Interpolation of Option Prices / Interpolation of Implied Volatilities

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This page illustrates that a naive interpolation of option prices (via the interpolation of the prices itself or the interpolation of implied (Black-Scholes) volatilities) may generate strange and even impossible probability densities. Some interpolation methods will generate prices which violate the no-arbitrage condition.

It serves as a companion to Chapter 6 of Mathematical Finance.

Predefined Scenarios

Your may step through some predefined scenarios.

Scenario 1
A flat implied volatlity curve. Nothing to see here.
Scenario 2
A linear interpolation of implied volatilites may lead to an no-arbitrage violation. This scenario also shows that a constant exrapolation of implied volatilites may lead to an arbitrage violation. If the same input data is used in a cubic spline interpolation of implied volatlites no arbitrage violation occurs.
Note that the input prices itself are arbitrage free. To check this change the interpolation method to linear interpolation of option prices. This generates spikes in the probability desity (it is equivalent to a density which is a convex combination of dirac deltas). If all the spikes are positve (and the prices are monotone) no-arbitrage violation exists in the input data.
Scenario 3
A cubic spline interpolation of implied volatlites, which (in this case) does not generate an arbitrge violation. The probability density looks nice and smooth. Like in Scenario 2 the linear interpolation of the same implied volatlites leads to no-arbitage violations.
Scenario 4
This scenario shows that a linear extrapolation of implied volatilites may violate the no-arbitrage condition. The last point is the linear extrapolation on the implied volatility curve. The price curve is no longer monotone. The integral under the density shown is in fact larger than 1. Note: In this example the cubic spline and the linear interpolation of the implied volatlities generates a positive density (in the intervall shown)
Scenario 5
The input data is arbitrage free. Both, linear and cubic spline interplation of the implied volatilities generate violations of the no-arbitrage condition.

© Copyright 2007,2013 Christian P. Fries