Notes on Foreign Exchange
spot price is the observable market price of unit of foeign currency. Let denote foreign currency and denote domestic currency:
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:
Denote domestic interest rate = . The price of domestic zero-coupon bond
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:
Non-deliverable currency has restricted exchange by local regulations. CIP does not hold since covered interest arbitrage is not possible. For example:
- CNY: China Yuan
- TWD: New Taiwan Dollar
- KRW: South Korean Won
- INR: India Rupee
- PHP: Philippine Piso
- IDR: Indonesia Rupiah
- MYR: Malaysian Ringgit
- 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 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 sub-contracts:
Note that the value of a swap contract is fairly insensitive to spot rate changes, comparing to that of a forward contract.
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 .
Visualizing the transactions on a foreign currency call:
Visualizing the transactions on a domestic currency put:
FX options also satisfy
To evaluate the price of the option:
- Assumptions on the stochastic nature of St
- 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 follows the
geometric Brownian motion:
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
Black-Scholes PDE equation characterize as such:
However, since the foreign exchange spot rate is not tradable, we need to tweak the B-S 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
Garman-Kohlhagen PDE equation can be derived:
Solving the PDF:
Freynman-Kac equation with additional derivation, we can conclude that s.t. the
arbitrage-free price of the contingent claim is unequivocally determined as the expected value of the discounted final payoff under , and obeys the stochastic differential equation:
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.
spot rate quote means:
- , i.e. higher the , stronger the .
- is the
base currencyand is set to 1 unit, whereas is the
numeraire currencywhich is used as the numeraire.
bid-offer spread means:
- The dealer is willing to buy for
- The dealer is willing to sell for
- The highest price YOU can sell is
- The lowest price YOU can buy is
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 market-maker would construct the
short forward as follow. Note that borrowing and lending correspond to selling a forward and therefore the
bid-side forward point.
execute a short spot contract
execute a long spot contract
This is the same as selling an outright forward 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:
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 :
execute a short forward contract:
execute a long forward contract:
This is the same as a buy/sell swap:
Example 3 From a
|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 points||pay at bid-side points|
Note(): 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 , 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.
|Contract||Transactions||FX Risk||IR Spread Risk|
There are four ways to express an option price:
|Price||in units||in units|
The meaning of can be different:
- : at the spot rate
- : at the forward rate (preferred by traders)
- : delta-neutral
Where a -delta option is an option with a delta of . Risk reversal can also denote the difference in
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
Under the Black-Scholes framework, delta-netural strike () options have the highest vega :
In addition, option gamma
- FINM 37301 Foreign Exchange: Markets, Pricing and Products, Anthony Capozzoli, University of Chicago