# Tag Info

### 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 \...
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### Difference between GARCH and Heston Volatility model

Heston gives an expression for the characteristic function, from which option prices can be computed. Therefore it can be calibrated (statically) on a set of vanilla option prices with different ...

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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 ...

### 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 ...

### Option Pricing Model Calibration In Practice

The typical approach is: you only use option data from the last day. Furthermore, you only include those points that are liquid enough. One approach to this is to weigh the modelling error of an ...

### 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 ...
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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 ...
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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 ...
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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 $\... 6 votes Accepted ### CIR Process from Ornstein–Uhlenbeck process I don't think that the statement you reference is correct for general$n \in \mathbb{R}$but only for$n \in \mathbb{N}$. The intuition behind this is that each Ornstein-Uhlenbeck (OU) process is ... 6 votes Accepted ### Intuition for the Effect of Vol of Vol in Heston Model on Volatility Surface Maybe it would help you to think of it the following way. The strike$\sigma^2(T)$of a fresh-start variance swap of maturity$T$in the Heston model only depends on parameters$(v_0,\theta,\kappa)$, ... 6 votes Accepted ### Heston: Variance of Integrated Variance Studying zero-coupon bond prices in the CIR (1985) short rate model,$\text{d}r_t=\kappa(\theta-r_t)\text{d}t+\xi\sqrt{r_t}\text{d}W_t$, Hirsa (2013, Section 1.2.6.2) states that the characteristic ... 5 votes Accepted ### derivation of heston pde in gatheral 1) Gatheral expresses everything in forward terms: forward value of the spot and of the call. Consider an asset$A$. You need to hold$A$at time$T$but since you don't need it now you don't want ... 5 votes Accepted ### parameters in Heston model and their impact on volatility smile Intuition: You can think of the vol smile as a reflection of the risk neutral distribution (compared to the Black Scholes Gaussian density). A fat tailed distribution creates the smile: fat tail -> ... 5 votes ### Option Pricing Model Calibration In Practice I know two papers explaining how to calibrate this kind of models, and one of them explain the impact of the quality of the fit on a pricing model: Aït-Sahalia, Y. (2002, January). Maximum likelihood ... 5 votes Accepted ### Analytical Solution for Heston Model This equation is unrelated to the Heston model. It is simply the value of a European call under the a constant coefficient geometric Brownian motion, i.e. the Black and Scholes (1973) model. Here$\nu$... 5 votes ### Terminal Variance in the Heston Model From the equations of the model it is clear that$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 ... 5 votes Accepted ### 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) $$\sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \... 5 votes Accepted ### Boundary conditions Heston's stochastic volatility model You can't really derive or prove boundary conditions. You impose them and try to economically motivate them. Let's consider a European-style call option and go through the boundary conditions step by ... 5 votes Accepted ### Calibration Heston Local Stochastic Volatility (LSV) Model Under Heston LSV (HLSV) dynamics, Gatheral's equality is:$$ \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,...
The Feller condition applies without modification. That is under the assumption that $v$ is square-root process with poisson-arrival jumps (as you wrote), and assuming the jump distribution is ...