# Tag Info

7

Quasi Random Numbers are more tricky than it might seem, using them as a black box like with PRNGs is risky. E.g. an unscrambled Sobol' sequence is uniform only asymptotically, while for realistic sample sizes there are subvolumes with significantly different densities. You often do not realize that because the convergence graph looks good anyway, it gives ...

5

Apart from numerical stability errors, Cholesky and PCA (without dim reduction) shall produce exactly the same distribution, they are two symmetric decomposition of the same covariance matrix and thus are equivalent for transforming a standard normal vector. Of course when doing different things with PCA components, such as in dim reduction or quasi Monte ...

4

Glassermann et al. have published an approach where the loss distribution is approximated by a quadratic function in the risk factors. Based on this estimation they can apply importance sampling and stratified sampling to reduce the variance of the monte carlo estimate. I have not implemented their technique, but their numerical results look very good. You ...

4

For a vanilla option, this is a very slow way to get the boundary, and it's somewhat unreliable for any option. In either a more standard grid scheme or in a LS solver, you obtain the boundary by finding two nodes such that one of them has option value equal to early exercise value, and its neighbor has option value above early exercise value. This gives ...

3

I would argue (this is also what Quartz already hinted at) that PRNGs are far easier to set up than a well functioning QMC and are thus generally user-friendlier Excel and R both offer a PRNG. (but not a QMC) Thus someone working with these software will be more likely to use a PRNG than to painstakingly implement a QMC. Also as Quartz explained one needs ...

3

Note: In your text you refer to the stock prices and using a normal inverse gaussian. This would correspond to a normal model. However the formula you write suggests a log-normal (Black-Scholes) like model (not sure what X is), i.e. using a normal inverse gaussian for the stock returns. For Excel: The spreadsheet at ...

3

In binomial tree models, there is no such a thing as a path. The binomial tree represents information about the distribution of the zero-curve at a given time and preserve enough information between different times to let you compute conditional expectations. Generally, you can not price path-dependant instruments in a model based on trees—because there is ...

3

Glassermans's book suggests the stock price as the default idea for a control variate. In this paper, "Efficient, almost exact simulation of the Heston stochastic volatility model", by Haastrecht and Pelsser, (2008) the authors claim that this approach also works well for the Heston model (see appendix A2). The book is very approachable and available ...

3

In fact you can calibrate $\theta(t)$ piecewise constant and $\alpha$ and $\sigma$ to bond prices only. You don't need the swaption prices in mM. If you let $\sigma(t)$ depend on $t$ (this is called the generalized Hull-White model) then you need information about the options market. For the model as you write it you don't necessarily need MC to calculate ...

2

One of the main things you give up is a simple halting condition for your estimation algorithm. With pseudorandom numbers, the algorithm can keep track of the standard error, and stop when it has passed a threshold: error_est = Inf n = 0 while not error_est < target_precision: n = n + 1 x = new_random_sample() samples.append( F(x) ) ...

2

Do $N$ MC simulations of $M$ samples, calculating your estimate of VaR for each one $\{\widehat{VaR}_i\}_{i=1}^N$ and you now have an IID sample! Take the sample (or unbiased) standard deviation for your estimate of VaR (this is probably what you mean by error) $SD(\widehat{VaR})=\sqrt{\frac{1}{N-1} \sum_{i=1}^N (\widehat{VaR}_i - \overline{VaR})^2}$ and of ...

2

There are a lot of methods for simulating such a process, the real problem here is to preserve positivity of the next simulated step as the Gaussian increment might result in negative value and then a non definite value for the next "square-root" step. An approach that might be suitable to your more general needs is the following where a ...

2

Generally speaking, if you have two or three sources of noise, you are still going to be much better off pricing American options on a lattice than via LSMC. Too often, LSMC becomes the refuge of academics lacking patience to learn proper lattice techniques. Now, you can frequently reduce the difficulty of pricing American options by considering the ...

1

The Papageorgiou paper is presumably referring specifically to quasi-random sequences used in path generation. Researchers had noticed that, in high dimensions, QR sequences tend to have good space coverage for the first couple of dimensions: but terrible coverage for the latter dimensions: (Plots here are points 101-200 from a 32-dimensional QR ...

1

Regarding your second question: one possible approach is to reduce the instrument you are trying to value to something simpler, for which an analytical solution are an alternative methodology does exist. You can then vary parameters and check that the valuation is behaving as expected. If you are using simulations because your price process is more ...

1

You can use the either, as both necessarily are symmetric positive definite; covariance is a personal preference. It's really just a matter of scaling, as $\mathcal{N}(0,\Sigma)$ is distributionally $\sqrt{\Sigma} \mathcal{N}(0,1)$. Correlation would require additional scaling (i.e. multiplication of every $\mathcal{N}(0,\rho)$ element by its respective ...

1

Let's say your return realization for path $i$ is $r_i = \beta\cdot f_i$, where $f_i=(f_{1i}, f_{2i}, f_{3i})$ - factors realizations, and $\beta$ - factor coefficients. So, your VaR is $VaR=percentile(r_i,\alpha)$, where $\alpha$ - confidence. The simplest Monte Carlo stopping criterion is to keep adding paths $i$ and computing VaR on the growing sample ...

1

A very simple approach could be the following: draw a random number for each day for each stock. If you refer to "average/mean" by return and to "standard deviation/variance" by volatility, you could use these for the distribution parameters of the random numbers per stock. If you dislike that values can go below zero, apply Euler's exponential function on ...

1

I would define the weights $w_1,\ldots,w_n$ as whatever number you want and the basket given by $$B_t = \sum_{i=1}^n \frac{w_i}{W}S_t^{(i)}\ , \qquad W = \sum_{i=1}^nw_i$$ so the weights always sum to one. This doesn't make much sense, however, because you are changing the product, not a market variable. This meaning that when the weights change, the ...

1

Once the single-factor Hull-White model is calibrated, you can compute zero-coupon bond prices in closed form (i.e., without running simulations). See http://en.wikipedia.org/wiki/Hull%E2%80%93White_model#Analysis_of_the_one-factor_model .

1

When estimating covariance matrices, you run into problems as the number of assets/risk factors approaches or exceeds the number of observations. Some eigenvalues will go to zero, or be very small. This will mean that the covariance matrix is positive semi-definite instead of positive definite. Since the Cholesky decomposition requires a positive definite ...

1

The approach of reflecting is expensive, since the $d$-simplex has $d$ maximal faces, all of which have to be checked for intersection at each step. Additionally, if the random walk moves into a corner, the number of moves which have to be discarded can become very high. Depending on the configuration of the constraints this could well be your best solution. ...

1

Different optimizations could help. Parallel computing makes even worse if each computation is fast enough due to overhead. Thus it may be better to use profiler to get what can be improved. Usually it helps to send larger problems to parallel computation cores. Matlab is very good at matrix operations and it could be better to treat different draws of MC ...

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