, for European calls, the minimum value is not the intrinsic value because these options cannot be exercised prior to expiration. The minimum value is the lower bound of Max(0, X(1 + r)-T – S0). The lower bound is below the intrinsic value because X(1 + r)-T < X. This point is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for a European Put
Put Price X X(1 + r)-T Pe(S0,T,X) X(1 + r)-T X Stock Price (S0) Prior to expiration, the European put is a non-linear function of the stock price and lies above the minimum value. As expiration approaches, the price converges to the minimum value, which itself converges to the intrinsic value. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Because American puts might be exercised early and their lower bound is the intrinsic value, while the lower bound for European puts is the minimum value, the American put price will be no lower than the European put price, that is, Pa(S0,T,X) ≥ Pe(S0,T,X). This point is illustrated in the next figure. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

The Price Curve for European and American Puts
Put Price X X(1 + r)-T Pa(S0,T,X) Pe(S0,T,X) X(1 + r)-T X Stock Price (S0) Prior to expiration, the American put price lies no lower than that of the European put price and both are bounded from below by their established minimum values. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing As with calls, dividends affect the price of a put option. Recall that we define the stock price minus the present value of the dividends as Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Then replace S0 with S0′
For a currency paying interest at the rate ρ Lower bound: Max(0, X(1 + r)-T - S0′ Lower bound: Max(0, X(1 + r)-T - S0(1 + ρ)-T Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing With puts, dividends do not trigger early exercise. In fact, they discourage it. If early exercise is justified, it will be done immediately after the stock goes ex-dividend. This results in capturing the benefit of the stock price decline that occurs as the stock goes ex-dividend. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put Option Pricing Two other factors affect a put option price:
Volatility makes a put option be worth more by increasing the payoffs in outcomes in which the stock price is low not decreasing the payoffs in outcomes in which the stock price is high Interest rates make a put option be worth less but the effect is not very strong and difficult to explain without an option pricing model. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Put-call parity expresses the relationship between call and put prices. It is a form of relative pricing, in which we can determine the price of the put in terms of the price of the call or vice versa. But we have to know either the put or the call price to know the other. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity To understand put-call parity, remember that at expiration European puts are worth X – ST if X > ST (in-the-money) and 0 if X ≤ ST (at- or out-of-the-money). European calls are worth ST – X if ST > X (in-the-money) and 0 if ST ≤ X (at- or out-of-the-money). The stock is worth ST regardless of whether ST is more than X or not. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Let us construct a portfolio, called Portfolio A, that consists of one share of stock and a European put. This is called a protective put. To buy this portfolio, we will have to pay the price of the stock, S0, and the price of the put, Pe(S0,T,X). At expiration, this portfolio will be worth the following. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Value at Expiration Instrument ST ≤ X ST > X Stock
Total X ST Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Now construct a portfolio, called Portfolio B, that consists of one call and a zero-coupon bond with face value set at the exercise price X. This is called a fiduciary call. To buy this portfolio, we will have to pay the price of the call, Ce(S0,T,X) and the price of the bond, X(1 + r)-T. At expiration, this portfolio will be worth the following. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Value at Expiration Instrument ST ≤ X ST > X Call
ST - X Bond X X Total X ST Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Regardless of which of the two outcomes occurs (ST ≤ X or ST ≤ X), Portfolios A and B produce the same result. Portfolios that produce the same result must require the same initial investment. Thus, their initial values must be the same: Pe(S0,T,X) + S0 = Ce(S0,T,X) + X(1 + r)-T This is put-call parity. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity What if Pe(S0,T,X) + S0 > Ce(S0,T,X) + X(1 + r)-T?
The left-hand side is too high and/or the right-hand side is too low, so Sell the instruments on the left-hand side and receive Pe(S0,T,X) + S0 Buy the instruments on the right-hand side and pay Ce(S0,T,X) + X(1 + r)-T You will have a net cash inflow of Pe(S0,T,X) + S0 - Ce(S0,T,X) - X(1 + r)-T at the start. Now look at what happens at expiration Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Value at Expiration Instrument ST ≤ X ST > X
Regardless of the outcome, we will neither pay nor receive anything at expiration. But we can keep the amount received at the start. Short Put -(X – ST) Short Stock -ST -ST Long Call ST - X Risk-free Bond X X Total Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Our cash flows look like this.
Pe(S0,T,X) + S0 Ce(S0,T,X) X(1 + r)-T +$$ $0 0 (start) T (expiration) Would anyone object to having this opportunity? Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Other investors will do this transaction, which will put downward pressure on the value of Portfolio A and upward pressure on the value of Portfolio B until equality is reached. If Portfolio A is worth less than Portfolio B, investors will do the opposite transactions. Equilibrium is then reached with put-call parity holding as stated. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Pe(S0,T,X) + S0 = Ce(S0,T,X) + X(1 + r)-T
Put-Call Parity Look at the equation for put-call parity: Note that positive signs mean long or buying, negative signs mean short or selling. So the above says (and we proved): Pe(S0,T,X) + S0 = Ce(S0,T,X) + X(1 + r)-T buy put + buy stock = buy call + buy bond (lending) Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Using algebra to rearrange the put-call parity equation, now look at some other combinations. Equation Meaning Pe(S0,T,X) = Ce(S0,T,X) – S0 + X(1 + r)-T buy put = buy stock, short call, buy bond Ce(S0,T,X) = Pe(S0,T,X) + S0 - X(1 + r)-T buy call = buy put, buy stock, issue bond (borrow) S0 = Ce(S0,T,X) + X(1 + r)-T - Pe(S0,T,X) buy stock = buy call, buy bond, sell put X(1 + r)-T = Pe(S0,T,X) + S0 - Ce(S0,T,X) buy bond = buy put, buy stock, sell call Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity Changing the signs, we can get some more combinations.
Equation Meaning -Pe(S0,T,X) = -Ce(S0,T,X) + S0 - X(1 + r)-T sell put = sell stock, buy call, sell bond (borrow) -Ce(S0,T,X) = -Pe(S0,T,X) - S0 + X(1 + r)-T sell call = sell put, sell stock, buy bond -S0 = -Ce(S0,T,X) - X(1 + r)-T + Pe(S0,T,X) sell stock = sell call, sell bond (borrow), buy put -X(1 + r)-T = -Pe(S0,T,X) - S0 + Ce(S0,T,X) sell bond (borrow) = sell put, sell stock, buy call Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity If you understand this material, you should be able to convince yourself that each of these eight results is true. Do this by constructing a table showing that the payoffs of the combination on the left-hand sides are the same for both outcomes (ST ≤ X and ST > X) as the payoffs of the respective combinations on the right-hand sides. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity When there are dividends, replace S0 with S0′:
For options on a currency: Pe(S0′,T,X) + S0′ = Ce(S0′,T,X) + X(1 + r)-T Pe(S0,T,X) + S0(1 + ρ)-T = Ce(S0,T,X) + X(1 + r)-T Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Put-Call Parity For American options, the rule is more complex and the equation is an inequality: Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

This completes the tutorial.
If you understand this material, you are now ready to begin learning option pricing models. Chance-Brooks, 8e © 2010 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.

Option Pricing Principles Tutorial: for Chapter 3 of An Introduction to Derivatives and Risk Management, 8th ed. D. Chance and R. Brooks What are the upper.

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Option Pricing Principles Tutorial: for Chapter 3 of An Introduction to Derivatives and Risk Management, 8th ed. D. Chance and R. Brooks What are the upper.

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Presentation on theme: "Option Pricing Principles Tutorial: for Chapter 3 of An Introduction to Derivatives and Risk Management, 8th ed. D. Chance and R. Brooks What are the upper."— Presentation transcript:

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