Notes on Foreign Exchange
Theoretical Pricing
FX Spot Contract
The spot price
is the observable market price of unit of foeign currency. Let denote foreign currency and denote domestic currency:
A FX spot contract
is an agreement where the buyer purchase units of foreign currency at a fixed rate at current time .
The contract value to the buyer is:
FX Forward Contract
Denote domestic interest rate = . The price of domestic zerocoupon bond
A FX forward contract
is an agreement where the buyer agree to purchase units of foreign currency at a fixed rate at future time :
The time value of a forward contract is:
We set to calculate the forward price
at time . The equation is also called the covered interest parity
, or CIP:
NonDeliverable forward
Nondeliverable
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.
We know that the exponential function can be expressed as the sum of the Maclaurin series:
Applying this to the forward rate:
FX Swap Contract
A FX swap contract
contains two FX forward contracts at time with opposite directions.
For example, a buy/sell
swap contract:
The present value of the swap contract is the sum of the present value of the two subcontracts:
Note that the value of a swap contract is fairly insensitive to spot rate changes, comparing to that of a forward contract.
FX Option
A FX option
conveys the right, but not the obligation, to exchange units of foreign currency for units of domestic currency, at a future date .
For example, the buyer of a foreign currency call strike at , have the right at maturity to buy unit of at even if .
This is equivalent to the the buyer of units of domestic currency put strike at , which grants the buyer the right at maturity to sell unit of at a rate of , even if the exchange rate falls below .
In formula:
Visualizing the transactions on a foreign currency call:
Visualizing the transactions on a domestic currency put:
FX options also satisfy putcall parity
:
GarmanKohlhagen
To evaluate the price of the option:
 Assumptions on the stochastic nature of S_{t}
 Create a “riskfree” hedge portfolio, in order to find a governing PDE for the option value, which also leads to an equivalent riskneutral probability measure
 Solve the PDE directly, with appropriate boundary conditions
We know that if a tradable asset follows the geometric Brownian motion
:
Applying Ito's formula
any value of a derivative contract :
Setting the drift term to be zero as the derivative contract is tradeable, we can derive the BlackScholes
PDE equation characterize as such:
However, since the foreign exchange spot rate is not tradable, we need to tweak the BS formula. Let and denote a bank account in domestic and foreign currencies, where and . Construct replicating portfolio and set the drift term to be , the GarmanKohlhagen
PDE equation can be derived:
Solving the PDF:
Using the FreynmanKac
equation with additional derivation, we can conclude that s.t. the arbitragefree
price of the contingent claim is unequivocally determined as the expected value of the discounted final payoff under , and obeys the stochastic differential equation:
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 for most currency pairs.
The spot rate quote
means:
 , i.e. higher the , stronger the .
 is the
base currency
and is set to 1 unit, whereas is thenumeraire currency
which is used as the numeraire.
The bidoffer spread
means:
 The dealer is willing to buy for
 The dealer is willing to sell for
Equivalently:
 The highest price YOU can sell is
 The lowest price YOU can buy is
FX Forward Contract
The forward point is commonly expressed in the unit pip
, or point in percentage, that is worth .
Example 1
When selling a forward for foreign currency , the bid side spot rate plus bid side forward points shall be equal to the bid side outright forward rate.
A marketmaker would construct the short
forward as follow. Note that borrowing and lending correspond to selling a forward and therefore the bidside
forward point.
Time  Transactions 

borrow execute a short spot contract lend 

receive execute a long spot contract pay 
This is the same as selling an outright forward contract:
Time  Transactions 

N/A  
receive pay 
FX Swap Contract
A FX swap contract intends to adjust the timing of cash flows from to 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:
A buy/sell
swap on means that it buys a forward on at and sells a forward on at . This correspond to borrowing and lending .
Example 2
A short outright forward position on can be thought of as a buy/sell swap on with a spot transaction at the near date and , similar to Example 1. Here :
Time  Transactions 

borrow execute a short forward contract: pay receive lend 

receive execute a long forward contract: pay receive pay 
This is the same as a buy/sell swap:
Time  Transactions 

recieve pay 

receive pay 
Example 3
From a marketmaker
perspective:
Contract  Swap Point  T1  T2 

Buy/Sell  offerside swap point  pay at bidside points  sell at offerside points 
Sell/Buy  bidside swap point  sell at bidside points  pay at bidside points 
Note(): because a swap has less interest rate risk than an outright forward, the marketmaker can easily construct a swap with bidside points for both near and far dates.
Example 4
Say the swap point is , then a party that buy/sell the foreign currency is paying
the swap point, because it is selling at a lower Far rate.
Conversely, a party that sell/buy 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  in units  in units 

Notional as  

Notional as  

Straddle
The meaning of can be different:
 : at the spot rate
 : at the forward rate (preferred by traders)
 : deltaneutral
Risk Reversal
Where a delta option is an option with a delta of . Risk reversal can also denote the difference in implied volatility
:
Butterfly
Note that butterfly is vega () 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
:
Under the BlackScholes framework, deltanetural strike () options have the highest vega :
In addition, option gamma
Reference:
 FINM 37301 Foreign Exchange: Markets, Pricing and Products, Anthony Capozzoli, University of Chicago