Merton’s Alternative Derivation of the Black–Scholes Equation

Summary

The document presents a question about an alternative derivation of the Black–Scholes model in Robert Merton’s rational option pricing work. It describes a setup in which a European warrant’s value depends on the stock price, riskless bond price, and time to expiration. The resulting partial differential equation includes second derivatives with respect to the stock and bond prices, a cross derivative, and a time derivative, with stock and bond volatilities and their correlation entering the coefficients.

The question then cites a change of variables and a boundary-condition representation for the warrant value, and asks how solving the original equation leads to that price. No answer or derivation is included. Thus, the material identifies the mathematical setup and the point of confusion, but offers no solution, worked example, or evidence for the claimed transformation. It is useful as a pointer to a joint stock-and-bond pricing PDE, while leaving the key analytical steps unresolved.

Key ideas

• The setup treats a European warrant value as a function of stock price, bond price, and time to expiration.
• The stated PDE includes stock and bond curvature terms, a mixed derivative, and a time derivative.
• The cross term reflects the correlation between stock and bond price changes.
• A change of variables and boundary conditions are cited as part of the route to a price representation.
• The document asks for the derivation but does not provide a solution or worked pricing example.

Tags

# Alternative derivation of Black Scholes by Merton


# Alternative derivation of Black Scholes by Merton












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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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.