Modeling Implied Volatility Alongside Simulated Stock Prices
Summary
The document asks how to simulate implied volatility (IV) together with stock prices in a Monte Carlo option analysis. The author generates stock paths with Geometric Brownian Motion using estimated daily return mean and standard deviation, then values options along those paths with the Bjerksund–Stensland 2002 model. They want to replace a constant IV assumption with simulated IV values.
The central modeling question is whether IV should follow an independent process or be linked to the underlying price, which often moves inversely to volatility. The author proposes using mean and standard deviation of daily VIX returns because symbol-specific IV data are unavailable. The document raises the trade-off between adding a volatility process and relying on an assumed relationship, but provides no answer, test, or evidence that either approach improves option simulations. Its proposed VIX proxy may also differ from the volatility behavior of a particular stock.
Key ideas
- The author simulates stock prices with Geometric Brownian Motion and values options along each path.
- They ask whether simulated implied volatility should be independent of the stock price or correlated with it.
- The proposed volatility input is based on daily VIX returns because stock-specific implied volatility data are unavailable.
- The document poses the modeling problem but does not provide a solution or empirical comparison.
Tags
Full text
# 63139 # Do simulated values for IV need to be linked to the simulated series of underlying prices when used together in a Monte Carlo Simulation? I've been using thousands of simulated stock price series generated with mean and standard deviation of daily returns and Geometric Brownian Motion, and then running these simulated price series through the Bjerksund Stensland 2002 option pricing model to generate thousands of simulations of option prices. Up to this point, I have assumed constant IV for the IV input in the option pricing model, though I would like to use a simulated series of values for IV in the same way I currently am with the price of the underlying (if this is indeed advantageous...perhaps it isn't). My question: if I am to do this, can the IV series be generated using its own mean and standard deviation in the same way I would generate prices for the underlying, or do the simulated values for IV have to be in lockstep with my simulated price series such that simulated price declines usually see an uptick in IV, rises in simulated price see a downturn in IV, etc.? In other words, can both simulated price and IV be on their own independent random walks, or does simulated IV have to be a function of underlying price to reflect this generally inverse relationship between the price of the underlying and IV? If it's relevant, my plan is to simply use mean / standard deviation of daily VIX returns for generating the simulated IV series as I don't have access to symbol-specific IV data. If this way of doing things is problematic and a waste of time, feel free to let me know!
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