Basics of Option Pricing and Options Greeks

Option pricing is the process of determining the fair premium that a buyer should pay for the right, but not the obligation, to buy or sell a currency at a predetermined strike price. In the Indian market, exchange‑traded currency options are European‑style, meaning they can be exercised only at expiry, which makes the Black‑Scholes‑Merton (BSM) model the standard valuation tool.

The BSM model incorporates five key inputs: the current spot rate (S), the strike price (K), the time to expiry (T), the volatility of the underlying (σ) and the risk‑free rates of both the domestic (r_d) and foreign (r_f) currencies. Each input reflects a market reality – for example, σ captures the expected fluctuation of the exchange rate, while r_d and r_f adjust for the cost of carry.

For the NISM exam, you will often be asked to identify which variable changes the option price most, to compute a price given numeric inputs, or to recognise the effect of using the wrong rate convention. Remember that the model assumes continuous compounding and that volatility is annualised.

The BSM formula for a European call on a foreign currency treats the foreign currency as an asset that pays a continuous dividend equal to the foreign risk‑free rate (r_f). This dividend‑adjusted approach yields the call price: C = S e^{-r_f T} N(d_1) - K e^{-r_d T} N(d_2). The put price follows from put‑call parity or can be written directly as P = K e^{-r_d T} N(-d_2) - S e^{-r_f T} N(-d_1).

The intermediate variables d_1 and d_2 capture the combined effect of the inputs: d_1 = \frac{\ln(S/K) + (r_d - r_f + \frac{\sigma^2}{2})T}{\sigma\sqrt{T}} and d_2 = d_1 - \sigma\sqrt{T}. N(·) denotes the cumulative standard normal distribution function.

Exam questions may present any subset of the inputs and ask you to solve for the missing one, or they may give you d_1 and d_2 and require you to compute the price. Always check whether the rates are quoted on an annual basis and whether they need to be converted to a continuous compounding format (use e^{-rT}).

The Greeks quantify how the option premium reacts to small changes in underlying variables. They are indispensable for risk management and are a frequent focus of NISM questions. Each Greek isolates one source of risk while holding other inputs constant.

Delta measures the price sensitivity to the underlying spot rate, Gamma captures the curvature of that relationship, Theta reflects time decay, Vega gauges sensitivity to volatility, and Rho indicates the impact of interest‑rate changes. Understanding the sign and magnitude of each Greek helps you predict profit‑loss scenarios for both buyers and sellers.

In the exam, you may be asked to match a Greek to its definition, compute a Greek given the BSM inputs, or interpret how a change in market conditions (e.g., rising volatility) will affect the option’s value. Remember that for currency options, the domestic and foreign rates affect Delta and Rho differently from equity options.

Delta (Δ) tells you how much the option price will move for a one‑unit change in the spot exchange rate, assuming all other inputs stay constant. For a call, Δ is positive because the option gains value when the underlying appreciates; for a put, Δ is negative.

In the BSM framework, Delta for a currency call is given by Δ_call = e^{-r_f T} N(d_1). The foreign risk‑free rate (r_f) reduces the effective exposure because the holder earns the foreign interest while holding the option.

Exam questions frequently present a Delta value and ask you to infer whether the option is deep‑in‑the‑money, at‑the‑money or out‑of‑the‑money, or they may require you to compute Delta from the BSM inputs.

Gamma (Γ) measures the curvature of the option’s price curve with respect to the spot rate. It tells you how Delta itself changes as the underlying moves. A high Gamma indicates that Delta can shift quickly, which is critical for hedging.

For currency options under BSM, Gamma is the same for calls and puts and is expressed as Γ = \frac{e^{-r_f T} N'(d_1)}{S \sigma \sqrt{T}} where N'(d_1) is the standard normal probability density function at d_1.

In the exam, you may be asked to compare Gamma values for at‑the‑money versus deep‑in‑the‑money options, or to compute Gamma to assess the stability of a Delta‑hedged position.

Theta (Θ) represents time decay – the amount by which the option premium erodes each day, assuming all else unchanged. For a long call, Θ is typically negative because the right to buy becomes less valuable as expiry approaches. The BSM expression is complex, but the key exam insight is that Theta is largest for at‑the‑money options and accelerates in the final weeks.

Vega (ν) measures sensitivity to volatility. Both calls and puts have positive Vega, meaning higher expected volatility raises the option’s value. Vega peaks for at‑the‑money options and declines as the option moves deep in or out of the money.

Rho (ρ) captures the effect of changes in the domestic risk‑free rate. For a call, ρ is positive (higher r_d ↑ price); for a put, ρ is negative. The foreign rate enters the pricing through the carry term and indirectly influences Delta, but Rho itself is defined only with respect to the domestic rate in the NISM syllabus.