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Heston model

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Heston model

In finance, the Heston model, named after Steven Heston, is a mathematical model describing the evolution of the volatility of an underlying asset.[1] 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.

Basic Heston model

The basic Heston model assumes that St, the price of the asset, is determined by a stochastic process:[2]

dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \,

where \nu_t, the instantaneous variance, is a CIR process:

d\nu_t = \kappa(\theta - \nu_t)\,dt + \xi \sqrt{\nu_t}\,dW^{\nu}_t \,

and {\scriptstyle dW^S_t, dW^{\nu}_t} are Wiener processes (i.e., random walks) with correlation ρ, or equivalently, with covariance ρ dt.

The parameters in the above equations represent the following:

  • μ is the rate of return of the asset.
  • θ is the long variance, or long run average price variance; as t tends to infinity, the expected value of νt tends to θ.
  • κ is the rate at which νt reverts to θ.
  • ξ is the vol of vol, or volatility of the volatility; as the name suggests, this determines the variance of νt.

If the parameters obey the following condition (known as the Feller condition) then the process \nu_t is strictly positive [3]

2 \kappa \theta > \xi^2 \, .

Extensions

In order to take into account all the features from the volatility surface, the Heston model may be a too rigid framework. It may be necessary to add degrees of freedom to the original model. A first straightforward extension is to allow the parameters to be time-dependent. The model dynamics are then written as:

dS_t = \mu S_t\,dt + \sqrt{\nu_t} S_t\,dW^S_t \, .

Here \nu_t, the instantaneous variance, is a time-dependent CIR process:

d\nu_t = \kappa_t(\theta_t - \nu_t)\,dt + \xi_t \sqrt{\nu_t}\,dW^{\nu}_t \,

and {\scriptstyle dW^S_t, dW^{\nu}_t} are Wiener processes (i.e., random walks) with correlation ρ. In order to retain model tractability, one may require parameters to be piecewise-constant.

Another approach is to add a second process of variance, independent of the first one.

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 \,
d\nu^1_t = \kappa^1(\theta^1 - \nu^1_t)\,dt + \xi^1 \sqrt{\nu^1_t}\,dW^{\nu^1}_t \,
d\nu^2_t = \kappa^2(\theta^2 - \nu^2_t)\,dt + \xi^2 \sqrt{\nu^2_t}\,dW^{\nu^2}_t \,

A significant extension of Heston model to make both volatility and mean stochastic is given by Lin Chen (1996). In the Chen model the dynamics of the instantaneous interest rate are specified by

dr_t = (\theta_t-r_t)\,dt + \sqrt{r_t}\,\sigma_t\, dW_t,
d \alpha_t = (\zeta_t-\alpha_t)\,dt + \sqrt{\alpha_t}\,\sigma_t\, dW_t,
d \sigma_t = (\beta_t-\sigma_t)\,dt + \sqrt{\sigma_t}\,\eta_t\, dW_t.

Risk-neutral measure

See Risk-neutral measure for the complete article

A fundamental concept in derivatives pricing is that of the Risk-neutral measure; this is explained in further depth in the above article. For our purposes, it is sufficient to note the following:

  1. 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.
  2. 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 price of each of the underlying assets is a martingale. See Girsanov's theorem.
  3. 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.
  4. By selecting certain values for the drifts described above, we may obtain an equivalent measure which fulfills the arbitrage-free condition.

Consider a general situation where we have n underlying assets and a linearly independent set of m Wiener processes. The set of equivalent measures is isomorphic to Rm, the space of possible drifts. Let us consider the set of equivalent martingale measures to be isomorphic to a manifold M embedded in Rm; initially, consider the situation where we have no assets and M is isomorphic to Rm.

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 M by one dimension. Hence we can see that in the general situation described above, the dimension of the set of equivalent martingale measures is m-n.

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 e^{-\rho t}S_t will be a martingale.

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.

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 swaps). Hence we could add a volatility-dependent asset; 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.

Implementation

A recent discussion of implementation of the Heston model is given in a paper by Kahl and Jäckel .[4]

Information about how to use the Fourier transform to value options is given in a paper by Carr and Madan.[5]

Extension of the Heston model with stochastic interest rates is given in the paper by Grzelak and Oosterlee.[6]

Derivation of closed-form option prices for time-dependent Heston model is presented in the paper by Gobet et al.[7]

Derivation of closed-form option prices for double Heston model are presented in papers by Christoffersen [8] and Gauthier. [9]

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.[10]

See also

References

  1. ^  
  2. ^ Wilmott, P. (2006), Paul Wilmott on quantitative finance (2nd ed.), p. 861 
  3. ^ Albrecher, H.; Mayer, P.; Schoutens, W.; Tistaert, J. (January 2007), Wilmott Magazine: 83–92,  
  4. ^ Kahl, C.; Jäckel, P. (2005). "Not-so-complex logarithms in the Heston model". Wilmott Magazine. pp. 74–103. 
  5. ^ Carr, P.; Madan, D. (1999). "Option valuation using the fast Fourier transform". Journal of Computational Finance 2 (4). pp. 61–73. 
  6. ^ Grzelak, L.A.; Oosterlee, C.W. (2011). "On the Heston Model with Stochastic Interest Rates". SIAM J. Fin. Math. 2. pp. 255–286. 
  7. ^ Benhamou, E.; Gobet, E.; Miri, M. (2009). "Time Dependent Heston Model". SSRN Working Paper.  
  8. ^ Christoffersen, P.; Heston, S.; Jacobs, K. (2009). "The Shape and Term Structure of the Index Option Smirk: Why Multifactor Stochastic Volatility Models Work so Well". SSRN Working Paper. 
  9. ^ Gauthier, P.; Possamai, D. (2009), "Efficient Simulation of the Double Heston Model", SSRN Working Paper 
  10. ^ Babak Mahdavi Damghani (2013). "De-arbitraging with a weak smile". Wilmott. http://www.readcube.com/articles/10.1002/wilm.10201?locale=en
  11. ^ Mahdavi Damghani, Babak (2013). "De-arbitraging With a Weak Smile: Application to Skew Risk".  
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