Stochastic Calculus

In 1978, Breeden and Litzenberger showed how under risk-neutral pricing, that the discounted Risk-Neutral Density (RND) function could be estimated directly from option prices.

In this tutorial we demonstrate how to use the Breeden-Litzenberger formula to derive the risk-neutral density function from European call options with stochastic volatilty under the Heston model. We price the call options using the semi-analytical solution for the Heston model using rectangular integration and compare this to the QuantLib implementation in Python. Once we have the option prices (or possibly market prices) we can implement the Breeden-Litzenberger formula easily using pandas dataframe shift functions. To utilise the estimated pdf within other calculations, we need to make use of scipy’s interpolation function.

In this tutorial we will be simulating Geometric Brownian Motion in Python. Geometric Brownian motion (GBM) S is defined by S0 > 0 and the dynamics as defined in the following Stochastic Differential Equation (SDE):

In this tutorial we will investigate the stochastic process that is the building block of financial mathematics. We will consider a symmetric random walk, scaled random walk and browian motion. The mathematic notation and explanations are from Steven Shreve’s book Stochastic Calculus for Finance II.
Important properties of brownian motion is that it is a martingale Markovprocess and that it accumulates quadratic variation at rate one per unit time.
Note: Quadratic Variation is perhaps what makes Stochastic Calculus so different from Ordinary Calculus.