Deriving the Currency Spot-Forward Relationship by No-Arbitrage
Summary
The document derives the currency forward price from the spot exchange rate and the interest rates in the two currencies. It defines the local currency as the numéraire and treats the foreign currency’s spot price as its value in local currency. With continuously compounded rates, it compares two portfolios that each begin with one unit of local currency.
One portfolio converts into foreign currency at spot and invests at the foreign risk-free rate; the other invests locally and enters a forward contract to buy foreign currency at maturity. Since both portfolios have equal initial value and are constructed to have the same terminal exposure, no-arbitrage requires their maturity values to agree. Equating them gives the spot rate multiplied by the exponential of the foreign-minus-local rate differential over the term. The proof assumes the stated risk-free rates and idealized tradable investments; it does not address transaction costs, market frictions, or complications in actual currency markets.
Key ideas
- The local currency is used as the numéraire, and the foreign currency is valued at the spot exchange rate.
- Compare investing locally with converting to and investing in the foreign currency while hedging the exchange rate forward.
- Equal initial values and no-arbitrage imply that the two portfolios must have equal maturity values.
- The forward price depends on spot and the difference between foreign and local continuously compounded rates.
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Full text
# Spot-Forward Relationship - Proof
# Spot-Forward Relationship - Proof
Does anyone know of a decent proof for the spot-forward relationship of a currency? I've been looking on Google for hours and I'm not getting anywhere. My lecture notes are useless in that they don't even tell us what the spot-forward relationship is. I'm presuming it is in a Black-Scholes setting with constant interest rates but who knows because my lecturer doesn't feel the need to include information like that.
This is the question: Write and prove in details the spot-forward relationship satisfied at date $t$ by the price of a currency $X(t)$ and the forward price $f(t,T)$.
That's all I have to go on. Any help will be gratefully received.
## Answer by Soumirai (score 2, accepted)
https://quant.stackexchange.com/a/38474
Let's go for a detailed and rigorous proof.
Let us define our local currency $Y$ as the numéraire, i.e. the asset in terms of whose price the relative prices of all other tradeables are expressed. $X$ is therefore the foreign currency, whose price in terms of $Y$ is $X(t)$ at any time $t$.
Let $r_X(t,T)$ be the risk-free interest rate in currency $X$ and $r_Y(t,T)$ the risk-free interest rate in currency $Y$, for maturity $T-t$ and at time $t$. Both are continuously coumpounded. Let two portfolios of value $1$ (in terms of currency $Y$) at time $t$: $P_1(t)=P_2(t)=1$.
The first portfolio $P_1$ consists in buying currency $X$ at spot price $X(t)$ and investing this amount in the risk-free rate of currency $X$. The final value at time $T$ in currency $Y$ is therefore: $P_1(T)=X(t)*e^{(T-t)r_X(t,T)}/X(T)$.
The second portfolio $P_2$ consists in buying currency $X$ in the future at time $T$, and at a forward price $f(t,T)$ determined at $t$. In the meanwhile, the $1$ is invested in the risk-free rate in currency $Y$. The final value at time $T$ in currency $Y$ is therefore: $P_2(T)=e^{(T-t)r_Y(t,T)}*f(t,T)/X(T)$.
Note that the two portfolios are riskless and have the same initial value. Hence, by no-arbitrage we must have $P_1(s)=P_2(s), \forall s\geq t$, and in particular:
$$P_1(T)=P_2(T)$$ $$X(t)*e^{(T-t)r_X(t,T)}/X(T)=e^{(T-t)r_Y(t,T)}*f(t,T)/X(T)$$ $$f(t,T)=X(t)*e^{(T-t)(r_X(t,T)-r_Y(t,T))}$$
Hope this helps!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.