6

Well, what you find is that the introduction of stochastic vol changes the delta of your options. So what does this mean? If the new delta reduces the variance of your hedged portfolio versus the pure local vol model , then it means that the introduction of stochastic vol has resulted in a better description of market dynamics versus the pure local vol ...


5

The following source contains detailed answers to your questions in a research paper from ETH Zürich. van der Weijst, Roel (2017). "Numerical Solutions for the Stochastic Local Volatility Model" http://resolver.tudelft.nl/uuid:029cbbc3-d4d4-4582-8be2-e0979e9f6bc3


5

Because vanilla derivatives with European exercise depend only on total variance , not on it's dynamics in time. If you have a simpler model (like interpolation of these total variances from your volatility surface) you don't have as much of unobservable parameters stochastic volatility models have. Having more parameters (which many times would need to be ...


5

This depends largely on the model as well as the market, so there is no one-size-fits-all approach. Let us take the Stochastic Alpha Beta Rho (SABR) model, which has four parameters, as an example: $\alpha$, the initial instantaneous volatility of an ATM option. This is calibrated frequently and often intra-day due to the importance of ATM options. $\beta$, ...


4

Maybe it would help you to think of it the following way. The strike $\sigma^2(T)$ of a fresh-start variance swap of maturity $T$ in the Heston model only depends on parameters $(v_0,\theta,\kappa)$, see related question here. More specifically \begin{align} \sigma^2(T) &= \Bbb{E}_0^\Bbb{Q}\left[ \frac{1}{T} \langle \ln S\rangle_T \right] \\ &= \...


3

I write down the solution for the Heston model. You can directly generalise the result. Let $f=f(t,s,v)\in C^{1,2,2}(\mathbb{R}_+^3)$ be a real-valued function (portfolio value) and consider the two-dimensional stochastic process $(S_t,v_t)$ with \begin{align*} \mathrm{d}S_t&=(r-q) S_t \mathrm{d}t+\sqrt{v_t} S_t \mathrm{d}W_{1,t}, \\ \mathrm{d}v_t&=\...


3

I agree with the comment made by will: for a given model, you can potentially compute a Delta according to any "stickiness rule" depending on which data you decide to bump vs. keep constant. That being said, if you look at the following quantity $$ \Delta = \left. \frac{\partial V}{\partial S_0} \right\vert_{\Theta} $$ that we could call the in-model Delta ...


2

Both @alexprice and @FunnyBuzer have some good points, and I have upvoted them. I think I have enough to add here that I'll make another answer entry. First off, @AFK was fairly correct that you do not need stochastic volatility for vanilla (European exercise) option pricing, since (as he says and alexprice elaborates) you just interpolate the surface of ...


2

I think that the main advantage of using a stochastic volatility model is to produce a consistent volatility smile. Let's consider the pricing formulas for the normal and lognormal volatilities: $$dS_t=\sigma dW_t\Rightarrow \mathbb{E}[(S_T-K)^+]=(S_t-K)\Phi\left(\frac{s-S_t}{\sigma\sqrt{\Delta t}}\right)+\sigma\sqrt{\Delta t}\phi\left(\frac{s-S_t}{\sigma\...


2

The relationship between the two models is described in details in Implied Volatility Formulas for Heston Models by Hagan et al. In particular an expansion of the implied volatility under the Heston model that matches the one of a SABR model is described. It gives an explicit correspondence between the parameters of each model.


2

First and foremost it is important to clarify that the underlying is not necessarily normal/lognormal but for the special cases of $\beta$ the underlying is normal/lognormal Conditioned on a realization of the volatility. As mentioned in the answer by @ilovevolatility. Simple stochastic calculus will show the properties you mentioned. For realized ...


2

You don't want to use the SABR (or an extension) to price equity options or FX options. The lag of mean-reversion in the model's volatility dynamics leads to explosive behavior and to a implied distribution that is absolutely not in line with empirics -- especially on longer time horizons. To my knowledge people use it mostly for interest rate derivatives. ...


2

I use Gatheral's notations. The SVI-Jump-Wings (SVI-JW) parameterization of the implied variance v (rather than the implied total variance The raw and natural parametrizations describe the total implied variance for one slice (fixed tenor). The SVI-JW describes the implied variance for one slice (fixed tenor). The total implied variance slice for a fixed ...


2

These papers are interested in modelling the stochastic volatility, so implicitely they model the dynamics of the forward price which has zero drift under the corresponding terminal measure. This makes the exposition much simpler. It is then easy to switch back to the stock price: european options: an option with expiry $T$ on the stock price $S_t$ is ...


2

volatility of the volatility controls convexity of the skew/smile => more vol of vol generates more convex function ( = more smile) mean revertion and correlation between brownian motions both control ATM skew. long term variance controls overall level of skew (moves whole skew graph higher) In practice these parameters are calibrated to market quotes of ...


2

It is a big topic but here is a simplistic recipe! The starting point would be to check the distribution of the historical returns. Histogram would give an idea of how the shifts are distributed. Have a look at the tails, if the tails are fat or don’t ‘tail-off’ then that would be indicative of jumps or non constant volatility. If you decide that a simple ...


2

Yes, that's what we wish to see from the correctly-specified model. Now, let me try to answer your 2nd and 3rd questions together as they are based on the same confusion. There are two different concepts: model-implied volatility and model-implied BSIV (Black-Scholes Implied Volatility). I think you are confused because of mixing them up. So yes, people ...


2

I solved from (2) to (4) by myself ! (2) My answer Use the result of (1) with keeping in mind that the following R.H.S is $\mathcal{F}_t $ measurable. \begin{eqnarray} V_t &=& E^{\mathbb{P}} \left[ \exp \left(- \int^T_t r_s ds \right) \cdot ( P(T,S) - K )^+ \middle| \mathcal{F}_t \right] \\ &=& P(t, T) E^{ \tilde{\mathbb{P}} } \left[ ( P(...


2

No, the simulation is not exact in general, precisely for the reason you mentioned. By "exact", it is meant that there is no discretization error in time. Of course, there will always be a Monte-Carlo sampling error. For the Black-Scholes model, the simulation is exact if you simulate the log asset, as it is a standard arithmetic Brownian motion, and then ...


1

Consider the model \begin{align*} \mathrm{d}S_t &=rS_t\mathrm{d}t+\sigma_tS_t\mathrm{d}W_{1,t}, \\ \mathrm{d}\sigma^2_t &= \alpha \sigma_t^2\mathrm{d}t+\xi\sigma_t^2\mathrm{d}W_{2,t}, \end{align*} where the Brownian motions $(W_{1,t})$ and $(W_{2,t})$ are independent. Denote the averaged variance by $$\bar{V}=\frac{1}{t}\int_0^t\sigma_s^2\mathrm{d}s.$...


1

(Cumulative Integration Formula Replacing $du$ and $dB_s$) I have developed formulas to solve this by myself! \begin{eqnarray} \int^t_0 \int^u_0 dB_s \ du &=& \int^t_0 \int^u_s du \ dB_s \\ \int^T_t \int^u_0 dB_s \ du &=& \int^T_0 \int^u_s du \ dB_s - \int^t_0 \int^u_s du \ dB_s \end{eqnarray} Therefore, we can use the following ...


1

(My answer) the Vasicek Bond Price and its Forward Price Recall the result of Exercise 5.2.(1) or Exercise 4.5.(10). \begin{eqnarray} P(t, T) &=& E \left[ \exp \left( - \int^T_t r_u du \right) \middle| \mathcal{F}_t \right] \\ &=& E \left[ \exp \left( - \int^T_t \left( e^{-bu} r_0 + \sigma \int^u_0 e^{-b(u-s)} dB_s \right) du\right) ...


1

I solved by myself. The following is this solution. Let $T-t=s$, one reaches the following equation. \begin{eqnarray} B'(s) + \beta B(s) + \frac{1}{2} \sigma^2 B(s)^2 =1 \end{eqnarray} One finds out it is the Riccatti equation because of $A(s)=0$. Therefore, one reaches the following equation. \begin{eqnarray} B' = - \frac{1}{2} \sigma^2 B^2 - \...


1

Please see, if the below serves as a counter-example - Consider, $\Sigma= \rho S\sigma - \rho K S\sigma^2 +S\sigma$ So, $\Sigma_\rho = S\sigma - K S \sigma^2$ There exists $K$, such that $K= \frac 1 \sigma$ where $\Sigma_\rho = 0$. Evaluating $\Sigma_{S \sigma}$ below - $\Sigma_S= \rho \sigma - \rho K \sigma^2 +\sigma $ $\Sigma_{S \sigma} = \rho - 2 ...


1

We need to categorise the types of models before we consider the term $\dfrac{\partial^2 \Sigma}{\partial S \partial \sigma}$. I will only consider local volatility models and stochastic volatility models. Local volatility models The local volatility function is, of course, $\sigma^2(K,T)=2 \dfrac{\partial_T C_{KT}}{\partial_{K}^2 C_{KT}}$ This can be ...


1

Given the main uses of the VaR relate to risk management such as limit management, and measurement of P&L volatility, it is usually calculated under the physical/real world measure. Reason being that the risk measure are normally used to predict or explain the P&L movements from one day to another, which one can relate to their historical movements. ...


1

Here is a practical hedge for forward volatility swaps using only straddles with a certain strike, and with a notional that is determined by the skew at that magic strike. The same method for spot/seasoned volatility swaps will be posted online in due course as well. https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3354408


1

The answer given so far, by Mats Lind, to the first question is not in the spirit of the paper. I am referring to Question: In the paper they say one should really integrate from $y$ to $\infty$... What they mean is that you should not use any information on what happens from $-\infty$ to $y$, which would correspond to the standard cumulative distribution ...


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