Forward Spot Rate Curves

Consider a vector of maturities $(m_{1},\ldots,m_{N})$ and corresponding spot (zero) rate vector at time $t\geq0$ (which could be obtained from bootstrapping for instance, or in the case of LIBOR rates already given):
$$\left(S_{t}^{(m_{1})},\ldots,S_{t}^{(m_{N})}\right).$$
This can be viewed as a stochastic process with state space $\mathbb{R}^{N}$ or as $N$ individual stochastic processes $\{S^{(m_{j})}_{t}\}_{t\geq0}$ for $m_{j}\in\{m_{1},\ldots,m_{N}\}$ with state space $\mathbb{R}$.  Corresponding to this process is a forward process
$$\left\{\left(f_{t}^{(m_{1})}(T),\ldots,f_{t}^{(m_{N})}(T)\right)\right\}_{t\geq0}$$
that has an extra (deterministic) dimension in the variable $T$ and therefore the forward process has state space $\mathcal{C}(\mathbb{R})^{N}$.  Let us isolate our attention to one component of this process, say $f_{t}^{(m_{j})}(T)$ for some $1\leq j\leq N$, the forward spot rate process corresponding to maturity $m_{j}$ and future time $T$.  How do we determine this process in terms of $S^{m_{j}}_{t}$ so that for each $T\geq0$ the usual arbitrage conditions are satisfied?  For a non-dividend paying stock, the process is given by
$$f_{t}(T)=S_{t}e^{r(T-t)}.$$
But there is no obvious analogue for interest rates.  In fact, $f_{t}^{(m_{j})}(T)$ will depend on two components of the spot rate process for each $T$.  The definition is
$$(1)\;\;\;\;f^{(m_{j})}_{t}(T)=\frac{S^{(T+m_{j})}_{t}(T+m_{j})-S^{(T)}_{t}T}{m_{j}}.$$
This is obtained by equating the cash flows of borrowing at $S^{(T+m_{j})}_{t}$ for $T+m_{j}$ years versus borrowing at $S^{(T)}_{t}$ for $T$ years and then rolling forward the returns at $f^{(m_{j})}_{t}(T)$ for $m_{j}$ years.  This of course restricts $T$ to the set $\mathcal{T_{j}}:=\{m_{i}\}_{1\leq i\leq N-k(j)}$ where $k(j)$ is such that $T+m_{j}\leq m_{N}$ for all $T\in\mathcal{T_{j}}$.

An equivalent and more commonly encountered way to express this is to write
$$(2)\;\;\;\;f^{(m_{j})}_{t}(T_{1},T_{2})=\frac{S^{(T_{2})}_{t}T_{2}-S_{t}^{(T_{1})}T_{1}}{T_{2}-T_{1}}.$$
In this case, we require $T_{2}-T_{1}=m_{j}$ and $m_{1}\leq T_{2}\leq m_{N}.$  Of course, if we are interested in a forward rate at time $T$ that is not compatable with the spot rate vector that we have available at time $t$, then we can interpolate the pot curve (although this could potentially lead to arbitrage opportunities if you (a bank, say) are using this forward curve to price fixed income derivatives such as a swaps, because it is conceivable that a potential arbitrageur could get a spot rate for that interpolated term date different from your interpolated value!).

If we compute (1) for all $T\in\mathcal{T}_{j}$ we get a vector of points that represents the forward curve for the spot rate $S^{(m_{j})}_{t}$ associated with term $m_{j}$ computed at time $t$.  If we then do this for each $1\leq j\leq N$, we get a set of forward curves, one for each spot rate term.  If we fix a $T$ and plot $f^{(m_{j})}_{t}(T)$ for each $m_{j}$, then we get a forward term structure curve corresponding to the forward time $T$.  The particular interpretation taken depends on the context; usually the former is used for evaluating swap derivatives and the ladder for evaluating bond derivatives.

Valuing European FX (Currency) Options

I. Overview

A currency option is a contract that gives the buying party the right, but not the obligation, to buy (call) or sell (put) a foreign currency at a pre-defined exchange rate (strike).

For a given currency pair (e.g. EURJPY), the call or put designation of the option contract always refers to the base currency; therefore, the foreign currency is the base currency and the domestic currency is the quoted currency.  For example, a Japanese company expects to make a payment of 1,000,000 Euros in six months; they can hedge this exposure (while also participating in favorable price movements) by purchasing a call option on EUR/JPY.  Similarly, if they expected to receive Euros, they'd enter into a put option on EUR/JPY.

Because of the nature of FX transactions, every currency option is technically two options: a call and a put.  So a call on EUR/JPY is a call (buy/receive) on EUR and a put (sell/give) on JPY for a specified exchange rate (the strike) and a put on EUR/JPY is a put (sell/give) on EUR and a call (buy/receive) on JPY.

Exercise.  Sometimes a currency pair is not actively quoted.  For example, if the domestic currency is EUR and the foreign currency is JPY, the pair underlying a given currency contract would be JPY/EUR, which is not actively quoted.  Of course, one could simply invert this currency pair and the above discussion applies with this exchange rate.  However, show that for currencies C1 and C2 that a call (put) on C1/C2 is a equivalent to a put (call) on C2/C1.  So if the European company (domestic currency) wants to buy a put on JPY/EUR (they want to sell JPY that they expect to receive sometime in the future for EUR), then they can buy an (equivalent) call on EUR/JPY.

As a final note, each currency has an associated risk free rate $r_{d}$ and $r_{f}$ (domestic and floating, respectively). 

 II. Valuation Theory

The Black-Scholes model on a dividend paying equity is easily adapted to the present situation by regarding the currency pair exchange rate as a price with an associated volatility and the foreign riskless rate as a dividend yield.  The justification for the ladder follows the same cost of carry argument as in the case of a dividend.

The usual formulas are
$$c(S,t)=Se^{-r_{f}(T-t)}\mathbb{N}(d_{1})-Ke^{-r_{d}(T-t)}\mathbb{N}(d_{2})$$
and
$$p(S,t)=Ke^{-r_{d}(T-t)}\mathbb{N}(-d_{2})-Se^{-r_{f}(T-t)}\mathbb{N}(-d_{1})$$
where of course
$$d_{1}=\frac{\log(S/K)+(r_{d}-r_{f}+\sigma^{2}/2)(T-t)}{\sigma\sqrt{T-t}}$$
and
$$d_{2}=d_{1}-\sigma\sqrt{T-t}.$$

Because $F_{t}=S_{t}e^{-(r_{d}r_{f})(T-t)}$, the forward price of the currency pair, the formulas above can be simplified to

Thus one only needs the discount rate in the domestic currency.

III. Example

Consider a call on