Volatility smiles and skews are fundamental to understanding how markets price options. They reveal how implied volatility (IV) varies across strike prices, reflecting real-world market expectations about extreme price movements. While the Black-Scholes-Merton (BSM) model assumes a lognormal distribution for asset prices, this assumption often diverges from actual market behavior.

In this blog, we’ll explore how using the lognormal distribution versus the implied risk-neutral distribution affects the pricing of deep out-of-the-money (OTM) call options through two real-world examples: Tesla stock (TSLA) and the EUR/USD foreign exchange (FX) pair.

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The Basics of Implied and Lognormal Distributions

Lognormal Distribution

• The BSM model assumes that the logarithm of the underlying asset price is normally distributed, resulting in a lognormal distribution for the actual price.
• Key Characteristics:
  • Positively skewed, with a thin left tail (prices cannot drop below zero).
  • A long right tail, reflecting the unbounded potential for price increases.
  • Assumes constant volatility, producing a flat implied volatility curve across strikes.

Implied Risk-Neutral Distribution

• Derived from observed volatility smiles or skews, the implied distribution adjusts for real-world market expectations.
• Key Characteristics:
  • Heavier left tail, reflecting a higher probability of large downward price moves.
  • Heavier or thinner right tail, depending on the asset class and market.
  • Reflects market-implied probabilities for extreme moves, unlike the fixed structure of the lognormal distribution.

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Volatility Smiles and Skews: Tesla Stock vs. EUR/USD FX

Case 1: Tesla Stock (Equities)

• Equity options typically exhibit a volatility skew (or “smirk”):
  • OTM puts have higher implied volatility due to hedging demand against sharp price declines (e.g., fear of a market crash or regulatory setbacks).
  • OTM calls often have lower demand, resulting in a flatter or less heavy right tail in the implied volatility curve.
• The implied distribution for Tesla reflects:
  • Fat left tail: Higher probability of significant price drops (e.g., Tesla missing earnings expectations or broader market downturns).
  • Less heavy right tail: Lower probability of extreme upward moves compared to the lognormal distribution.

Case 2: EUR/USD Foreign Exchange (FX)

• FX options often exhibit a volatility smile:
  • Implied volatility is higher for both deep OTM puts and calls, reflecting the possibility of large price swings in either direction (e.g., central bank decisions, geopolitical events).
• The implied distribution for EUR/USD reflects:
  • Heavier tails on both sides: Greater probabilities of extreme upward and downward moves compared to the lognormal distribution.

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Impact on Deep OTM Call Option Pricing

For Tesla Stock (Equities):

• Lognormal Distribution:
  • Assumes a longer right tail, overestimating the probability of extreme upward moves (e.g., Tesla hitting an unprecedented price due to unforeseen positive news).
• Implied Distribution:
  • Reflects market skepticism about extreme upward moves, resulting in a less heavy right tail.
• Impact: Using the lognormal distribution causes deep OTM Tesla call options to be overpriced compared to the implied distribution. The lognormal exaggerates the likelihood of Tesla experiencing a sharp rally.

For EUR/USD FX Options:

• Lognormal Distribution:
  • Assumes thinner tails, underestimating the probabilities of large moves in either direction.
• Implied Distribution:
  • Reflects higher probabilities of both extreme upward and downward moves due to real-world market dynamics.
• Impact: Using the lognormal distribution causes deep OTM EUR/USD call options to be underpriced compared to the implied distribution. The lognormal fails to capture the heightened likelihood of significant FX price swings.

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Why Does This Happen?

Lognormal Distribution Assumptions:

• The lognormal distribution assumes constant volatility and a symmetric view of price changes, which oversimplifies market behavior.
• It does not reflect real-world concerns like asymmetric hedging demand or macroeconomic volatility.

Implied Distribution Adjustments:

• The implied distribution, derived from observed volatility smiles or skews, captures:
  • Market participants’ fear of downside risks (e.g., sharp declines in Tesla’s stock price).
  • Acknowledgment of significant upside risks in FX markets, such as currency devaluation or central bank interventions.

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Real-World Examples

Tesla Stock Example:

Imagine Tesla’s stock is trading at $800, and you’re evaluating a deep OTM call option with a strike price of $1,200:

• Lognormal Pricing: The BSM model assumes a longer right tail, overestimating the probability that Tesla will exceed $1,200.
• Implied Pricing: The implied distribution adjusts for market skepticism, reflecting a lower probability of such a large upward move.
• Result: The lognormal model overprices the deep OTM Tesla call option.

EUR/USD FX Example:

Now consider EUR/USD trading at 1.10, and you’re evaluating a deep OTM call option with a strike price of 1.25:

• Lognormal Pricing: The BSM model underestimates the likelihood of EUR/USD reaching 1.25 due to its thinner right tail.
• Implied Pricing: The implied distribution accounts for the higher probability of extreme moves, reflecting geopolitical or economic uncertainty.
• Result: The lognormal model underprices the deep OTM EUR/USD call option.

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Has Tesla’s IV Skew Been Observed?

Yes, Tesla’s options have consistently exhibited an implied volatility skew. Here’s why:

1. Hedging Demand for OTM Puts:
   • Institutional investors purchase OTM puts to protect against sharp price declines, increasing demand and implied volatility for OTM puts.
2. Relatively Lower Demand for OTM Calls:
   • While speculative traders might bid up OTM calls, the market generally assumes Tesla’s upside potential is already priced in, reducing implied volatility for OTM calls.
3. Event-Driven Volatility:
   • Tesla’s earnings announcements, product launches, and regulatory developments lead to spikes in IV for downside protection, amplifying the skew.

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Takeaways for Traders

1. Implied Distributions Reflect Market Realities:
   • Volatility smiles and skews provide a more accurate representation of market expectations than the flat IV curve implied by the lognormal distribution.
2. Lognormal Mispricing:
   • For equities like Tesla, the lognormal distribution overprices deep OTM call options by overestimating the likelihood of extreme upward moves.
   • For FX pairs like EUR/USD, the lognormal distribution underprices deep OTM call options by underestimating the likelihood of large price swings.
3. Tail Behavior Drives Pricing Differences:
   • The implied distribution’s fat tails ensure more accurate pricing for deep OTM options, adjusting for the real-world probabilities of extreme moves.
4. Practical Implications for Traders and Risk Managers:
   • Using implied distributions helps traders price deep OTM options more accurately, leading to better risk management and hedging strategies.

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By understanding the nuances of implied and lognormal distributions, traders can align their pricing models with real-world market behavior, avoiding the pitfalls of oversimplified assumptions.