slides15 - 15 More about options • Black-Scholes •...

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Unformatted text preview: 15. More about options • Black-Scholes • Employee stock options • Warrants • Convertible and/or callable bonds Black-Scholes Model An alternative to the binomial tree approach to option valuation is the Black-Scholes option pricing model. Fischer Black and Myron Scholes derived no-arbitrage pricing formula for European call and put options based on the followng assumptions: • the distribution of returns is log normal • there are no costs or benefits associated with ownership of the underlying asset (e.g., no dividends) • no transaction costs • the risk-free rate is known and constant over the life of the option. • trading takes place continuously Note: Robert Merton also contributed. Scholes and Merton won the Nobel prize in 1997 for their work (Black died in 1995). 2 Black-Scholes — continued The derivation of the option pricing formula is based on the same idea as in the binomial examples discussed previously. The investor forms a hedge portfolio that is risk-free for very small movements in the stock price. The hedge portfolio must be adjusted every time the stock price moves (dynamic hedging) . If this is done properly, it is possible to perfectly hedge, so the return on the portfolio must be equal to the risk-free rate. Black-Scholes tells us • the price of the option implied by the model • the hedge ratio 3 Black-Scholes — continued Caveat: In practice, the assumptions underlying the model are not perfectly satisfied , therefore actual option prices may differ slightly from the price implied by the model. But, the model price is generally close to actual prices, and pro- vides a good basis to start from when trying to determine a fair price in practice. Although the BS formula is for European options, most traded op- tions are American. The American option will be worth slightly more than the equivalent European option. We refer to the differ- ence as the early exercise premium . It is generally small. 4 Black-Scholes — continued The BS price for a call option is: C t = S t N ( d 1 )- Ke- R f ( T- t ) N ( d 2 ) d 1 = ln( S t /K ) + ( R f + σ 2 / 2)( T- t ) σ √ T- t d 2 = d 1- σ √ T- t where C t = Value of call option at time t S t = Asset price at time t K = Strike price T = Exercise date...
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This note was uploaded on 03/31/2008 for the course FNCE 3010 taught by Professor Donchez,ro during the Fall '07 term at Colorado.

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slides15 - 15 More about options • Black-Scholes •...

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