What Is Phi?
Phi (Φ) is the first derivative of option value with respect to the foreign risk-free rate in FX options (partial V / partial rf). It is the FX-options analog of rho. Phi appears in the Garman-Kohlhagen FX-options model alongside rho (the domestic-rate sensitivity). Phi is the structural sensitivity for currency-options carry trades and cross-currency hedging.
What Is Phi in Options?
Phi captures sensitivity of an FX option's value to changes in the foreign currency's risk-free rate. The FX option payoff depends on both rates: the domestic rate (where the option is denominated) and the foreign rate (where the underlying currency is "borrowed" in the standard FX-options framework). Phi tracks the foreign-rate sensitivity; rho tracks the domestic-rate sensitivity.
Two intuitions. First, in Garman-Kohlhagen the foreign rate plays the role of a continuous dividend yield on the foreign currency: a higher foreign rate increases the carry cost of holding the foreign currency, so call phi is negative (calls on the foreign currency lose value when the foreign rate rises) and put phi is positive (puts gain). The signs mirror epsilon, not the domestic-rate rho. Second, FX-options pricing requires both rates because cross-currency carry is the embedded structure: holding the foreign currency earns the foreign rate; the option's strike represents a forward cross-rate that depends on the rate differential.
Worked Example
EUR/USD at 1.10, 1-year ATM call (K=1.10), domestic (USD) rate 4%, foreign (EUR) rate 2%, IV 8%. Garman-Kohlhagen gives:
- Call phi (per unit of EUR rate change) = approximately -0.55 (sign and magnitude depending on payoff convention)
- Per-1%-rate scaling: -0.0055 per unit notional
If the EUR rate rises from 2% to 3% (1pp increase), the option value changes by approximately -$0.0055 per unit notional. For a $10M-notional call, the impact is approximately $55,000.
The relative size compared to rho (USD rate sensitivity) depends on the rate differential and the option's moneyness. For ATM options, |phi| is typically comparable to |rho| for ATM options when rates are similar; the two move in opposite directions for the same option.
Why Phi Matters
Phi is operationally important only for FX-options trading; it has no analog for equity options where there is only one rate. For currency-options desks, phi is one of the four core Greeks (alongside delta, gamma, vega) that gets tracked. Three operational uses:
- Carry-trade hedging: FX-options strategies often involve carry exposure. Phi is the analytical Greek that quantifies sensitivity to changes in the carry differential.
- Cross-currency hedging: hedging FX-options books against changes in the foreign rate requires knowing aggregate phi.
- Rate-differential trading: trades that bet on the convergence or divergence of two countries' rates have natural FX-options expressions; phi vs rho dispersion is the analytical signal.
How Pricing Models Compute Phi
- Garman-Kohlhagen: the standard FX-options model. Closed-form phi parallel to BS rho.
- Heston (extended to FX): phi computed by Fourier inversion. Heston-FX models extend the variance dynamics to currency markets.
- Jump diffusion (FX): jumps capture currency-pair regime breaks; phi computed via standard finite difference on jump-augmented pricing.
Special Cases
- Long-dated FX options: phi is meaningful. Hedge with foreign-currency rate instruments.
- Short-dated FX options: phi is small. Often ignored in retail FX-options books.
- ATM FX options: phi is approximately equal in magnitude to rho but opposite sign.
- FX options with large rate differentials: phi and rho asymmetric. Strong carry direction.
Related Greeks
Phi is one of three rate Greeks. Rho is the domestic-rate first-order sensitivity. Epsilon is the dividend-yield sensitivity (the equity-options analog of phi). Together, rho, phi, and epsilon describe the carry-and-discount Greeks of an option across underlying classes.
Related Concepts
Rho · Epsilon · Black-Scholes · All 17 Greeks
References & Further Reading
- Garman, M. and Kohlhagen, S. (1983). "Foreign Currency Option Values." Journal of International Money and Finance, 2(3), 231-237.
- Castagna, A. (2010). FX Options and Smile Risk. Wiley. Practitioner reference for FX-options hedging and smile risk.
This page is part of the 17 Greeks reference covering every options Greek with formula, intuition, worked example, and how each pricing model computes it. Browse the full documentation.