A company's stock is selling for 4. The company has no...

A company's stock is selling for $$\$ 4$$. The company has no outstanding debt. Analysts consider the liquidation value of the company to be at least $$\$ 300,000$$ and there are 100,000 shares outstanding. What volatility smile would you expect to see?

Key Concepts

Asymmetric Risk and Volatility Skew

When an asset’s return distribution is skewed—due, for example, to a lower bound on losses from liquidation—the risk is not evenly distributed. This leads to a situation where options with strikes on one side (often the deep out-of-the-money puts) are priced differently from their counterparts, creating an asymmetric pattern in implied volatilities known as a volatility skew. Such skews highlight the market’s recognition of asymmetric risks in the asset's potential outcomes.

Option Pricing under Non-Lognormal Distributions

Standard option pricing models like Black?Scholes assume that the underlying asset follows a lognormal distribution, with support over all positive values. When additional information such as a required minimum value or other structural features is present, the actual distribution deviates from lognormality. This necessitates adjustments in implied volatility across different strikes to capture the true risk reflected by the altered probability distribution.

Implied Volatility Smile

This is the phenomenon where implied volatility, as backed out from option prices, varies with the strike price, rather than remaining constant as assumed in the Black?Scholes model. It typically forms a U-shaped curve indicating that options which are deep in- or out-of-the-money have higher implied volatilities, reflecting market adjustments for extreme outcomes and deviations from normality in the underlying asset’s return distribution.

Default Risk and Limited Liability

Limited liability implies that a firm’s equity holders are not responsible for losses beyond their investment. When there is a firm-specific minimum value—such as a liquidation value—this creates a floor in the payoffs to shareholders, thereby altering the risk profile. Investors price options differently because the worst-case outcome is bounded, which can affect the pricing of put options and contribute to a skewed or modified volatility smile.

Transcript

Please give Ace some feedback