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Pricing a Two-Asset Digital Option with a Change of Numeraire

Article Quant Q&A · Author: Imela
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Summary

The document considers a cash-or-nothing claim that pays when one of two risky assets exceeds the other at maturity. Both assets are modeled with geometric Brownian motion driven by the same Wiener process, alongside a risk-free asset. The proposed pricing route is to use the first asset as numeraire, assuming it is tradable and pays no dividends.

Under that measure, the ratio of the second asset to the first is a martingale and follows a lognormal process, reducing the event to a digital option on that ratio. Its price can be expressed using the cumulative distribution function of a normal variable. The response gives the conceptual method but no formula details or numerical example. The shared single risk driver and the no-dividend assumption are material limits; the setup would need adjustment for other correlation structures or asset income.

Key ideas

  • The claim pays according to which of the two assets has the higher value at maturity.
  • Using the first asset as numeraire makes its relative price dynamics easier to analyze.
  • The second asset divided by the numeraire is a martingale under the associated measure.
  • The payoff becomes a digital option on a lognormal ratio, with pricing expressed through a normal cumulative distribution.

Tags

OptionsDerivatives pricingMulti-asset
Full text
# How to price options that depend on two assets in continuous time?


# How to price options that depend on two assets in continuous time?












Let $S_1$ and $S_2$ be two risky assets. The market also has a riskfree asset, and only one driving Wiener process. The parameters are as in the Black SCholes, with $\mu_1, \mu_2, \sigma_1, \sigma_2$.

$dS_i = \mu_1S_idt + \sigma_i S_idW^P$

I wish to price the option at time $t$ which gives us $1$ if $S_1 > S_2$ at time $T$, otherwise $0$.

My method: calculate $S_1(T) - S_2(T)$ explicitly. Then we need the $Q$-probability that this is strictly positive. However, assuming this is the correct method, I get stuck on calculating this $Q$-probability.

EDIT: I got an answer, but it depends on the cumulative distribution of a normal distribution. Does this look right guys?

## Answer by jimifiki (score 1)

https://quant.stackexchange.com/a/32044

You should work in the numeraire of $S_1$ (if it is a tradable which doesn't pay dividends). In this numeraire $S_1$ has no drift and $S_2/S_1$ is a martingale.

You should get a digital option on a log normal process which actually has a pricing formula in terms of the cumulative distribution of a normal distribution.

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.