Hot answers tagged stochastic-volatility
8
There is another reason why Stoc Vol Models should be usually prefered 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 ...
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
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 ...
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 ...
3
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 ...
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
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 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
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.
1
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) ...
1
Dealing with model error under stochastic volatility (in a more formal way) you could use the UVM (Uncertain Volatility Framework). Here are what i think are the most seminal references:
Avellenada et al (1995) Pricing And Hedging Derivative Securities In Markets With Uncertain Volatilities
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.50.3736
...
1
For the univariate case, consider X which is the log prices of some stock. First, fit X with an AR(p) model and collect the residuals. Next, fit a Garch(p,q) model and collect the conditional standard deviations. Scale the initial residuals by the conditional standard deviations to produce a new series that has mean of 0 and variance 1. For the sake of ...
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