# Tag Info

24

Yes, there is such a rule and it is not too hard to grasp. Consider the 3-element correlation matrix $$\left(\begin{matrix} 1 & r & \rho \\ r & 1 & c \\ \rho & c & 1 \end{matrix}\right)$$ which must be positive semidefinite. In simpler terms, that means all its eigenvalues must be nonnegative. Assuming that $\rho$ and $r$...

12

Well, actually it's designed to use whatever discretization you throw at it---but for the time being only Euler discretization is implemented, mostly for lack of time or interest on the part of contributors. If you want to use exact numerics with a process, you can just code the corresponding discretization class (you'll have to inherit from ...

12

There was a proxy called the ECU. You should be able to use the weights on the Wikipedia page to get a time series back to 1979. Alternatively, the St. Louis FRED also provides this time series.

11

Centroidal Voronoi methods you mean? i.e. approximating a continous space with discrete points (generators) and for the sake of modeling evaluate the neighborhood around each generator as having the same value? Example. Here is a guy who encodes images with unicode in twitter. He is quantizing in both the spacial and color spaces. http://www.flickr.com/...

9

This is in fact a tricky matter. As you say one way is to calculate delta by an analytic formula, i.e. calculate the first derivative of the option pricing formula you are using with respect to the underlying's spot price. The second way is to do it numerically, i.e. change the spot price by a small value $dS$, calculate the value of the option and then ...

9

Except in highly unusual cases, financial PDEs lack analytic solutions. The mathematical tools used are Monte Carlo, plus the usual ones for solving PDEs on grids, almost always one of the following: Trees, for very simple cases Explicit finite differencing, for throwaway projects or very specific cases Implicit or Crank-Nicolson finite differencing for ...

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 jump-...

8

Not so fast! I think it is of the utmost importance to first examine whether the data points are real outliers, i.e. noise that is contaminating the data, or perhaps the most important pieces of the time series! For example when you look at US stock market data of the last 50 years and remove only the ten biggest moves because they are outliers you get a ...

7

You may want to look into these two open source projects: QuantLib which is aimed at providing a comprehensive software framework for quantitative finance. This is written in C++. JQuantLib the 100% Java implementation based on the first project.

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

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

Fastest method is a pre-generated lookup table with carefully selected in-memory structure so you don't get too many CPU cache misses (avoiding the memory latency). If you want an absolute speed, you also can go for a hardware specific implementation (GPU, FPGA).

5

As far as PDEs (deterministic) are concerned we have the notion of a "strong solution" (directly solving the differential operator in the strong formulation of the problem) and the "weak solution" that deals with a weak formulation of the problem. For the strong formulation, finite differences are the way to go since they are the natural discretization of ...

5

FDMs represent PDEs over a simple grid shape; the different implementations are just different recurrence relations to approximate the solutions to the PDE between boundary values (e.g., for options pricing, $T=[t_\mathrm{now},t_\mathrm{maturity}]$ and $S=[\mathrm{deep\_itm},\mathrm{deep\_otm}])$. FEM is a general name for a lot of different ...

5

Cubature (of a given order) is a general method that allows you to do some approximate integration by being exact on a subset of integrand. If you are given a measure $M$ over for example $\mathbb R^n$ then will approach $M$ by (typically) a discrete measure $M^d=\sum_{i=1}^m \lambda_i\delta(x_i)$ such that polynomials $P$ of degree less or equal to $\gamma$...

5

Here is a website devoted to optimal quantization methods for numerical probability and mathematical finance specifically: http://www.quantize.maths-fi.com You can find a wide bibliography on the subject on the web site, as well as a database of pre-computed quantization grids.

5

When you decide if the performance improvement is worth it you can add these to the downside ow using single precision: the result of your basic B-S pricer will eventually need to be multiplied with a notional and maybe a discount factor; For a sufficiently large notional you will see different results than the one calculated using double precision. Is ...

5

By definition the fair value of an option is given by an expectation value of the payoff, $\mathbf{E}\left[\textrm{payoff}(\textit{paths})\right]$. The probability distribution of the paths is the risk neutral measure. This is just an integral expression of the form you wrote. This applies to all option prices. Many options are, of course, special in the ...

4

As far as I know, differential equations such as the Black-Scholes PDE are solved once analytically and then the result is used directly. If a given derivatives-pricing differential equation could not be solved analytically, it would probably be better to model it numerically using Monte Carlo methods than to derive a complicated PDE which must then be ...

4

C is not used for any particular reason in numerical optimizations other than for legacy reasons. However, there are areas where C is preferred over C++ though even C is not the preferred language of choice. To mind comes programming FPGAs. Though VHDL and Verilog are by far the standards. But "behavioral synthesis" allows to utilize C or C relatives such as ...

4

who told you that ? I am used to create new trade systems in C++ to make the customers requirements feasible. CERN used C++ to prove higgs boson particle. I see people using C to program embedded like microwaves or fridges :D but it is just my opnion, I would like to hear others.

4

Working on trigonometric polynomial decomposition, the first step is to take a big look at Fourier transformation. It is very powerfull, well documented and probably well implemented on your favorite language. It will give you the decomposition of your time series. You can remove highest frequencies, which correspond to noise, to have a good estimation.

4

Here are a few more papers about MCMC and alike methods for derivative pricing and co. : Blanchet-Scalliet, Patras - Counterparty risk valuation for CDS Jasra, Del Moral - Sequential Monte Carlo Methods for Option Pricing Frey, Schmidt - Filtering and Incomplete Information in Credit Risk Peters, Briers, Shevchenko, Doucet - Calibration and Filtering for ...

4

The method described in Hallerbach (2004) always worked well for me. We derive an estimator for Black-Scholes-Merton implied volatility that, when compared to the familiar Corrado & Miller [JBaF, 1996] estimator, has substantially higher approximation accuracy and extends over a wider region of moneyness.

3

It could be much more simple: if you use the method of moments (MM) then you estimate the mean and the variance and for example the kurtosis of your sample. Then you fit the parameters to these statistics. Alternatively you use maximum-likelihood (MLE). For MM: from wikipedia you get the mean and the variance. In your notation you can fit $b = \bar{r}$ so $... 3 Building upon +Imorin answer, you should have a look specifically at discrete cosine transforms. It's a standard approach when trying to express finite sequences as a sum of cosines. I would start from there, especially as it's implemented in every common language (R, Matlab, Python for starters). Only then evaluate if you need more. 3 The Papageorgiou paper is presumably referring specifically to quasi-random sequences used in path generation. Researchers had noticed that, in high dimensions, QR sequences tend to have good space coverage for the first couple of dimensions: but terrible coverage for the latter dimensions: (Plots here are points 101-200 from a 32-dimensional QR ... 3 There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable ... 3 I am not sure if I understood your question correctly but I will try to answer it anyway. If you have a standard normal random vector$z \sim N(\mathbb{0},I_n)$(where$z,0 \in \mathbb{R}^{n\times1}$and$I_n \in \mathbb{R}^{n\times n}$is the identity matrix) and you want to transform it into a multivariate normal$x \sim N(\mu,\Sigma)$you do it the ... 3 You might want to set$a= \epsilon - d$and write$\epsilon>0$as a constraint. I guess$\textbf{lsqnonlin}$is the suitable fonction for what you intend to do. I personnally like to use and play around with$\textbf{fmincon}\$, which allows more flexibility and performs well, if you are willing to provide Jacobian and/or Hessian in algorithms options

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