Using Parallelization for Heston Model and Sampling from $\chi^2$ Law

The Heston Model

The Heston model is a stochastic volatility model due to Heston ('92), who modeled the volatility as Cox, Ingersoll and Ross did for modeling interest rates ('85).

Definition

$$ \begin{align*} dS_t &= rS_t dt + \sqrt{V_t} S_t dW_t \\ dV_t &= -k(V_t - V^0) dt + \sigma \sqrt{V_t} dZ_t \end{align*} $$

• $V_t$ is the instantaneous variance
• $\sigma$ is called the "volatility of volatility"
• $W_t$, $Z_t$ are two Brownian motions correlated with correlation $\rho$

Remark: For the implementation, we used the linear discretisation:

$$ \begin{align*} S_{t+\Delta t} &= S_t + r S_t \Delta t + \sqrt{v_t}S_T\sqrt{\Delta t}(\rho G_1 + \sqrt{1-\rho^2} G_2) \\ v_{t+\Delta t} &= k (v_0 - v_t)\Delta t + \sigma \sqrt{f(v_t)}\sqrt{\Delta t } G_1 \end{align*} $$

I : Monte Carlo Heston Pricing

Using the provided Monte Carlo Black Scholes pricing code, we adapted it to simulate the Heston model dynamics by replacing the square root in the volatility process by 0 if it's negative.

Number of scenarios	CPU execution time	GPU execution time
1000	0.75s	0.176 ms
100000	8.79s	0.514 ms
1000000	82.962s	4.196 ms

More experiments can be conducted using the notebook: Question1.ipynb and the CUDA file MC.cu.

II : Sampling from Decentred $\chi^2$

Methodology

1. Defined multiple device functions: GS_star, GKM1, GKM2, GKM3 which perform gamma simulation, and then we simulate using generate_gamma and chi2 as described in the paper.
2. We use the kernel function that:
   • Initializes a random number generator (curandState).
   • Generates a Poisson random variable to add the 'non-centrality' aspect to the chi-squared distribution.
   • Runs device functions as intended on the threads using the curand states in a way that ensures statistical independence.

Performance Comparison

Performance on GPU (Nvidia T4)

Number of samples	Time taken
10000	0.005807s
1000000	0.048577s
10000000	0.341024s
100000000	2.452472s

Performance on CPU (AMD Ryzen 9 5900HS)

Number of samples	Time taken
10000	0.512911s
1000000	51.430150s

More experiments can be conducted using the notebook: Question2.ipynb and the CUDA file chi2.cu.

III : The Heston Model Without Euler

Methodology: 3 Steps

1. We compute the variance using the conditional law:
   • We fix $V_u$ and we use Chi2 from the previous question to compute $V_t$ given $V_u$ with the parameters given in the article.
2. We simulate the integral of variance using the simulated variance:
   • We start by coding the characteristic function of the integral given $V_t$ and $V_u$.
   • We use this characteristic function to compute the cumulative distribution function. In fact, this leads to an infinite sum. However, we select an integer $N$ at which the summation can be terminated. The higher $N$ is, the more accurate our computation becomes.
   • Then we use the (Inverse) Distribution Function Method for sampling. And since we don't have access to a closed form of $F^{-1}$, we use Newton's method to compute $x$ such that $F(x) = U$.
3. We generate the next instance of the asset price using the computed elements and a formula provided in the article.

Link for our code for this question: Code for Question 3.