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16

There are a wide variety of models (by which I mean the theoretical / mathematical formulation of how the underlying financial variable(s) of interest behave). The most popular ones differ depending on the asset class under consideration (though some are mathematically the same and named differently). Some examples are: Black-Scholes / Black / ...


16

Monte Carlo is most useful when you lack analytic tractability or when you have a highly multidimensional problem. For example, even using simple lognormal and poisson models, there exist path-dependent payoffs or multi-asset computations such that no analytic solution exists and such that any PDE finite difference solution would require 3 or more ...


11

The difference between the two is that the first will lead you to a discretization scheme of the process. So you will have to simulate a whole (approximate) trajectory of (meaning by that $X'_{t_0},...,X'_{t_n}$) up to time $T$ (the expiry of your vanilla option) to get to $X'_T$ which is then only an approximation of $X_T$. The second method is exact and ...


8

I believe this is a nice paper for you to start with. Check out what references it cited and who cited it. Markov Chain Monte Carlo Analysis of Option Pricing Models "Use the Markov Chain Monte Carlo (MCMC) method to investigate a large class of continuous-time option pricing models. These include: constant-volatility, stochastic volatility, price ...


8

The problem is that you are creating a new random number generator for each iteration. Move new MersenneTwister() out of the loop: MersenneTwister mtsign = new MersenneTwister(); MersenneTwister mt = new MersenneTwister(); for(int i = 0; i<= NumberOfTrials-1; i++ ) { // use mtsign and mt here ... } Furthermore, you don't need two generators, you ...


8

MCMC can be used for Bayesian inference of other models with hidden variables. Gibbs sampling, for example, is used in Hidden Markov Models. Here is a paper that discuss the differences between MCMC and the more classical approach using the EM algorithm. The question is: Are HMMs a useful model in finance? Some academics argue that they have predictive ...


8

As far as I know MCMC and also (PMCMC) can be usefull for (bayesian) estimation of parameters of some Hidden process like in the Heston Model case based on observations of the Stock (filtering). But the problem here is that those estimates are not matching those based on calibration of vanilla options of the Risk Neutral measure. So as an econometric tool it ...


8

In this scenario, the "joint dynamics" are trivially computed since the option value is a known deterministic function of stock price. For example, the mean of the option value for time $\tau$ is $$ \mu_O = \int_0^\infty BSM( S_\tau ) p(S_\tau) dS_\tau $$ which is best computed using quadrature as available in standard numerical libraries like scipy. The ...


8

You need to compute your greeks as finite differences, but the full procedure may be pretty tricky. I will use vega $\aleph$ as the example here. Let's begin by designating your Monte Carlo estimator as a function $V(\sigma,s,M)$ where $\sigma$ is the volatility as usual, $s$ is the seed to your random number generator, and $M$ is the sample count. To ...


7

The main component of that option premium is (forward-looking) volatility $\sigma$. The very simplest formula you could use for ATM options is the Bachelier model \begin{equation} \text{Call}_T = \sigma S \sqrt{\frac{T}{2\pi}} \end{equation} where the time to expiration is $T$ and $S$ is the current underlying price. This formula is "wrong" strictly ...


7

Okay just to wind things down here, I think an important clarification is needed if readers might come and seek to a similar solution. The Geometric Brownian Motion (GBM) is a model of asset prices dynamics which is usually given as follows: $$ dS_t = \mu S_t dt + \sigma S_t dB_t$$ where $B_t$ is a standard brownian motion which has several important ...


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 ...


6

Fundamentally this is no different from other simulation-based estimation---see this little experiment in R: R> set.seed(42) R> rowMeans(replicate(200,sapply(1:6, +> FUN=function(x) mean(rnorm(10^x)), simplify=TRUE))) [1] -2.47827e-02 -9.46800e-03 2.38226e-03 -1.08650e-03 9.41395e-05 1.06759e-05 R> We are calculating the mean of ...


6

You are typically interested in evaluating $E\left[ f(X_T)-f(\bar{X}_T^{(n)}) \right]$ (refered as the weak convergence) $X_t$ the solution of the sde : $dX_t^x=b(X_t^x)dt+\sigma(X_t^x)dW_t$ $\bar{X}_t^{(n)}=b(\underline{t},X_{\underline{t}}^{(n)})\cdot (t-\underline{t})+\sigma(\underline{t},X_{\underline{t}}^{(n)})\cdot (W_{\underline{t}}-W_t)$ is your ...


6

For such high-dimensional path problems you will want to use the Morokov technique (you can find the paper online), which takes QR samples for the "important" dimensions and then reverts to pseudorandom for the less important dimensions in an interest rate problem remarkably similar to yours. (Similar principles apply to using QR sequences in factor model ...


6

Check this document out: link to pdf file Also, if you are concerned with actual performance of your code and want to implement efficient code then gsl libraries would be the first place look at: link. It's got everything you need.


6

You have the right intuition but the approach is not quite right. The issuer has the right to call back the bond at a pre-defined call price. So your decision criterion is "call when the value of the bond >= contractual call price". We are comparing prices in the decision rule, not the YTM of the callable bond with the coupon of the bond. Note that ...


5

The best I have seen so far is William Wheaton's work in this area. I don't know how much is described in his papers but he and Torto created a system that combined factor models for things like local and national price indexes with specific economics of commercial real estate ventures (such as balloon payments on construction milestones and the like). The ...


5

The very easiest change you can make is to switch to quasirandom sampling. I favor the Niederreiter sequence, for which you can find implementations in most languages around the web. You can also get a (sometimes tremendous) speed boost by running using a control variate. Even a swap would probably reduce your variance somewhat. I don't recall the CIR ...


5

Since both $ER$ and $S$ are gaussian random, why not just assume their dependence is captured by their covariance, and make your draws from the bivariate normal distribution? It is hard to construct any other way of making two marginal gaussians cointegrated. Even if the variables were not gaussian, you would probably find yourself relating them using a ...


5

Tools from the field of stochastic optimization are best suited for these problems. In particular, attached is a paper on non-parametric density estimation for stochastic optimization that describes an algorithm if state variables can be associated with draws from the predictive distribution. Here's another approach by Kuhn. These are all one-period ...


5

In quantitative finance, we sometimes find ourselves choosing a new stochastic model for what market variables are random, and how. For example, someone might decide that they like the SDE \begin{equation} dS = \mu\ S\ dt + \left( \frac{S_0}{S} \right)^{\frac32} \sigma\ S\ dW \end{equation} because they want to capture a leverage effect. Now, this SDE ...


5

I would also look into pricing models based upon models other than lognormal (Black-Scholes). Do some research on "fat tailed" or stable distributions. There can also be known by their specific distribution names as Levy, Levy-Poisson, or Cauchy. http://en.wikipedia.org/wiki/Fat_tail


5

I'm guessing, and correct me if I'm wrong, you want to create a number of possible paths the stock price could follow with the local volatilty given by GARCH depending on the simulated history, or in pseudocode: N <- numberOfPaths T <- numberOfSteps for (i in 1:N) { newSeries <- pastPrices for (t in 1:T) { epsilon <- normrnd(0,1) ...


5

For completeness, let's restate that the discrete case goes like this: $$\Delta S_t = S_{t+\Delta t}- S_t = \mu S_t \Delta t + \sigma \sqrt{\Delta t} Z_t $$ with $Z_t \sim \mathcal{N}(0,1)$. What you are doing in your case (although there is a typo in your formula) is to use the exact solution of the SDE to model the move between two points of $S$. ...


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

Those people citing copulas are actually answering a different question, because they are leading you to a solution whose transformed distribution function has the requested correlation. You have two distributions $P_1$ and $P_2$. Let me begin by pointing out that this problem is not actually solvable in the general case. That's because either $P_1$ or ...


4

This is essentially the same question as your previous question and the issue is still the same: variability just does not go away just because you use 100 million draws once. Compare the distribution of results of $N$ Monte Carlo simulations at $n_1 = 1,000,000$ with those for $n_2 = 10,000,000$. You will see a reduction but that does not imply that every ...


4

I think, as with many machine learning approaches to investing decision support, it depends largely on the data. With a good selection of features, yes dynamic models like you're talking about will probably do better than a simple linear regression; but then again, with a good selection of features, linear regression will probably work reasonably well, too. ...


4

A probabilistic view on your full scale simulation. In the steps 1-3 you calculate the 0.99 quantile of the lognormal distribution with parameters $\ln N(\ln S_0 +(\mu - \frac{\sigma^2}{2})t,\sigma^2 t^2)$. The cdf of lognormal distribution is $\Phi(\frac{\ln x-\mu}{\sigma})$ Thus, you can calculate $V_p$ through $V_p=e^{\ln S_0 +(\mu - ...



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