*Learning objectives: Define a volatility smile and volatility skew. Explain the implications of put-call parity on the implied volatility of call and put options. Compare the shape of the volatility smile (or skew) to the shape of the implied distribution of the underlying asset price and to the pricing of options on the underlying asset. Describe characteristics of foreign exchange rate distributions and their implications on option prices and implied volatility.*

**Questions:**

23.6.1. The risk-free rate is 5.0% per annum with continuous compounding. The current price of a non-dividend-paying stock is $37.00. A six-month call option (on this stock) with a strike price of $40.00 trades at a price of $2.30 with an implied volatility of 30.0%. A six-month put option (on the same stock) with a strike price of $40.00 trades at a price of $3.80. Which of the following is the

**best**trade?

a. There is no obvious trade here

b. Buy the put, sell the call, and buy the stock

c. Sell the put, buy the call, and short the stock

d. Buy the put, buy the call, and short the stock

23.6.2. The graph below plots an implied volatility skew for six-month options on a single, liquid (underlying) equity asset; for example, the S&P 500 index. The horizontal axis is the ratio of strike price to current stock price, K/S(0). The methodology includes inferring implied volatilities via the Black-Scholes option pricing model (BSM OPM) across a vector of currently trading option prices at various strike prices (and the current asset price, of course). The dashed horizontal line is located at Y = 21.4% which is the average of implied volatilities, for reference:

Each of the following statements is true

**EXCEPT**which is false?

a. This shape is best explained by positive correlation between the equity asset price and volatility

b. The implied volatility is higher for deep in-the-money calls than for deep in-the-money puts

c. The implied volatility is higher for deep out-of-the-money puts than for deep out-of-the-money calls

d. This skew's shape implies the asset price's probability distribution has a heavy left tail but a light (aka, thin) right tail in comparison to the lognormal distribution

23.6.3. An equity asset's current price is $100.00. Below are the implied volatilities of three options on the equity asset at three different strike prices ($70.00, $100,00, and $130.00) with identical three-month maturities (T = 0.25 years):

The plot above includes three specific implied volatilities:

- Deep in-the-money (ITM) call option with a strike-to-price ratio, K/S(0) = 0.70 has an implied volatility of 44.0%.
- At-the-money option (ATM) call option with strike-to-price ratio, K/S(0) = 1.00 has an implied volatility is 25.0%.
- Deep out-of-the-money (OTM) call option with strike-to-price ratio, K/S(0) = 1.30 has an implied volatility of 11.3%.

**TRUE**?

a. We expect a deep in-the-money (ITM) put option with a strike price of $130.00 to have an implied volatility of 44.0%

b. Because the implied volatility of the OTM call option is too low (by almost 8.0%), the arbitrage trade is to buy the OTM call, sell an ITM put, and short the stock

c. For a deep out-of-the-money (OTM) put, the assumption of a 25.0% volatility input into the Black-Scholes-Merton will return a model price that understates its market price

d. For a deep in-the-money (ITM) put, the assumption of a 25.0% volatility input into the Black-Scholes-Merton will return a model price that understates its market price

**Answers here:**

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