Interpolate Implied Volatility Before Pricing an Unquoted Option
Summary
The document compares two ways to estimate the price of a European call at a strike and maturity absent from the market: interpolate neighboring option prices directly, or interpolate model inputs such as implied volatility and then reprice with Black-Scholes. Its recommendation is to interpolate in volatility space, on the grounds that volatility is an input to the pricing model and produces a smoother, more coherent estimate than averaging nonlinear prices.
The illustration uses three equally spaced strikes with the same Black-Scholes implied volatility. Interpolating the outer option prices would set the middle price to their average, which the answer says can create a butterfly arbitrage; interpolating volatility and applying the pricing formula avoids that particular problem. This is a brief conceptual argument, not a full interpolation framework. It does not specify surface coordinates, arbitrage constraints across maturities, or how to handle dividends and other inputs when constructing a complete volatility surface.
Key ideas
- For an unquoted option, the proposed approach is to interpolate implied volatility and then reprice with Black-Scholes.
- Option prices are nonlinear functions of model inputs, so direct price interpolation can distort relationships among strikes.
- The equal-spacing example shows how price averaging may create a butterfly arbitrage.
- The discussion does not provide a complete arbitrage-free volatility-surface construction method.
Tags
Full text
# Interpolating on the BS parameters and injecting in the BS formula vs interpolating directly on option prices # Interpolating on the BS parameters and injecting in the BS formula vs interpolating directly on option prices Let's consider a simple European call option. In practice, the way the Black-Scholes formula is used to price it is by injecting all of the parameters and paying special attention to the volatility and the dividends where their implied values are used. This gives then, by definition, the market price of the corresponding call option. Now, suppose one wants to price a call option with maturity $T$ and strike $K$ that isn't found in the market (and hence has no corresponding implied volatility or dividends). One could either interpolate between the prices of the options with the closest maturities and strikes, or one could interpolate between the implied volatilities and dividends corresponding to the closest maturities and strikes and then inject into the BS formula. Which one is the best approach ? ## Answer by onlyvix.blogspot.com (score 2) https://quant.stackexchange.com/a/24872 The best way is to interpolate in volatility space. The reason is because it is closer to the intrinsic pricing of the option, and it is less likely to produce an arbitrage. Like Alex C noted in the comment - prices are nonlinear function of inputs, and interpolating in them does not make sense. Inputs are "free", and interpolated value of inputs will likely be somewhere in the smoothed area. Really simple example - consider 3 european options with consecutive equidistant strikes, with the same black-scholes IV. If you don't know the middle option's price, you should interpolate vols, and then price the middle option. If you were to interpolate prices, then your middle option will be the average of the other two, and will create a free butterfly, which will be an arbitrage.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.