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3

There are lots of papers online and here are a few I would suggest math.umn riskworx G. Dimitroff, J. de Kock Nowak, Sibetz I you have matlab there is an step step example to calibrate SABR model. Since it uses the financial toolbox of matlab for a few functions I dont think you can replicate it in any other language. There must be C++ code available ...


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1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset $A$. You need to hold $A$ at time $T$ but since you don't need it now you don't want to buy it now. Instead you enter a forward contract with someone that says that at time $T$ you will pay the amount $K$ and get the asset in exchange. What ...


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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 ...


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About the integration problem: Your integrand is highly oscillatory, and the adaptive quadrature of Matlab doesn't handle such integrands very well. In general, I would recommend Mathematica when Matlab's standard procedures don't perform well. In this case, a Levin-type method would perform much better. The reason that quadv produces NaN values is because ...


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Well if you think that this model represents reality more accurately than the Black-Scholes assumptions. A lot of people do indeed think so. But I wouldn't say you're "tweaking" Black-Scholes... you're just assuming another model altogether and you will use risk-neutral pricing to compute the fair value of the option at time $t$, just like BS. Frankly, I'm ...


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It is a Wiener integral as your integrand is a deterministic function of time. It is known that the Wiener integral is stationary gaussian process with independent increments. So $z(t) \sim \mathcal N\left(0, \int_0^te^{-2k(t-s) }~ds\right)$ and $(z(t)-z(s)) \amalg z(u), \ \forall u,s,t \in \mathbb R_+ \text{ such that }u\leq s, s\leq t $ or alternatively ...


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CRR is just a numerical approximation to Black--Scholes. Its main use is in getting American option price. There is no real difference other than slight inaccuracy when using it for Europeans. So no it wouldn't do what you ask. Your questions are philosophical. What is the purpose of the model? if you estimate the volatility from a time series then you can ...


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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 ...


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Historical returns are not to be used 'untreated' for the calculation of option prices. The expectation that you will be using in Monte Carlo will take the form $$ C(K,T) = E^Q\{D(T)\ \max[0, S_T-K, 0]\} $$ where $T$ is the maturity, $K$ is the strike price, $S$ is the stock price and $D$ is the discount factor. But the expectation is taken under the 'risk ...


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I think you need to ask your question differently to get better answers than mine. Your Black Scholes part has two problems. First positive infinity should be negative infinity. Second, you are assuming zero dividends in Black Scholes but you are assuming a possibly positive div yield q in the CEV part. If the div yield q is sufficiently positive in the ...


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Not sure your question is about having a process for covariance or to have multivariate GARCH. The standard viewpoint on a stochastic volatility for covariance is to use a Whishart process. See for instance Philipov, A. and M. E. Glickman (2006, July) Multivariate stochastic volatility via wishart processes. Journal of Business & Economic Statistics 24 ...


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Actually BS model is still applicable in the market where the upwards/downwards move is much more probable than move in the opposite direction. The Black-Scholes price process model has the form: $\frac{dS}{S} = \mu dt + \sigma dW$ And with significantly non-zero $\mu$ (called drift) it will capture just what you are talking about. Quite surprisingly, the ...


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Look at Gilli & Schumann's paper. They provide a Bates' model estimates set, the way to improve such estimates calibrating those ones using an Heuristic model and, lastly, the relative codes in matlab, in order to be able to replicate the model. Unfortunately, there are not available the relative call prices estimated time series; I think that noone ...


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I think you're looking for multivariate GARCH models of which this is an overview paper. Multivariate GARCH models have one big drawback: they are pretty hard to estimate due to the number of correlations. This paper by Caporin and McAleer might be of interest in that regard.


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alpha + beta < 1 is the stationary condition for GARCH. If alpha and beta are low that means volatility of the stock does not have clustering behaviors. I think you can have a look at ADF and PACF of Return^2 time series first. If the first order autocorrelation is very significant but alpha is not, then perhaps you can check on the parameter calibration. ...


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The Feller condition applies without modification. That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is strictly positive and initial level $v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ...


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Show that the discounted expectation price of the new security is the same as the solution of the PDE. Once this is done all three assets have discounted price processes which are martingales so there can be no arbitrage.


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I cannot guarantee that it is error-free, but this paper (appendix A) has a relatively straightforward derivation of the Heston price for a european call.


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You might want to have a look at the stochvol vignette (http://cran.r-project.org/web/packages/stochvol/vignettes/article.pdf), where this process is described in detail in Algorithm 1. In particular, if I understand you correctly, what you need is step 4b. Now to your code: 1) It's not really a rolling forecast, because you estimate the model only once. ...


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I would suggest that you use a more 'modern' method to recover option prices from characteristic functions. The approach of this papers (for practical calculations of option prices) is somewhat outdated. The backbone of affine models (such as SVJJ) is the characteristic function $\psi(u)$ of the log-price distribution, which is known in closed form. The ...


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GBM is defined as $$ S_t = S_{t-1}\exp\left( \left(\mu - \frac{\sigma^2}{2} \right)dt + \sigma dW_t\right)$$ So, in your notation, assuming your daily parameters: $$ S_{new} = S_{previous}\cdot\exp\left( \left({drift} - \frac{{volatility}^2}{2} \right)days + volatility \,\sqrt{days}\,N(0,1)\right)$$ So your formula was incorrect. The youtube you quote is ...


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To calibrate BS you compute volatility $\sigma$, to calibrate SABR you compute implied $\alpha$, the volvol and $\beta$, the skewness. These parameters does not play the same role. So you can't really use the parameters of one models to calibrate another. But you can build equivalent parmaters, i.e. compute an equivalent vol under SABR to use BS pricing ...



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