Chapter 4: Exchange Traded Currency Options

A currency option is a derivative financial contract that gives the buyer the right, but not the obligation, to buy or sell a specified quantity of foreign currency at a predetermined exchange rate on or before an expiration date. Currency options provide asymmetric payoff profiles, making them indispensable instruments for asymmetric hedging and volatility trading.

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4.1 Basics of Options

Call Options vs. Put Options

1. Call Option (Right to Buy):
   • Gives the buyer the right to buy the underlying currency at the strike price.
   • Purchased when the trader anticipates the base currency will appreciate (e.g., expecting USD/INR to rise).
2. Put Option (Right to Sell):
   • Gives the buyer the right to sell the underlying currency at the strike price.
   • Purchased when the trader anticipates the base currency will depreciate (e.g., expecting USD/INR to fall).

Core Participants: Buyer vs. Writer (Seller)

In every options contract, the rights and obligations are fundamentally asymmetric:

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4.2 Differences Between Futures and Options

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4.3 Styles of Options

1. European Style:
   • The option can be exercised only on the expiration date.
   • All Exchange-Traded Currency Options in India (USD/INR, EUR/INR, GBP/INR, JPY/INR, EUR/USD, GBP/USD, USD/JPY) are European style.
2. American Style:
   • The option can be exercised on any business day up to and including the expiration date.
   • Commonly found in OTC markets and US equity options.
3. Bermudan Style:
   • The option can be exercised on specified predetermined dates prior to expiration.

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4.4 Moneyness of an Option

Moneyness describes the relationship between the strike price ($) of the option and the current spot exchange rate ($) of the underlying currency:

Intrinsic Value vs. Time Value

Total Option Premium is comprised of two distinct components: 8075\text{Option Premium} = \text{Intrinsic Value} + \text{Time Value}8075

1. Intrinsic Value ($):
   • The immediate financial payoff if the option were exercised today: 8075\text{Call Intrinsic Value} = \max(0, S - X)8075 8075\text{Put Intrinsic Value} = \max(0, X - S)8075
   • Intrinsic value can never be negative (it is either positive or zero).
2. Time Value (Extrinsic Value):
   • The premium amount in excess of the intrinsic value: 8075\text{Time Value} = \text{Option Premium} - \text{Intrinsic Value}8075
   • Reflects the probability that the option will move further into the money before expiration.
   • Time Decay: Time value steadily declines as the expiration date nears, reaching exactly zero at expiration.

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4.5 Basics of Option Pricing and Option Greeks

Six Factors Governing Currency Option Pricing

The Option Greeks

Option Greeks measure the sensitivity of the option's price to changes in fundamental pricing variables:

1. Delta ($\Delta$):
   • Sensitivity of option price ($) to a unit change in the underlying spot exchange rate: 8075\Delta = \frac{\partial V}{\partial S}8075
   • Call Delta: Ranges between *bash$ and $+1. ATM Call Delta $\approx +0.50$.
   • Put Delta: Ranges between *hBc1$ and bash. ATM Put Delta $\approx -0.50$.
   • Hedge Ratio: Delta signifies the number of currency futures needed to create a delta-neutral hedge.
2. Gamma ($\Gamma$):
   • Rate of change of Delta with respect to changes in the spot price: 8075\Gamma = \frac{\partial^2 V}{\partial S^2} = \frac{\partial \Delta}{\partial S}8075
   • Measures the stability of Delta. Gamma is highest for ATM options and approaches zero for deep ITM and deep OTM options.
3. Theta ($\Theta$):
   • Rate of change of option price with respect to the passage of time (Time Decay): 8075\Theta = \frac{\partial V}{\partial t}8075
   • Theta is virtually always negative for long option positions (loss of value with time) and positive for option sellers.
   • Time decay accelerates dramatically in the final 30 days before expiration.
4. Vega ($ u$):
   • Sensitivity of option price to a 1% change in implied volatility ($\sigma$): 8075 u = \frac{\partial V}{\partial \sigma}8075
   • Both Call and Put options have positive Vega. Vega is highest for ATM options and longer-dated expiries.
5. Rho ($\rho$):
   • Sensitivity of option price to changes in risk-free interest rates:
     • Rho Domestic ($\rho_d$): Sensitivity to domestic interest rate ($). Calls have positive $\rho_d$; Puts have negative $\rho_d$.
     • Rho Foreign ($\rho_f$): Sensitivity to foreign interest rate ($). Calls have negative $\rho_f$; Puts have positive $\rho_f$.

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4.6 Option Pricing Methodology & Put-Call Parity

Garman-Kohlhagen Model (1983)

The Black-Scholes-Merton model was adapted for currency options by Garman and Kohlhagen by treating the foreign currency as an asset providing a continuous dividend yield equal to the foreign risk-free interest rate ($):

8075C = S \cdot e^{-r_f T} \cdot N(d_1) - X \cdot e^{-r_d T} \cdot N(d_2)8075 8075P = X \cdot e^{-r_d T} \cdot N(-d_2) - S \cdot e^{-r_f T} \cdot N(-d_1)8075

Where: 8075d_1 = \frac{\ln(S / X) + (r_d - r_f + \sigma^2 / 2) T}{\sigma \sqrt{T}}8075 8075d_2 = d_1 - \sigma \sqrt{T}8075

• (d)$ denotes the cumulative standard normal distribution function.

Put-Call Parity for Currency Options

Put-Call Parity establishes an absolute equilibrium relationship between European Call and Put options of identical strike and expiration: 8075C + X \cdot e^{-r_d T} = P + S \cdot e^{-r_f T}8075

• If this parity is violated, arbitrageurs execute Conversions or Reversals to earn riskless profits until equilibrium is restored.

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4.7 Implied Volatility (IV) and Volatility Surface

• Historical Volatility (HV): Statistical annualized standard deviation of past daily exchange rate returns.
• Implied Volatility (IV): The volatility figure backed out from current market option prices using the pricing model. Represents the market's forward-looking expectation of volatility.
• Volatility Smile / Skew: In foreign exchange markets, OTM Puts and OTM Calls typically trade at higher implied volatilities than ATM options, forming a "smile" or "skew" due to fat tails and perceived tail-risk (jump risk).

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4.8 Contract Specifications: Exchange-Traded Currency Options

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Key Exam Takeaways for NISM Series I

• All currency options traded on Indian exchanges are European style (exercisable only on expiry day).
• In India, currency options are Cash Settled in Indian Rupees; no physical currency delivery occurs.
• The option pricing model applicable to currency options is the Garman-Kohlhagen model.
• An option buyer has limited risk (premium paid) and unlimited profit potential; an option seller has limited profit (premium received) and unlimited risk.
• Vega is positive for both Calls and Puts, and reaches its maximum for At-The-Money (ATM) options.
• Theta represents time decay; ATM options experience accelerated time decay in the final 30 days prior to expiry.
• Put-Call Parity: + X \cdot e^{-r_d T} = P + S \cdot e^{-r_f T}$.