# bsderive - Math 473/Numerical Analysis/Fall 2009 Derivation...

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Unformatted text preview: Math 473/Numerical Analysis/Fall 2009 Derivation of the Black-Scholes 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 risk-free 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 50-1000 51 20-1000 52 40-1000 53 60-1000 54 80-500 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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bsderive - Math 473/Numerical Analysis/Fall 2009 Derivation...

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