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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 ...

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

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 ...

4

You have intense academic research on orderbook dynamics simulations, just cite: Econophysics: Empirical facts and agent-based models, by Anirban Chakraborti, Ioane Muni Toke, Marco Patriarca, Frederic Abergel (Arxiv 2010) High Frequency Simulations of an Order Book: a Two-Scales Approach by: Charles-Albert Lehalle, Olivier Guéant, Julien Razafinimanana, ...

4

You can use empirical distribution and use Mean-CVaR as a target function. CVar ("Expected shortfall") is considered a better risk metrics than VaR if we depart from the light-tailed normal distribution. The code below is in R and is taken from the book "Portfolio Optimization with R/Rmetrics" By Diethelm Wuertz, Yohan Chalabi, William Chen, Andrew Ellis. ...

4

The use of risk-neutral measure is based on the ability to arbitrage away the instantaneous risk of contingent claims. Although for forward contracts the hedge quantity is 1.0, in the general contingent claims case we must assume it varies instantaneously with the market state. The Girsanov Theorem tells us what the difference is, instantaneously, between ...

3

the risk neutral drift is needed for pricing of derivatives. For a $100\%$ equity portfolio you can take the real world drift - sometimes a good guess is a drift of zero. For fixed-income you could do the same and might need more sophistication for the variance term. If you have short-dated bonds then you will need a special model for the pull-to-par. For ...

3

Yes. The risk neutral and the real path share the same volatility, so the difference is in the drift rate, where the risk-neutral path drifts with the risk-free rate r. You may want to check out Paul Willmots book, esp. ch. 26, for applications.

3

Note: There is a typo in your third equations. Instead of $S(u)$ it should be $S(t_{i})$ and in place of $S(t)$ there should be $S(t_{i+1})$. In fact, given $S(t_{i})$ we have that $$S(t_{i+1}) = S(t_{i}) \exp\left( (\mu - \frac{1}{2} \sigma^2) (t_{i+1} - t_{i}) + \sigma (W(t_{i+1}) - W(t_{i})) \right)$$ is the exact solution of the SDE. Hence, the ...

3

You can use Michaud's Resampled Efficient Frontier as a technique, or Atillio Meucci's Entropy Pooling. In Michaud's approach you can sample returns with replacement for each of your assets. Based on these draws you can calculate the expectations, variances, and covariances for each simulation. You can then construct, say, a 1,000 efficient frontiers and ...

3

I highly recommend you to stick with the error function (RMSE) value minimization approach. I love MC techniques for this and related problem solving and thus do not recommend you to use anything else because of its simplicity and transparency. It comes down to using the right discretization function and to possibly implement variance reduction approaches. ...

3

Doesn't the Heston model have some Fourier transform formulae for pricing vanillas? I think one could use those to calibrate to the vanillas. Can't provide references at this moment, on the road. Edit: check out http://www.visixion.com/dok/Visixion_Calibrating_Heston.pdf -- I haven't read this closely but it sounds familiar

3

Look here for multivariate distribution on the positive quadrant ... quite difficult. http://xianblog.wordpress.com/tag/multivariate-analysis/ I have been thinking about this for weeks and months in the context of credit risk (modelling default intensities jointly) and I think this does not work.

3

Normally, one uses MC methods when: Analytical solutions do not exist PDE style solutions also don't work (they are usually still faster than MC) You need to price some exotic, but computation time does not matter (MC methods are easy(-ier) and fast to code-up) Note: Using MC is not free of assumptions: you always assume a distribution for the driving ...

3

To answer the more general question that seems to be giving you trouble, Ito's lemma is the stochastic version of the chain rule of standard calculus. What is it useful for? That's like asking what the chain rule is useful for. Calculus is useful in quantitative finance, and in particular, for stochastic processes, you need to use the stochastic version ...

2

Fourier Transform seems a good method for option pricing by take advantage of Fast Fourier Transform technique, such as the following paper written by Peter Carr and Dilip B. Madan: http://portal.tugraz.at/portal/page/portal/Files/i5060/files/staff/mueller/FinanzSeminar2012/CarrMadan_OptionValuationUsingtheFastFourierTransform_1999.pdf

2

I am going to supply an answer that is quite similar to SRKX's (which is very very good) because I want to discuss in more detail a few important things. First, you cannot use a stochastic volatility model for the SDE that you've provided as that's GBM with constant diffusion. However, based on what you've said it's obvious you wish to model a discretized ...

2

You can find the derivation of the Heston characteristic function (its Fourier Transform) in Gatheral (2006). Using the characteristic function, you can optimize the model on the prices. There are multiple approaches to optimize, among others pattern search (which is very slow) and stochastic optimization (randomly jump around and stop after n iterations), ...

2

Freddy has already answered it and my answer had an assumption in it so clarifying - If payoff of basket with underlined securities A,B and C are $$P_b = C_1*P_A + C_2*P_B + C_3*P_C$$ Where $$C_1 , C_2 ,C_3$$ are contants then portfolio delta is $$\delta_b = C_1*\delta_a+C_2*\delta_b+C_3*\delta_c$$ In short as Freddy Said , and I assumed if the ...

1

With respect to your first question: Yes. The regression has to determine the conditional expectation of the continuation value, i.e., the (discounted) value of the future cash flows including the exercise criteria(s) you have determined for the remaining future exercise times, conditional to the assumption that you did not exercise at or prior the current ...

1

Here's a decent study of calibration performance using fast fourier transforms versus other techniques. It concludes Gaussian quadrature works better than other techniques. http://www.frankfurt-school.de/dms/publications-cqf/CPQF_Arbeits6.pdf Edit: AZhu points out the link above is dead and that a working link is ...

1

All you need is to use the discretization to implement the MC approach. The following links should get you started: http://www.lcy.net/files/BDT_Seminar_Paper.pdf http://www-2.rotman.utoronto.ca/~hull/TechnicalNotes/TechnicalNote23.pdf ...

1

The method described in Brandt & Santa-Clara (2006) is somewhat different then that of the current answers and requires a bit more work but might be of interest nonetheless. Their ideas are further developed in a number of other papers. Their approach relies on sample moments of the long-horizon returns of the expanded set of assets in a ...

1

Monte Carlo generates random walk which is z ~N(0,1) which you dont want to use so that is out of the way . If you are looking at portfolio optimization where returns are non normal I also suggest that you have a look at this paper. if you want to look at CVar then you can look at this one

1

If we have some function $f(a,b,c,...)$, where $a,b,c,...$ can be stochastic or otherwise, then Ito's lemma is used to find $df(a,b,c,...)$. 1) You can simply do raw Monte Carlo. Consider a contingent claim maturing in $6$ months. Then for each $i$-th simulation you can calculate: $S(T)_i = S(t)e^{(r-q-\frac12 \sigma^2)0.5 + \sigma \sqrt{0.5}z_i)}$ ...

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