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Deriving the Product SDE for Correlated Assets with Itô’s Rule

Article Quant Q&A · Author: Animesh Saxena
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Summary

The document addresses how to derive the stochastic differential equation for the product of two assets, including when each asset already has a known explicit solution. The key method is Itô’s product rule: combine each asset’s change with the other asset’s current level, and include the cross-variation term. That final term captures the effect of correlation between the two Brownian motions and changes the product’s drift.

The answers affirm that multiplying explicit solutions is valid, but the resulting dynamics must agree with the Itô-derived equation. The excerpt does not work through the full derivation or resolve the questioner’s specific drift discrepancy. It also contains inconsistent volatility notation in one answer, so readers should check each asset’s diffusion coefficient and the convention used for the Brownian cross-variation when applying the method.

Key ideas

  • Itô’s product rule includes a cross-variation term when deriving the dynamics of two multiplied processes.
  • Correlation between the assets’ Brownian motions contributes to the product’s drift.
  • Multiplying explicit solutions is valid when the resulting process is interpreted consistently with Itô’s rule.
  • The excerpt leaves the specific drift error unresolved and includes inconsistent coefficient notation.

Tags

StatisticsDerivatives pricingOptions
Full text
# Why can't I multiply two SDE Solutions?


# Why can't I multiply two SDE Solutions?












SDE 1 is S1 = S10 exp( (r1-sigma^2/2) * dt + sigma dW1 )

S2 = S20 exp( (r2-sigma2^2/2) * dt + sigma2 dW2 )

E[dW1 dW2] = rho

I want to price an option on S1 x S2 I know I need to use the SDE's to find the SDE for d(S1 S2) using Ito...but what if I use this

S = S1 x S2 using two analytical solutions of SDE? Why the drift term comes out incorrectly? Why can't I multiply two SDE solutions to get an equation for the third....?

## Answer by wsw (score 1)

https://quant.stackexchange.com/a/15752

You can use Ito's product rule $d(X \, Y) = dX \, Y + X \, dY + dX \, dY$. In your case, you have $$ dS_{1,t} = S_{1,t} \left( r_1 dt + \sigma_1 dW_{1,t} \right) $$ and $$ dS_{2,t} = S_{2,t} \left( r_2 dt + \sigma_1 dW_{2,t} \right) $$

## Answer by Ulysses (score 0)

https://quant.stackexchange.com/a/15738

By all means, if $dS_i = \mu(S^i_t)\mathrm dt + \sigma(S^i_t)\mathrm dW^i_t$ for $i=1,2$ and you know explicit formulas for $S_i$ then their product satisfies the SDE you derive using the Ito lemma for a product. Could you elaborate, on what is the issue you have with the drift term?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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