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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 ...


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 ...


4

In general these are the two basic approaches to QuantFinance: Sell side (market maker, risk neutral): You use risk-neutral probabilities ("$\mathbb{Q}$") e.g. in option pricing (to e.g. calculate your greeks and hedge your portfolio), so that you live on the spread. Buy side (market/risk taker): You use real-world probabilites ("$\mathbb{P}$") for e.g. ...


3

EQ1 is uni-variate case. EQ2 is multivariate case, in which you have to use correlated $X_t$. His way of doing is making $Y_t$ independent so that you can simulate freely. He does so by finding PC on $\Delta$. Alternatively, you could generate correlated $X_t$ in your simulation. To benchmark your model / code, you should first test and reproduce a given ...


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

The Black-Scholes price of this option is approximately $14.8$. When I run a Monte Carlo simulation with $10000$ paths and "exact" time stepping, I get results very close to this value. You are simulating the terminal asset price with the first-order Euler approximation over multiple time steps: $$S(t+\Delta t)= S(t) + rS(t)\Delta t + \sigma ...


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 ...


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 ...


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

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

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 ...


2

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 ...


2

You have the correct approach. (1) The simulation generates sampled portfolio values, $P_1,P_2, \dots, P_n$ at time $t=T$. VaR is specified as a left-tail percentile. Order the sample as $$P_{(1)} \leq P_{(2)} \leq \dots \leq P_{(n)}.$$ If you are considering $VaR_\alpha$ at the $100(1-\alpha) \% $ confidence level , then choose the smallest integer $k$ ...


2

You can find a brief but useful explanation of Brownian bridge techniques in Andersen and Piterbarg (page 125), which includes references for further reading. It's probably the best place to start. They discuss valuing barrier options specifically, and discuss the performance issues mentioned here. Later (pg 647), they use Brownian bridges in constructing ...


2

To my knowledge the real world drift plays a crucial role in risk management. The reason being that one is not interested in risk adjusted paths but in real-world scenarios that might actually occure. Still you should be aware that "the real world drift" is a somewhat controversial topic in quant circles. Nobody knows exactly how to get it. Mostly you end ...


1

This depends on your method to generate the normal random numbers. The problem with normal cdf is that the direct inverse $\phi^{-1}(Z)$ is hard to solve for directly. There are some other methods to generate $N(\mu, \sigma^2)$ from $U(0,1)$. Two notable methods are: Box-Muller method Marsaglia polar method For most purposes you can use the above methods ...


1

I agree with @encor that it isn't an issue to include some logic to avoid errors. I imagine that most non-uniform random number generators already include that. I don't think I've used a pseudo-random number generator that's given a 0. I'm not an expert on the topic, but a common implementation (linear congruential generator) relies on modular arithmetic. ...


1

Indeed for computational purposes, best you can do is use a uniform distribution on another interval $[10^{-10},1-10^{-10}]$, or just discard all occurences of $Z=0,1$. Discarding $Z=0,1$ is justified, since for continuous distributions $P(Z=0)=P(Z=1)=0$.


1

You can also include variance reduction techniques in you monte carlo simulations, such as control variates or antithetic variates. Both aim at reducing the variability of your simulated option price and are very popular for monte carlo simulations. http://en.wikipedia.org/wiki/Antithetic_variates http://en.wikipedia.org/wiki/Control_variates Both are ...


1

If you mean by fat tails just fatter tails than the gaussian distribtuion, i.e. a distribution with finite variance, for instance the Student's t-distribution has fatter tails than the normal distribution. If you mean distributions with infinite variance, you have to have a look at Lévy distribution. In a first attempt you could just substitute the standard ...


1

1) Brownian Bridge is used in Quasi Monte Carlo pricing of asian options to reexpress paths in a basis where few selected components/subspaces bring the most contribution, so as to align these to the best distributed dimensions/subspaces of a low discrepancy sequence. This allows for better coverage and thus faster convergence and paths amount reduction. ...


1

American options pricing (swaption is just a kind of option) is a bit tricky due to the early exercise. Here is a page listing possible approaches, including some numeric methods, and some close form approximation formula. As I understand, lattice methods (tree, PDE discretization such as forward shooting) are fine to price American options. There're ...


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 .



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