Notes for Lecture 5 (February 28)

Transcription

1 Midterm 7:40 9:00 on March 14 Ground rules: Closed book. You should bring a calculator. You may bring one 8 1/2 x 11 sheet of paper with whatever you want written on the two sides. Suggested study questions for midterm: The questions and problems at the end of the chapters. I cannot give you any better examples. The answers are in Hull s Solutions Manual. 1

2 Option pricing will occupy us for most of the rest of the course. Black-Scholes is the dominant theory of option pricing. Black-Scholes assumes that stock prices follow geometric Brownian motion. This assumption is at best approximately true. So we have a lot to learn: Facts and insights about options pricing that do not depend on the assumption of geometric Brownian motion (Chapters 7, 8). The Black-Scholes theory (Chapters 10, 11). How to do Black-Scholes calculations (Chatpers 9, 11, DerivaGem). What to do about the inexactitude of Black-Scholes in practice (Chapters 13, 17). 2

3 Chapter 6 (Today) Options Markets Important institutional information. I will hurry over this chapter today, and I am not assigning homework from it, but it will be on the exams. Chapter 7 (Today) Properties of Stock Option Prices This chapter, which I will cover thoroughly in lecture today, emphasizes properties of option prices that depend only on arbitrage arguments. These properties are more reliable than the Black-Scholes formula, because they do not depend on the assumption of geometric Brownian motion. Chapter 8 (March 7) Trading Strategies Involving Options This chapter is mainly concerned with how other derivatives can be constructed using the call and put options that are traded on option exchanges. Chapter 10 (March 14, before midterm) Model of the Behavior of Stock Prices This chapter introduces the mathematical foundation of the Black-Scholes theory (geometric Brownian motion). This topic is too mathematical to cover in depth in this course. But I will cover the most important points in a lecture on the evening of the midterm. 3

4 Chapter 9 (March 28) Introduction to Binomial Trees This chapter introduces a commonly used computational method for solving the Black-Scholes equation. Chapter 11 (April 4) The Black-Scholes Model This challenging chapter covers the basic theory of Black-Scholes, including the Black-Scholes differential equation and the Black-Scholes pricing formula. Chapter 12 (April 11) Options on Stock Indices, Currencies and Futures This chapter considered only options on stocks. This chapter extends the theory to other options. Chapter 13 (April 18) The Greek Letters This chapter considers how options that are priced using Black-Scholes can then be hedged. Chapter 14 (April 25) Value at Risk This chapter is concerned with market risk, not with options. Chapter 17 (May 2) Volatility Smiles and Alternatives to Black-Scholes What to do when geometric Brownian motion is violated so grossly that Black-Scholes does not come close. 4

5 Put-Call Parity; No Dividends c + Xe -rt = p + S 0 Consequences: Bound on price of European call: c S 0 Xe -rt Bound on price of European put: p Xe -rt S 0 Fuller story: S 0 C = c S 0 Xe -rt X P p Xe -rt S 0 Arbitrage is possible if one of these conditions is violated. (More precisely: if it is violated by a margin exceeding transaction costs.) (In the case of no dividends, early exercise of an American call is never advantageous; this is why C = c.) 5

6 Put-Call Parity; No Dividends c + Xe -rt = p + S 0 --Portfolio A: European call + present value of strike price --Portfolio B: European put + the stock Both are worth max(s 0,X) at maturity. So both are worth the same today. Is this an arbitrage argument? If c + Xe -rt > p + S 0, then you sell portfolio A and buy portfolio B: At time 0: --Go short in the call and borrow present value of strike price --Buy the put and the stock --Pocket the arbitrage profit (c + Xe -rt ) (p + S 0 ). At time T: If X > S T, --Counterparty will not exercise call. --Use put to sell stock for X; pay off loan. If S T > X, --Your put is worthless, but counterparty will exercise call. --Sell stock to counterparty for X; pay off loan. 6

7 --Portfolio A: European call + present value of strike price --Portfolio B: European put + the stock If p + S 0 > c + Xe -rt, then you buy portfolio A and sell portfolio B: At time 0: --Go short in the put and the stock. --Buy the call and lend the present value of the strike price. --Pocket the arbitrage profit (p + S 0 ) (c + Xe -rt ). At time T: If X > S T, --Your call is worthless, but counterparty will exercise put. --Buy stock from put counterparty for X, covering the short position in the stock. If S T > X, --Counterparty will not exercise put. --Use stock from call counterparty for X, covering the short position in the stock. 7

8 Put-Call Parity; No Dividends c + Xe -rt = p + S 0 c p = S 0 Xe -rt Consequence: The European call and European put have the same price when the strike price X is chosen so that Xe -rt = S 0 or X = S 0 e rt This is the forward price!! The European call and European put with a given maturity have the same price when the strike price is chosen to equal the stock s forward price. 8

9 This example is on pages c = 3 S 0 = 20 T = 1 r = 10% X = 18 D = 0 S 0 Xe -rt = 20 18e -0.1 = 3.7 c S 0 Xe -rt is violated! What is the arbitrage opportunity? The call is too cheap relative to the stock. So we buy the call and short the stock. Now: Shorting one share of the stock yields $20. We spend $3 to buy one call, and we put the other $17 in the bank. One year from now: Our $17 has become $17e.1 = $ If S 1 > 18, we exercise the call at a cost of $18, in order to close out our short position in the stock. Net profit: $.79 If S 1 18, we buy a share of the stock in the market to close out the short position, for a greater net profit. 9

10 This example is on page 173. p = 1 S 0 = 37 T = 0.5 r = 5% X = 40 D = 0 Xe -rt S 0 = 40e -0.05x0.5 37= 2.10 p Xe -rt S 0 is violated! What is the arbitrage opportunity? The put and the stock are too cheap relative to the strike price. So we buy the put and the stock. Now: Borrow $38. Buy one put for $1. Buy one share of stock for $37. Six months from now: We owe $38e 0.05x0.5 = $ If S , we exercise the put obtaining $40 for our stock share, for a net profit of $1.04 If S 0.5 > 40, we sell our share in the market, for an even greater profit. 10

11 This example is on page 174. c = 3 p = 2.25 S 0 = 31 T = 0.25 r = 10% X = 30 D = 0 c + Xe -rt = p + S 0 = c + Xe -rt = p + S 0 is violated! What is the arbitrage opportunity? The call is too cheap relative to the put and the stock. So buy the call and short the put and stock. Now: Shorting the put and stock yields $ $31 = $ The call costs $3. Bank the remaining $ Three months from now: Our $30.25 has become $30.25e 0.1x0.25 = $ If S , the put is exercised by the counterparty. If S 0.25 > 30, we exercise the call. Either way, we spend $30 to buy the share to cover our short stock position, leaving us a net profit of $

12 Question 7.1 An investor buys a call with strike price X and writes a put with the same strike price. Describe the investor s position. 12

13 Question 7.1: An investor buys a call with strike price X and writes a put with the same strike price. Describe the investor s position. Answer: If S T > X, then the counterpary will not exercise the put, but the investor will exercise the call to make a profit of S T X. If X > S T, then the investor will not exercise the call, but the counterparty will exercise the put to make a profit of X S T, which represents a loss of S T X. for the investor. So the investor s payoff is S T X in any case. By the way: This is the same as the long position in forward contract with delivery price X. When X is equal to the forward price, the long forward contract is worth zero. So the call and put are equally valuable in this case. 13

14 Question 7.2 Why is an American option worth at least as much as a European option on the same asset with the same strike price and exercise date? Question 7.3 Why is an American option always worth at least as much as its intrinsic value? The intrinsic value of a call option at strike price X is max(s X,0) where S is the current value of the stock. 14

15 European Options; No Dividends: Put-Call Parity: c + Xe -rt = p + S 0 or c p = S 0 Xe -rt American Options; No Dividends: Put-Call Relationship: S 0 X C P S 0 Xe -rt Problem 7.19 asks you to prove this. To get C P S 0 Xe -rt, use c p = S 0 Xe -rt C = c P p To get S 0 X C P, use C = c and compare: Portfolio A: One European call plus cash X (This costs c + X, or C + X.) Portfolio B: One American put plus one share (This costs P + S 0.) We need to show that A is worth more than B whether the put option is exercised early or not. If the put option is exercised early, say at time τ, then B is worth X at that time, and A is worth at least Xe rτ. If the put option is not exercised early, then B is worth max(s T,X), while A is worth max(s T X,0) + Xe rt, or max(s T,X) + (Xe rt X). 15

16 Put-Call Parity: c + Xe -rt = p + S 0 For American Options: S 0 X C P S 0 Xe -rt Example 7.3 (page 178) The risk-free rate is 10%. ECI is a non-dividend paying stock. The current price per share of ECI is $19. An American call option with exercise price $20 and maturity in 5 months is worth $1.50. Question: What is the value of a European put with exercise price $20 and maturity in 5 months? Answer: By put-call parity, p = c + Xe -rt S 0 = e -0.1x5/12 19 = $1.68. Question: What can you say about the value of an American put with exercise price $20 and maturity in 5 months? Answer: By put-call parity, S 0 X C P S 0 Xe -rt or P 19 20e -0.1x5/12 or $1.68 P $

17 No Dividend Dividend with present value D Put-Call Parity c + Xe -rt = p + S 0 c + D + Xe -rt = p + S 0 c p = S 0 Xe -rt Lower Bound on Price of European Call c S 0 Xe -rt c S 0 D Xe -rt Lower Bound on Price of European Put p Xe -rt S 0 p D + Xe -rt S 0 Relationship for American Options S 0 X C P S 0 Xe -rt S 0 D X C P S 0 Xe -rt 17