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Answer: The price of the option on ABC stock would be relatively low and the price of the option on USD/GBP FX rate would be relatively high compared to those computed from the lognormal counterparts.
## Explanation **B is correct.** This question examines the differences in volatility smiles between equity options and foreign exchange options, and how these differences affect the pricing of deep out-of-the-money call options when using implied risk-neutral distributions versus lognormal distributions. ### Key Concepts: **1. Equity Options Volatility Smile:** - For equity options, the implied volatility smile typically shows higher implied volatilities for deep out-of-the-money puts than for deep out-of-the-money calls - This creates an **asymmetric volatility smile** with a heavier left tail and less heavy right tail compared to the lognormal distribution - The implied distribution has **fatter left tail** (higher probability of large price declines) and **thinner right tail** (lower probability of large price increases) **2. Foreign Exchange Options Volatility Smile:** - For FX options, the volatility smile is typically **symmetric** - Both tails are heavier than the lognormal distribution - The implied distribution has **fatter tails on both sides** ### Impact on Deep Out-of-the-Money Call Options: **For ABC Stock (Equity):** - Using the implied distribution (which has thinner right tail) results in **lower prices** for deep out-of-the-money call options compared to the lognormal distribution - The thinner right tail means lower probability of large price increases, making deep OTM calls less valuable **For USD/GBP FX Rate:** - Using the implied distribution (which has fatter tails) results in **higher prices** for deep out-of-the-money call options compared to the lognormal distribution - The fatter right tail means higher probability of large price movements, making deep OTM calls more valuable Therefore, the correct relationship is: - **ABC Stock option price: Relatively LOW** (compared to lognormal) - **USD/GBP FX option price: Relatively HIGH** (compared to lognormal) This understanding is crucial for risk managers when pricing exotic options or when using volatility surface models for risk measurement.
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A risk committee of the board of company ABC is discussing the difference between pricing deep out-of-the-money call options on ABC stock and pricing deep out-of-the-money call options on the USD/GBP foreign exchange (FX) rate using the Black-Scholes-Merton model. The committee considers pricing each of these two options based on two distinct probability distributions of underlying asset prices at the option expiration date: a lognormal probability distribution, and an implied risk-neutral probability distribution obtained from the volatility smile for each aforementioned option of the same maturity and the same moneyness. If the implied risk-neutral probability distribution is used instead of the lognormal distribution, which of the following is correct?
A
The price of the option on ABC stock would be relatively high and the price of the option on USD/GBP FX rate would be relatively low compared to those computed from the lognormal counterparts.
B
The price of the option on ABC stock would be relatively low and the price of the option on USD/GBP FX rate would be relatively high compared to those computed from the lognormal counterparts.
C
The price of the option on ABC stock would be relatively low and the price of the option on USD/GBP FX rate would be relatively low compared to those computed from the lognormal counterparts.
D
The price of the option on ABC stock would be relatively high and the price of the option on USD/GBP FX rate would be relatively high compared to those computed from the lognormal counterparts.