3

Imagine that space Z is exposed to the FX risk (i.e., currency exchange rate risk ), and we aim to provide a hedging solution for that. One choice is to consider a currency-forward contract. I wonder how I can derive the value of the forward contract when the spot domestic and foreign rates are a stochastic process, for example, following a Vasicek model. How should I discount the payoff of the forward contract in order to obtain a fair price? I think the final price should be a function of the forward rate.

If my understanding is correct, for the payoff function, we have something like this. Denote $S_T$ the spot FX rate at time T, K the strike rate at which we exchange the currencies. Then we have that

payoff= $S_T - K$

or I should consider

payoff= $S_T -F(t, T)$

where $F(t, T)$ stands for the forward exchange rate.

nbbo2
  • 11,262
  • 3
  • 21
  • 33
user53249
  • 419
  • 2
  • 9
  • 1
    When you enter into the contract the value of $F(t,T)$ (the forward exchange rate) is written down and assigned to the variable $K$ and then at time T the payoff occurs. The two expressions are basically equivalent. It would be clearer if you write $S_T-F(t_0,T)$ in the second where $t_0$ is the (fixed) time you entered the contract. – nbbo2 Jun 04 '21 at 12:36
  • In this case, how I should discount each term because here we are working under two different rates which are stochastic. By the way, $F(t_0, T)$ is a random variable again?

    Thank you very much for your comments in advance.

     – user53249 Jun 04 '21 at 12:45
  • $F(t,T)$, a random variable, is the ratio of the prices of two Zero Coupon Bonds, one in EUR and the other in USD. So there is one stochastic interest rate in the numerator and another in the denominator. HTH. – nbbo2 Jun 04 '21 at 12:58
  • Yes, I know what you mean. $F(t, T) = \frac{P_t^f(t, T)}{P_t^d(t, T)}X_t$. So for pricing, according to the arbitrage-free pricing method, we have that

    \begin{equation} V(t, T) = E^Q\Big[D\big(t, T)(X_T- F(t, T)\big)|\mathcal{F}_t\Big] \end{equation}

     – user53249 Jun 04 '21 at 13:04
  • 1
    Now, what is the discount factor? The measure should be under domestic risk-neutral measure or foreign risk-neutral measure? – user53249 Jun 04 '21 at 13:05
  • Given the information at time t, we know the value of the forward rate because the zero-coupon bond prices under two currencies are a function of interest rates which are known at time t, and we already know what the value of $X_t$ is at time t. So, normally the $F(t, T)$ should be a deterministic function of time. Am I right? – user53249 Jun 04 '21 at 13:10
  • Not sure what you exactly ask. The value at initiation will be zero. If you value it after initiation, you look at the strike (agreed forward rate) vs market forward at day of MtM and discount with ccy2 (e.g. if EURUSD, discount is USD).

    If you use stochastic rates, you will still use covered interest rate parity (so spot, and two rates give you forward). You only discount with ccy2 interest rate (just like Garman Kohlhagen)

     – AKdemy Jun 04 '21 at 13:10
  • 1
    I do not fully understand what you mentioned. If it is possible, please feel free to answer or not, I would be grateful if you could provide me with your solution or write a few lines to see how you proceed with this matter. – user53249 Jun 04 '21 at 13:13
  • By the way, I should declare that I am looking for the present value of the forward contract. – user53249 Jun 04 '21 at 13:14

2 Answers2

2

Using CCY1CCY2 (e.g. EURUSD quoted in units of domestic currency per unit of foreign currency. Here EUR is foreign, USD domestic). To get FWD rate at initiation: $$f(s,ccy1,ccy2,t) = s*exp^{(r_{ccy2}-r_{ccy1})*t}$$

  • Does not matter if r is stochastic or not
  • notional in ccy1, and value/premium in ccy2 (all else is a transformation as shown here)

At initiation of the FWD, you have zero value at the prevailing forward rate ($f(s,ccy1,ccy2,t) = K$).

Afterwards, for Mark to Market, you use that rate and compare it to the current FWD rate in the market (or what you model, but unless you are a market maker, I am not sure what the benefit of this will be). In other words, the forward value observed at t of a T maturity FWD contract is simply the PV of the difference in foreign exchange prices. $$N_{EUR}*(F_t -K)*xp^{−_{2}}$$

If notional is not in CCY1 (EUR), you multiply by K to get the equivalent CCY2 (USD) notional.

AKdemy
  • 9,269
  • 1
  • 23
  • 90
  • so, even if we have a stochastic interest rate, we will have the same equation. How Is should find K? How about $r_{ccy2}$ and $r_{ccy1}$? These are constant and defined regarding domestic and foreign currencies. – user53249 Jun 04 '21 at 13:49
  • Well with stochastic rates, these are modelled. However, FX forwards are super liquid and unless you either have such a long run that there are no quotes, or you are a market maker and want to use stochastic rates, I would simply use market forwards and ignore stochastic rates. I thought the question was about the discount factor. So you are interested in the implementation of a stochastic model? – AKdemy Jun 04 '21 at 13:55
  • Yes, If I want to do pricing under a stochastic process, then how I can deal with different parts in the expectation operator. I mean what is the discount factor? I am a bit confused by the rate we are supposed to discount the cash flow. – user53249 Jun 04 '21 at 14:37
2

Let's denote by:

  • $P(s,e)$: the zero coupon bond price at $s$ with maturity $e$
  • $d$ and $f$ superscripts: the domestic and foreign currency (of your FX rate).

The FX forward contract with strike $K$ and delivery at $T$ pays the following payoff at $T$ (in $d$ currency): $$ Payoff(T) =(S(T) - K) $$ So, its price at $t$ is the discounted payoff under the (domestic) risk-neutral measure: $$ Price(t) = \mathbb{E} \left[ e^{-\int_t^Tr^d(u)du}(S(T) - K) | \mathcal{F}_t \right] $$ Here, it's convenient to switch to the (domestic) T-forward measure $\mathbb{Q}_T^d$ (associated with numéraire $P^d(u,T)$: $$ Price(t) = P^d(t, T) \mathbb{E}^T \left[ S(T) - K | \mathcal{F}_t \right] $$

Now, no product inside the expectation, only the FX is left inside. We can write: $$ S(T) = S(T)\frac{P^f(T, T)}{P^d(T, T)} $$

The numerator is a tradeable asset. So, expressed in the numéraire $P^d(u,T)$ it is a $\mathbb{Q}_T^d$-martingale, and we get: $$ Price(t) = P^d (t, T) \left( S(t)\frac{P^f(t, T)}{P^d(t, T)} - K \right)\\ $$

In financial terms, this term is what you call the FX forward rate: $$ F(t, T) = S(t)\frac{P^f(t, T)}{P^d(t, T)} $$ and the price of the forward contract with strike $K$ is the discount difference between this FX forward and the strike: $$ Price(t) = P^d(t, T) \left(F(t, T) - K \right) $$

byouness
  • 2,210
  • 2
  • 13
  • 28