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| In finance, the '''Heston model''', named after [[Steven L. Heston|Steven Heston]], is a [[mathematical model]] describing the evolution of the [[volatility (finance)|volatility]] of an [[underlying]] asset.<ref>"A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options", by Steven L. Heston, ''The Review of Financial Studies'' 1993 Volume 6, number 2, pp. 327–343 [http://www.jstor.org/pss/2962057]</ref> It is a [[stochastic volatility]] model: such a model assumes that the volatility of the asset is not constant, nor even deterministic, but follows a [[random process]].
| | The author's title is Christy. Credit authorising is how she tends to make a residing. Doing ballet is something she would by no means give up. For many years he's been residing in Alaska and he doesn't strategy on changing it.<br><br>Feel free to visit my web site - [http://www.youronlinepublishers.com/authWiki/AdolphvhBladenqq free psychic] |
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| == Basic Heston model ==
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| The basic Heston model assumes that ''S<sub>t</sub>'', the price of the asset, is determined by a stochastic process:<ref name=Wilm2006>{{citation
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| | last1 = Wilmott| first1 = P.
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| | year = 2006
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| | title = Paul Wilmott on quantitative finance
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| | edition = 2nd
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| | page= 861
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| }}</ref>
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| :<math>
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| dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \,
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| </math>
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| where <math>\nu_t</math>, the instantaneous variance, is a [[CIR process]]:
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| :<math>
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| d\nu_t = \kappa(\theta - \nu_t)\,dt + \xi \sqrt{\nu_t}\,dW^{\nu}_t \,
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| </math>
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| and <math>{\scriptstyle dW^S_t, dW^{\nu}_t}</math> are [[Wiener process]]es (i.e., random walks) with correlation ρ, or equivalently, with covariance ρ dt.
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| The parameters in the above equations represent the following:
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| * μ is the rate of return of the asset.
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| * θ is the '''long variance''', or long run average price variance; as ''t'' tends to infinity, the expected value of ν<sub>''t''</sub> tends to θ.
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| * κ is the rate at which ν<sub>''t''</sub> reverts to θ.
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| * ξ is the '''vol of vol''', or volatility of the volatility; as the name suggests, this determines the variance of ν<sub>''t''</sub>.
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| If the parameters obey the following condition (known as the Feller condition) then the process <math>\nu_t</math> is strictly positive <ref name=Albr2007>{{citation
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| | last1 = Albrecher| first1 = H.
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| | last2 = Mayer| first2 = P.
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| | last3 = Schoutens| first3 = W.
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| | last4 = Tistaert| first4 = J.
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| | year = 2007
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| | journal = Wilmott Magazine
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| | pages= 83–92
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| |month= January
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| | id = {{citeseerx|10.1.1.170.9335}}
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| }}</ref>
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| :<math>
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| 2 \kappa_t \theta > \xi^2 \, .
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| </math>
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| == Extensions ==
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| In order to take into account all the features from the volatility surface, the Heston model may be a too rigid framework.{{Citation needed|date=November 2010}} It may be necessary to add degrees of freedom to the original model.
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| A first straightforward extension is to allow the parameters to be time-dependent.{{Citation needed|date=November 2010}} The model dynamics are then written as:
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| :<math>
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| dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \, .
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| </math>
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| Here <math>\nu_t</math>, the instantaneous variance, is a time-dependent [[CIR process]]:
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| :<math>
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| d\nu_t = \kappa_t(\theta_t - \nu_t)\,dt + \xi_t \sqrt{\nu_t}\,dW^{\nu}_t \,
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| </math>
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| and <math>{\scriptstyle dW^S_t, dW^{\nu}_t}</math> are [[Wiener process]]es (i.e., random walks) with correlation ρ. In order to retain model tractability, one may require parameters to be piecewise-constant.{{Citation needed|date=November 2010}}
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| Another approach is to add a second process of variance, independent of the first one.{{Citation needed|date=November 2010}}
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| :<math>
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| dS_t = \mu S_t\,dt + \sqrt{\nu^1_t} S_t\,dW^{S,1}_t + \sqrt{\nu^2_t} S_t\,dW^{S,2}_t \,
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| </math>
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| :<math>
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| d\nu^1_t = \kappa^1(\theta^1 - \nu^1_t)\,dt + \xi^1 \sqrt{\nu^1_t}\,dW^{\nu^1}_t \,
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| </math>
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| :<math>
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| d\nu^2_t = \kappa^2(\theta^2 - \nu^2_t)\,dt + \xi^2 \sqrt{\nu^2_t}\,dW^{\nu^2}_t \,
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| </math>
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| A significant extension of Heston model to make both volatility and mean stochastic is given by Lin Chen (1996). {{Citation needed|date=November 2010}} In the [[Chen model]] the dynamics of the instantaneous interest rate are specified by
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| :<math> dr_t = (\theta_t-r_t)\,dt + \sqrt{r_t}\,\sigma_t\, dW_t,</math>
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| :<math> d \alpha_t = (\zeta_t-\alpha_t)\,dt + \sqrt{\alpha_t}\,\sigma_t\, dW_t,</math>
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| :<math> d \sigma_t = (\beta_t-\sigma_t)\,dt + \sqrt{\sigma_t}\,\eta_t\, dW_t.</math>
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| == Risk-neutral measure ==
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| :''See [[Risk-neutral measure]] for the complete article''
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| A fundamental concept in derivatives pricing is that of the [[Risk-neutral measure]];{{Citation needed|date=November 2010}} this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:
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| #To price a derivative whose payoff is a function of one or more underlying assets, we evaluate the expected value of its discounted payoff under a risk-neutral measure.
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| #A risk-neutral measure, also known as an equivalent martingale measure, is one which is equivalent to the real-world measure, and which is arbitrage-free: under such a measure, the discounted prices of each of the underlying assets is a martingale. See [[Girsanov's theorem]].
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| #In the Black-Scholes and Heston frameworks (where filtrations are generated from a linearly independent set of Wiener processes alone), any equivalent measure can be described in a very loose sense by adding a drift to each of the Wiener processes.
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| #By selecting certain values for the drifts described above, we may obtain an equivalent measure which fulfills the arbitrage-free condition.
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| Consider a general situation where we have <math>n</math> underlying assets and a linearly independent set of <math>m</math> Wiener processes. The set of equivalent measures is isomorphic to '''R'''<sup>m</sup>, the space of possible drifts. Let us consider the set of equivalent martingale measures to be isomorphic to a manifold <math>M</math> embedded in '''R'''<sup>m</sup>; initially, consider the situation where we have no assets and <math>M</math> is isomorphic to '''R'''<sup>m</sup>.
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| Now let us consider each of the underlying assets as providing a constraint on the set of equivalent measures, as its expected discount process must be equal to a constant (namely, its initial value). By adding one asset at a time, we may consider each additional constraint as reducing the dimension of <math>M</math> by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is <math>m-n</math>.
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| In the Black-Scholes model, we have one asset and one Wiener process. The dimension of the set of equivalent martingale measures is zero; hence it can be shown that there is a single value for the drift, and thus a single risk-neutral measure, under which the discounted asset <math>e^{-\rho t}S_t</math> will be a martingale.{{Citation needed|date=November 2010}}
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| In the Heston model, we still have one asset (volatility is not considered to be directly observable or tradeable in the market) but we now have two Wiener processes - the first in the Stochastic Differential Equation (SDE) for the asset and the second in the SDE for the stochastic volatility. Here, the dimension of the set of equivalent martingale measures is one; there is no unique risk-free measure.{{Citation needed|date=November 2010}}
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| This is of course problematic; while any of the risk-free measures may theoretically be used to price a derivative, it is likely that each of them will give a different price. In theory, however, only one of these risk-free measures would be compatible with the market prices of volatility-dependent options (for example, European calls, or more explicitly, [[variance swap]]s). Hence we could add a volatility-dependent asset;{{Citation needed|date=November 2010}} by doing so, we add an additional constraint, and thus choose a single risk-free measure which is compatible with the market. This measure may be used for pricing.
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| == Implementation ==
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| A recent discussion of implementation of the Heston model is given in a paper by Kahl and Jäckel
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| .<ref name=Kahl2005>{{citation
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| | last1 = Kahl | first1 = C.
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| | last2 = Jäckel | first2 = P.
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| | year = 2005
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| | title = Not-so-complex logarithms in the Heston model
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| | journal = Wilmott Magazine
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| | pages = 74–103
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| | url = http://www.math.uni-wuppertal.de/~kahl/publications/NotSoComplexLogarithmsInTheHestonModel.pdf
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| }}</ref>
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| Information about how to use the Fourier transform to value options is given in a paper by Carr and Madan.<ref name=Carr1999>{{citation
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| | last1 = Carr | first1 = P.
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| | last2 = Madan | first2 = D.
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| | year = 1999
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| | title = Option valuation using the fast Fourier transform
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| | journal = Journal of Computational Finance
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| | volume = 2
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| | issue = 4
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| | pages = 61–73
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| | url = http://www.math.nyu.edu/research/carrp/papers/pdf/jcfpub.pdf
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| }}</ref>
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| Extension of the Heston model with stochastic interest rates is given in the paper by Grzelak and Oosterlee.<ref name=GO09>{{citation
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| | last1 = Grzelak | first1 = L.A.
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| | last2 = Oosterlee | first2 = C.W.
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| | year = 2011
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| | title = On the Heston Model with Stochastic Interest Rates
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| | journal = SIAM J. Fin. Math.
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| | volume = 2
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| | pages = 255–286
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| | url = http://scitation.aip.org/getpdf/servlet/GetPDFServlet?filetype=pdf&id=SJFMBJ000002000001000255000001&idtype=cvips&doi=10.1137/090756119&prog=normal
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| }}</ref>
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| Derivation of closed-form option prices for time-dependent Heston model is presented in the paper by Gobet et al.<ref name=BGM1999>{{citation
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| | last1 = Benhamou | first1 = E.
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| | last2 = Gobet | first2 = E.
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| | last3 = Miri | first3 = M.
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| | year = 2009
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| | journal = SSRN Working Paper
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| | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1367955
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| }}</ref>
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| Derivation of closed-form option prices for double Heston model are presented in papers by Christoffersen
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| <ref name=CHJ2009>{{citation
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| | last1 = Christoffersen | first1 = P.
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| | last2 = Heston | first2 = S.
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| | last3 = Jacobs| first3 = K.
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| | year = 2009
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| | journal = CREATES Research Paper
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| | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1447362
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| }}</ref>
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| and also Gauthier.
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| <ref name=GP2009>{{citation
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| | last1 = Gauthier | first1 = P.
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| | last2 = Possamai | first2 = D.
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| | year = 2009
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| | journal = SSRN Working Paper
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| | url = http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1434853
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| }}</ref>
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| There exist few known parametrisation of the volatility surface based on the heston model (Schonbusher, SVI and gSVI) as well as their de-arbitraging methodologies.<ref name=damghani>{{cite journal | author=Babak Mahdavi Damghani | title=De-arbitraging with a weak smile | publisher=Wilmott | year = 2013}}http://www.readcube.com/articles/10.1002/wilm.10201?locale=en</ref>
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| == See also ==
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| *[[Stochastic volatility]]
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| *[[Risk-neutral measure]] (another name for the equivalent martingale measure)
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| *[[Girsanov's theorem]]
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| *[[Martingale (probability theory)]]
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| *[[SABR Volatility Model]]
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| == References ==
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| <references/>
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| {{Stochastic processes}}
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| {{DEFAULTSORT:Heston Model}}
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| [[Category:Derivatives (finance)]]
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| [[Category:Mathematical finance]]
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| [[Category:Stochastic processes]]
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