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Unformatted text preview: Math 473/Numerical Analysis/Fall 2009 Derivation of the BlackScholes equation Terminology. Consider a European call option. t time T expiry S ( t ) value of underlying security V ( S,t ) value of option μ drift of underlying security σ volatility of underlying security r riskfree interest rate Gearing of options. Suppose currently a share of stock costs $50, and we expect that in one month the price will substantially rise. Consider the following two strategies for investing $1000: (1) (Buy the stock) We buy 20 shares and sell them at the end of the month. (2) (Buy an option) We can buy a call option at $2 per share with a strike price of $53. At the end of the month our profit is given by the following table: Stock price Stock Option 501000 51 201000 52 401000 53 601000 54 80500 55 100 56 120 500 57 140 1000 58 160 1500 59 180 2000 60 200 2500 Delta hedging. Consider a portfolio consisting of the option and an amount Δ of a short position in the underlying security. At time t...
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 Spring '10
 CAREY
 $2, one month, $50, $53, $1000

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