I am currently reading the Theory of Rational Option Pricing (1973) by Robert Merton. In the paper, I encountered a section under the title "An Alternative Derivation of the Black- Scholes Model". I am having a lot of trouble understanding the intuition behind this alternate derivation and some of the notation.

In the paper, it is assumed that the option price is a function of the stock price, the riskless bond price, and the length of time to expiration as such $H(S,P,\tau;E)$. After some manipulation (can be seen in the original paper "Theory of Rational Option Pricing") the second-order linear partial differential equation is found, as seen below.

$$\frac 12 [\sigma^2S^2H_{11}+2\rho\sigma\delta SPH_{12}+\delta^2 P^2H_{22}]-H_3=0$$

By change of variables and using the boundary conditions for European warrants the price for any European warrant must satisfy $$H(S,P,\tau ,E)=EP(\tau)y [S/EP(\tau), \int^\tau_0 V^2(s)ds] $$ A lot of work has been put into deriving the Black- Scholes differential equation, however, I have not been able to find anything on Merton's alternative derivation. So my question is, how do I solve the PDE (shown first) to get to the price of the European warrant?


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.