FX derivatives: models and methods

Last updated 10 November 2026

This page describes how GIAnalyzer and the Gamma Impact API value FX derivatives and measure their risk: the instruments covered, the model behind each, the market data each takes, and what the models do not do. It describes GIAnalyzer version 5.12.0; the Gamma Impact API covers the same instruments and models. It is provided for information and is not a contractual specification.

The function-level detail (every function, its arguments and outputs) is in the function reference.

Dates and conventions

Each valuation uses four dates:

• Valuation date: the date the value is for.
• Spot date: the settlement date of a spot trade on the valuation date.
• Expiry date: the last date the option can be exercised or its barrier observed.
• Delivery date: the date the cash flows settle.

Time to expiry, used with volatility, runs in calendar days from the valuation date to expiry, divided by 365.

Discounting runs from the spot date to the delivery date, in each currency's own day-count convention. Values are therefore present values at the spot date.

Day-count conventions: Actual/Actual (ISDA and ICMA), Actual/360, Actual/365, NL/365, and the 30/360 family (30/360, 30E/360, 30US/360, 30G/360, 30I/360, 30/360 ISDA).

Compounding: interest rates may be given with annual, semi-annual, quarterly, monthly, simple or continuous compounding. They are converted to continuous compounding over the period of the calculation before use.

Market data

Interest rates are given as a flat rate per currency or as a zero-rate curve per currency. A curve is interpolated at the delivery date. Interpolation is linear in the zero rate by default; log-linear, exponential, cubic spline and log-cubic spline are available, with flat extrapolation beyond the first and last points.

Forward points can replace one of the two interest-rate inputs: the missing currency's rates are implied from the points and the other currency's rates through covered interest parity.

Volatility is given either as a single flat volatility or as a volatility surface:

• The surface has expiries in rows and delta pillars in columns (for example 10- and 25-delta puts, at-the-money, 25- and 10-delta calls). Risk-reversal and butterfly quotes are also accepted and converted to pillar volatilities in the usual market way: at-the-money plus butterfly, plus or minus half the risk reversal.
• The volatility for a given strike is found on the surface by converting strikes to deltas. The at-the-money pillar is taken as the 50-delta point. Deltas are spot deltas and are not premium-adjusted.
• Between pillars the surface is interpolated in delta and in time, and it is held flat beyond the outermost pillars and expiries.

Instruments and models

FX forwards and swaps

The forward rate comes from covered interest parity: spot, adjusted by the two currencies' interest rates from the spot date to the delivery date, each in its own compounding and day count. Forward points are the difference between the forward rate and spot.

The value of a forward is the difference between the two legs, each discounted at its own currency's rate from the spot date to delivery, and expressed in either currency. Swaps are valued as their two forward legs.

Further outputs: implied interest rates from forward points or forward prices, cross rates and cross forward points through USD, and the rate sensitivities described under Risk measures.

Vanilla options

European FX options are valued with the Garman-Kohlhagen model (Black-Scholes with a foreign and a domestic interest rate). Volatility applies over the time to expiry; discounting applies from spot to delivery, so the model separates the expiry and the delivery date.

The value can be expressed as an amount, a percentage of notional or in pips, and in either currency. Implied volatility is found from a given premium by an iterative search.

Digital options

European digital options paid at delivery are valued in closed form: a cash-or-nothing payout in the counter currency, an asset-or-nothing payout in the base currency (Reiner and Rubinstein, 1991).

One-touch and no-touch options

One-touch options, paid at the hit or at expiry, are valued with the closed-form formulas for continuously monitored touch options given by Wystup (2006). A no-touch is valued as the complement of the one-touch paid at expiry. Payouts in the base currency are valued through the symmetry between the two currencies.

Single-barrier options

Knock-in and knock-out options with an up or down barrier are valued in closed form:

• Continuously monitored barriers ("American" barriers): the Merton (1973) and Reiner-Rubinstein (1991) formulas. Knock-in options are valued directly and knock-out options as the vanilla minus the knock-in.
• Barriers observed at expiry only ("European" barriers): replicated with vanilla options and a digital option.

If spot is already beyond the barrier on the valuation date, a knock-out is worth zero and a knock-in is worth the vanilla.

Double-barrier options

Options with a lower and an upper barrier, both continuously monitored, are valued with a series solution of the method-of-images type (Kunitomo and Ikeda, 1992), summed until further terms no longer change the value. Knock-in options are valued as the vanilla minus the knock-out.

KIKO options

An option with one knock-in and one knock-out barrier is valued as the difference between a single-barrier knock-out option and a double-barrier knock-out option, both from the formulas above.

Asian options

Discrete arithmetic average-rate options with a fixed strike are valued with a moment-matching lognormal approximation of the average (Levy, 1992; Haug, Haug and Margrabe, 2003). Fixings already observed are taken into account: they are averaged and the strike is adjusted for the remaining fixings. Fixings are assumed to be evenly spaced up to expiry.

Dual currency deposits

A dual currency deposit is valued as a deposit with its coupon, minus a vanilla option sold by the depositor, struck at the conversion rate, with expiry at the deposit's maturity.

Target redemption forwards (TARF)

TARFs are valued by Monte Carlo simulation. The exchange rate follows geometric Brownian motion from one fixing date to the next, with the drift given by the two currencies' interest rates and the volatility given flat or as a term structure by fixing date. At each fixing the settlement is computed, including the leverage on the losing side; gains accumulate towards the target, and the trade ends when the target is reached, with the last payment full, capped or none, as specified. Fixings already observed are taken into account. The value is the average of the discounted path values. The number of paths and the random seed are set by the user.

Vanna-Volga adjustment for exotic options

For the touch, barrier, double-barrier, KIKO and digital options, a smile-adjusted value is available by the Vanna-Volga method (Castagna and Mercurio, 2007). The exotic's vega, vanna and volga are matched by three instruments from the volatility surface: an at-the-money option and the 25-delta put and call. The cost of the smile on those three instruments is carried over to the exotic. The adjustment can be weighted by the probability that the barrier is not hit, as is common market practice.

Risk measures

All sensitivities are computed by revaluing the instrument with a small change in one input (finite differences), not from analytic formulas.

Greeks are given per unit of notional or as amounts. Deltas are not premium-adjusted.

Value at Risk for FX positions (historical, Monte Carlo and hybrid historical) is described on the Value at Risk page.

What the models do not do

Stating the limits plainly is part of a methodology. The FX models:

• use no stochastic or local volatility model. The smile enters through the volatility surface for vanilla options and through the Vanna-Volga adjustment for exotic options. The exotic functions that read a surface without Vanna-Volga use its at-the-money volatility;
• treat barriers as continuously monitored, with no correction for discrete monitoring, and have no rebates;
• value Asian options with evenly spaced fixings, arithmetic average-rate only;
• give no Monte Carlo standard error for the TARF value, and no TARF Greeks;
• do not adjust deltas for the premium.

References

• Castagna, A. and Mercurio, F. (2007). The vanna-volga method for implied volatilities. Risk, January, 106–111.
• Garman, M. and Kohlhagen, S. (1983). Foreign currency option values. Journal of International Money and Finance 2(3), 231–237.
• Haug, E. G. (2007). The Complete Guide to Option Pricing Formulas, 2nd edition. McGraw-Hill.
• Haug, E. G., Haug, J. and Margrabe, W. (2003). Asian pyramid power. Wilmott Magazine.
• Kunitomo, N. and Ikeda, M. (1992). Pricing options with curved boundaries. Mathematical Finance 2(4), 275–298.
• Levy, E. (1992). Pricing European average rate currency options. Journal of International Money and Finance 11(5), 474–491.
• Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science 4(1), 141–183.
• Reiner, E. and Rubinstein, M. (1991). Breaking down the barriers. Risk 4(8), 28–35; and Unscrambling the binary code. Risk 4(9), 75–83.
• Wystup, U. (2006). FX Options and Structured Products. Wiley.

Document history