How Volatility Skew Affects a Binary Option’s Fair Value
Summary
The document explains why a positive volatility skew can lower the fair value of a binary option, despite a distribution with fatter tails seeming to imply a higher chance of a far out-of-the-money call finishing in the money. The first explanation represents the binary payoff as a narrow put spread: if skew makes the farther out-of-the-money put more expensive, the spread costs less.
A second explanation focuses on the option’s local volatility and its changing exposure. The binary is long volatility and gamma while out of the money, but becomes short those exposures when it is in the money; higher volatility farther into the tail can therefore hurt its hedged value. The discussion distinguishes physical probabilities from risk-neutral pricing and gives intuition rather than a formal derivation. Its explanation depends on the described skew and payoff region, so it should not be read as a universal claim that skew always reduces every binary’s value.
Key ideas
- A binary can be approximated by a narrow spread of vanilla options.
- A more expensive far out-of-the-money put can reduce the cost of that spread.
- A binary’s volatility and gamma exposures change sign as it moves from out of the money to in the money.
- A higher chance of finishing in the money under a physical distribution does not by itself establish a higher risk-neutral fair value.
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Full text
# Intuitive explanation for the value of a binary option being lower when volatility skew is positive? # Intuitive explanation for the value of a binary option being lower when volatility skew is positive? According to the formula for pricing binary options with a volatility skew, it appears that the value of the binary option for a given strike gets lower, the higher the volatility skew at that strike. Why is this, intutively? I (think I) understand the derivation, but intuitively it seems like a positive volatility skew should make far OTM binary call options more expensive, not cheaper, because it should create "fatter tails" in the implied PDF of the price of the underlying at that expiry. ## Answer by cholzer68 (score 3) https://quant.stackexchange.com/a/61513 You can replicate the payout of a binary with a put spread with strike prices which are very close to one another. Higher skew makes the further out of the money put more expensive, which makes the put spread cheaper. ## Answer by Mats Lind (score 0) https://quant.stackexchange.com/a/61507 Just on pure intuition (close to a guess); the binary does not pay you for the fat tail. All you want is high volatility locally around the strike to add to the chances for the option to expire in the money. Just narrowly in the money counts as much as far out on the tail. Edit: The put spread in the other answer above could provide a good starting point to deepen the intuition. It buys volatility nearer the current underlying price and sells it back farther out on the tail. Narrow it down to (an out of the money, OTM) binary: it is long volatilty OTM and short volatility (vola) in the money all (ITM) the way out on the tail. The fat-tail distribution has relatively low vola when we are OTM and want vola to put us ITM. It has high vola when we are ITM and want low vola to stay there. The takeaway here I guess is not to look at the expected return (P-probability): even though the chances for the very far OTM binary to expire ITM may increase with the fatness of the tail, its fair price will not! The fair price is what you would pay to break-even when you carry the option risklessly with a dynamic delta-hedge in the underlying until expiry (risk neutral Q-probability). And where you are OTM you are long vola and long gamma and reap high rebalancing profits when vol is high. Where you are ITM and out on the tail it is the reverse, you lose in rebalancing when the large high vola moves producing the fatness even further out occur. Forget the (non-risk adjusted P-probability) expected return and look at how the vola skew matches where you will be long and short vola and gamma!
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.