Quant Explains Risk-Neutral Option Pricing
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Anytime we create a model we are to suffer from the non-stationarity of reality: model mispecification and misparameterization
We still need to develop a price for derivative contracts, some may think the price will go to the moon, some may think it'll sell for pennies on the dollar, in any case, we need a consistent framework for pricing that mitigates (not entirely, but in a controlled way) this raw speculation and capacity for mispecification and misparameterization
Black-Scholes produce a portfolio replication arugment showing in a complete market with randomness that is hedgable the price of maintaining a continuous hedge should be the price of the option
Clearly, there is no risk, so the portfolio by a no arbitrage assumption must earn the risk-free rate, this is why when we simulate underlying dynamics to derive a price we must include a drift term equivalent to the risk-free rate otherwise we are underpricing the option relative to this hedging framework (e.g. hedging removes the risk premium compensation for bearing the risk but we still earn the risk-free rate)
In complete markets, there is only risk-neutral measure, so its relatively easy to produce a price
In reality, markets are not compelete, there is not just one risk-neutral measure but infinite to choose from
Instead of wandering around infinity we infer the risk-neutral measure and the premium from unhedgable risk by calibrating models to the market liquid instruments, this is what produces skew and the implied volatility surface
Ironically, we are back to the mispecification and misparameterization problem only now in the risk-neutral world, we can quote prices too high for our clients, its a framework, not a solution, but the math powers trillions of dollars of transactions in the financial markets
I hope you enjoyed!
Roman
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📖 Chapters:
00:00 - Should Different Outlooks Mean Different Prices?
03:16 - European Option Contracts and the Pricing Problem
05:48 - Stochastic Modeling of Underlying Dynamics
09:11 - MLE of Geometric Brownian Motion
10:00 - Selecting a Model Parameterization (P-Measure)
12:35 - Model Mispecification and Misparameterization
17:54 - Risk-Neutral Pricing and Change of Measure
21:04 - Justification of Risk-Neutral Pricing
24:04 - The Reality of the Academic Literature
26:25 - Extrapolating Prices in Incomplete Markets
30:18 - Inferring Risk Premiums and Risk-Neutral Measures
34:22 - TL;DW Executive Summary
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🗣️ Shout Outs
A special thank you to my members on YouTube for supporting my channel and enabling me to continue to create videos just like this one!
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Dr. Jason Pirozzolo
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