Importance Sampling

The goal of this program is to optimize the drift parameter of the geometric brownian motion for out-the-money options. The model of the underlying asset movement accounts for both continuous diffusion and discrete jumps in asset prices; however, the jump related parameters are set to zero in this program.

Usage

$ git clone git@github.com:mlian031/importance-sampling.git
$ cd importance-sampling
$ python -m venv venv && source venv/bin/activate
$ pip install -r requirements.txt
$ python main.py

Background

Importance sampling

In Monte Carlo simulation, importance sampling is a variance reduction technique based on changing the probability measure. The core idea is: Given expectation $E[h(X)]$ where $X$ has density $f$, we can rewrite:

$$ \alpha = E[h(X)] = \int h(x)f(x)dx $$

For any other density $g$ satisfying $f(x) > 0 \Rightarrow g(x) > 0$, we can represent this as:

$$ \alpha = \int h(x)\frac{f(x)}{g(x)}g(x)dx = \tilde{E}\left[h(X)\frac{f(X)}{g(X)}\right] $$

where $\tilde{E}$ indicates expectation under $g$. This leads to the importance sampling estimator:

$$\hat{\alpha}_g = \hat{\alpha}_g(n) = \frac{1}{n}\sum_{i=1}^n h(X_i)\frac{f(X_i)}{g(X_i)}$$

with $X_1,\ldots,X_n$ drawn from $g$. The ratio $f(X_i)/g(X_i)$ is called the likelihood ratio or Radon-Nikodym derivative.

From Glasserman, Monte Carlo Methods in Financial Engineering

A jump diffusion model

The Merton Jump Diffusion model extends geometric Brownian motion by adding a compound Poisson process to model sudden price jumps. The stochastic differential equation is:

$$ \frac{dS(t)}{S(t)} = \mu dt + \sigma dW(t) + dJ(t) $$

where:

• $S(t)$ is the asset price at time $t$
• $\mu$ is the drift
• $\sigma$ is the volatility
• $W(t)$ is a standard Brownian motion
• $J(t)$ is a compound Poisson process given by:

$$ J(t) = \sum_{j=1}^{N(t)} (Y_j - 1) $$

Here $N(t)$ is a Poisson process with intensity $\lambda$ and $Y_j$ are independent jump sizes, typically lognormally distributed:

$$ \log(Y_j) \sim N(a, b^2) $$

The solution to the SDE is:

$$ S(t) = S(0)e^{(\mu-\frac{1}{2}\sigma^2)t+\sigma W(t)}\prod_{j=1}^{N(t)} Y_j $$

For option pricing under the risk-neutral measure, the drift is adjusted to:

$$ \mu = r - \lambda m $$

where $r$ is the risk-free rate and $m = E[Y_j - 1]$ compensates for the jumps.

From Glasserman, Monte Carlo Methods in Financial Engineering

Importance sampling implementation

The optimal $\lambda$ is determined by maximizing the variance reduction ratio, defined as:

$$ \text{Variance Reduction} = \left(\frac{\text{Standard MC Standard Error}}{\text{IS Standard Error}}\right)^2 $$

This is achieved by minimizing the negative of this ratio:

$$ \text{Objective Function: } -\text{Variance Reduction} $$

The bounds for $\lambda$ are:

$$ \lambda_{\text{lower}} = \max(0.001, r - 2\sigma), \quad \lambda_{\text{upper}} = r + 2\sigma $$

These bounds are centered around the risk-free rate $r$, corresponding to the mean $\mu$ under the risk-neutral measure, adjusted by $\pm 2\sigma$, encompassing approximately 95% of the distribution in a normal context.

To enhance the likelihood of finding the global optimum, the algorithm tests multiple initial values for $\lambda$ within these bounds.

The optimization uses scipy.optimize.minimize with the L-BFGS-B method. This algorithm iteratively computes the gradient of the objective function and updates $\lambda$ (distinct from the Poisson intensity parameter $\lambda_j$) using an approximate Hessian. The process continues until the change in the objective function or the gradient norm is below the default tolerance.

In essence, the algorithm identifies the optimal drift adjustment by:

1. Simulating with an initial $\lambda$.
2. Optimizing $\lambda$ to maximize variance reduction.
3. Iteratively refining the simulation and selecting the $\lambda$ with the highest variance reduction ratio across multiple initial points.

Sample simulations

The average variance reduction for this particular simulation is:

$$ \frac{\text{Mean Var MC}}{\text{Mean Var IS}} = 14.605 $$

Results Summary:
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|   Paths (N) |   Analytical |   MC Price |   MC StdErr |   MC RMSE |   MC Time(s) | MC Rel.Err   |   IS Price |   IS StdErr |   IS RMSE |   IS Time(s) | IS Rel.Err   | Var.Red   |
+=============+==============+============+=============+===========+==============+==============+============+=============+===========+==============+==============+===========+
|       10000 |       3.2475 |     3.24   |      0.0866 |    0.0655 |            0 | 0.23%        |     3.2464 |      0.0227 |    0.0169 |            0 | 0.03%        | 14.55x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10116 |       3.2475 |     3.2458 |      0.0862 |    0.0673 |            0 | 0.05%        |     3.2472 |      0.0226 |    0.019  |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10234 |       3.2475 |     3.2451 |      0.0856 |    0.0715 |            0 | 0.07%        |     3.249  |      0.0224 |    0.0177 |            0 | 0.05%        | 14.57x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10353 |       3.2475 |     3.2542 |      0.0853 |    0.0677 |            0 | 0.21%        |     3.2472 |      0.0223 |    0.0181 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10473 |       3.2475 |     3.2402 |      0.0845 |    0.0656 |            0 | 0.22%        |     3.2481 |      0.0222 |    0.0161 |            0 | 0.02%        | 14.54x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10595 |       3.2475 |     3.2509 |      0.0843 |    0.066  |            0 | 0.10%        |     3.2484 |      0.022  |    0.0168 |            0 | 0.03%        | 14.65x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10718 |       3.2475 |     3.251  |      0.0837 |    0.0619 |            0 | 0.11%        |     3.2484 |      0.0219 |    0.0183 |            0 | 0.03%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10843 |       3.2475 |     3.2485 |      0.0833 |    0.0643 |            0 | 0.03%        |     3.2477 |      0.0218 |    0.0181 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       10969 |       3.2475 |     3.2471 |      0.0828 |    0.0694 |            0 | 0.01%        |     3.2482 |      0.0217 |    0.0156 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11097 |       3.2475 |     3.2504 |      0.0824 |    0.0673 |            0 | 0.09%        |     3.249  |      0.0215 |    0.0181 |            0 | 0.05%        | 14.65x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11226 |       3.2475 |     3.2493 |      0.0819 |    0.0676 |            0 | 0.06%        |     3.2478 |      0.0214 |    0.0166 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11357 |       3.2475 |     3.2485 |      0.0815 |    0.068  |            0 | 0.03%        |     3.2466 |      0.0213 |    0.018  |            0 | 0.03%        | 14.67x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11489 |       3.2475 |     3.2463 |      0.0809 |    0.0656 |            0 | 0.04%        |     3.2486 |      0.0212 |    0.0173 |            0 | 0.03%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11623 |       3.2475 |     3.2515 |      0.0805 |    0.066  |            0 | 0.12%        |     3.2471 |      0.0211 |    0.0167 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11758 |       3.2475 |     3.2474 |      0.08   |    0.0637 |            0 | 0.00%        |     3.2467 |      0.0209 |    0.017  |            0 | 0.02%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       11895 |       3.2475 |     3.2494 |      0.0795 |    0.0644 |            0 | 0.06%        |     3.2463 |      0.0208 |    0.0164 |            0 | 0.04%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12033 |       3.2475 |     3.246  |      0.0791 |    0.0652 |            0 | 0.04%        |     3.2492 |      0.0207 |    0.0163 |            0 | 0.05%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12173 |       3.2475 |     3.2442 |      0.0785 |    0.0639 |            0 | 0.10%        |     3.2484 |      0.0206 |    0.0156 |            0 | 0.03%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12315 |       3.2475 |     3.2471 |      0.0781 |    0.0603 |            0 | 0.01%        |     3.2478 |      0.0204 |    0.0158 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12458 |       3.2475 |     3.2492 |      0.0777 |    0.0616 |            0 | 0.05%        |     3.2476 |      0.0203 |    0.0166 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12603 |       3.2475 |     3.2432 |      0.0772 |    0.0576 |            0 | 0.13%        |     3.2465 |      0.0202 |    0.0165 |            0 | 0.03%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12750 |       3.2475 |     3.2501 |      0.0768 |    0.057  |            0 | 0.08%        |     3.2498 |      0.0201 |    0.0156 |            0 | 0.07%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       12898 |       3.2475 |     3.242  |      0.0762 |    0.0593 |            0 | 0.17%        |     3.2465 |      0.02   |    0.0157 |            0 | 0.03%        | 14.54x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13049 |       3.2475 |     3.2478 |      0.0759 |    0.0604 |            0 | 0.01%        |     3.2478 |      0.0199 |    0.0155 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13200 |       3.2475 |     3.2481 |      0.0755 |    0.0589 |            0 | 0.02%        |     3.2477 |      0.0198 |    0.0156 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13354 |       3.2475 |     3.2467 |      0.0751 |    0.0637 |            0 | 0.02%        |     3.2476 |      0.0196 |    0.0153 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13509 |       3.2475 |     3.2536 |      0.0747 |    0.0605 |            0 | 0.19%        |     3.2475 |      0.0195 |    0.0167 |            0 | 0.00%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13667 |       3.2475 |     3.2512 |      0.0743 |    0.062  |            0 | 0.12%        |     3.2472 |      0.0194 |    0.0162 |            0 | 0.01%        | 14.65x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13826 |       3.2475 |     3.2495 |      0.0737 |    0.0563 |            0 | 0.06%        |     3.2493 |      0.0193 |    0.0163 |            0 | 0.06%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       13987 |       3.2475 |     3.2514 |      0.0733 |    0.0564 |            0 | 0.12%        |     3.2476 |      0.0192 |    0.0153 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14149 |       3.2475 |     3.2517 |      0.073  |    0.0546 |            0 | 0.13%        |     3.2461 |      0.0191 |    0.0143 |            0 | 0.04%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14314 |       3.2475 |     3.2449 |      0.0725 |    0.0573 |            0 | 0.08%        |     3.2466 |      0.019  |    0.0149 |            0 | 0.03%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14481 |       3.2475 |     3.2484 |      0.0721 |    0.0595 |            0 | 0.03%        |     3.2467 |      0.0189 |    0.015  |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14649 |       3.2475 |     3.2457 |      0.0716 |    0.0587 |            0 | 0.06%        |     3.2472 |      0.0188 |    0.0152 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14820 |       3.2475 |     3.2479 |      0.0712 |    0.0566 |            0 | 0.01%        |     3.2482 |      0.0186 |    0.0146 |            0 | 0.02%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       14992 |       3.2475 |     3.2491 |      0.0709 |    0.0567 |            0 | 0.05%        |     3.2486 |      0.0185 |    0.0148 |            0 | 0.03%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       15167 |       3.2475 |     3.2513 |      0.0705 |    0.055  |            0 | 0.12%        |     3.2468 |      0.0184 |    0.0135 |            0 | 0.02%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       15343 |       3.2475 |     3.2462 |      0.0699 |    0.0525 |            0 | 0.04%        |     3.2466 |      0.0183 |    0.0156 |            0 | 0.03%        | 14.56x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       15522 |       3.2475 |     3.2459 |      0.0696 |    0.0562 |            0 | 0.05%        |     3.248  |      0.0182 |    0.0141 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       15702 |       3.2475 |     3.2496 |      0.0693 |    0.0548 |            0 | 0.06%        |     3.2471 |      0.0181 |    0.0144 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       15885 |       3.2475 |     3.2491 |      0.0688 |    0.0539 |            0 | 0.05%        |     3.249  |      0.018  |    0.0145 |            0 | 0.05%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       16070 |       3.2475 |     3.245  |      0.0684 |    0.0558 |            0 | 0.08%        |     3.2479 |      0.0179 |    0.0149 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       16257 |       3.2475 |     3.2478 |      0.068  |    0.0569 |            0 | 0.01%        |     3.2487 |      0.0178 |    0.0145 |            0 | 0.04%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       16446 |       3.2475 |     3.2468 |      0.0676 |    0.0564 |            0 | 0.02%        |     3.2467 |      0.0177 |    0.014  |            0 | 0.02%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       16638 |       3.2475 |     3.2502 |      0.0672 |    0.0572 |            0 | 0.08%        |     3.2484 |      0.0176 |    0.0139 |            0 | 0.03%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       16831 |       3.2475 |     3.2468 |      0.0668 |    0.054  |            0 | 0.02%        |     3.2463 |      0.0175 |    0.0134 |            0 | 0.04%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       17027 |       3.2475 |     3.2501 |      0.0665 |    0.053  |            0 | 0.08%        |     3.2474 |      0.0174 |    0.0134 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       17225 |       3.2475 |     3.25   |      0.0661 |    0.0551 |            0 | 0.08%        |     3.2473 |      0.0173 |    0.0137 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       17426 |       3.2475 |     3.2429 |      0.0656 |    0.0512 |            0 | 0.14%        |     3.2482 |      0.0172 |    0.0135 |            0 | 0.02%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       17629 |       3.2475 |     3.2479 |      0.0653 |    0.0527 |            0 | 0.01%        |     3.2477 |      0.0171 |    0.0142 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       17834 |       3.2475 |     3.2504 |      0.065  |    0.048  |            0 | 0.09%        |     3.2467 |      0.017  |    0.0139 |            0 | 0.03%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       18041 |       3.2475 |     3.2439 |      0.0645 |    0.0508 |            0 | 0.11%        |     3.246  |      0.0169 |    0.0135 |            0 | 0.04%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       18251 |       3.2475 |     3.2447 |      0.0641 |    0.0478 |            0 | 0.09%        |     3.2463 |      0.0168 |    0.0139 |            0 | 0.04%        | 14.57x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       18464 |       3.2475 |     3.2502 |      0.0638 |    0.0483 |            0 | 0.09%        |     3.248  |      0.0167 |    0.0135 |            0 | 0.02%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       18679 |       3.2475 |     3.2427 |      0.0634 |    0.0527 |            0 | 0.15%        |     3.2463 |      0.0166 |    0.0131 |            0 | 0.04%        | 14.57x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       18896 |       3.2475 |     3.2433 |      0.063  |    0.0483 |            0 | 0.13%        |     3.2469 |      0.0165 |    0.0138 |            0 | 0.02%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       19116 |       3.2475 |     3.2457 |      0.0627 |    0.0497 |            0 | 0.06%        |     3.2473 |      0.0164 |    0.0142 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       19338 |       3.2475 |     3.2497 |      0.0624 |    0.0511 |            0 | 0.07%        |     3.2467 |      0.0163 |    0.0139 |            0 | 0.02%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       19563 |       3.2475 |     3.2457 |      0.062  |    0.0499 |            0 | 0.06%        |     3.247  |      0.0162 |    0.0128 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       19791 |       3.2475 |     3.2473 |      0.0616 |    0.0482 |            0 | 0.01%        |     3.2459 |      0.0161 |    0.013  |            0 | 0.05%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       20022 |       3.2475 |     3.2471 |      0.0613 |    0.0491 |            0 | 0.01%        |     3.2472 |      0.016  |    0.0128 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       20255 |       3.2475 |     3.2513 |      0.0609 |    0.0479 |            0 | 0.12%        |     3.2473 |      0.0159 |    0.0132 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       20490 |       3.2475 |     3.2462 |      0.0606 |    0.0475 |            0 | 0.04%        |     3.2479 |      0.0159 |    0.0129 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       20729 |       3.2475 |     3.2507 |      0.0603 |    0.0507 |            0 | 0.10%        |     3.2471 |      0.0158 |    0.0132 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       20970 |       3.2475 |     3.2449 |      0.0598 |    0.047  |            0 | 0.08%        |     3.2481 |      0.0157 |    0.012  |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       21214 |       3.2475 |     3.2443 |      0.0595 |    0.049  |            0 | 0.10%        |     3.2483 |      0.0156 |    0.0122 |            0 | 0.03%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       21461 |       3.2475 |     3.245  |      0.0592 |    0.0489 |            0 | 0.08%        |     3.248  |      0.0155 |    0.0116 |            0 | 0.02%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       21711 |       3.2475 |     3.2483 |      0.0589 |    0.047  |            0 | 0.03%        |     3.2475 |      0.0154 |    0.0122 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       21963 |       3.2475 |     3.2481 |      0.0585 |    0.0461 |            0 | 0.02%        |     3.2479 |      0.0153 |    0.0119 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       22219 |       3.2475 |     3.2482 |      0.0582 |    0.0457 |            0 | 0.02%        |     3.246  |      0.0152 |    0.0124 |            0 | 0.05%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       22478 |       3.2475 |     3.2498 |      0.0579 |    0.0487 |            0 | 0.07%        |     3.2465 |      0.0151 |    0.0118 |            0 | 0.03%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       22739 |       3.2475 |     3.2511 |      0.0576 |    0.0443 |            0 | 0.11%        |     3.2487 |      0.015  |    0.0117 |            0 | 0.04%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       23004 |       3.2475 |     3.2496 |      0.0572 |    0.0437 |            0 | 0.07%        |     3.2477 |      0.015  |    0.0123 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       23272 |       3.2475 |     3.2473 |      0.0569 |    0.0439 |            0 | 0.00%        |     3.2469 |      0.0149 |    0.0112 |            0 | 0.02%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       23542 |       3.2475 |     3.2471 |      0.0565 |    0.0443 |            0 | 0.01%        |     3.2473 |      0.0148 |    0.0117 |            0 | 0.00%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       23816 |       3.2475 |     3.2446 |      0.0561 |    0.0449 |            0 | 0.09%        |     3.2479 |      0.0147 |    0.0119 |            0 | 0.01%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       24094 |       3.2475 |     3.247  |      0.0559 |    0.0447 |            0 | 0.02%        |     3.2464 |      0.0146 |    0.012  |            0 | 0.03%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       24374 |       3.2475 |     3.2441 |      0.0555 |    0.046  |            0 | 0.10%        |     3.2481 |      0.0145 |    0.0112 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       24658 |       3.2475 |     3.2456 |      0.0552 |    0.0459 |            0 | 0.06%        |     3.2488 |      0.0145 |    0.0117 |            0 | 0.04%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       24945 |       3.2475 |     3.2509 |      0.055  |    0.042  |            0 | 0.11%        |     3.2481 |      0.0144 |    0.0109 |            0 | 0.02%        | 14.64x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       25235 |       3.2475 |     3.2461 |      0.0546 |    0.0427 |            0 | 0.04%        |     3.2496 |      0.0143 |    0.0113 |            0 | 0.07%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       25529 |       3.2475 |     3.2475 |      0.0543 |    0.0456 |            0 | 0.00%        |     3.2477 |      0.0142 |    0.0105 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       25826 |       3.2475 |     3.2454 |      0.054  |    0.0435 |            0 | 0.06%        |     3.2487 |      0.0141 |    0.011  |            0 | 0.04%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       26126 |       3.2475 |     3.2488 |      0.0537 |    0.038  |            0 | 0.04%        |     3.2477 |      0.014  |    0.0109 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       26430 |       3.2475 |     3.2444 |      0.0533 |    0.043  |            0 | 0.09%        |     3.2488 |      0.014  |    0.0118 |            0 | 0.04%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       26738 |       3.2475 |     3.2501 |      0.0531 |    0.0439 |            0 | 0.08%        |     3.2474 |      0.0139 |    0.0109 |            0 | 0.00%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       27049 |       3.2475 |     3.2497 |      0.0527 |    0.0446 |            0 | 0.07%        |     3.2473 |      0.0138 |    0.0111 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       27364 |       3.2475 |     3.2502 |      0.0524 |    0.0403 |            0 | 0.08%        |     3.2477 |      0.0137 |    0.0105 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       27682 |       3.2475 |     3.2468 |      0.0521 |    0.0435 |            0 | 0.02%        |     3.2463 |      0.0136 |    0.0107 |            0 | 0.03%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       28005 |       3.2475 |     3.2419 |      0.0517 |    0.0403 |            0 | 0.17%        |     3.2473 |      0.0136 |    0.0111 |            0 | 0.00%        | 14.57x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       28330 |       3.2475 |     3.2414 |      0.0514 |    0.0408 |            0 | 0.19%        |     3.2473 |      0.0135 |    0.011  |            0 | 0.01%        | 14.56x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       28660 |       3.2475 |     3.2493 |      0.0512 |    0.0408 |            0 | 0.05%        |     3.2479 |      0.0134 |    0.011  |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       28994 |       3.2475 |     3.2507 |      0.051  |    0.0403 |            0 | 0.10%        |     3.2478 |      0.0133 |    0.0102 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       29331 |       3.2475 |     3.2458 |      0.0506 |    0.0397 |            0 | 0.05%        |     3.2474 |      0.0133 |    0.0106 |            0 | 0.00%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       29673 |       3.2475 |     3.2497 |      0.0504 |    0.0381 |            0 | 0.07%        |     3.2464 |      0.0132 |    0.0103 |            0 | 0.03%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       30018 |       3.2475 |     3.2494 |      0.0501 |    0.0402 |            0 | 0.06%        |     3.2477 |      0.0131 |    0.0102 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       30367 |       3.2475 |     3.2491 |      0.0498 |    0.0382 |            0 | 0.05%        |     3.2463 |      0.013  |    0.0103 |            0 | 0.04%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       30721 |       3.2475 |     3.2484 |      0.0495 |    0.0391 |            0 | 0.03%        |     3.2476 |      0.0129 |    0.0104 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       31078 |       3.2475 |     3.2472 |      0.0492 |    0.04   |            0 | 0.01%        |     3.2462 |      0.0129 |    0.0105 |            0 | 0.04%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       31440 |       3.2475 |     3.2456 |      0.0489 |    0.0392 |            0 | 0.06%        |     3.2478 |      0.0128 |    0.01   |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       31806 |       3.2475 |     3.247  |      0.0486 |    0.037  |            0 | 0.01%        |     3.2474 |      0.0127 |    0.01   |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       32176 |       3.2475 |     3.2428 |      0.0483 |    0.0397 |            0 | 0.15%        |     3.2473 |      0.0127 |    0.0107 |            0 | 0.00%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       32550 |       3.2475 |     3.2495 |      0.0481 |    0.0391 |            0 | 0.06%        |     3.2473 |      0.0126 |    0.0097 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       32929 |       3.2475 |     3.2485 |      0.0478 |    0.0393 |            0 | 0.03%        |     3.2477 |      0.0125 |    0.0095 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       33312 |       3.2475 |     3.2507 |      0.0476 |    0.0395 |            0 | 0.10%        |     3.2476 |      0.0124 |    0.0099 |            0 | 0.00%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       33700 |       3.2475 |     3.2436 |      0.0472 |    0.0373 |            0 | 0.12%        |     3.2466 |      0.0124 |    0.0094 |            0 | 0.03%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       34092 |       3.2475 |     3.2482 |      0.047  |    0.0377 |            0 | 0.02%        |     3.2468 |      0.0123 |    0.0098 |            0 | 0.02%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       34489 |       3.2475 |     3.2492 |      0.0467 |    0.0366 |            0 | 0.05%        |     3.2469 |      0.0122 |    0.0097 |            0 | 0.02%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       34891 |       3.2475 |     3.25   |      0.0465 |    0.0363 |            0 | 0.08%        |     3.2469 |      0.0121 |    0.0098 |            0 | 0.02%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       35297 |       3.2475 |     3.2479 |      0.0461 |    0.036  |            0 | 0.01%        |     3.2471 |      0.0121 |    0.0101 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       35707 |       3.2475 |     3.2496 |      0.0459 |    0.0357 |            0 | 0.06%        |     3.2476 |      0.012  |    0.0096 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       36123 |       3.2475 |     3.2496 |      0.0457 |    0.0373 |            0 | 0.07%        |     3.2475 |      0.0119 |    0.0095 |            0 | 0.00%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       36543 |       3.2475 |     3.2471 |      0.0454 |    0.0355 |            0 | 0.01%        |     3.2471 |      0.0119 |    0.0095 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       36969 |       3.2475 |     3.2469 |      0.0451 |    0.0356 |            0 | 0.02%        |     3.247  |      0.0118 |    0.0093 |            0 | 0.02%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       37399 |       3.2475 |     3.249  |      0.0448 |    0.0358 |            0 | 0.05%        |     3.2471 |      0.0117 |    0.0093 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       37834 |       3.2475 |     3.2452 |      0.0446 |    0.0361 |            0 | 0.07%        |     3.2466 |      0.0117 |    0.0089 |            0 | 0.03%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       38274 |       3.2475 |     3.2502 |      0.0443 |    0.0327 |            0 | 0.09%        |     3.248  |      0.0116 |    0.0089 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       38720 |       3.2475 |     3.2488 |      0.0441 |    0.035  |            0 | 0.04%        |     3.2475 |      0.0115 |    0.0088 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       39171 |       3.2475 |     3.2437 |      0.0438 |    0.0336 |            0 | 0.12%        |     3.2478 |      0.0115 |    0.0089 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       39626 |       3.2475 |     3.251  |      0.0436 |    0.035  |            0 | 0.11%        |     3.2477 |      0.0114 |    0.0089 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       40088 |       3.2475 |     3.2482 |      0.0433 |    0.032  |            0 | 0.02%        |     3.2473 |      0.0113 |    0.0088 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       40554 |       3.2475 |     3.2485 |      0.0431 |    0.0351 |            0 | 0.03%        |     3.2472 |      0.0113 |    0.0095 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       41026 |       3.2475 |     3.2473 |      0.0428 |    0.0354 |            0 | 0.01%        |     3.248  |      0.0112 |    0.0089 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       41504 |       3.2475 |     3.248  |      0.0426 |    0.0346 |            0 | 0.02%        |     3.2487 |      0.0111 |    0.009  |            0 | 0.04%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       41987 |       3.2475 |     3.2446 |      0.0423 |    0.0297 |            0 | 0.09%        |     3.2467 |      0.0111 |    0.0091 |            0 | 0.03%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       42475 |       3.2475 |     3.2509 |      0.0421 |    0.0335 |            0 | 0.11%        |     3.2477 |      0.011  |    0.0087 |            0 | 0.01%        | 14.63x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       42970 |       3.2475 |     3.2466 |      0.0418 |    0.0333 |            0 | 0.03%        |     3.2473 |      0.0109 |    0.0085 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       43470 |       3.2475 |     3.2448 |      0.0416 |    0.0327 |            0 | 0.08%        |     3.247  |      0.0109 |    0.0086 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       43976 |       3.2475 |     3.25   |      0.0414 |    0.0331 |            0 | 0.08%        |     3.2479 |      0.0108 |    0.0081 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       44487 |       3.2475 |     3.2447 |      0.0411 |    0.0312 |            0 | 0.09%        |     3.247  |      0.0108 |    0.0086 |            0 | 0.01%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       45005 |       3.2475 |     3.2473 |      0.0409 |    0.0326 |            0 | 0.00%        |     3.2475 |      0.0107 |    0.0087 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       45529 |       3.2475 |     3.2477 |      0.0406 |    0.0321 |            0 | 0.01%        |     3.2477 |      0.0106 |    0.0086 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       46059 |       3.2475 |     3.2506 |      0.0404 |    0.0317 |            0 | 0.09%        |     3.2478 |      0.0106 |    0.0088 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       46595 |       3.2475 |     3.2481 |      0.0402 |    0.0313 |            0 | 0.02%        |     3.247  |      0.0105 |    0.0086 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       47137 |       3.2475 |     3.2493 |      0.0399 |    0.033  |            0 | 0.06%        |     3.2475 |      0.0105 |    0.0084 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       47686 |       3.2475 |     3.2461 |      0.0397 |    0.0303 |            0 | 0.04%        |     3.2478 |      0.0104 |    0.008  |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       48241 |       3.2475 |     3.2494 |      0.0395 |    0.0317 |            0 | 0.06%        |     3.2479 |      0.0103 |    0.0084 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       48802 |       3.2475 |     3.2466 |      0.0392 |    0.0305 |            0 | 0.03%        |     3.2477 |      0.0103 |    0.0082 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       49370 |       3.2475 |     3.2482 |      0.039  |    0.0301 |            0 | 0.02%        |     3.2474 |      0.0102 |    0.0079 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       49945 |       3.2475 |     3.248  |      0.0388 |    0.0303 |            0 | 0.01%        |     3.2475 |      0.0102 |    0.0079 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       50526 |       3.2475 |     3.2455 |      0.0386 |    0.0315 |            0 | 0.06%        |     3.2478 |      0.0101 |    0.0084 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       51114 |       3.2475 |     3.2481 |      0.0383 |    0.0327 |            0 | 0.02%        |     3.2476 |      0.01   |    0.0086 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       51709 |       3.2475 |     3.2468 |      0.0381 |    0.0301 |            0 | 0.02%        |     3.2476 |      0.01   |    0.0083 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       52310 |       3.2475 |     3.2487 |      0.0379 |    0.0295 |            0 | 0.04%        |     3.2474 |      0.0099 |    0.0079 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       52919 |       3.2475 |     3.2503 |      0.0377 |    0.0295 |            0 | 0.09%        |     3.248  |      0.0099 |    0.0076 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       53535 |       3.2475 |     3.2456 |      0.0375 |    0.0305 |            0 | 0.06%        |     3.2464 |      0.0098 |    0.0078 |            0 | 0.03%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       54158 |       3.2475 |     3.248  |      0.0373 |    0.0287 |            0 | 0.02%        |     3.2474 |      0.0098 |    0.0076 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       54789 |       3.2475 |     3.2488 |      0.0371 |    0.0284 |            0 | 0.04%        |     3.2474 |      0.0097 |    0.0081 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       55426 |       3.2475 |     3.2453 |      0.0368 |    0.0285 |            0 | 0.07%        |     3.247  |      0.0096 |    0.0074 |            0 | 0.02%        | 14.58x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       56071 |       3.2475 |     3.247  |      0.0366 |    0.0288 |            0 | 0.01%        |     3.2479 |      0.0096 |    0.0078 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       56724 |       3.2475 |     3.2465 |      0.0364 |    0.029  |            0 | 0.03%        |     3.2475 |      0.0095 |    0.0076 |            0 | 0.00%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       57384 |       3.2475 |     3.2476 |      0.0362 |    0.0289 |            0 | 0.00%        |     3.2474 |      0.0095 |    0.0074 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       58052 |       3.2475 |     3.2479 |      0.036  |    0.0293 |            0 | 0.01%        |     3.2479 |      0.0094 |    0.0077 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       58727 |       3.2475 |     3.2481 |      0.0358 |    0.0287 |            0 | 0.02%        |     3.2482 |      0.0094 |    0.0075 |            0 | 0.02%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       59411 |       3.2475 |     3.2475 |      0.0356 |    0.0292 |            0 | 0.00%        |     3.2477 |      0.0093 |    0.0069 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       60102 |       3.2475 |     3.2475 |      0.0354 |    0.0287 |            0 | 0.00%        |     3.2476 |      0.0093 |    0.008  |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       60802 |       3.2475 |     3.2472 |      0.0352 |    0.027  |            0 | 0.01%        |     3.2478 |      0.0092 |    0.0075 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       61509 |       3.2475 |     3.246  |      0.035  |    0.0274 |            0 | 0.04%        |     3.2479 |      0.0092 |    0.0078 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       62225 |       3.2475 |     3.2481 |      0.0348 |    0.0275 |            0 | 0.02%        |     3.248  |      0.0091 |    0.0075 |            0 | 0.02%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       62949 |       3.2475 |     3.2456 |      0.0346 |    0.0294 |            0 | 0.06%        |     3.2474 |      0.009  |    0.0067 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       63682 |       3.2475 |     3.248  |      0.0344 |    0.0256 |            0 | 0.02%        |     3.2474 |      0.009  |    0.0073 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       64423 |       3.2475 |     3.2471 |      0.0342 |    0.027  |            0 | 0.01%        |     3.2476 |      0.0089 |    0.0073 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       65173 |       3.2475 |     3.2489 |      0.034  |    0.0283 |            0 | 0.04%        |     3.2474 |      0.0089 |    0.0071 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       65931 |       3.2475 |     3.2468 |      0.0338 |    0.0279 |            0 | 0.02%        |     3.2477 |      0.0088 |    0.0071 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       66699 |       3.2475 |     3.2483 |      0.0336 |    0.0268 |            0 | 0.02%        |     3.2473 |      0.0088 |    0.007  |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       67475 |       3.2475 |     3.2479 |      0.0334 |    0.0251 |            0 | 0.01%        |     3.2473 |      0.0087 |    0.0065 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       68260 |       3.2475 |     3.2482 |      0.0332 |    0.0277 |            0 | 0.02%        |     3.2476 |      0.0087 |    0.007  |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       69055 |       3.2475 |     3.2475 |      0.033  |    0.0258 |            0 | 0.00%        |     3.2473 |      0.0086 |    0.0073 |            0 | 0.00%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       69858 |       3.2475 |     3.2484 |      0.0328 |    0.026  |            0 | 0.03%        |     3.2472 |      0.0086 |    0.0073 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       70671 |       3.2475 |     3.248  |      0.0326 |    0.0263 |            0 | 0.02%        |     3.2467 |      0.0085 |    0.0069 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       71494 |       3.2475 |     3.2458 |      0.0324 |    0.0256 |            0 | 0.05%        |     3.2472 |      0.0085 |    0.0067 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       72326 |       3.2475 |     3.2473 |      0.0322 |    0.0265 |            0 | 0.00%        |     3.248  |      0.0084 |    0.0067 |            0 | 0.02%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       73168 |       3.2475 |     3.2513 |      0.0321 |    0.0254 |            0 | 0.12%        |     3.248  |      0.0084 |    0.0064 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       74019 |       3.2475 |     3.2474 |      0.0319 |    0.0241 |            0 | 0.00%        |     3.2479 |      0.0083 |    0.0065 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       74881 |       3.2475 |     3.2478 |      0.0317 |    0.0254 |            0 | 0.01%        |     3.2472 |      0.0083 |    0.0066 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       75752 |       3.2475 |     3.2486 |      0.0315 |    0.0262 |            0 | 0.03%        |     3.2474 |      0.0082 |    0.0069 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       76634 |       3.2475 |     3.248  |      0.0313 |    0.0252 |            0 | 0.02%        |     3.2467 |      0.0082 |    0.0064 |            0 | 0.02%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       77525 |       3.2475 |     3.2472 |      0.0311 |    0.0246 |            0 | 0.01%        |     3.2483 |      0.0082 |    0.0068 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       78428 |       3.2475 |     3.2462 |      0.031  |    0.0241 |            0 | 0.04%        |     3.2472 |      0.0081 |    0.0065 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       79340 |       3.2475 |     3.2461 |      0.0308 |    0.0255 |            0 | 0.04%        |     3.2471 |      0.0081 |    0.0057 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       80264 |       3.2475 |     3.2482 |      0.0306 |    0.0231 |            0 | 0.02%        |     3.2472 |      0.008  |    0.0061 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       81198 |       3.2475 |     3.2484 |      0.0304 |    0.0236 |            0 | 0.03%        |     3.248  |      0.008  |    0.0063 |            0 | 0.02%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       82143 |       3.2475 |     3.2467 |      0.0303 |    0.0245 |            0 | 0.03%        |     3.2474 |      0.0079 |    0.0063 |            0 | 0.00%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       83099 |       3.2475 |     3.2439 |      0.0301 |    0.0226 |            0 | 0.11%        |     3.2471 |      0.0079 |    0.0063 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       84066 |       3.2475 |     3.2495 |      0.0299 |    0.0227 |            0 | 0.06%        |     3.2476 |      0.0078 |    0.0064 |            0 | 0.00%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       85044 |       3.2475 |     3.2459 |      0.0297 |    0.0234 |            0 | 0.05%        |     3.2479 |      0.0078 |    0.0064 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       86034 |       3.2475 |     3.2467 |      0.0296 |    0.0236 |            0 | 0.02%        |     3.2469 |      0.0077 |    0.006  |            0 | 0.02%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       87035 |       3.2475 |     3.2479 |      0.0294 |    0.023  |            0 | 0.01%        |     3.2476 |      0.0077 |    0.0059 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       88048 |       3.2475 |     3.2462 |      0.0292 |    0.0243 |            0 | 0.04%        |     3.2473 |      0.0076 |    0.0066 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       89073 |       3.2475 |     3.2483 |      0.0291 |    0.0241 |            0 | 0.03%        |     3.247  |      0.0076 |    0.0062 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       90110 |       3.2475 |     3.246  |      0.0289 |    0.0229 |            0 | 0.05%        |     3.2471 |      0.0076 |    0.0059 |            0 | 0.01%        | 14.59x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       91158 |       3.2475 |     3.2473 |      0.0287 |    0.0221 |            0 | 0.01%        |     3.2477 |      0.0075 |    0.006  |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       92219 |       3.2475 |     3.2494 |      0.0286 |    0.0222 |            0 | 0.06%        |     3.2471 |      0.0075 |    0.0057 |            0 | 0.01%        | 14.62x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       93293 |       3.2475 |     3.2486 |      0.0284 |    0.0226 |            0 | 0.03%        |     3.2483 |      0.0074 |    0.0057 |            0 | 0.03%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       94378 |       3.2475 |     3.2467 |      0.0282 |    0.0219 |            0 | 0.02%        |     3.2477 |      0.0074 |    0.006  |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       95477 |       3.2475 |     3.2484 |      0.0281 |    0.0229 |            0 | 0.03%        |     3.2474 |      0.0073 |    0.0059 |            0 | 0.00%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       96588 |       3.2475 |     3.2462 |      0.0279 |    0.0239 |            0 | 0.04%        |     3.2478 |      0.0073 |    0.0057 |            0 | 0.01%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       97712 |       3.2475 |     3.2463 |      0.0277 |    0.0226 |            0 | 0.04%        |     3.248  |      0.0073 |    0.0058 |            0 | 0.02%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|       98849 |       3.2475 |     3.2486 |      0.0276 |    0.0213 |            0 | 0.03%        |     3.2477 |      0.0072 |    0.0057 |            0 | 0.01%        | 14.61x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+
|      100000 |       3.2475 |     3.248  |      0.0274 |    0.0205 |            0 | 0.02%        |     3.2473 |      0.0072 |    0.0059 |            0 | 0.00%        | 14.60x    |
+-------------+--------------+------------+-------------+-----------+--------------+--------------+------------+-------------+-----------+--------------+--------------+-----------+

Average Variance Reduction: 14.605x

Citations

P. Glasserman, Monte Carlo Methods in Financial Engineering, vol. 53. New York, NY: Springer New York, 2003. doi: 10.1007/978-0-387-21617-1.