Chapter 17 Option Pricing Framework n Background q

Chapter 17 Option Pricing

Framework n Background q q q n One-period analysis q q q n Put-call parity, Arbitrage bound, American call option Black-Scholes Formula q q 2020/12/7 Definition and payoff Some features about option strategies Price using discount factor Derive Black-Scholes differential equation Asset Pricing 2

Background (1) n n n n Option; Call/Put; Strike Price Expiration Date Underlying Asset European/ American Option Payoff/Profit 2020/12/7 Asset Pricing 3

Background (2) 2020/12/7 Asset Pricing 4

Background (3) n Some Interesting Features of Options q High Beta (High Leverage) - q Shaping Distribution of Returns: - n Trading Hedging OTM Put + Stock But Short OTM Put Option and Long Index q q q Return Distribution Extremely Non-normal The Chance of Beating the Index for one or even five years is extremely high, but face the catastrophe risk So what kind of investments can and cannot be made is written in the portfolio management contracts. 2020/12/7 Asset Pricing 5

Background (4) n Strategies q q q 2020/12/7 By combining options of various strikes, you can buy and sell any piece of the return distribution. A complete set of option is equivalent to complete markets. Forming payoff that depends on the terminal stock price in any way Asset Pricing 6

One-period analysis n The law of one price q n No arbitrage q n n n existence of a discount factor existence of positive discount factor How to pricing option Put-Call Parity Arbitrage Bounds Discount Factors and Arbitrage Bounds Early Exercise 2020/12/7 Asset Pricing 7

Put-call parity n In the book of John C. Hull, Strategies q q 2020/12/7 (1) hold a call, write a put , same strike price (2) hold stock, borrow strike price X Asset Pricing 8

Put-call parity n According to the Law of One Price, applying to both sides for any m, We get：�� 2020/12/7 Asset Pricing 9

Arbitrage bounds n Portfolio A dominates portfolio B n Arbitrage portfolio 2020/12/7 Asset Pricing 10

Arbitrage bounds C Call value Today Call value in here X/Rf 2020/12/7 S Stock value today Asset Pricing 11

Discount factors and arbitrage bounds This presentation of arbitrage bound is unsettling for two reasons, First, you many worry that you will not be clever enough to dream up dominating portfolios in more complex circumstances. Second, you may worry that we have not dream up all of the arbitrage portfolios in this circumstance. 2020/12/7 Asset Pricing 12

Discount factors and arbitrage bounds • This is a linear program. In situations where you do not know the answer, you can calculate arbitrage bounds. (Ritchken(1985)) • The discount factor method lets you construct the arbitrage bounds 2020/12/7 Asset Pricing 13

Early exercise? n By applying the absence of arbitrage, we can never exercise an American call option without dividends before the expiration date. payoff price n S-X is what you get if you exercise now. the value of the call is greater than this value, because you can delay paying the strike, and exercising early loses the option value 2020/12/7 Asset Pricing 14

Black-Scholes Formula (Standard Approach Portfolio Construction: Review) 2020/12/7 Asset Pricing 15

Black-Scholes Formula (Standard Approach Risk Neutral Pricing: Review ) Where: 2020/12/7 Asset Pricing 16

Black-Scholes Formula (Discount Factor) n Write a process for stock and bond, then use to price the option. the Black-Scholes formula results, q (1) solve for the finite-horizon discount factor and find the call option price by taking the expectation q 2020/12/7 (2) find a differential equation for the call option and solve it backward. Asset Pricing 17

Black-Scholes Formula (Discount Factor) n n The call option payoff is The underlying stock follows The is also a money market security that pays the real interest rate In continuous time, all such discount factors are of the form: 2020/12/7 Asset Pricing 18

Method 1: price using discount factor n Use the discount factor to price the option directly: Where 2020/12/7 and are solutions to Asset Pricing 19

n How to find analytical expressions for the solutions of equations of the form (17. 2) 2020/12/7 Asset Pricing 20

Applying the Solution to (17. 2) Ignoring the term of And Proof Later We get: 2020/12/7 Asset Pricing 21

Evaluate the call option by doing the integral 2020/12/7 Asset Pricing 22

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Proof: 2020/12/7 not Affect Asset Pricing 25

Where: This is the integral under the normal distribution, with mean of and, standard variance of 1, so the integral is 1. we multiply both sides without any change. 2020/12/7 Asset Pricing 26

Method 2: derive Black-Scholes Differential Equation n Guess that solution for the call option is a function of stock price and time to expiration, C=C(S, t). Use Ito’s lemma to find derivatives of C(S, t) 2020/12/7 Asset Pricing 27

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• This is the Black-Scholes differential equation for the option price • This differential equation has an analytic solution, one standard way to solve differential equation is to guess and check, and by taking derivatives you can check that (17. 7) does satisfy (17. 8). 2020/12/7 Asset Pricing 29

Thanks Your suggestion is welcome! 2020/12/7 Asset Pricing 30