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16 votes

Pricing VIX Futures

Heston - Change of measure Consider the following Heston dynamics written under the real world measure $\Bbb{P}$ \begin{gather} \frac{dS_t}{S_t} = \mu_t dt + \sqrt{v_t} dW_S^{\Bbb{P}}(t),\ S(0) = S_0 \...
Quantuple's user avatar
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11 votes
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Delta of an option under Heston model

Bad news: Your calculation is not quite correct As you say, the initial price of a European call option is $$C(S_0;K,T)= S_0e^{-qT}\Pi_1-Ke^{-rT}\Pi_2. \tag{$\star$}$$ However, the exercise ...
Kevin's user avatar
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10 votes
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Modeling Call Price w.r.t. Strike w Models that Capture Vol Smile

The way that I understand your question is that you are looking to fit the market prices of European plain vanilla options of a single maturity and then back out the corresponding implied probability ...
LocalVolatility's user avatar
10 votes

Book/ Articles recommendation for Volatility models

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 ...
Jesper Tidblom's user avatar
10 votes
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Deriving the solution for European call option in the Heston Model

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 + \...
Kevin's user avatar
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7 votes

Heston Model Integration Oscillations

There has been a huge amount of work on this. Generally a Fourier transform approach is used. First, be careful to use the form of the characteristic function that does not wind about zero in order ...
Mark Joshi's user avatar
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7 votes
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CIR Process from Ornstein–Uhlenbeck process

I don't think that the statement you reference is correct for general $n \in \mathbb{R}$ but only for $n \in \mathbb{N}$. The intuition behind this is that each Ornstein-Uhlenbeck (OU) process is ...
LocalVolatility's user avatar
7 votes
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Interpretation and intuition behind the Put-Call symmetry under the Heston Model

This is a consequence of transforming a Put on $S_T$ with strike $K$ into a Call on $(K S_0)/S_T$ with strike $S_0$ under the stock measure. The new set of parameters $r_p$, $q_p$, $\kappa_p$, ... etc ...
Antoine Conze's user avatar
7 votes
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What is the Radon-Nikodym derivative in the Heston model?

Let \begin{align*} \mathrm{d}S_t&=\mu S_t\mathrm{d}t+\sqrt{v_t}S_t\mathrm{d}B_{S,t}, \\ \mathrm{d}v_t&=\kappa(\bar{v}-v_t)\mathrm{d}t+\xi\sqrt{v_t}\mathrm{d}B_{v,t}, \end{align*} where $\...
Kevin's user avatar
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7 votes
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Introductory material for getting started with local and stochastic volatility modelling

If you are looking for a short introduction into various concepts used in volatility modeling without too much mathematical derivations (although written by a mathematician), I would recommend 'Smile ...
BEQuant's user avatar
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6 votes
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Jim Gatheral's assertion on ATM implied volatility vs. square root variance

Below are my 2 cents only, but this was too long for a comment. As he shows in the next lines (see also Variance Swaps chapter of Bergomi's book) $$ \sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \tilde{\...
Quantuple's user avatar
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6 votes
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Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface

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 $(v_0,\theta,\kappa)$, ...
Quantuple's user avatar
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6 votes
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Most accurate Fourier transform method for extreme OTM options

I'll give it a start and stick with Fourier methods. The approaches from Carr and Madan (1999) and Fang and Oosterlee (2009) are indeed known to be inaccurate for highly OTM options. I'd suggest to ...
Kevin's user avatar
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6 votes
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Boundary conditions Heston's stochastic volatility model

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 ...
Kevin's user avatar
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6 votes

Introductory material for getting started with local and stochastic volatility modelling

You may find A Short Note on Volatility Models an interesting summary providing bird's-eye overview of general ideas in volatility modeling. I would highly recommend SABR and SABR LIBOR Market Models ...
Hasek's user avatar
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5 votes
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About the Feller Condition in Heston Calibration

You should not use the Feller condition as a constraint. In many cases its violation will be required for a good fit to the market data.
q.t.f.'s user avatar
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5 votes
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Analytical Solution for Heston Model

This equation is unrelated to the Heston model. It is simply the value of a European call under the a constant coefficient geometric Brownian motion, i.e. the Black and Scholes (1973) model. Here $\nu$...
LocalVolatility's user avatar
5 votes

Terminal Variance in the Heston Model

From the equations of the model it is clear that $v_t$ is the instantaneous variance of the log-returns, not the terminal annualised variance of the log-asset price. Put differently, you are you ...
Quantuple's user avatar
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5 votes

Is pricing options using the volatility surface implied by the Heston model equivalent to pricing using the Heston model directly for all options?

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 ...
Antoine Conze's user avatar
5 votes
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Calibration Heston Local Stochastic Volatility (LSV) Model

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,...
ir7's user avatar
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5 votes
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Heston: Variance of Integrated Variance

Studying zero-coupon bond prices in the CIR (1985) short rate model, $\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic ...
Kevin's user avatar
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4 votes

Heston Model Integration Oscillations

I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT
Kiwiakos's user avatar
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4 votes

Heston ITM and OTM options pricing

There are a number of tricks. My favourite is to use the Black--Scholes price as a control. The integrals become much better behaved. You compute the difference of the Heston price from the BS price ...
Mark Joshi's user avatar
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4 votes
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Example of complex structured products on FX market?

Here are a few FX structured product examples: All of these can be notes or swaps, notes will pay back the notional at the end and carry no credit risk (and are normally set so that they are worth ...
will's user avatar
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4 votes
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Simulation of Heston process Quantlib-Python

The snippets below will generate spot and vol paths from QuantLib's HestonProcess, and generate the plots shown. Notice that in the vol histogram, we see a peak ...
StackG's user avatar
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4 votes
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Short Maturity Implied Volatility in Heston Model

I was naively taking for granted that MATLAB's implementation of Jäckel's method for implying volatilities is precise for each strike and maturity. This is not the case! I also overlooked the fact ...
ffbzona's user avatar
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