Deriving the Two-Factor Convenience-Yield Pricing PDE

# Generalized Black Scholes PDE in a Two Factor model


# Generalized Black Scholes PDE in a Two Factor model












I'm reading the book of Clewlow and Strickland on Energy derivatives. In the section about the two-factor model, an equation, similar to B&S PDE is presented, but the proof is not presented.

Spot dynamics: $$ dS=(r-\delta)S dt + \sigma S dW $$ Convenience yield dynamics: $$ d\delta= \alpha(\hat{\delta}-\delta) dt + \sigma_\delta dW_{\delta}$$ $$ dW dW_{\delta} = \rho dt $$ The book says: The joint process for the spot and the convenience yield lead to the following differential equation for contingent claim prices: $$ 1/2 C_{SS} S^2\sigma^2 + 1/2 C_{\delta \delta} \sigma^2_{\delta} + C_{S \delta} S \rho \sigma \sigma_\delta + C_S S(r-\delta) + C_{\delta}( \alpha(\hat{\delta}-\delta_t) - \lambda_\delta \sigma_\delta) + C_t = 0 $$ Where $\lambda_\delta $ is the market price of risk per unit of convenience yield.

I have the feeling that a term in the equation is missing ($-rC $), but I can't find anywhere on the web proof of this equation! Could you please provide me a reference book or paper where I can find the proof of it?

Thank you in advance

## Answer by ShaftSinker (score 1)

https://quant.stackexchange.com/a/80227

This question is a bit dated and unfortunately has not received any answers so in the interest of others who stumble onto this questions I can provide a bit of background for this model along with some sources. I'll avoid the algebraic manipulations but I hope that the framework of the model and the derivation of the PDE will be clear.

First of all, this model was first published (to my knowledge) in the paper titled "Stochastic Convenience Yield and the Pricing of Oil Contingent Claims" by Gibson & Schwartz in 1990.

As to the method used to derive this equation, the source provided by Gibson & Schwartz is a paper by Brennan & Schwartz titled "A continous Time Approach to Pricing Bonds" published in 1979.

The main idea of the derivation is to construct a risk free portfolio using two other arbitrary derivatives. Two derivatives are necessary to obtain a risk free portfolio as the model has two sources of uncertainty (price and convenience yield). Consider a derivative whose price is determined by $B(t, \delta, S)$. We can then consider two other arbitrary derivatives $F_1(t, \delta, S)$ and $F_2(t, \delta, S)$. We then look for coefficents $x_1$ and $x_2$ such that the portfolio short one unit of $B$, long $x_1$ units of $F_1$ and long $x_2$ units of $F_2$ is risk free. The value of this porfolio is given by:

$x_1F_1(t, \delta, S)+x_2F_2(t, \delta, S)-B(t, \delta, S)$

This will provide three equations, one for each factor (diffusion terms) which are set to zero (meaning the porfolio is "risk free"). The last equation is from the drift term which, following an arbitrage argument, must now grow at the risk free rate (as it is risk free). This system of linear equations (in $x_1$ and $x_2$) has two coefficients and three equations and hence a solution does not exist in general. An equivalent condition to the existence of the solution is found in the paper (equation 6 in Brennan & Schwartz) and is used as the pricing equation.

Lastly to the question on the term $rC$ missing from the equation, I believe that you are correct and it is indeed missing. In fact, this term stems from setting the drift term to be growing at the risk free rate in the derivation.

Happy to provide more details if there are questions.