Simulating Stock Prices with Normal and Normal Inverse Gaussian Returns
Summary
The document distinguishes modeling a stock price directly with a normal process from modeling returns with a normal inverse Gaussian distribution and exponentiating those returns to obtain prices. It explains that a formula of the form S(t) = S(0) exp(X) implies a log-price or return model, while the normal-price case uses an additive process. It also describes an Euler update for the normal case, where drift and volatility increments are added to the previous price.
The answer points to a spreadsheet example for Black–Scholes Monte Carlo simulation and sketches the corresponding normal-process update. It notes that the normal Euler scheme is exact, while for a log-normal process it is preferable to apply the scheme to log prices and then exponentiate. The excerpt does not provide a full sampling procedure or parameter calibration for a normal inverse Gaussian distribution, and its equations should be checked for notation and parameter consistency before implementation.
Key ideas
- An additive normal stock-price model differs from a log-normal model built from returns.
- Exponentiating a random variable models multiplicative price changes when that variable represents log returns.
- The normal-process Euler update adds drift and a volatility-scaled random increment to the previous price.
- For a log-normal process, applying the scheme to log prices and exponentiating is preferable.
- The document does not explain how to sample or calibrate normal inverse Gaussian returns.
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# Stock prices using a monte carlo simulation with a normal inverse gauss distribution # Stock prices using a monte carlo simulation with a normal inverse gauss distribution I am supposed to model daily stock prices with a normal inverse gauss distribution in excel. I feel like I am misssing some basics because I cant transform the information from the academic papers into an excel formula. Does anyone have any experience with this distribution? How do I go from the PDF to the St = So * exp (X) formula? ## Answer by Christian Fries (score 3) https://quant.stackexchange.com/a/8355 Note: In your text you refer to the stock prices and using a normal inverse gaussian. This would correspond to a normal model. However the formula you write suggests a log-normal (Black-Scholes) like model (not sure what X is), i.e. using a normal inverse gaussian for the stock returns. For Excel: The spreadsheet at http://www.christian-fries.de/finmath/spreadsheets/ does a Monte-Carlo Simulation of a Black-Scholes model and the corresponding risk neutral valuation of a derivative. To convert the log-normal process to a normal one: convert the Euler scheme ``` =D18+$C$6*D18*$C$8+$C$7*D18*NORMINV(RANDOM();0;1)*SQRT($C$8) ``` you have ``` =D18+$C$6*$C$8+$C$7*NORMINV(RANDOM();0;1)*SQRT($C$8) ``` where this is cell E18 and - D18 denotes the previous cell / realization of the stock - \$C\$6 is the drift r - \$C\$8 is the time step dt - \$C\$7 is the volatility $\sigma$ and each row then corresponds to a Monte-Carlo path. (Remark: In the normal case case, the Euler scheme is the exact solution. In the log normal case it is much better to use the Euler scheme for log(S), i.e. $S(t+\Delta t) = S(t) * \exp(r \Delta t - 0.5 \sigma \Delta t + \Delta W(t))$ - I assume you can guess the corresponding Excel formula.
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