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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?

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