© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-1 Properties of Option Prices (cont’d) Early exercise for American options Calls on a non-dividend-paying stock Early exercise is not optimal if the price of an American call prior to expiration satisfies C Amer (S t, K, T – t) > S t – K If this inequality holds, you would lose money by early- exercising (receiving S t – K ) as opposed to selling the option (receiving C Amer (S t, K, T – t) > S t – K).

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-2 Properties of Option Prices (cont’d) Proof From the put-call parity, we have Since C Amer  C Eur, we have 

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-3 Properties of Option Prices (cont’d) Alternative Proof From the put-call parity, we have 

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-4 Properties of Option Prices (cont’d) Early-exercising has the following effects: 1.Throw away the implicit put protection should the stock later move below the strike price. 2.Accelerate the payment of the strike price. 3.(No early-exercise) The possible loss from deferring receipt of the stock. However, when there is no dividends, we lose nothing by waiting to take physical possession of the stock.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-5 Properties of Option Prices (cont’d) Exercising calls just prior to a dividend If the stock pays dividends, the parity relationship is The early exercise is possible if If dividends do make early exercise rational, it will be optimal to exercise at the last moment before the ex- dividend date.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-6 Properties of Option Prices (cont’d) Early exercise for puts The put will never be exercised as long as P > K – S. Supposing that the stock pays no dividends, parity for the put is The no-exercise condition, P > K – S, then implies The early exercise is possible if the call is sufficiently valueless.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-7 Properties of Option Prices (cont’d) Time to Expiration –An American option (both put and call) with more time to expiration is at least as valuable as an American option with less time to expiration. This is because the longer option can easily be converted into the shorter option by exercising it early. –A European call option on a non-dividend-paying stock will be at least as valuable as an otherwise identical option with a shorter time to expiration. This is because a European call on a non-dividend-paying stock has the same price as an otherwise identical American call.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-8 Properties of Option Prices (cont’d) –European call options on dividend-paying stock and European puts may be less valuable than an otherwise identical option with less time to expiration. This is because the one with less time to expiration can receive the dividend earlier.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved.10-9 Properties of Option Prices (cont’d) Different strike prices (K 1 < K 2 < K 3 ), for both European and American options –A call with a low strike price is at least as valuable as an otherwise identical call with a higher strike price: –A put with a high strike price is at least as valuable as an otherwise identical call with a low strike price : –The premium difference between otherwise identical calls with different strike prices cannot be greater than the difference in strike prices:

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Properties of Option Prices (cont’d) –The premium difference for otherwise identical puts also cannot be greater than the difference in strike price: –Premiums decline at a decreasing rate for calls with progressively higher strike prices. The same is true for puts as strike prices decline (Convexity of option price with respect to strike price):

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Proof: Recall asymmetry butterfly spread for which implies (*)

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved –These statements are all true for both European and American options. Properties of Option Prices (cont’d)

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Properties of Option Prices (cont’d) Exercise and Moneyness –If it is optimal to exercise an option, it is also optimal to exercise an otherwise identical option that is more in-the- money. Example Suppose a call option on a dividend-paying stock has a strike price of $50, and the stock price is $70. Also suppose that it is optimal to exercise the option. The option must sell for $70 – $50 = $20. What can we say about the premium of a 40-strike option?

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Properties of Option Prices (cont’d) Since the 40-strike call is optimal to exercise.

Chapter 10 Binomial Option Pricing: Basic Concepts

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Introduction to Binomial Option Pricing The binomial option pricing model enables us to determine the price of an option, given the characteristics of the stock or other underlying asset. The binomial option pricing model assumes that the price of the underlying asset follows a binomial distribution—that is, the asset price in each period can move only up or down by a specified amount. The binomial model is often referred to as the “Cox- Ross-Rubinstein pricing model”.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved A One-Period Binomial Tree Example –Consider a European call option on the stock of XYZ, with a $40 strike and 1 year to expiration. –XYZ does not pay dividends, and its current price is $41. –The continuously compounded risk-free interest rate is 8%. –The following figure depicts possible stock prices over 1 year, i.e., a binomial tree $60 $41 $30

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Computing the Option Price Next, consider two portfolios: –Portfolio A: buy one call option. –Portfolio B: buy 2/3 shares of XYZ and borrow $ at the risk-free rate. Costs –Portfolio A: the call premium, which is unknown. –Portfolio B: 2/3  $41 – $ = $8.871.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Computing the Option Price (cont’d) Payoffs: –Portfolio A:Stock Price in 1 Year $30$60 Payoff 0$20 –Portfolio B:Stock Price in 1 Year $30$60 2/3 purchased shares$20$40 Repay loan of $ – $20 –$20 Total payoff 0 $20

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Computing the Option Price (cont’d) Portfolios A and B have the same payoff. Therefore –Portfolios A and B should have the same cost. Since Portfolio B costs $8.871, the price of one option must be $ The idea that positions that have the same payoff should have the same cost is called the law of one price.

© 2013 Pearson Education, Inc., publishing as Prentice Hall. All rights reserved Computing the Option Price (cont’d) –There is a way to create the payoff to a call by buying shares and borrowing. Portfolio B is a synthetic call. –One option has the risk of 2/3 shares. The value 2/3 is the delta () of the option: the number of shares that replicates the option payoff.