# Tag Info

10

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 ...

9

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: $$... 9 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 ... 7 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 ... 6 I would say Start with Black Scholes to look at accuracy. In particular, you have a closed formula and you know what the characteristic function for lognormal is. Running FFT and comparing FFT pricing with the closed formula will give you an idea of what are the convergence issues, what is the behaviour at the boundaries (extreme strikes) etcetera. Then ... 6 First, to make that clear: The Heston model does not generate negative volatility, but - for example - an Euler discretization of the Heston model may generate negative volatility (or variance). It is not a problem of the model. It is a problem of the numerical scheme. If you use an Euler scheme which generates negative volatility and then use any of the ... 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 ... 5 You can find the derivation of the Heston characteristic function (its Fourier Transform) in Gatheral (2006). Using the characteristic function, you can optimize the model on the prices. There are multiple approaches to optimize, among others pattern search (which is very slow) and stochastic optimization (randomly jump around and stop after n iterations), ... 5 I highly recommend you to stick with the error function (RMSE) value minimization approach. I love MC techniques for this and related problem solving and thus do not recommend you to use anything else because of its simplicity and transparency. It comes down to using the right discretization function and to possibly implement variance reduction approaches. ... 5 From this abstract: The Heston stochastic volatility process is a degenerate diffusion process where the degeneracy in the diffusion coefficient is proportional to the square root of the distance to the boundary of the half-plane. The generator of this process with killing, called the elliptic Heston operator, is a second-order, degenerate-elliptic ... 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 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 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 ... 4 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 ... 4 Expanding a bit on chrisaycock's answer, and noting in particular from the abstract In mathematical finance, solutions to obstacle problem for the elliptic Heston operator correspond to value functions for perpetual American-style options on the underlying asset. we can see that this would be used to price those few rare cases of perpetual options. ... 4 It is not necessarily something that must be wrong with your model. Inherent in the Heston discretization methods of its continuous time dynamics is the possibility of negative values in the variance process. Here are couple solutions you can look at in order to "fix" your problem: Usage of different Euler schemes, such as the Full Truncation scheme. ... 4 I'd use FFT or similar rather than direct integration. Here is an old paper with Heston example: Option pricing using fractional FFT 4 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... 4 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 is the constant volatility and \mu is the risk-neutral drift of the asset. For a stock you could for example have \mu = r - q where r is the risk-free ... 4 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 ... 4 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 ...

4

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] \\ &= \...

3

To check your results, you might try "The Heston Model: A Practical Approach with Matlab Code" by Nimalin Moodley, http://math.nyu.edu/~atm262/fall06/compmethods/a1/nimalinmoodley.pdf , in particular the www.ingber.com open source C++ code for Adaptive Simulated Annealing (+ SWIG to wrap/parse it to the language you are using)

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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 ...

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