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# P2.T5.410. Implied volatility smile and the implied asset distribution

#### Nicole Seaman

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AIMs: Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility. Describe the volatility smile for equity options and foreign currency options and give possible explanations for its shape. Describe alternative ways of characterizing the volatility smile.

Questions:

410.1. Suppose most market participants think that foreign currency exchange rates are lognormally distributed, such that they are comfortable using the same (constant) volatility to value options with the Black-Scholes option pricing model (BSM OPM). Risk Analyst Sally has performed an analysis, however, and she knows that the lognormal assumption is not good: rather, she expects a actual distributions with heavier tails, both left and right, than the lognormal. Which trade most takes advantage of this situation?

c. Long strangle

410.2. Suppose the volatility is uncertain and positive correlated with the underlying asset price. Relative to prices implied by a constant volatility, which of the following scenarios is most plausible?

a. Lower price for out-of-the-money (OTM) put options and higher price for in-the-money (ITM) put options
b. Lower price for out-of-the-money (OTM) call options and higher price for in-the-money (ITM) call options
c. Lower price for at-of-the-money (ATM) call options and higher price for at-the-money (ATM) put options
d. Lower price for in-the-money (ITM) put options and higher price for out-of-the-money (OTM) put options

410.3. Suppose the implied distribution of a certain option asset class exhibits, relative to the lognormal distribution, a lighter left-tail and a heavier right-tail. Further, define risk reversal (RR) as the difference in implied volatility (utilizing the BSM OPM) between a 25 delta call and -25 delta put: RR = implied volatility [0.25 delta call option] - implied volatility [-0.25 delta put option]. Which of the following is mostly likely true of the risk reversal (RR) value?

a. Negative risk reversal
b. Zero risk reversal
c. Positive risk reversal
d. Risk reversal equal approximately to implied volatility of 50 delta call; i.e., ATM call

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