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The lognormal distribution is the assumed probability distribution of asset prices in the Black-Scholes model. It implies that returns are normally distributed, prices cannot go negative, and upside is theoretically unlimited while downside is bounded at zero.
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Key takeawayReal returns are not lognormal. They have fatter tails and negative skew. Premium sellers who size positions using lognormal assumptions (standard Black-Scholes) underestimate tail risk. Add a 20-30% buffer to your expected worst-case scenarios to account for real-world fat tails.
💡 Why It Matters
The lognormal distribution is the foundation assumption of Black-Scholes, and its failure to match reality is the reason premium sellers have an edge. Real returns have fatter tails and negative skew, so options priced under lognormality systematically underprice tail risk while the market overprices it.
⚙ How It Works
Under lognormality, the logarithm of price returns follows a normal distribution. This ensures prices cannot go negative, the distribution is right-skewed (matching the positive drift of stocks), and multiplicative returns compound correctly. All Black-Scholes Greeks and probabilities assume this distribution.
📋 Example
Under lognormal assumptions, the probability of a 3-sigma daily SPX move (approximately 4.5% when VIX is 24) is 0.27%. In reality, 3-sigma daily moves occur about 1-2% of the time, or roughly 5x more frequently than predicted. A premium seller sizing at the 3-sigma level needs to prepare for this higher-than-modeled frequency.
⚠ Common Mistakes
Traders accept Black-Scholes probability of profit calculations at face value. The lognormal assumption makes these calculations optimistic for OTM option sellers. A position showing 85% probability of profit under lognormality might have only 78-80% real-world probability once fat tails are considered.
❓ FAQ
What is Lognormal Distribution?
The lognormal distribution is the assumed probability distribution of asset prices in the Black-Scholes model. It implies that returns are normally distributed, prices cannot go negative, and upside is theoretically unlimited while downside is bounded at zero.
Why does Lognormal Distribution matter for premium sellers?
Real returns are not lognormal. They have fatter tails and negative skew. Premium sellers who size positions using lognormal assumptions (standard Black-Scholes) underestimate tail risk. Add a 20-30% buffer to your expected worst-case scenarios to account for real-world fat tails.
How does Lognormal Distribution work?
Under lognormality, the logarithm of price returns follows a normal distribution. This ensures prices cannot go negative, the distribution is right-skewed (matching the positive drift of stocks), and multiplicative returns compound correctly. All Black-Scholes Greeks and probabilities assume this distribution.
What are common mistakes with Lognormal Distribution?
Traders accept Black-Scholes probability of profit calculations at face value. The lognormal assumption makes these calculations optimistic for OTM option sellers. A position showing 85% probability of profit under lognormality might have only 78-80% real-world probability once fat tails are considered.
Related Terms
Bachelier Model
The Bachelier model (normal model) assumes the underlying follows an arithmetic Brownian motion with normally distributed returns, rather than the lognormal ...
Barone-Adesi-Whaley Model
The Barone-Adesi-Whaley (BAW) model is a quadratic approximation for pricing American options that decomposes the American option price into its European val...
Binomial Option Pricing
A discrete-time pricing model that builds a recombining tree of possible price paths.
Bjerksund-Stensland Model
The Bjerksund-Stensland model provides a closed-form approximation for pricing American options by treating early exercise as a flat barrier problem.
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