Valuing a Foreign Currency Forward as Two Zero-Coupon Bonds

# How to regard foreign currency forward as foreign and domestic bonds on VaR


# How to regard foreign currency forward as foreign and domestic bonds on VaR












In John Hull's book `Options, Futures and Other Derivatives 9th` `page 507`

We want to calculate the VaR of a `forward contract of a foreign currency` and we should spread forward into two bonds. It's said that

`a forward contract to buy a foreign currency. Suppose the contract matures at time T. It can be regarded as the exchange of a foreign zero-coupon bond maturing at time T for a domestic zero-coupon bond maturing at time T.`

We know that the payment at $T$ is $$Q_T - K$$ it's easy to understand regarding $K$ as a domestic zero-coupon bond maturing at time $T$ with principle $K.$ But how to regard $Q_T$ as the `foreign zero-coupon?`

## Answer by Gordon (score 1)

https://quant.stackexchange.com/a/35771

Let $X_t$ be the exchange rate from one unit foreign currency to units domestic currency. Moreover, let $K$ be the forward rate set at the contract inception time, which is usually prior to the VaR calculation date $t_0=0$. Then the value at time $t$, where $t_0 \le T \le T$, of the forward contract is given by \begin{align*} B_t^d\,\mathbb{E}\left(\frac{X_T-K}{B^d_T} \,\big|\, \mathcal{F}_t\right), \end{align*} where $B_t^d$ is the domestic money-market account value, $P^f(t, T)$ is the value at time $t$ of the foreign zero coupon bond with maturity $T$, and $\mathbb{E}$ is the expectation operator under the domestic risk-neutral measure $\mathbb{Q}^d$.

As $P^f(t, T)X_t$ and $P^d(t, T)$ are values at time $t$ of domestic tradable assets. the processes $\left\{\frac{P^f(t, T)X_t}{B_t^d}, \, t_0\le t \le T \right\}$ and $\left\{\frac{P^d(t, T)}{B_t^d}, \, t_0\le t \le T \right\}$ are martingales under the domestic risk-neutral measure. Then, the value at $t$ of the forward contract is given by \begin{align*} B_t^d\,\mathbb{E}\left(\frac{X_T-K}{B^d_T} \,\big|\, \mathcal{F}_t\right) &= B_t^d\,\mathbb{E}\left(\left(\frac{P^f(T, T)X_T}{B^d_T} -K\frac{P^d(T, T)}{B^d_T}\right) \,\big|\, \mathcal{F}_t\right)\\ &=P^f(t, T)X_t - K P^d(t, T). \end{align*} That is, the value of the exchange of certain amount of foreign zero-coupon bond with certain amount of domestic zero-coupon bond.

> Addendum

Here, we provide a mathematical derivation that the process $\left\{\frac{P^f(t, T)X_t}{B_t^d}, \, 0 \le t \le T \right\}$ is a martingale under the domestic risk-neutral measure $\mathbb{Q}^d$.

Let $\mathbb{Q}^f$ be the foreign risk-neutral measure, and $\mathbb{E}^f$ be the corresponding expectation operator. Note that the process $\left\{\frac{P^f(t, T)}{B_t^f}, \, 0 \le t \le T \right\}$ is a martingale under the foreign risk-neutral measure. We denote by $\eta_t$ the Radon–Nikodym derivative \begin{align*} \frac{d\mathbb{Q}^d}{d\mathbb{Q}^f}\big|_t = \frac{B_t^d X_0}{B_t^f X_t}. \end{align*} Then, for $0 \le s \le t \le T$, \begin{align*} \mathbb{E}\left(\frac{P^f(t, T)X_t}{B_t^d}\, \big| \, \mathcal{F}_s\right) &= \mathbb{E}^f\left(\frac{\eta_t}{\eta_s}\frac{P^f(t, T)X_t}{B_t^d}\, \big| \, \mathcal{F}_s\right)\\ &=\mathbb{E}^f\left(\frac{B_t^d}{B_t^f X_t} \frac{B_s^f X_s}{B_s^d}\frac{P^f(t, T)X_t}{B_t^d}\, \big| \, \mathcal{F}_s\right)\\ &=\frac{B_s^f X_s}{B_s^d}\mathbb{E}^f\left(\frac{P^f(t, T)}{B_t^f}\, \big| \, \mathcal{F}_s\right)\\ &=\frac{P^f(s, T)X_s}{B_s^d}. \end{align*} That is, $\left\{\frac{P^f(t, T)X_t}{B_t^d}, \, 0 \le t \le T \right\}$ is a martingale under the domestic risk-neutral measure.