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Unformatted text preview: 15. More about options • BlackScholes • Employee stock options • Warrants • Convertible and/or callable bonds BlackScholes Model An alternative to the binomial tree approach to option valuation is the BlackScholes option pricing model. Fischer Black and Myron Scholes derived noarbitrage 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 riskfree 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 BlackScholes — 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 riskfree 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 riskfree rate. BlackScholes tells us • the price of the option implied by the model • the hedge ratio 3 BlackScholes — 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 BlackScholes — 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.
 Fall '07
 DONCHEZ,RO
 Corporate Finance, Options, Valuation

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