Option Valuation
Option Valuation
Chapter 21
Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price – exercise price for a put option: exercise price – stock price
the intrinsic value for out-the-money or at-themoney options is equal to 0 time value of an option = difference between actual call price and intrinsic value as time approaches expiration date, time value goes to zero
21-2
Determinants of Option Values
Call + – + + + – Put – + + + – +
Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend rate of stock
21-3
Binomial Option Pricing consider a stock that currently sells at S0 the price an either increase by a factor u or fall by a factor d (probabilities are irrelevant) consider a call with exercise price X such that dS0 < X < uS0 hence, the evolution of the price and of the call option value is uS0 Cu = (uS0 – X) C S0 dS0 Cd = 0
21-4
Binomial Option Pricing (cont.) now, consider the payoff from writing one call option and buying H shares of the stock, where Cu − Cd uS0 − X H= = uS0 − dS0 uS 0 − dS0 the value of this investment at expiration is Up Down Payoff of stock HuS0 HdS0 Payoff of calls –(uS0 – X) 0 Total payoff HdS0 HdS0
21-5
Binomial Option Pricing (cont.) hence, we obtained a risk-free investment with end value HdS0 arbitrage argument: the current value of this investment should be equal to its present discounted value using the risk-free rate H is called the hedge ratio (the ratio of the range of call option payoffs and the range of the stock price) the argument is based on perfect hedging, or replication (the payoff of the investment replicates a risk-free bond)
21-6
Binomial Option Pricing – Algorithm
1. given the end of period stock prices, uS0 and dS0, calculate the payoffs of the call option, Cu and Cd 2. find the hedge ratio H = (Cu – Cd)/(uS0 – dS0) 3.
Chapter 21
Intrinsic and Time Value intrinsic value of in-the-money options = the payoff that could be obtained from the immediate exercise of the option for a call option: stock price – exercise price for a put option: exercise price – stock price
the intrinsic value for out-the-money or at-themoney options is equal to 0 time value of an option = difference between actual call price and intrinsic value as time approaches expiration date, time value goes to zero
21-2
Determinants of Option Values
Call + – + + + – Put – + + + – +
Stock price Exercise price Volatility of stock price Time to expiration Interest rate Dividend rate of stock
21-3
Binomial Option Pricing consider a stock that currently sells at S0 the price an either increase by a factor u or fall by a factor d (probabilities are irrelevant) consider a call with exercise price X such that dS0 < X < uS0 hence, the evolution of the price and of the call option value is uS0 Cu = (uS0 – X) C S0 dS0 Cd = 0
21-4
Binomial Option Pricing (cont.) now, consider the payoff from writing one call option and buying H shares of the stock, where Cu − Cd uS0 − X H= = uS0 − dS0 uS 0 − dS0 the value of this investment at expiration is Up Down Payoff of stock HuS0 HdS0 Payoff of calls –(uS0 – X) 0 Total payoff HdS0 HdS0
21-5
Binomial Option Pricing (cont.) hence, we obtained a risk-free investment with end value HdS0 arbitrage argument: the current value of this investment should be equal to its present discounted value using the risk-free rate H is called the hedge ratio (the ratio of the range of call option payoffs and the range of the stock price) the argument is based on perfect hedging, or replication (the payoff of the investment replicates a risk-free bond)
21-6
Binomial Option Pricing – Algorithm
1. given the end of period stock prices, uS0 and dS0, calculate the payoffs of the call option, Cu and Cd 2. find the hedge ratio H = (Cu – Cd)/(uS0 – dS0) 3.