Option pricing formulas were developed by Bachelier [#!kn:BachelierASENS1900!#], Kruizenga [#!kn:KruizengaMIT1956!#], Sprenkle [#!kn:SprenkleYale1960!#] and Boness [#!kn:BonessJPE1964!#] 1.1 . Black-Scholes [#!kn:BlackScholesJPE1973!#] showed how to derive a formula of the same form as Boness [#!kn:BonessJPE1964!#] using an equilibrium approach. Boness and the others had been unable to come up with the right discount rate and expected rate of return, the risk free rate in both cases, these were discovered and proven by Black-Scholes . Black-Scholes report a no-arbitrage derivation based on a suggestion by Merton who showed the Boness type formula with the Black-Scholes parameter restrictions for equity options also obtains if interest rates are normally distributed. Black-Scholes showed how to derive a partial differential equation of the McKean-Samuelson type [#!kn:McKeanIMR1965!#] [#!kn:SamuelsonIMR1965!#]. Merton [#!kn:MertonBJEMS1973!#] extended that equation to include a second random source from interest rates for pricing equity options only. Cox-Ross [#!kn:CoxRossJFE1976!#] showed how to interpret the Black-Scholes solution in terms of risk neutral probability and to price options with other stock price processes. This was an economic interpretation of the mathematics of Black-Scholes and Merton which was already risk neutral probability.
The general approach to arbitrage in a single currency was done by Garman [#!kn:GarmanBerkeley1976!#] at the same time as Richard developed it for just random interest rates [#!kn:RichardJFE1978!#].1.2 The Garman approach was for any type of security contingent on any type of random variable.