Pricing a Weighted Call with a Change of Numeraire
Summary
The document prices a terminal payoff equal to A_T times the positive part of R_T minus a strike. The response changes probability measure using A as the numeraire, so the asset-weighted option payoff becomes an ordinary call payoff on R under the new measure. Girsanov’s theorem adjusts the drift of R to account for its correlation with A, and the expectation is then reduced to a Black-style option pricing expression multiplied by the initial value and accumulated growth of A.
The derivation assumes correlated geometric Brownian motions with deterministic or constant drifts and volatilities, and uses a Brownian-motion decomposition to handle correlation. The response notes that if A represents the money-market account, its growth corresponds to the short interest rate. It omits some details and does not fully specify discounting conventions or every condition required for the measure change. Applying the result therefore requires checking that A is a valid numeraire and that model assumptions match the instruments and payoff being valued.
Key ideas
- Factoring the payoff as A_T times a call payoff on R motivates using A as the numeraire.
- The numeraire change transforms the weighted expectation into an expectation of a standard call payoff under a new measure.
- Correlation between the assets shifts R’s drift under the new measure.
- The derivation assumes geometric Brownian dynamics with deterministic or constant parameters and omits some technical details.
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# How to use a change of numeraire to price this option?
# How to use a change of numeraire to price this option?
I recently asked this question regarding how to price an option with payoff:
$$\text{Payoff}_T = (A_TR_T - A_T \lambda)^+ $$
Let's assume for generality that $A_t$ and $R_t$ are GMB's:
$$dA_t = \mu_{A,t} A_t dt + \sigma_{A,t} A_t dW_{A,t}$$ $$dR_t = \mu_{R,t} R_t dt + \sigma_{R,t} R_t dW_{R,t}$$
This comment mentioned that because the payoff can be seen as
$$\text{Payoff}_T = A_T(R_T - \lambda)^+ $$
we could use a change of numeraire approach to price it with Black's formula.
I'm not really familiar with this method, could somebody show me how it would work?
## Answer by Gordon (score 7, accepted)
https://quant.stackexchange.com/a/21375
Note: your previous question assumed log-normality instead of normality.
By Cholesky decomposition, we assume that, under measure $P$, \begin{align*} \frac{dR_t}{R_t} &= \mu_{R,t} dt + \sigma_{R,t}\, dW^1_t\\ \frac{dA_t}{A_t} &= \mu_{A,t} dt + \sigma_{A,t}\, d\left(\rho W^1_t + \sqrt{1-\rho^2} W^2_t\right), \end{align*} where $W^1$ and $W^2$ are two independent standard Brownian motions. Here, we assume that $\mu_{R,t}$, $\mu_{A,t}$, $\sigma_{R,t}$, and $\sigma_{A,t}$ are deterministic or constants. Define the probability measure $\widetilde{P}$ such that we have the Radon Nykodym derivative \begin{align*} \frac{d\widetilde{P}}{dP}\big|_t &= \frac{A_t}{A_0}\frac{1}{e^{\int_0^t\mu_ {A,s}ds}}\\ &= \exp\left(-\frac{1}{2}\int_0^t\sigma_{A,s}^2 ds + \int_0^t\sigma_{A,s}\,d\left(\rho W^1_s + \sqrt{1-\rho^2} W^2_s\right)\right). \end{align*} By Girsanov theorem, \begin{align*} \widetilde{W}^1_t &= W_t^1 - \rho \int_0^t \sigma_{A,s} ds \,\, \mbox{ and}\\ \widetilde{W}^2_t &= W_t^2 - \sqrt{1-\rho^2} \int_0^t \sigma_{A,s} ds \end{align*} are two independent standard Brownian motions under $\widetilde{P}$. Moreover, under $\widetilde{P}$, \begin{align*} \frac{dR_t}{R_t} &= \left(\mu_{R,t}+\rho \sigma_{A,t} \sigma_{R,t}\right) dt + \sigma_{R,t}\, d\widetilde{W}^1_t. \end{align*} Note also that \begin{align*} \frac{dP}{d\widetilde{P}}\big|_t &= \frac{A_0}{A_t}e^{\int_0^t\mu_ {A,s}ds}. \end{align*} Then, \begin{align*} E_P(A_T(R_T - \lambda)^+) &= E_{\widetilde{P}}\left(\frac{dP}{d\widetilde{P}}\big|_T A_T (R_T - \lambda)^+ \right)\\ &= A_0\, e^{\int_0^T\mu_{A, s} ds }\,E_{\widetilde{P}}\left( (R_T - \lambda)^+ \right), \end{align*} which reduces to Black-Scholes' formula. Some details are omitted here.
If $\mu_{A,t}$ is the short interest rate at time $t$, then $e^{\int_0^t\mu_ {A,s}ds}$ is the money-market account value at time $t$.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.