Volatility Surface Dynamics: Sticky Moneyness and Sticky Strike

Summary

The note distinguishes fitting an implied volatility surface at the current time from specifying how that surface changes as time passes and the underlying price moves. A cross section indexed by maturity and strike can be fitted to current market data, but a pricing model also needs rules for the future surface conditional on the evolving underlying price.

Two illustrative dynamics are given. Under sticky moneyness, volatility depends on time to maturity and log strike relative to spot, so a move in the underlying shifts the relevant moneyness. Under sticky strike, volatility depends on time to maturity and the fixed strike, so the surface stays attached to strike levels. The excerpt does not fully compare the Ad Hoc Black–Scholes and deterministic volatility function models, estimate either model, or provide empirical evidence. It highlights that choosing a functional fit to today's surface does not by itself determine the volatility dynamics needed for valuation.

Key ideas

• A fitted implied volatility surface describes market quotes at a particular time.
• Pricing over time requires a rule for how the surface responds to changes in spot and time.
• Sticky moneyness links volatility to strike relative to the underlying price.
• Sticky strike keeps volatility associated with fixed strike levels.
• The examples clarify dynamics but do not fully characterize the two named model classes.

Tags

# Difference between Deterministic Volatility Function approach and Ad Hoc Black Scholes?


# Difference between Deterministic Volatility Function approach and Ad Hoc Black Scholes?












I am thoroughly confused after reading Dumas, Fleming & Whaley (1998) "Implied Volatility Functions: Empirical Tests". Both the Ad Hoc BS Model and the Deterministic Volatility Function approaches in the paper seem to posit a structure for a function determining volatility and then run a regression to determine the coefficients. What is the difference between the two approaches? What am I overlooking?

## Answer by M. Jeunesse (score 1)

https://quant.stackexchange.com/a/26417

At $t=0$, you have a vol surface $(T,K)\to\sigma(t=0,S_t=S_0,T,K)$

the hard question is which dynamics i.e $\sigma$ seen as $\sigma: (t,S_t)\to ((T,K)\to \sigma(t,S_t,T,K))$

and even, if you imagine that behind this, there is a deterministic function, you still have to suppose a dependence with respect to $S_t$

Here are two examples

1) $\sigma(t=0,S_t=S_0,T,K)=f(T-t,\ln(K/S_t))$ where $f$ is a deterministic function fitting your data at $t=0$ (kind of sticky moneyness)

2) $\sigma(t=0,S_t=S_0,T,K)=g(T-t,K)$ where $g$ is a deterministic function fitting your data at $t=0$ (kind of sticky strike)

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.