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16

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 \\ dv_t = \kappa(\theta-v_t)dt + \xi \sqrt{v_t} dW_v^{\Bbb{P}}(t),\ v(0) = v_0 \\ d\langle W_S^\Bbb{P}, W_v^\Bbb{P} \rangle_t = \rho dt \end{gather} In order to ...


14

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 strikes and maturities. Hence this produces risk neutral parameters that can be used to price other more exotic products. However, it is a pain to estimate the ...


11

To recover the Black-Scholes pricing equation, you should first express the standard normal cdf in terms of its characteristic function analogous to the Heston solution: $$ N(x) = \frac{1}{2} - \frac{1}{\pi} \int_0^{\infty} Re [\frac{e^{-i\phi x} f(\phi)}{i\phi}] d\phi $$ where $f(\phi)$ is the characteristic function of the standard normal distribution: $$ ...


10

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 density function. There are multiple ways that you could approach your problem. 1) Modelling the Market Prices The market prices of European plain vanilla ...


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

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 probabilities $\Pi_1$ and $\Pi_2$ depend on the stock price $S_0$ too! Thus, you need the product rule and the chain rule to differentiate the option price with respect ...


7

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 option by its bid-ask spread and vega. Using data from multiple days is not a good approach, because you might have options with the same strike but different ...


7

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 to avoid having to count the normal of windings. Second, using contour shifts can make the integral much better behaved. eg integrate along the line with $0.5$...


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 ...


7

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 try out one the following three alternatives. The first one seems to be the most relevant one. Saddlepoint Method I begin to cite Hirsa (2013): The saddlepoint ...


6

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 normally distributed. Thus the sum of $n$ squared OU processes is chi-squared distributed with $n$ degrees of freedom. Define $X$ to be a $n$-dimensional vector ...


6

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] \\ &= \...


6

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 $\mathrm{d}B_{S,t}\mathrm{d}B_{v,t}=\rho\mathrm{d}t$. The market price of risk (or Girsanov kernel or Sharpe ratio) is ${\varphi}_t=\left(\frac{\mu-r}{\sqrt{v_t}},\...


5

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 estimation of discretely sampled diffusions: A closed-form approximation approach. Econometrica 70 (1), 223-262. Azencott, R., Y. Gadhyan, and R. Glowinski (...


5

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 to buy it now. Instead you enter a forward contract with someone that says that at time $T$ you will pay the amount $K$ and get the asset in exchange. What ...


5

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 -> higher prob of exercise than Gaussian with constant stdev -> higher option price than BS with ATM vol -> higher implied vol for given strike. Skewed distributions ...


5

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 confusing $$ v_t \approx \text{var}(\ln(S_{t+\delta t}/S_t))/\delta t $$ with $$\text{var}(\ln(S_t))/t$$ presumably because in the Black-Scholes framework these ...


5

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 . are those that correspond to the Heston dynamics for the process $((K S_0)/S_t, v_t)$ under the stock measure. General results on that kind of symmetry can ...


5

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) $$ \sigma_{VS}^2(T) = \int_{-\infty}^{+\infty} \tilde{\sigma}^2(z,T) \phi(z) dz \tag{0} $$ where $\sigma_{VS}(T)$ denotes the volatility of a fresh-start variance swap of maturity $T$; $\phi(\cdot)$ the standard ...


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

There is a qualitative shift in the shape of the density. When V is small it is monotone decaying. When V is large it looks more like a Gaussian. Another reason he uses two schemes is that he wants match two moments of the density. When V is small, the moment matching equations for the quadratic Gaussian are unsolvable. When V is large they are unsolvable ...


4

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 strictly positive and initial level $v_0>0$. The reason is, conditional on no jumps occuring, the process is just a square root process, for which the references ...


4

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


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