Option Pricing Methodology

This sub‑topic explains how the premium of an exchange‑traded currency option is calculated. It is a core part of the NISM Series I exam because pricing directly influences trading decisions, risk management and profitability. Understanding the methodology also helps you answer scenario‑based questions that involve option valuation.

Learning Objectives

1Identify the key inputs required for pricing a currency option
2Apply the Garman‑Kohlhagen model to compute call and put premiums
3Use put‑call parity to check consistency of calculated prices
4Interpret how volatility, time and interest‑rate differentials affect option value

What is Option Pricing?

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 rate on or before expiry. In the Indian market, these options are listed on exchanges such as NSE and BSE, and the pricing must conform to the model prescribed by SEBI and NISM.

The premium reflects the expected benefit of the option plus a compensation for the risk taken by the writer. Because currency options are European‑style on Indian exchanges, the valuation assumes that exercise can only occur at expiry, which simplifies the mathematical treatment.

For the exam, you will often be given the spot rate, strike, time to maturity, domestic and foreign risk‑free rates, and volatility. You must know which of these are inputs, how they interact, and which formula to apply.

Key Inputs in Currency Option Valuation

Spot exchange rate (S₀) – the current market rate expressed as INR per unit of foreign currency (e.g., INR/USD). It is the starting point for any valuation.

Strike price (K) – the agreed‑upon rate at which the holder can buy (call) or sell (put) the foreign currency. The relationship between S₀ and K determines intrinsic value.

Domestic risk‑free rate (r_d) – typically the RBI repo rate or the yield on a government security of comparable maturity, expressed annually.

Foreign risk‑free rate (r_f) – the yield on a risk‑free instrument in the foreign currency’s jurisdiction, such as the US Treasury rate for USD options.

Time to expiry (T) – measured in years (e.g., 6 months = 0.5). The longer the time, the higher the premium because of greater uncertainty.

Volatility (σ) – the annualised standard deviation of the log returns of the exchange rate. It captures market uncertainty and is the most sensitive input in the model.

ℹ️Exam Trap – Mixing Up r_d and r_f

Students often place the domestic rate in the foreign‑discount factor and vice‑versa. Remember: discount the strike with r_d and the spot with r_f in the Garman‑Kohlhagen formula.

Garman‑Kohlhagen Model

The Garman‑Kohlhagen model is the currency‑specific adaptation of the Black‑Scholes model. It incorporates two risk‑free rates – one for the domestic currency and one for the foreign currency – to reflect the cost of carry in FX markets.

The model assumes log‑normal distribution of exchange rates, no arbitrage, continuous trading, and that the option is European. These assumptions align with the regulatory framework for exchange‑traded currency options in India.

Two intermediate variables, d₁ and d₂, capture the combined effect of spot, strike, rates, volatility and time. Once d₁ and d₂ are computed, the call and put premiums are obtained by weighting the discounted spot and strike with the cumulative normal distribution values N(d₁) and N(d₂).

∑Formula: Call Premium – Garman‑Kohlhagen

C = S_{0} \times e^{-r_{f} T} \times N(d_{1}) \; - \; K \times e^{-r_{d} T} \times N(d_{2})

Where:

C= Premium of a European call option in INR per unit of foreign currency

S_{0}= Current spot exchange rate (INR per unit of foreign currency)

K= Strike price of the option (INR per unit of foreign currency)

r_{d}= Domestic risk‑free interest rate (annual, expressed as decimal)

r_{f}= Foreign risk‑free interest rate (annual, expressed as decimal)

T= Time to expiry in years

\sigma= Annualised volatility of the exchange rate (decimal)

N(\cdot)= Cumulative standard normal distribution function

d_{1}= Intermediate variable defined below

d_{2}= Intermediate variable defined below

Worked Example

Given S_{0}=74.5, K=75, r_{d}=0.06, r_{f}=0.02, \sigma=0.12, T=0.5 years: Step 1: Compute d_{1}=\frac{\ln(S_{0}/K)+(r_{d}-r_{f}+0.5\sigma^{2})T}{\sigma\sqrt{T}} = \frac{\ln(74.5/75)+(0.06-0.02+0.5\times0.0144)\times0.5}{0.12\sqrt{0.5}} = 0.1993 Step 2: d_{2}=d_{1}-\sigma\sqrt{T}=0.1993-0.0849=0.1144 Step 3: N(d_{1})≈0.579, N(d_{2})≈0.544 Step 4: Discount factors: e^{-r_{f}T}=e^{-0.01}=0.99005, e^{-r_{d}T}=e^{-0.03}=0.97045 Step 5: C = 74.5×0.99005×0.579 - 75×0.97045×0.544 = 42.70 - 39.63 = 3.07 INR Verification: 74.5×0.99005×0.579 - 75×0.97045×0.544 = 3.07

∑Formula: Put Premium – Garman‑Kohlhagen

P = K \times e^{-r_{d} T} \times N(-d_{2}) \; - \; S_{0} \times e^{-r_{f} T} \times N(-d_{1})

Where:

P= Premium of a European put option in INR per unit of foreign currency

S_{0}= Spot exchange rate (INR per unit of foreign currency)

K= Strike price (INR per unit of foreign currency)

r_{d}= Domestic risk‑free rate (decimal)

r_{f}= Foreign risk‑free rate (decimal)

T= Time to expiry in years

\sigma= Annualised volatility (decimal)

N(\cdot)= Cumulative standard normal distribution

d_{1}= Same as in call formula

d_{2}= Same as in call formula

Worked Example

Using the same inputs as the call example: Step 1: N(-d_{1}) = 1 - N(d_{1}) = 1 - 0.579 = 0.421 Step 2: N(-d_{2}) = 1 - N(d_{2}) = 1 - 0.544 = 0.456 Step 3: P = 75×0.97045×0.456 - 74.5×0.99005×0.421 = 33.22 - 31.07 = 2.15 INR Verification: 75×0.97045×0.456 - 74.5×0.99005×0.421 = 2.15

Put‑Call Parity for Currency Options

Put‑call parity links the prices of European call and put options with the same strike, expiry and underlying. For currency options the relationship is: C - P = S_{0}e^{-r_{f}T} - Ke^{-r_{d}T}.

This equation is useful in the exam to verify whether a given set of premiums is arbitrage‑free. If the equality does not hold, a trader could construct a risk‑free profit by buying the cheaper side and selling the expensive side.

Remember that the parity uses the discounted spot (foreign rate) and discounted strike (domestic rate). Mixing the rates will cause a mismatch and is a frequent source of errors in NISM questions.

Comparison of Call and Put Premium Components (Illustrative)

Component	Call (INR)	Put (INR)
Discounted Spot (S₀e^{-r_f T})	73.755	73.755
Discounted Strike (Ke^{-r_d T})	72.784	72.784
N(d₁) / N(-d₁)	0.579 / 0.421	0.579 / 0.421
N(d₂) / N(-d₂)	0.544 / 0.456	0.544 / 0.456
Premium (C / P)	3.07	2.15

Effect of Volatility and Time to Maturity

Volatility (σ) is the single most influential input. As σ rises, both d₁ and d₂ move closer to zero, increasing N(d₁) and N(d₂). The net effect is a higher call premium because the option becomes more likely to finish in‑the‑money.

Time to maturity (T) works similarly. A longer T raises the discount factor for the spot (e^{-r_f T}) less than it raises the volatility term (σ√T). Consequently, the option’s time value grows, especially when the option is at‑the‑money.

Exam questions often present two scenarios with different σ or T and ask which premium is higher. Use the intuition that higher σ or longer T → higher premium, all else equal.

Call Premium vs. Volatility (Other inputs fixed)

Worked Example – INR/USD Call Option

Example: Hedging a Future USD Purchase

Scenario

An Indian importer expects to buy USD 100,000 in three months to pay for raw material. The current INR/USD spot is 74.5. To lock in the rate, the importer buys a European call option with strike 75, expiry in 0.25 years, domestic rate 6% p.a., foreign rate 2% p.a., and implied volatility 12% p.a.

Solution

Step 1: Convert the exposure to per‑unit terms. Use S₀=74.5, K=75, r_d=0.06, r_f=0.02, σ=0.12, T=0.25.\nStep 2: Compute d₁ = [ln(74.5/75)+(0.06-0.02+0.5×0.0144)×0.25]/(0.12√0.25) = 0.141.\nStep 3: d₂ = d₁ - σ√T = 0.141 - 0.06 = 0.081.\nStep 4: N(d₁)≈0.556, N(d₂)≈0.532.\nStep 5: Discount factors: e^{-r_f T}=e^{-0.005}=0.9950, e^{-r_d T}=e^{-0.015}=0.9851.\nStep 6: Call premium per USD = 74.5×0.9950×0.556 - 75×0.9851×0.532 = 41.28 - 39.33 = 1.95 INR.\nStep 7: Total premium = 1.95 × 100,000 = 195,000 INR.\nThe importer pays this premium now and, if the spot exceeds 75 at expiry, can exercise the option to buy USD at 75, effectively capping the cost.

Conclusion

The example shows how each input feeds into the Garman‑Kohlhagen formula and how the premium is scaled to the transaction size. Remember to convert the per‑unit premium to the total exposure for exam calculations.

⚠️Volatility Estimation Pitfall

Never use a single‑day price swing as σ. The exam expects you to use either historical annualised volatility (standard deviation of daily log returns × √252) or the implied volatility quoted by the exchange.

⭐Exam Takeaways

Option pricing for currency options uses the Garman‑Kohlhagen model, which incorporates both domestic (r_d) and foreign (r_f) risk‑free rates.
Key inputs are spot (S₀), strike (K), time (T), volatility (σ), r_d and r_f; each must be expressed in consistent annual terms.
Call premium: C = S₀e^{-r_f T}N(d₁) – Ke^{-r_d T}N(d₂); Put premium: P = Ke^{-r_d T}N(-d₂) – S₀e^{-r_f T}N(-d₁).
Put‑call parity (C‑P = S₀e^{-r_f T} – Ke^{-r_d T}) is a quick check for arbitrage‑free pricing.
Higher volatility or longer time to expiry always increase the option premium, ceteris paribus.
Use annualised σ and convert T to years; discount factors use e^{-rate×T}.
Common mistake: swapping r_d and r_f in the discount factors – keep domestic rate with the strike and foreign rate with the spot.
When given a scenario, compute the per‑unit premium first, then scale it to the transaction size for the final answer.

Practice Questions

8 questions on Option Pricing Methodology

Which of the following is NOT listed as a key input for pricing a currency option?

In the Garman‑Kohlhagen call premium formula, which rate is used to discount the strike price K?

For the example with S₀=74.5, K=75, r_d=0.06, r_f=0.02, σ=0.12 and T=0.5, which discount factor is applied to the spot rate?

If all inputs remain unchanged but volatility (σ) increases, what is the expected effect on the call premium?

According to put‑call parity for currency options, C‑P equals which of the following expressions?

If a candidate mistakenly swaps r_d and r_f in the call formula, the computed premium will most likely be:

All else equal, how does increasing the time to expiry (T) affect the premium of an at‑the‑money European currency call?

In the worked example with T=0.25 years, what is the discount factor applied to the foreign risk‑free rate?

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