Sanity check: Option prices with market parameters

[KumarRishabh]

Due to some unknown phenomenon while conducting simulation experiments, the previous implementations weren't working as required. Currently, the Julia functions weighted_heston() and weighted_heston_M2() don't work similarly, but they are supposed to work the same when the M parameter in the former functions is set to 2. The real check would be the accuracy of the option parameters, for the given parameter sets in the paper.

[KumarRishabh]

Zhenyu Cui et al. used the following parameters to get European option prices using the CTMC method. Given the Heston SDE is defined as follows (alternate to Mike's formulation of the Heston SDE):

$$ dS_t = \mu S_t dt + \sqrt(V_t) S_t dW^{(1)}_t $$

$$ dV_t = \eta (\theta - V_t) dt + \sigma_v \sqrt(V_t) dW^{(2)}_t $$

where the following parameter values were used:

• $S_0 = 10$ (Initial stock price)
• $v_0 = 0.04$ (Initial variance)
• $T = 1$ (Time to maturity)
• $K = 4$ (Strike price)
• $\rho = -0.75$ (Correlation between the Brownian motions of the stock price and the variance)
• $\sigma_v = 0.15$ (Volatility of the variance process)
• $\eta = 4$ (Rate of mean reversion for the variance process)
• $\vartheta = 0.035$ (Long-term variance mean)
• $r = 0$ (Risk-free interest rate) The table shows the European option prices for the Heston model using various grid sizes. Here is the summary of the prices for different values of $N$ and $M$:

For different values of $N$:

• For $N = 50$, the CTMC price is 5.9831 with a relative error of $2.82 \times 10^{-3}$ and a rate of 1.89.
• For $N = 70$, the CTMC price is 5.9911 with a relative error of $1.49 \times 10^{-3}$ and a rate of 1.94.
• For $N = 90$, the CTMC price is 5.9945 with a relative error of $9.12 \times 10^{-4}$ and a rate of 1.92.
• For $N = 110$, the CTMC price is 5.9963 with a relative error of $6.20 \times 10^{-4}$ and a rate of 2.00.
• For $N = 130$, the CTMC price is 5.9974 with a relative error of $4.43 \times 10^{-4}$ and a rate of 2.00.

The benchmark price (BM) for the Heston model is 6.0000.

For different values of $M$:

• For $M = 50$, the CTMC price is 5.9831 with a relative error of $2.82 \times 10^{-3}$ and a rate of 1.83.
• For $M = 70$, the CTMC price is 5.9909 with a relative error of $1.52 \times 10^{-3}$ and a rate of 1.85.
• For $M = 90$, the CTMC price is 5.9943 with a relative error of $9.59 \times 10^{-4}$ and a rate of 1.87.
• For $M = 110$, the CTMC price is 5.9961 with a relative error of $6.43 \times 10^{-4}$ and a rate of 1.90.
• For $M = 130$, the CTMC price is 5.9974 with a relative error of $4.68 \times 10^{-4}$ and a rate of 1.92.

The benchmark price (BM) for the Heston model is 6.0000.