The Easiest Way to Derive Black-Scholes
So we start with just two assumptions
1) The underlying asset follows a lognormal random walk
2) Arbitrage arguments allow us to use a risk-neural valuation approach (Cox-Rubinstein’s proof is easiest here), discounting the expected payoff of the option at expiration by the riskless rate and assuming the underlying’s return is the risk free rate
Derivation of Black-Scholes for a European call option c with strike K, discount rate r, on stock S, with time to maturity t, and expectations operator E.
Equation 1: The definition of a call option
Equation 4: substitute into R an exponential and its normal distribution, where f(u) is the normal density function with a mean of μt=(ln(r)- ½ σ2)t and volatility σ√t
2013-02-16 05:55:02
Source: http://falkenblog.blogspot.com/2013/02/the-easiest-way-to-derive-black-scholes.html
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