Deriving the Black Equation for Options on Futures

Summary

This exchange concerns transforming the Black–Scholes equation from an underlying spot price to a forward or futures price to obtain an equation for options on futures. The question gives the relationship between spot and forward prices and presents the corresponding partial differential equations, asking how the drift term changes under the transformation.

The response sketches an attempt to apply the chain rule, with the intended result that the spot-price drift term cancels in the forward-price formulation. However, the displayed differentiation contains a notation error: it reuses the same time derivative on both sides and does not clearly account for holding the forward price fixed when taking a partial derivative. The exchange therefore signals the key idea but is not a reliable step-by-step derivation. It also does not specify contract conventions, carry assumptions, or boundary conditions.

Key ideas

• The Black equation for a futures option can be obtained by changing variables from spot price to forward price.
• The transformation changes how the drift term appears in the pricing equation.
• The response intends to show cancellation of the spot drift term through the chain rule.
• The written derivation is incomplete and contains ambiguous time-derivative notation.

Tags

# Option on Futures - Black Equation Derivation


# Option on Futures - Black Equation Derivation












How to derive Generalised Black equation for Option on Future using generalised Black Scholes Equation

$$F=exp(r(T−t))S$$

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2F^2\frac{\partial^2 V}{\partial F^2} -rV = 0$$

From $$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} -rV = 0$$

## Answer by Vinay Dwivedi (score 1)

https://quant.stackexchange.com/a/35236

Update:Will's link helped Derivation: I applied following Chain rule V=(F,t) and F = (S, t) (F = Forward Price, S = Underlying Price)

I get

$$\frac{\partial V}{\partial t} = \frac{\partial V}{\partial F}\frac{\partial F}{\partial t} + \frac{\partial V}{\partial t}\frac{\partial t}{\partial t} $$

= $$\frac{\partial V}{\partial t} = \frac{\partial V}{\partial F}\frac{\partial F}{\partial t} + \frac{\partial V}{\partial t}$$

gets me $$\frac{\partial V}{\partial t} = \frac{\partial V}{\partial t} - rF\frac{\partial V}{\partial t}$$

Substituting in Black Scholes equation, RS term gets cancelled out w.r.t F.

Future price has No value at the time of Purchase!

Thanks a lot!

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