In this tutorial we will be implementating a simple slow and fast binomial pricing model in python. We will treat binomial tree as a network with nodes (i,j) with i representing the time steps and j representing the number of ordered price outcome.
One of the main benefits of Monte Carlo simulations is to price options under multiple factors. By this I refer to multiple underlying asset prices or stochastic volatility or even changing interest rates. In this tutorial we will explore the pricing of a European Spread Call Option on the difference between two stock indices following a more general stochastic process. The SDE’s will be have stochastic volatility as described by the Heston Model (1993):
Trading Hypothesis: If Interest Rates Rise, Technology Companies valuations will be impacted worse than the broader stock market.
How can I benefit?
Today we look at how combining variance reduction methods like antithetic, delta and gamma-based control variates can help to substantially reduce the standard error around your theoretical option price during monte carlo simulations.
In this tutorial we will investigate ways we can reduce the variance of results from a Monte Carlo simulation method when valuing financial derivatives by using Delta-based control variates.
In this tutorial we will investigate ways we can reduce the variance of results from a Monte Carlo simulation method when valuing financial derivatives.
Monte Carlo simulations is a way of solving probabilistic problems by numerically simulating many possible scenarios so that you may calculate statistical properties of the outcomes, such as expectations, variances of probabilities of certain outcomes. In the case of Financial Derivatives, this gives us a handy tool for which to price complex derivatives for which and analytical formulae is not possible.
The Heston model is a useful model for simulating stochastic volatility and its effect on the potential paths an asset can take over the life of an option. It’s popular because of:
– easy closed-form solution for European option pricing
– no risk of negative variances
– incorporation of leverage effect
This allows for more effective modeling than the Black-Scholes formula allows due to its restrictive assumption of constant volatility.
One of the nice things about the Heston model for European option prices is that there is a closed-form solution once you have the characteristic function. So, discretisation of the SDE is not required for valuing a European option, however if you would like to value other option types with complex features using the Heston model than you can use the following code:
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):
