The Easiest Way to Derive Black-Scholes

Black-Scholes is perhaps the most famous equation in finance. It was originally derived by Fischer Black and Myron Scholes by using arbitrage to create a partial differential equation, which turned out to be the well-known heat equation from physics. Solving differential equations is hard, for me anyway (it doesn't come up a lot, so like my French, je sais un peu). Cox and Rubinstein showed how to use a binomial model to prove risk neutrality, and that proof is a lot easier. From there you can derive the result in a relatively simple way.

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 2: End of period stock price as a function of its return by definition, where R is the gross rate of return

Equation 3: rewriting eq(1) in integral form, where h() is the lognormal density function, and labeling S(0) as simply S, (note K and k are the same below, I'm too lazy to change them)

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