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10

There is another reason why Stoc Vol Models should be usually preferred to Local Vol Models, this reason is explained in the Hagan et al. paper "Managing Smile Risk" about SABR process and is in simple terms the fact that "smile dynamics" is poorly predicted by local vol models leading to bad Hedging of exotic options. Anyway Local Vol models have the good ...


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 ...


6

Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the ...


6

For pricing and hedging a portfolio of vanilla options, stochastic volatility is almost always preferable to local volatility since empirically it more accurately captures the evolution of the smile.


6

To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$ N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi $$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$ ...


5

Let $dS_t = \mu_tS_tdt + \sigma_tS_tdW_t$ be the underlying GBM (Geometric Brownian Motion)-like dynamics as in the question. Let $B_t$ a Brownian motion such that $d[B,W]_t = \rho dt$, $\rho\in[-1,1].$ CIR (Cox-Ingersoll-Ross) for $\sigma_t^2$ (when combined with GBM-like underlying dynamics, it is the popular Heston SV model) $$d\sigma_t^2 = ...


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) ...


4

You cannot derive the probability distribution you require, because for VaR you need a real-world probability distribution. From the options prices, it is only possible to obtain a risk-neutral distribution. Now, if you are willing to assume some kind of parametric relationship between the risk-neutral and real-world distributions, then you might find the ...


4

You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior. For this reason, the fact that intraday variance may, ahem, vary around ...


4

The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.


4

The model is similar to the Barndorff-Nielsen - Shephard model. But this model is much more general. On the other hand in this paper by Heston it is exactly your form that is used. Already Scott in 1987 considered a model of your form (see this) Finally in this thesis you find the names Hull-White model (of course there is the interest rate model too) and ...


3

You could read it like this: The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving. Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in ...


3

Well, the main intuition of the Merton model is that a company's equity can be treated as a call option on its assets, thus allowing for the application of Black-Scholes option pricing methods. Let's consider a company that has assets $A_{t}$ financed by equity $E_{t}$ and a zero-coupon debt $B_{t}$ with face value K, and maturity T. At time of maturity T, ...


3

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 ...


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 ...


3

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 ...


2

Unless it is due to random chance, there seems to be a bias in your estimation method for $\kappa$, and this bias appears to depend on the size of the sample. This may be revealing a deeper underlying problem with your technique that will ultimately make it clearer what the tradeoff is between accuracy and sample size. I do not believe it should be the ...


2

Bermudan swaptions (often on interest rates) are typically valued with a model that incorporates mean-reversion parameters. This might be as naive as Black-Karasinski, but more often is somewhat more sophisticated, for example Generalized Vasicek. Calibrating the model involves choosing model parameters that "best" fit the observed bermudan swaption ...


2

It would help if you made your question more clear (or included a link to a copy of the paper). If you know call (or put) option prices for a single stock across all strikes $K$ for common expiry $T$, you can then derive the distribution of the stock price at time $T$ in the $T$-forward measure (in which the numeraire is the zero-coupon bond with maturity ...


2

For pricing, there are a few products whose prices are sensitive to the forward smile and when you compute that with just local vol, it is not realistic. So if you are a seller, you go to the next church and find something that looks kindof reasonable, and that kind of can reconstruct a reasonnable forward smile structure. The game in pricing is to not ...


2

I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as $$ \frac{dS_t}{S_t} = \sqrt{v_t}dW_t $$ $$ dv_t = -\kappa(v_t-\eta)dt + \theta \sqrt{v_t}dZ_t $$ with $d\langle W,Z\rangle_t = \rho dt$ and $v_0 = \sigma_0^2$. Using Itô's lemma we can derive $$ \log\left(\frac{S_t}{S_0}\right) = \int_0^t ...


2

In fact, even your VIX dynamics are not exact, since you can only obtain dynamics for an approximation of the actual VIX calculation (I presume you are just running the variance variable through a square root using Ito's rule). SKEW is even less tractable here since its calculation roughly goes as the third moment of the return distribution. I doubt you ...


2

GMM method is a powerful method to calibrate historically, only. Also, the historical Calibration is used in the banking industry for forecast an asset’s performances and not for replicating them. Mathematically, it's known that historical vs options calibration is equivalent to observing an asset through two different probabilities (historical vs the ...


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

Consider the following analogy: you can hedge a derivative in a deterministic-volatility model using either futures, or spot underlying. The hedge ratio will change, but all the mathematics to effectively eliminate stochastic portfolio PL is the same, and must work out to be equivalent. A similar situation applies here: any triangle of (nontrivial) ...


2

I am not an expert in this field, but it would be best to consider a full multivariate GARCH model. This paper by Engle and Sheppard should be a good start. I think the constant correlation matrix approach is covered to a certain extent too. I hope this helps.


2

The equation stated in the question is not at the core of Merton's credit model, (Not saying you claimed it is) but is a simple device in helping to solve the system of linear equations. The equation given simply establishes a relationship between the volatility of equity and the volatility of the assets and it follows from the application of Black Scholes ...


2

The standard realized volatility calculation assumes an underlying model: geometric Brownian motion with constant drift and volatility. Then realized vol squared is an unbiased estimator of the process volatility squared. If you want to move beyond Black Scholes then you have two possibilities: look at a different formal model and the estimators for its ...


2

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 ...


2

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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