title: “📖 Notes on MSFM - Foreign Exchange”
date: 2019-12-04
tags: notes
mathjax: true

home_only: true

Theoretical Pricing

FX Spot Contract

The spot price $S_t$ is the observable market price of $1$ unit of foeign currency. Let $\mathbb{F}$ denote foreign currency and $\mathbb{D}$ denote domestic currency:

$1\mathbb{F} = S_t \mathbb{D}$

A FX spot contract is an agreement where the buyer purchase $1$ units of foreign currency at a fixed rate $R$ at current time $t$ .

$Buyer \; \dfrac{\leftarrow 1\mathbb{F}}{R\mathbb{D} \rightarrow} \; Seller$

The contract value to the buyer is:

$1\mathbb{F} - R\mathbb{D} = (S_t - R)\mathbb{D}$

FX Forward Contract

Denote domestic interest rate = $r^{d}$ . The price of domestic zero-coupon bond $P^d(\tau) = e^{-r^{d}\tau}$

A FX forward contract is an agreement where the buyer agree to purchase $1$ units of foreign currency at a fixed rate $R$ at future time $T$ :

$Buyer \; \dfrac{\leftarrow \text{1}\mathbb{F}}{R\mathbb{D} \rightarrow} \; Seller$

The time- $t$ value of a forward contract is:

$\begin{align} PV_t^{forward} &= PV_t (1\mathbb{F} - R\mathbb{D}) \\ &= P^{f} \mathbb{F} - P^{d} R\mathbb{D} \\ &= P^{f} S_t \mathbb{D} - P^{d} R\mathbb{D} \\ &= [S_t e^{-r^f(T - t)} - R e^{-r^{d}(T - t))}] \mathbb{D} \\ \end{align}$

We set $PV_t^{forward}=0$ to calculate the forward price $F_t$ at time $t$ . The equation is also called the covered interest parity, or CIP:

$F_t = S_t e^{(r^{d} - r^f)(T - t)}$

Non-Deliverable forward

Non-deliverable currency has restricted exchange by local regulations. CIP does not hold since covered interest arbitrage is not possible. For example:

Asia

CNY: China Yuan
TWD: New Taiwan Dollar
KRW: South Korean Won
INR: India Rupee
PHP: Philippine Piso
IDR: Indonesia Rupiah
MYR: Malaysian Ringgit

Latin America:

COP: Colombian Peso
VEB: Venezuelan Bolívar
BRL: Brazilian Real
PEN: Peru Sol
UYU: Uruguayan Peso
CLP: Chilean Peso
ARS: Argentine Peso

Europe, Middle East and Africa:

EGP: Egyptian Pound
KZT: Kazakhstani Tenge

Given CIP, we can calculate the implied yield, which is the foreign interest rate implied by the forward rate, domestic spot rate and domestic interest rate.

$r^f_{implied} = r^{d} - \dfrac{\log{F_t/S_t}}{T - t}$

We know that the exponential function $e^x$ can be expressed as the sum of the Maclaurin series:

$e^x = \sum_{n = 1}^{\infty} \dfrac{x^n}{n!} = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} ...$

Applying this to the forward rate:

$\begin{align} F(t, T) &= S_t[1 + (r^{d} - r^f)(T - t) + O([(r^{d} - r^f)(T - t)]^2)] \\ &\approx S_t + S_t (r^{d} - r^f)(T - t) \end{align}$

FX Swap Contract

A FX swap contract contains two FX forward contracts at time $T_1, T_2$ with opposite directions.

For example, a buy/sell swap contract:

$At\; t = T_1, Buyer \; \dfrac{\leftarrow 1\mathbb{F}}{R_1\mathbb{D} \rightarrow} \; Seller \\ At\; t = T_2, Buyer \; \dfrac{\leftarrow R_2\mathbb{D}}{1\mathbb{F} \rightarrow} \; Seller$

The present value of the swap contract is the sum of the present value of the two sub-contracts:

$\begin{align} PV_t^{swap} &= PV_t^1 - PV_t^2 \\ &= (P^{f1}S_t - P^{d1}R_1)\mathbb{D} - (P^{f2}S_t - P^{d2}R_1)\mathbb{D} \\ &= [ S_t (e^{-r^{f1}(T_1 - t)} - e^{-r^{f2}(T_2 - t)}) - R_1e^{r^{d1}(T_1 - t)} + R_2e^{r^{d2}(T_2 - t)}]\mathbb{D} \\ \end{align}$

Note that the value of a swap contract is fairly insensitive to spot rate changes, comparing to that of a forward contract.

$\begin{align} \dfrac{\partial PV^{swap}}{\partial S} &= [(e^{-r^{f1}(T_1 - t)} - e^{-r^{f2}(T_2 - t)})] \\ &\approx r^f (T_2 - T_1) \;\;[\text{assuming } r^{f1} \approx r^{f2}] \\\\ \dfrac{\partial PV^{forward}}{\partial S} &\approx 1 \\ \end{align}$

FX Option

A FX option conveys the right, but not the obligation, to exchange $1$ units of foreign currency for $K$ units of domestic currency, at a future date $T$ .

For example, the buyer of a foreign currency call strike at $K$ , have the right at maturity to buy $1$ unit of $\mathbb{F}$ at $K$ even if $S_T > K$ .

This is equivalent to the the buyer of $K$ units of domestic currency put strike at $1/K$ , which grants the buyer the right at maturity to sell $K$ unit of $\mathbb{D}$ at a rate of $1/K$ , even if the exchange rate $1/S_T$ falls below $1/K$ .

In formula:

$Call^{\mathbb{F}}(Strike = K\mathbb{D}) = KPut^{\mathbb{D}}(Strike = (1/K)\mathbb{F})$

Visualizing the transactions on a foreign currency call:

$\begin{align} At\;time\;t=0 &: Buyer \; \xrightarrow{Price(Call^{\mathbb{F}})} \; Seller \\ At\;time\;t=T &: Buyer \; \xleftarrow[max(\;0, (S_T - K)\mathbb{D}\;) ]{} \; Seller \\ \end{align}$

Visualizing the transactions on a domestic currency put:

$\begin{align} At\;time\;t=0&: Buyer \; \xrightarrow{KPrice(Put^{ \mathbb{D}})} \; Seller \\ At\;time\;t=T&: Buyer \; \xleftarrow[K max(\;0, (1/K - 1/S_T)\mathbb{F}\;) ]{} \; Seller \\ \end{align}$

FX options also satisfy put-call parity:

$\begin{align} Call - Put & = P^{d}[F_t - K] \\ &= S_te^{-r^{f}(T-t)} - Ke^{-r^{d}(T-t)} \\ \end{align}$

Garman-Kohlhagen

To evaluate the price of the option:

Assumptions on the stochastic nature of S_t
Create a “risk-free” hedge portfolio, in order to find a governing PDE for the option value, which also leads to an equivalent risk-neutral probability measure
Solve the PDE directly, with appropriate boundary conditions

We know that if a tradable asset $S_t$ follows the geometric Brownian motion:

$dS_t = rS_tdt + \sigma S_tdW^{\mathbb{P}}$

Applying Ito's formula any value of a derivative contract $V(S_t, t)$ :

$dV(S_t, t) = \dfrac{\partial V}{\partial t}dt + \dfrac{\partial V}{\partial S_t}dS_t + \dfrac{1}{2}\dfrac{\partial^2 V}{\partial S_t^2}(dS_t)^2$

Setting the drift term to be zero as the derivative contract is tradeable, we can derive the Black-Scholes PDE equation characterize $V$ as such:

$rV = V_t + rS_tV_S + \dfrac{1}{2}\sigma^2S_t^2V_{SS}$

However, since the foreign exchange spot rate $S_t$ is not tradable, we need to tweak the B-S formula. Let $B^d$ and $B^f$ denote a bank account in domestic and foreign currencies, where $dB^d = r^dB^d \;dt$ and $dB^f = r^fB^f \;dt$ . Construct replicating portfolio and set the drift term to be $0$ , the Garman-Kohlhagen PDE equation can be derived:

$r^dV = V_t + (r^d - r^f) S_tV_S + \dfrac{1}{2}\sigma^2S_t^2V_{SS}$

Solving the PDF:

$Call^{G-K}_t = P^d [F_t\Phi(d_1) - K\Phi(d_2)] \\ where \; d_{1,2} = \dfrac{log\dfrac{F_t}{K} \pm \dfrac{1}{2}\sigma^2(T - t)}{\sigma \sqrt{T - t}}$

Using the Freynman-Kac equation with additional derivation, we can conclude that $\exists \; \mathbb{Q}$ s.t. the arbitrage-free price of the contingent claim $V$ is unequivocally determined as the expected value of the discounted final payoff under $\mathbb{Q}$ , and $S_t$ obeys the stochastic differential equation:

$dS = (r^d - r^f)Sdt + \sigma SdW^{\mathbb{Q}}$

Practical Pricing

FX Spot Contract

The trade date is when the terms of the transaction are agreed, and the value date is when transaction occurs, which is trade date $+2$ for most currency pairs.

The spot rate quote $EURUSD = 1.2$ means:

$1\;EUR = 1.2\;USD$ , i.e. higher the $EURUSD$ , stronger the $EUR$ .
$EUR$ is the base currency and is set to 1 unit, whereas $USD$ is the numeraire currency which is used as the numeraire.

The bid-offer spread $EURUSD = 1.199 / 1.201$ means:

The dealer is willing to buy $1\;EUR$ for $1.199\;USD$
The dealer is willing to sell $1\;EUR$ for $1.201\;USD$

Equivalently:

The highest price YOU can sell $1\;EUR$ is $1.199\;USD$
The lowest price YOU can buy $1\;EUR$ is $1.201\;USD$

FX Forward Contract

$\begin{align} \text{Forward Point} &= Forward \;\text{(outright)} - Spot \\ &= S_t (e^{(r^d - r^f)T}-1) \end{align}$

The forward point is commonly expressed in the unit pip, or point in percentage, that is worth $0.01\%$ .

Example 1 When selling a forward for foreign currency $\mathbb{F}$ , the bid side spot rate plus bid side forward points shall be equal to the bid side outright forward rate.

A market-maker would construct the short $\mathbb{F}$ forward as follow. Note that borrowing $\mathbb{F}$ and lending $\mathbb{D}$ correspond to selling a forward and therefore the bid-side forward point.

Time	Transactions
$t = 0$	borrow $e^{-r^f_{offer}T}\mathbb{F}$ execute a short $\mathbb{F}$ spot contract lend $S_te^{-r^f_{offer}T}\mathbb{D}$
$t = T$	receive $S_te^{(r^d_{bid}-r^f_{offer})T}\mathbb{D}$ execute a long $\mathbb{F}$ spot contract pay $1\mathbb{F}$

This is the same as selling an outright forward contract:

Time	Transactions
$t = 0$	N/A
$t = T$	receive $F_t\mathbb{D}$ pay $1\mathbb{F}$

FX Swap Contract

A FX swap contract intends to adjust the timing of cash flows from $T_1$ to $T_2$ and alter the value date on an existing trade. The near rate should be consistent with the market forward rate for the near date, and the same goes for the far rate. The swap point is equal to:

$\begin{align} \text{Swap Point} &= \text{Far Rate} - \text{Near Rate} \\ &= S_te^{(r^d - r^f)(T_2-T_1)} \end{align}$

A buy/sell swap on $\mathbb{F}$ means that it buys a forward on $\mathbb{F}$ at $T_1$ and sells a forward on $mathbb{F}$ at $T_2$ . This correspond to borrowing $\mathbb{F}$ and lending $\mathbb{D}$ .

Example 2 A short outright forward position on $\mathbb{F}$ can be thought of as a buy/sell swap on $\mathbb{F}$ with a spot transaction at the near date and $T_1=0$ , similar to Example 1. Here $\tau = T_2 - T_1$ :

Time	Transactions
$t = T_1$	borrow $e^{-r^f_{offer}\tau}\mathbb{F}$ execute a short $\mathbb{F}$ forward contract: $\;\;-\;\;$ pay $e^{-r^f_{offer}\tau}\mathbb{F}$ $\;\;-\;\;$ receive $F_{T_1}e^{-r^f_{offer}\tau}\mathbb{D}$ lend $F_{T_1}e^{-r^f_{offer}\tau}\mathbb{D}$
$t = T_2$	receive $F_{T_1}e^{(r^d_{bid}-r^f_{offer})\tau}\mathbb{D}$ execute a long $\mathbb{F}$ forward contract: $\;\;-\;\;$ pay $F_{T_1}e^{(r^d_{bid}-r^f_{offer})\tau}\mathbb{D}$ $\;\;-\;\;$ receive $(1/F_{T_2})F_{T_1}e^{(r^d_{bid}-r^f_{offer})\tau}\mathbb{F}$ pay $1\mathbb{F}$

This is the same as a buy/sell swap:

Time	Transactions
$t = T_1$	recieve $e^{-r^f_{offer}\tau}\mathbb{F}$ pay $F_{T_1}e^{-r^f_{offer}\tau}\mathbb{D}$
$t = T_2$	receive $F_{T_2}\mathbb{D}$ pay $1\mathbb{F}$

Example 3 From a market-maker perspective:

Contract	Swap Point	T1	T2
Buy/Sell	offer-side swap point	pay at bid-side points	sell at offer-side points
Sell/Buy	bid-side swap point	sell at bid-side $\ast$ points	pay at bid-side points

Note( $\ast$ ): because a swap has less interest rate risk than an outright forward, the market-maker can easily construct a swap with bid-side points for both near and far dates.

Example 4 Say the swap point is $-0.01$ , then a party that buy/sell the foreign currency $\mathbb{F}$ is paying the swap point, because it is selling at a lower Far rate.

Conversely, a party that sell/buy $\mathbb{F}$ is earning the swap point.

Risk Characteristics

Contract	Transactions	FX Risk	IR Spread Risk
Spot	1	Yes	No
Forward (Outright)	1	Yes	Yes
Swap	1	No	Yes

FX Option

There are four ways to express an option price:

Price $\rightarrow$	in $\mathbb{D}$ units	in $\mathbb{F}$ units
Notional as $1\mathbb{F}$	$P_{numccy}$ $=\text{Garman-Kohlhagen}$ $\rightarrow\mathbb{D}\text{ pips}$	$P_{baseccy\%}$ $=P_{numccy}/S_t$ $\rightarrow\mathbb{F}\text{ %}$
Notional as $1\mathbb{D}$	$P_{numccy\%}$ $=P_{numccy}/K$ $\rightarrow\mathbb{D}\text{ %}$	$P_{baseccy}$ $=P_{numccy\%}/S_t$ $\rightarrow\mathbb{F}\text{ pips}$

Straddle

$\text{Straddle} = \text{ATM Call} + \text{ATM Put}$

The meaning of $ATM$ can be different:

$ATMS$ : at the spot rate
$ATMF$ : at the forward rate (preferred by traders)
$DNS$ : delta-neutral

Risk Reversal

$\text{Risk Reversal} = \text{25-Delta Call} - \text{25-Delta Put}$

Where a $25$ -delta option is an option with a delta of $\pm25\%$ . Risk reversal can also denote the difference in implied volatility:

$\text{Risk Reversal} = \sigma_{\text{25-Delta Call}}- \sigma_{\text{25-Delta Put}}$

Butterfly

$\begin{align} \text{Butterfly} &= \text{25-Delta Call} + \text{25-Delta Put} - \text{Straddle} \\ &= \text{Strangle} - \text{Straddle} \end{align}$

Note that butterfly is vega ( $\partial V/\partial \sigma$ ) neutral, e.e. the strangle notional is usually larger than the straddle notional to create equal and offestting vega . BF can also denote the difference in implied volatility:

$\text{BF} = Avg(\sigma_{\text{25-Delta Call}}, \;\sigma_{\text{25-Delta Put}})- \sigma_{Straddle}$

Under the Black-Scholes framework, delta-netural strike ( $K=Se^{\sigma^2/2}$ ) options have the highest vega $\mathcal{V}$ :

$\begin{align} \mathcal{V} = \partial V/\partial \sigma &= Se^{-q\tau}\phi(d_1)\sqrt{\tau} \\ &= Ke^{-r\tau}\phi(d_2)\sqrt{\tau} \end{align}$

In addition, option gamma $\Gamma = \partial^2 V/\partial S^2 = \mathcal{V}/(S^2\sigma T)$