# Tag Info

11

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 ...

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 ...

5

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: $$... 4 You have to ask yourself what the ultimate purpose of this parameterization is. In their case, they imply the "end-goal is martingale pricing or maximum-likelihood estimation", both of which are ultimately about capturing long-period dynamics rather than intraday or interday behavior. For this reason, the fact that intraday variance may, ahem, vary around ... 3 The model is similar to the Barndorﬀ-Nielsen - Shephard model. But this model is much more general. On the other hand in this paper by Heston it is exactly your form that is used. Already Scott in 1987 considered a model of your form (see this) Finally in this thesis you find the names Hull-White model (of course there is the interest rate model too) and ... 3 You could read it like this: The typical change in equity value is equal to the typical change in asset value, adjusted for the probability of the assets surviving. Note that the formula is not specific to Merton models, it's also true for regular options and their underlyings. It's just that volatility of option prices isn't typically a concern in ... 3 Well, the main intuition of the Merton model is that a company's equity can be treated as a call option on its assets, thus allowing for the application of Black-Scholes option pricing methods. Let's consider a company that has assets A_{t} financed by equity E_{t} and a zero-coupon debt B_{t} with face value K, and maturity T. At time of maturity T, ... 3 You cannot derive the probability distribution you require, because for VaR you need a real-world probability distribution. From the options prices, it is only possible to obtain a risk-neutral distribution. Now, if you are willing to assume some kind of parametric relationship between the risk-neutral and real-world distributions, then you might find the ... 2 I assume no interest rates to clarify the approach. The Heston model is written under the risk-neutral probability as$$ \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 think a sketch of the proof would look like this Let's say you start from $$S_t = S_t \odot (\mu_t dt + \sigma_t dW_t)$$ where $S$ is an vector valued process of your $n$ risky assets prices, $W$ a standard $k$-dimensionnal brownian motion under the historic probability, $\sigma_t$ an $n \times k$ matrix valued process and $\odot$ is the Hadamard ...

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

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 = ...

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 ...

1

Black Scholes makes the assumption of deterministic (time varying) volatility of the underlying asset. Also, the volatility input to the option pricing model is implied by nature and does not rest on realized historical volatilities. Think about it, the whole notion of being able to price a derivative with contingent future payoff rests to large degree on ...

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 ...

Only top voted, non community-wiki answers of a minimum length are eligible