Stratégies Options - Forex Options - Garman

The Garman Kohlhagen model is an adaptation of the Black Scholes pricing model in order to be used on forex markets.

I - Black & Scholes Model

Variables
t the starting date
S the spot

Parameters
K the strike
r continuously compounded annualized interest rate
q continuously compounded annualized dividend rate
T expiry (in years)
σ the annualized volatility

For a maturity of τ = T - t ,

C = exp ( - q.τ ) . S . N( d₁ ) - exp ( - r.τ ) . K . N( d₂)

With
d₁ = [ Ln( S/K ) + ( ( r-q + 0.5σ² ).τ )] / ( σ√τ )
d₂ = [ Ln( S/K ) + ( ( r-q - 0.5σ² ).τ )] / ( σ√τ ) = d₁ - ( σ√τ )
N(.) Cumulative Normal Denstity
N( d₁ ) = ∫ [ ((1 / ( √2п )) . exp( -z²/2 ) ] dz, derived between –inf and d₁

It follows for the put,

P = - exp ( - q.τ ) . S . N( - d₁ ) + exp ( - r.τ ) . K . N( - d₂ )

II - Garman Kohlhagen Model

Currencies as for lots of assets enable to earn money by just holding them.
If one has to short a currency, one needs borrow it before. It has a cost as for every loan.
If one buys a pair, one buys a currency and sell short another one.

To take this fact into account, Garman Kohlhagen in 1983 made a adjustment on the Black Scholes Model.

Therefore,

A call with a maturity τ = T - t , r is the sold currency interest rate, rf is the bought currency interest rate.

C = exp ( - rf.τ ) . S . N( d₁ ) - exp ( - r.τ ) . K . N( d₂ )

With
d₁ = [ Ln( S/K ) + ( ( r-rf + 0.5σ² ).τ )] / ( σ√τ )
d₂ = [ Ln( S/K ) + ( ( r-rf - 0.5σ² ).τ )] / ( σ√τ ) = d₁ - ( σ√τ )
N(.) Cumulative Normal Density.
N( d₁ ) = ∫ [ ((1 / ( √2п )) . exp( -z²/2 ) ] dz, intégral derived between –inf and d₁

The put is,

P = - exp ( - rf.τ ) . S . N( - d₁ ) + exp ( - r.τ ) . K . N( - d₂ )

III - Example

A GBP / EUR call (call on GBP/put on EUR) struck at 1.80 with an underlying spot at 1.60 ( 1.60 EUR has to be sold in order to buy 1 GBP), EUR interest rate = 8%, GBP interest rate= 11%, maturity 182.5 days (one earn 11% and pay 8% if "long" GBP/EUR).

This call is worth EUR 0.02136.

Previous : Black & Scholes : Une Première Approche

Pdf connexes :
- Understanding N(d1) and N(d2) : Risk-Adjusted Probabilities in the Black-Scholes Model
- Black-Scholes Option Pricing Model

OPTIONS PRICING MODEL - INDEX
OPTIONS PRICING MODEL - INDEX
OPTIONS PRICING MODEL - CHAPTER I
OPTIONS PRICING MODEL - CHAPTER II
OPTIONS PRICING MODEL - CHAPTER III

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