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Subsections

Foreign Currency Options

European options.

To value european options, all that is necessary is a slight adjustment of the Black-Scholes equation to adjust for the interest-rate differential.
• : exchange rate.
• : exercise price.
• : interest rate in domestic currency.
• : interest rate in foreign currency.
• : volatility of exchange rate.
• : time to maturity.

where

Computer Algoritm, European Currency Option

// file currency_opt_euro_call.cc
// author:  Bernt A Oedegaard

#include <cmath>
#include "normdist.h"      // define the normal distribution function

double currency_option_price_call_european( double S,      // exchange_rate,
double X,      // exercise,
double r,      // r_domestic,
double r_f,    // r_foreign,
double sigma,  // volatility,
double time)   // time to maturity
{
double sigma_sqr = sigma*sigma;
double time_sqrt = sqrt(time);
double d1 = (log(S/X) + (r-r_f+ (0.5*sigma_sqr)) * time)/(sigma*time_sqrt);
double d2 = d1 - sigma * time_sqrt;
return S * exp(-r_f*time) * N(d1) - X * exp(-r*time) * N(d2);
};


References.

The original formulations of European foreign currency option prices are in Garman and Kohlhagen (1983) and Grabbe (1983).

See (Hull, 1993, Ch 11.3) for a textbook treatment.

American options.

For American options, the usual method is approximation using binomial trees, checking for early exercise due to the interest rate differential.

Computer Algoritm, Binomial Currency Option

// file currency_opt_bin_call.cc
// author: Bernt A Oedegaard

#include <cmath>
#include <algorithm>
#include <vector>

double currency_option_price_call_american_binomial(
double S, double X, double r, double r_f, double sigma,
double time, int no_steps) {
vector<double> exchange_rates(no_steps+1);
vector<double> call_values(no_steps+1);
double t_delta= time/no_steps;
double Rinv = exp(-r*(t_delta));
double u = exp(sigma*sqrt(t_delta));
double d = 1.0/u;
double uu= u*u;
double pUp   = (exp((r-r_f)*t_delta)-d)/(u-d); // adjust for foreign int.rate
double pDown = 1.0 - pUp;
exchange_rates[0] = S*pow(d, no_steps);
int i;
for (i=1; i<=no_steps; ++i) {
exchange_rates[i] = uu*exchange_rates[i-1]; // terminal tree nodes
}
for (i=0; i<=no_steps; ++i) call_values[i] = max(0.0, (exchange_rates[i]-X));
for (int step=no_steps-1; step>=0; --step) {
for (i=0; i<=step; ++i)   {
exchange_rates[i] = d*exchange_rates[i+1];
call_values[i] = (pDown*call_values[i]+pUp*call_values[i+1])*Rinv;
call_values[i] = max(call_values[i], exchange_rates[i]-X); // check for exercise
};
};
return call_values[0];
};


Next: Bond Algoritms. Up: Foreign Currency Algoritms Previous: Foreign Currency Algoritms   Contents   Index
Bernt Arne Odegaard
1999-09-09