Changing Numeraire for an Option on Two Risky Assets

Summary

The document asks how to price a European payoff equal to the positive part of the difference between two risky assets in a constant coefficient Black–Scholes market. It starts from ordinary risk neutral valuation and considers expressing the payoff using the ratio of the assets, which would make it resemble a call option with a unit strike.

The central issue is the measure change: the author has obtained a measure under which the asset ratio is a martingale, but is unsure whether it is the measure required for the proposed numeraire pricing identity. The document gives the model and the desired pricing transformation, but no solution, derivation, or numerical evidence. Its lesson is to distinguish the measure induced by changing numeraire from a separately chosen measure that merely makes a price ratio a martingale; the standard numeraire change must be derived consistently with the chosen numeraire.

Key ideas

• The payoff is a call like option on the difference between two risky assets.
• Ordinary risk neutral valuation discounts the expected payoff under the money market measure.
• The payoff can be expressed using the ratio of the two assets and the second asset as numeraire.
• A measure that makes the ratio a martingale must be consistent with the numeraire change used in pricing.

Tags

# Changing numeraire in Margrabes formula


# Changing numeraire in Margrabes formula












Consider a Black Scholes market with constant coefficients, a bond and two risky assets: $$dB_{t}=r B_{t}dt \\ dS_{t}^{i}=S_{t}^{i}(b_{i}dt+\sigma_{i,1}dW_{t}^{1}+\sigma_{i,2}dW_{t}^{2})$$ where $i=1,2$ and $W_{t}^{1},W_{t}^{2}$ are two independent Brownian Motions.

I want to determine the fair price of the option with payoff at maturity $T$: $$X(T)=(S_{T}^{1}-S_{T}^{2})^{+}$$

I have applied Girsanov´s Theorem twice, first to find a risk-neutral measure $Q$, then to find another measure $Q'$ such that the quotient process $S^{1}/S^{2}$ is a martingale under $Q'$.

Using risk-neutral pricing, I know that the fair price must be: $$E_{Q}[X(T)]e^{-rT}$$ I now want to change numeraire, to arrive at: $$E_{Q}[X(T)]e^{-rT}=E_{Q''}[(\frac{S^{1}_{T}}{S_{T}^{2}}-1)^{+}]S_{0}^{2}$$ which is the Black-Scholes price of a call on $S^{1}/S^{2}$ with strike price $K=1$ and maturity $T$ (and for which I have the classical formula). But this of course only holds, if $Q''$ is a risk-neutral measure for $S^{1}/S^{2}$.

And unfortunately $Q' \neq Q''$ ... how can I conclude?

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.