This repository contains a code basis for quantitative methods in finance, coded in C++ and visualized using Python.
As a first example of the methods in the repo, we can calibrate the Black-Scholes-Merton implied volatility to an option price surface. In this case, we are considering a European call option on an underlying with spot price $175, strikes ranging from $155 to $192.5 and maturities ranging from 1 to 10 weeks. The risk free rate and dividend yield are assumed to be 4.5% and 0.7% respectively. We use the Adam optimizer (a variant of gradient descent) for calibration of the model. Below, the price surface and corresponding volatility surface are plotted.
| European call price surface | European call BSM impled Vol |
|---|---|
The Black-Scholes-Merton implied volatility varies for different strikes and maturities. No single volatility parameter is sufficient to accurately replicate the price surface, which is a clear sign of the limitations of the BSM model (the above price surfaces is artificial, but the effect can be even more pronounced for real options). If we calibrate the BSM model to the above price surface, we get a best fit with a vol of 0.62 leading to a mean relative squared error of 0.04 on the prices. A more flexible model than BSM is the Heston model, in which the volatility itself follows its own stochastic process (which may be correlated to the price). Contrary to the single vol of the BSM model, the Heston model contains 5 adjustable parameters. Calibrating a Heston model to the above price surface, we can achieve a mean relative squared error of 0.02, half of the BSM error. Below, we show the price surfaces generated by the two models.
| BSM calibrated price surface | Heston calibrated price surface |
|---|---|
Both models have a hard time fitting the exact curvature of the true surface, but the slope of the curve can be seen to be more accurate in the Heston model, particularly for large times to maturity. For this plot, the BSM surface is calibrated using the analytical pricing formula. No analytical formula is available for the Heston model, but the prices can be approximated, e.g. via the characteristic function and Fast Fourier Transform, which is used in the calibration.
The repository also contains tools to simulate the stochastic processes modelling stock prices, which can be useful for Monte-Carlo based techniques. The example plot below shows three paths of a Heston model on the left and three paths of a Variance Gamma model on the right.
| Heston model paths | Variance Gamma model paths |
|---|---|
The Heston model models stock prices with a stochastic volatility process. Among other parameters, it involves a reversion rate, enabling analysts to incorparate typical stock behaviours like mean reversion or momentum in the model. It does not incorporate jumps however: like in the BSM model, the process is almost surely continuous. The Variance Gamma model on the other hand includes random discontinuous jumps. It can be used to model gap-ups and gap-downs or quasi instantaneuous price jumps due to unexpected and crucial news on a company.
Canonical models for derivative pricing and Hedging are included in the repository. The below plot shows the development of the fair price, Delta and Gamma of a european call option with a strike of $100 in the Black-Scholes-Merton model as time to maturity increases (left). The right plot shows the distribution of stock price in different models with similar parameters.
| Greeks with BSM model | Stock price distribution with different models |
|---|---|
It can be clearly seen that the choice of model affects the forecasts. For instance, both very high and very low returns are more likely in the Variance Gamma model than in the BSM model. The distribution is thus said to have "fat tails" in comparison to the log-normal distribution. In the Heston model very low returns are the likliest. For example, the conditional value at risk at the 5% level is approximately $31 for the BSM model and $37 for the Heston model.