Pricing a Two-Asset Put with a Change of Numeraire

# Pricing of European options on two underlying assets


# Pricing of European options on two underlying assets












Is anybody able to give the solution to the following problem?

Suppose we have two assets, each of which follows a GBM process, and where $dW_S$ and $dW_X$ are correlated $(dW_SdW_X=\rho)$.

$dS=\mu_s S \hspace{0.5mm}dt +\sigma_s S \hspace{0.5mm}dW_s $

$dX=\mu_X X \hspace{0.5mm}dt +\sigma_X X \hspace{0.5mm}dW_X $

Determine the price of an european put option $V$ with payoffs depending on the terminal value of both assets described by

$ V_T=X_T\max\{K-S_T,0\}$

## Answer by siou0107 (score 6, accepted)

https://quant.stackexchange.com/a/68905

Your statement of the problem is not very detailed. Are $\mu_{S/X}$ constant ? What about interest rates? In the classic exchange option problem, where the payoff is $(X_T - S_T)^+$, they actually do not matter since all the risk is related to $S$ and $X$, no cash is involved (up to selling your replicating portfolio at maturity to pay immediately the requested amount, which does not involve interest rates); here, without specific assumption you do have interest rate risk, and your market is incomplete in general, which involves having to pick a risk-neutral measure based on some parameterisation. I will try to give you a general methodology for such pricing problems, explaining which assumption I make at each step. You should then be able to adapt the reasoning to your specific problem.

First thing to remark is that since $X$ and $S$ are assets, their discounted values must be martingales under any risk-neutral measure $\mathbb{Q}$ by the first Fundamental Theorem of Asset Pricing, if they do not pay dividends. If they do, you can just normalise by some dividend factor $D_t^{S/X}$ that represents the reinvestment of dividends, e.g. $\exp{-\int_0^t{q_s^{S/X} \mathrm{d}s}}$ if the dividend stream is continuous at rate $q^{S/X}$.

Let us assume for now that $D_T \equiv 1$, i.e. that $S$ and $X$ are non-dividend paying. We can thus write : \begin{align} & dX_t = r_t X_t dt + \sigma_X X_t dW_t^X \\ & dS_t = r_t S_t dt + \sigma_S S_t dW_t^S = r_t S_t dt + \sigma_S S_t \left(\rho \, dW_t^X + \sqrt{1 - \rho^2} dW_t^\perp\right) \end{align} The drift has to be $r_t$ by the FTAP I. Note that these dynamics induce that both $S$ and $X$ are positive at all times.

Picking a specific risk-neutral measure $\mathbb{Q}$, the corresponding price for your derivative is $$ V_0 = \mathbb{E}^\mathbb{Q} \left[e^{-\int_0^T{r_t\mathrm{d}t}} X_T \left(K - S_T\right)^+\right] $$ Now, observe the following : since $X e^{-\int_0^\cdot{r_t\mathrm{d}t}}$ is a martingale under $\mathbb{Q}$, $\frac{X_T}{X_0} e^{-\int_0^T{r_t\mathrm{d}t}}$ is a positive random variable with expected value 1. It can thus define a new probability $\mathbb{Q}^X$ equivalent to $\mathbb{Q}$ by $$ \frac{\mathrm{d}\mathbb{Q}^X}{\mathrm{d}\mathbb{Q}} := \frac{X_T}{X_0} e^{-\int_0^T{r_t\mathrm{d}t}} $$

This corresponds to changing numéraire from the riskless asset $e^{\int_0^\cdot{r_t \mathrm{d}t}}$ to $X$, hence our notation. We then have \begin{align} V_0 & = X_0 \mathbb{E}^\mathbb{Q} \left[e^{-\int_0^T{r_t\mathrm{d}t}} \frac{X_T}{X_0} \left(K - S_T\right)^+\right] \\ & = X_0 \mathbb{E}^{\mathbb{Q}^X} \left[\left(K - S_T\right)^+\right] \end{align} Now, all you have to do is determine the distribution of $S_T$ under $\mathbb{Q}^X$. To do so, you must compute the drift change induced by the change of probability. The Girsanov theorem allows you to do so. Using Itō's lemma (or, equivalently, solving the SDE for $X$ that I have given above), you have that $$ X_T = X_0 e^{\int_0^T{\left(r_t - \frac{\sigma_X^2}{2}\right)\mathrm{d}t} + \int_0^T{\sigma_X\mathrm{d}W_t^X}} $$ hence $$ e^{-\int_0^T{r_t\mathrm{d}t}} \frac{X_T}{X_0} = e^{- \frac{\sigma_X^2}{2}T + \sigma_X W_T^X} $$ By the Girsanov theorem, $$ \widehat{W}_t^X := W_t^X - \sigma_X t $$ is a Brownian motion under $\mathbb{Q}^X$. The dynamics under $\mathbb{Q}^X$ are thus \begin{align} dX_t & = \left(r_t + \sigma_X^2\right) X_t dt + \sigma_X X_t d\widehat{W}_t^X \\ dS_t & = r_t S_t dt + \sigma_S S_t \left(\rho dW_t^X + \sqrt{1 - \rho^2} d W_t^\perp\right) \\ & = \left(r_t + \rho \sigma_X \sigma_S\right) S_t dt + \sigma_S S_t \left(\rho d\widehat{W}_t^X + \sqrt{1 - \rho^2} d W_t^\perp\right) \\ & = \left(r_t + \rho \sigma_X \sigma_S\right) S_t dt + \sigma_S S_t d\widehat{W}_t^S \end{align} Because $W^\perp$ is independent from $W^X$, it is also a $\mathbb{Q}^X$-Brownian motion independent from $\widehat{W}^S$, hence $d \langle \widehat{W}^X, \widehat{W}^S\rangle_t = \rho \, dt$.

Assuming that rates are deterministic, you have two sources of randomness ($W^S$ and $W^X$) and two assets to hedge them: you market is complete, and the replication price is an expectation under the unique risk-neutral measure $\mathbb{Q}$, or under the equivalent martingale measure $\mathbb{Q}^X$. By observing that the dynamics we just derived induces a lognormal distribution for $S_T$ under $\mathbb{Q}^X$, you can use a generalized Black formula (i.e. $\mathbb{E} \left[\left(K - X\right)^+\right]$ with $X$ lognormally distributed).

NOW, if $D_T^{X/S} \not \equiv 1$, i.e. $\mu^{X/S} \neq r_t$, the problem can quickly become more complex ; notably, as you have randomness on the dividends, you have a dividend risk and your market is not complete. One assumption that allows to solve the problem quite simply is that the dividend flows are continuous at deterministic rates $q^{S/X}$. Then, you have necessarily $\mu_X = r_t - q_t^X$ and $\mu_S = r_t - q_t^S$ under $\mathbb{Q}$, and the extension is straightforward.