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12

Let \begin{align*} Y_t = e^{(a+\frac{c^2}{2})t-cW_t}. \end{align*} Then \begin{align*} dY_t = Y_t\left[\big(a+c^2\big)dt -c dW_t \right]. \end{align*} Moreover, \begin{align*} d(X_tY_t) &= Y_t dX_t + X_t dY_t + d\langle X, Y\rangle_t\\ &=abY_tdt. \end{align*} That is, \begin{align*} X_t = Y_t^{-1}\left(X_0 + ab\int_0^t Y_sds\right). \end{align*}

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I've seen that Gordon answer is more concise and to the point. Take this as a complementary answer. This is a general approach that will work for all this type of linear SDEs, not just this one. Assume we have the following linear SDE $$dX_t = (F_t X_t +f_t)dt + (G_t X_t +g_t)dB_t \tag*{(1)}$$ where $F, G, f$ and $g$ are Borel measurable bounded ...

7

I have also currently started to learn about the subject. This is some of the material I have encountered: Many people recommend the book "The Volatility Surface: A Practitioner's Guide" by Jim Gatheral. It is a standard reference in the area (even though I personally found it a bit confusing and a bit unclear at some parts). The author also have ...

5

I'll answer both of your questions in one go: Your ideas are correct. If the Black-Scholes model was true, the implied volatility surface would be flat but it is not in real life. Thus, the geometric Brownian motion as stock price model is misspecified and we need more sophisticated models (sto vol, jumps etc), in particular if we want to price more ...

5

Intuitively, in a (log)-space homogenous diffusion model $$S_t \propto S_0, \forall t \geq 0$$ such that implied volatilities will only depend on the moneyness level and not on the absolute spot level, which is precisely the definition of sticky delta. Mathematically, consider a (log)-space homogeneous diffusion model (be it stochastic or not) $$\frac{... 4 Local volatility models capture skew today but not dynamics tomorrow. Stochastic vol captures dynamics tomorrow but not necessarily skew today (how well does your calibrated vol surface match observation?). To answer your question: if you're pricing exotic options that are path dependent, stochastic local vol is more accurate. If you're pricing vanilla ... 4 Gonzalez-Perez (2015) Model-free volatility indexes in the financial literature: A review makes some remarks on this topic in section 2.2. Andersen, Bondarenko & Gonzalez-Perez (2013) identify a new error source in VIX that generates a significant number of jumps in the volatility index unconnected with the underlying volatility process and that ... 4 That is not the SDE for the Heston model - it violates the affine property in the drift term. In other words, the paper has a typo. The correct SDE is:$$ d v_t = \kappa (m-v_t) dt + \nu \sqrt{v_t} dw_t $$where v_t := \sigma_t^2 is the variance. Let \xi_t^T := \mathbb{E}_t [ v_T] denote the forward variance and see that$$ \begin{align} \xi_{t}^{T} &...

3

You can compute the SV - LV price difference and see if it is substantial or not. This is easily done and will give you an indication of whether your product can be safely priced with LV only. start with a pure SV model: choose $\sigma(t,S)=1$ and do a rough calibration of the parameters that drive $u_t$, to historical data for instance Price your ...

3

Jumps are an attempt to solve a math mistake in Modern Portfolio Theory. In the 19502-70s, economists were working on solving the variance-mean tradeoff. Furthermore, they needed to do so with punchcard computing. That radically restricted the set of computable, potential solutions. Both the normal distribution and the log-normal distribution are ...

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

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 ...

3

Suppose that you are riskless asset with return $r_{ft}$ and a risky asset with return $r_t$ and conditional volatility $\sigma_t(r_t) := \sqrt{V_t(r_t)}$. We build a portfolio using weights $(w_1, w_2) \in \mathbb{R}$, or as you wrote it $w_t := w_{1t}$, $w_{2t} := 1 - w_t$. This portfolio will have a time $t$ return of $r_{pt}$. Its volatility is given by $... 3 Let me venture a guess. If I had to design a system from scratch, I would probably prefer GARCH processes to properly stochastic conditional volatility processes. The fact that one step ahead, the conditional volatility process is known makes filtering both trivial and faster. Moreover, this class of option pricing model affords me all the flexibility of ... 3 What not to do What you are asking us, without knowing, is related to how to price a variance swap. Well, under a general diffusion process, variance swaps can be priced by forming a suitably weighted portfolio of options over a continuum of strike prices with the entire portfolio maturing on a given date. The intuition is that your exposure to volatility ... 3 The simplest long vol strategy is to be long an ATM straddle and delta hedge it, the problem is that when it is no longer ATM the exposure to vol weakens. You could then sell that straddle and enter another ATM one. Another solution is the vol swap or variance swap mentioned by Stephane below. It gives constant exposure no matter what the level of S&P. ... 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 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 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 ... 2 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$. Therefore, one reaches the following equation. \begin{eqnarray} B' = - \frac{1}{2} \sigma^2 B^2 - \... 2 I solved from (2) to (4) by myself ! (2) My answer Use the result of (1) with keeping in mind that the following R.H.S is$\mathcal{F}_t $measurable. \begin{eqnarray} V_t &=& E^{\mathbb{P}} \left[ \exp \left(- \int^T_t r_s ds \right) \cdot ( P(T,S) - K )^+ \middle| \mathcal{F}_t \right] \\ &=& P(t, T) E^{ \tilde{\mathbb{P}} } \left[ ( P(T,... 2 Yes. You should use that function to calculate the implied volatility - market convention is to always quote implied volatility using the Black-Scholes model. Traders may execute a trade simply by agreeing a level of implied volatility combined with the use of the corresponding Bloomberg option pricing page. Someone once said, "it is the wrong number in ... 2 The Heston model can have that property. If you make the correlation negative between the Brownian motions in the$dS_{t}$process and the$d\nu_{t}$process you imply that price is negatively correlated with variance. 2 As you said, you estimated the$\mathcal{P}$parameters but for option pricing, one needs the$\mathcal{Q}$parameters. But there exists a transformation. Under$\mathcal{P}, Heston (1993) assumes \begin{align*} \mathrm{d}S_t &= \mu S_t\mathrm{d}t + \sqrt{v_t}S_t\mathrm{d}W_{1,t}^\mathcal{P}, \\ \mathrm{d}v_t &= \kappa(\theta-v_t)\mathrm{d}t + \xi\... 2 I do not mean to discourage you, but it sounds like you're a wee bit late for this round of volatility games, for two reasons: You are still trying to figure out how to implement a long vol strategy. The market has already priced the risk in, i.e. buying volatility is already expensive. However, never too late to learn and prepare for a next time. My ... 2 I am going to try to answer your more general question "Do all SV models generate a smile?" which you put in one of the comments. (Maybe edit also the title of your question if you want, if my answer is satisfactory.) I will take zero correlation between asset and the volatility process to start with. The generalisation to non-zero correlation is ... 2 Any stochastic volatility model will be incomplete. Asset price under no arbitrage satisfies an SDEdS_t = r(t, S_t) dt + \sigma(t, S_t) dW_t$, where$r(t, S_t)$and$\sigma(t, S_t)\$ can be stochastic. Second Fundamental Theorem of Asset Pricing states that a model is incomplete if and only if the associated equivalent martingale measure is not unique. In ...

1

Yeah this is often called Spot-Vol correlation and is well known. Most people take this into account. I think if you just google spot-vol correlation you will come up with many example/models.

1

I'm not aware of any books about this topic, but the rough volatility network is a good starting point. There you will find nearly all the relevant papers. https://sites.google.com/site/roughvol/home/risks-1

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Apparently your heston model parameters should define the surface. You're fitting to options quoted in the market, thus a minimization exercise. Not like local vol, where it needs a abitrage free implied vol surface to garantee uniqueness

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