10

Great question. Let me try provide some insights and thoughts regarding your points and questions raised. It may not be a full answer but hopefully it helps to connect the contents in the paper/book with some trading intuition: From a theoretical perspective, I don't see any mistake in your thinking regarding skew decay but two questions arise on my end: ...


10

Itô's Lemma The standard version of Itô's Lemma applies to a single Itô process $\text{d}X_t=\mu(t,X_t)\mathrm{d}t+\sigma(t,X_t)\mathrm dW_t$. Then, $$\mathrm{d}f(t,X_t) = \left(f_t+\mu(t,X_t)f_x + \frac{1}{2}\sigma(t,X_t)^2f_{xx}\right)\mathrm{d}t+\sigma(t,X_t)f_x\mathrm dW_t.$$ Let $\text{d}Y_t=m(t,Y_t)\mathrm{d}t+s(t,Y_t)\mathrm dW_t^{(2)}$ be a second ...


8

Stochastic-Local Vol (SLV) is an attempt to mix the strengths and weaknesses of both Stochastic Vol and Local Vol models. Below, I'll quickly summarise each model and their strengths and weaknesses, and then discuss how SLV tries to improve things. Although there are many stochastic vol models, I limit the discussion here to the Heston model to keep things ...


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

You can't really derive or prove boundary conditions. You impose them and try to economically motivate them. Let's consider a European-style call option and go through the boundary conditions step by step. $S=0$ When the underlying asset's value is zero, then the option to buy this asset is worthless. Thus, $$C(t,S=0,v)=0.$$ $S\to\infty$ As the underlying ...


4

Usually multidimensional objective function of calibration error of stochastic volatility models (Heston , bergomi etc) have many local minima, thus you would get similar calibration error for very different set of parameters. Some ways to deal with it: specify paramter range your are comfortable with. let's say you want your vol of vol to be in the region ...


4

Isn't this model just a bunch of classical Heston volatility processes, driven by the same Brownian motion? In this case, you can use some common schemes like Milstein. At least as a starter to toy with the model. If speed/accuracy is an issue, there probably exist some clever solutions as well.


4

Maybe this deck by Jim Gatheral would help get the intuition, see slides 10 and following. The dynamics you mentioned is obtained by: Looking at the Bergomi dynamics for the forward variance process; Assuming there is only one factor driving the dynamics; Noticing a similarity with a rough process when replacing the Bergomi exponential kernel by a Riemann-...


4

The numerical approximation of the call option price in the Heston model is notoriously unstable and can easily lead to imprecise answers for extreme parameter. Several different formulas exist for computing the price with some being more stable than others. The formula you are using is arguably one of the worst ones. The most precise algorithm I know of is ...


3

Zhu makes sense to me. The vega cash in Black-Scholes corresponds to a shift of the vol surface by 1%. If you bump only $v_0$ in Heston, you bump only the short maturities, and if your structure is also dependent on the long maturities, the vega will be vastly underestimated. So you need to bump the $\theta$ as well. I think it implicitly assumes that the ...


3

Consider the Heston model and the Local Volatility model with local volatility built (using Dupire) from the Heston reconstructed vanilla options implied volatility. The price of any European payoff will be the same under both models. The price of exotic options will usually not be the same.


3

In general, any European payoff (eg. European option) can be priced by integrating the payoff over the density of the log-returns of the underlying. However, the SVJ model does not omit any closed-form density and therefore we cannot derive any analytical pricing formula. But, it does have a closed-form characteristic function, which can be used to recover ...


3

Yes it should preserve positivity. However due to numerical noise you may observe very small negative values on the edges of the lattice, that you can truncate to zero. If you solve using Fokker-Planck you may want to start from $t=\delta t$ using a gaussian approximation for the density on the first step, so as to start from a smooth density. An alternative ...


3

In an incomplete market, vanilla options are independent assets like stocks or bonds. So the best way of thinking about how they are priced is the same way equilibrium prices in those markets occur: If too many people try to buy an option at a given strike then they push the price of those options up and we see that as the implied volatility increasing. The ...


3

The stochastic volatility model is calibrated to (a subset of) vanilla option prices. When the implied volatility is shifted to calculate vega, the model is calibrated again. Although pure stochastic volatility models can only match a few vanilla prices, a local volatility component is usually included nowadays. This allows the calibration to hit the whole ...


3

A similar question posted after this one has a very good answer. Especially the section that explains that the calibrated leverage surface is typically observed to flatten with maturity (a shortcoming of LV). Therefore the forward volatility smile will be less convex than on the initial pricing date and you will not be pricing deals properly which are ...


3

Here is a snip that will create and plot a Heston vol surface import numpy as np import QuantLib as ql from matplotlib import pyplot as plt from mpl_toolkits.mplot3d import Axes3D # Utility function to plot vol surfaces (can pass in ql.BlackVarianceSurface objects too) def plot_vol_surface(vol_surface, plot_years=np.arange(0.1, 2, 0.1), plot_strikes=np....


3

Here is something I did, maybe it helps: https://colab.research.google.com/drive/1M1YJncdswd-A9SgIOAjw6g6Se7NHU9mG?usp=sharing


3

Under Heston LSV (HLSV) dynamics, Gatheral's equality is: $$ \sigma_{LV}^{HLSV}(S_t,t) = \sqrt{E^{HSLV}\left[V_tL(S_t,t)^2 | S_t \right]} = L(S_t,t)\sqrt{E^{HSLV}\left[V_t | S_t \right]}, $$ as $L(S_t,t)$ is $\sigma(S_t)$-measurable, where superscript $HSLV$ is meant to remind us what is our dynamics we started with (in particular the joint probability ...


2

Any stochastic volatility model will be incomplete. Asset price under no arbitrage satisfies an SDE $dS_t = r(t, S_t) dt + \sigma(t, S_t) dW_t$, 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 ...


2

You generally need one tradable asset per source of risk in your model that is someway dependent on that noise. So in a world where you can trade a single stock, but have two sources of variance your model would be incomplete as there would be no way to fix the market price of volatility risk. However, if some other asset was tradable (say some reference ...


2

This is an interesting question. Peter A is correct that SV is typically combined with LV these days to get the so called SLV (stochastic local vol model). There is no obvious definition for Greeks here as there are no closed form solutions. Depending on the implementation, it will likely be based on a finite-difference solver of the PDE or Monte Carlo (MC) ...


2

You're implementing the formula $$C=S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2$$ where \begin{align*} \Pi_1&=\frac{1}{2}+\frac{1}{\pi}\int_0^\infty \text{Re}\left(\frac{e^{-iu\ln(K)}\varphi_{\ln(S_T)}(u-i)}{iu\varphi_{\ln(S_T)}(-i)}\right)\text{d}u,\\ \Pi_2&=\frac{1}{2}+\frac{1}{\pi}\int_0^\infty \text{Re}\left(\frac{e^{-iu\ln(K)}\varphi_{\ln(S_T)}(u)}{iu}\right)\...


2

If ${\frac {\partial^2 C} {\partial K ^2}}$ was zero, then the price-strike curve would just be a straight sloping-downwards line, and it would cost the same to buy either two call options at strike $K$ (portfolio A), or one option each at strike $K-1$ and strike $K+1$ (portfolio B). If you think about the payoffs at expiry where spot=$S_t$ of these two ...


2

Starting with $$dS_t = rS_tdt +\sigma_t S_tdW_t,$$ Ito Lemma in two steps gives: $$d\log S_t = S_t^{-1} dS_t - 2^{-1}S_t^{-2} (dS_t)^2 \; \; \; (*)$$ $$d\log S_t = (r-2^{-1}\sigma^2_t) dt + \sigma_t dW_t \; \; \; (**)$$ From (**) (and starting SDE) we get $$ d[\log S]_t = (d\log S_t)^2 = \sigma_t^2 dt = S_t^{-2} (dS_t)^2 $$ From (*) we then get: $$ d\log ...


1

Vanna-Vega-Volga is a cute way of interpolating the fx implied volatility surface. The problem is that the vols coming from that interpolation do not match the interbank quoted volatilities, you can't match the 10 delta wings without some weird contortions and using that methodology for first generation exotics results in the incorrect market price. To be ...


1

I found the following paper which answers my question somehow. https://www.scaillet.ch/pdfs/asymptotics.pdf


1

For the second question: The implied density is the density function we integrate call payoffs against to match market call prices, denoted $f$ here. So, ignoring discount factors, the answer comes from Dirac delta function's properties: $$ \mathbf{E}[\delta(S-K)] = \int_{-\infty}^\infty \delta(S-K)f(S) dS = f(K) $$ Alternatively: $$ C = C(K) = \int_K^\infty ...


1

Yes, a stochastic volatility SDE can be coupled with any underlying SDE (GBM, diffusion, mean reverting, LMM, etc.). Once stochastic volatility is present, the model earns the right to be labeled 'SV model'. In its name, one may want to specify the names of both SDE's, like in the SABR LMM example found here, or just call it LMM with SV extension. Similarly, ...


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