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What is the point of fitting curves to the implied smile in the market? (Other than pricing exotics where the hedging instruments are vanillas). How does fitting a vol curve help you trade/market make vanillas? Isn't it rather tautological to use the market's implied vol as your theoretical value when trading?

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    $\begingroup$ You would be given implied vol/vanilla option prices for a finite set of maturities and strikes. The options you would have on the book or the options you are planning to trade will usually not have the strike and maturity, so to price these, even if they are vanilla, you still need some form of interpolation. Many people call pricing glorified interpolation, in some ways that’s true. $\endgroup$ – Magic is in the chain Nov 22 '19 at 20:12
  • $\begingroup$ I understand the use in pricing instruments that aren't quoted (exotics, strikes/maturities, etc) through interpolation. What I am asking is is there a use besides this? How does a model like this help me trade the strikes and maturities I am calibrated to? $\endgroup$ – roz Nov 23 '19 at 1:36
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    $\begingroup$ Buying or selling an option is usually the easiest part, it’s post transaction management that is the trickiest. You will need to manage the financial reporting side, P&L, collateral, and risk. So you will need to compute price and Greeks for all these purposes. Even if the options were initially standard quoted instruments, the strike and maturity will move away, and same strike /maturity will no longer be liquid so you need some sort of interpolation for all these things $\endgroup$ – Magic is in the chain Nov 23 '19 at 13:05
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If you want to compare quotes across markets or over time it can be useful to use fixed points: eg the 110%/90% points to compute skew or the +/-25 delta points for risk-reversal. You can't rely on quotes existing at exactly those points so you would want to interpolate.

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Pricing of vanillas is basically interpolation of existing (or past) quotes. It is easier to interpolate in implied volatility space , than in price space. Reasons are we need to interpolate in multidimensional space (maturity, strike,forward, etc) and satisfy non-arbitrage conditions.

Using Black-scholes formula is convenient mapping which would also simplify satisfying the non-arbitrage conditions (positive density (convexity) , monotonicity vs strike).

In price space parametrisations would get more complex than in implied volatility terms (you can get good match in say some FX markets with just simple quadratic implied vol interpolation in log(K/F)/sqrt(T) strike (with some simple wings extrapolation).

Also implied vol parametrisations have advantage that parameters usually have easily understood effect to implied vol smile/skew.

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