question regarding relation between expectations on different measures

Question

I am a beginner to the theory of stochastic calculus and measure change. I have derived an equation related to expectations on different measures. I wanted some expert opinion on whether this is true or if I am making a mistake.

Let $X(t) = \frac{USD}{EUR}(t)$ be a FX spot. Let $B^{USD}(t) = e^{\int^t_0 r_{USD}(u) du}$ be the bank account numeraire which gives the USD risk neutral measure. Similarly let $B^{EUR}(t) = e^{\int^t_0 r_{EUR}(u) du}$ be the bank account numeraire which gives the EUR risk neutral measure. Now by arbitrage pricing theorem we have,

$$\frac{X(0)}{B^{USD}(0)} = \mathbb{E}^{Q_{USD}}\big[ \frac{X(T)}{B^{USD}(T)}\big]$$

also considering $1/X(t)$ to be asset we have,

$$\frac{1}{X(0) \cdot B^{EUR}(0)} = \mathbb{E}^{Q_{EUR}}\big[ \frac{1}{X(T) \cdot B^{EUR}(T)}\big]$$

Now since $B^{USD}(0) = B^{EUR}(0) = 1$ we can multiply both the equations to obtain the following

$$1 = \mathbb{E}^{Q_{USD}}\big[ \frac{X(T)}{B^{USD}(T)}\big] \cdot \mathbb{E}^{Q_{EUR}}\big[ \frac{1}{X(T) \cdot B^{EUR}(T)}\big]$$

Is this derivation correct or am I making a mistake? I have not found this in any book, I just derived it myself and since I am beginner I would like some expert opinion on whether this is correct or not.

Thanks a lot in advance.

Answer 1

I think your derivations are correct. The way I look at those problems, is thinking about the numeraire as a discounting asset. Then a measure change can be applied very easily. In this way, you don't have to compare $X(t)$ and $\frac{1}{X(t)}$ with each other under the different measures. For forex this might not be a problem, however, for stock prices the reciprocal interpetation may become difficult.

Let $B_{\text{EUR}}(t)$ be the value process of a risk-free investment in the currency EUR (for instance a bank account in euros), and let $B_{\text{USD}}(t)$ be the process governing the value of a risk-free investment in the USD. Let $X(t)$ be the process governing the FX spot rates. Furthermore, let $\mathbb{Q}_{\text{EUR}}$ be the risk-neutral measure under the risk-free EUR investment, such that the asset $X(t)$ discounted by the risk-free investment $B_{\text{EUR}}(t)$ is a martingale (I skip everything on existence and such). On the other hand, let $\mathbb{Q}_{\text{USD}}$ be the risk-neutral measure with respect to the risk-free investment $B_{\text{USD}}(t)$. Define the reltation $$ \dfrac{\text{d}\mathbb{Q}_{\text{USD}}}{\text{d}\mathbb{Q}_{\text{EUR}}} = \dfrac{B_{\text{USD}}(T) B_\text{EUR}(t)}{B_{\text{USD}}(t)B_\text{EUR}(T)}, $$ for $t < T$.

Let $\mathcal{F}_t$ be the information of the process $X(t)$ up to time $t$. Rewriting the conditional expectation \begin{align} B_\text{EUR}(t) \mathbb{E}^{\mathbb{Q}_\text{EUR}}\left(\left.\dfrac{X(T)}{B_\text{EUR}(T)} \right| \mathcal{F}_t\right) & = B_\text{EUR}(t) \int_{\mathbb{R}} \dfrac{X(T)}{B_\text{EUR}(T)} \text{d}\mathbb{Q}_\text{EUR} \\ & = B_\text{EUR}(t) \int_{\mathbb{R}} \dfrac{X(T)}{B_\text{EUR}(T)} \dfrac{B_{\text{USD}}(t)B_\text{EUR}(T)}{B_{\text{USD}}(T) B_\text{EUR}(t)} \text{d}\mathbb{Q}_{\text{USD}} \\ & = \int_{\mathbb{R}} X(T) \dfrac{B_{\text{USD}}(t)}{B_{\text{USD}}(T)}\text{d}\mathbb{Q}_{\text{USD}} \\ & = B_{\text{USD}}(t) \mathbb{E}^{\mathbb{Q}_\text{USD}}\left(\left. \dfrac{X(T)}{B_\text{USD}(T)}\right| \mathcal{F}_t\right) \end{align} Note that the discounted process under the new measure needs to be a martingale under that new measure, as all discounted assets under the new measure need to be martingales under the new measure. In this way the dynamics of the asset $X(t)$ can be found under the new measure.

This approach can also help you find prices discounted under other measures, such as the $T$-forward measure. The $T$-forward measure is often used when the interest rate is assumed to be stochastic. The discounting asset, or numeraire, under the $T$-forward measure is the zero-coupon bond.

Remark:

$\dfrac{\text{d}\mathbb{Q}_{\text{USD}}}{\text{d}\mathbb{Q}_{\text{EUR}}}$ is called the Radon-Nikodym derivative, I am not an expert on that measure theoretical concept, however, you can find plenty of info about it via googling the subject. For this post, it can be taken for granted. Finding the Radon-Nikodym derivative corresponding to the measure change you want to apply is constructing it in such a way the discount terms of the old measure cancel out.

I hope you find it usefull.