Stochastic volatility models at ρ=±1 as second class constrained Hamiltonian systems

https://doi.org/10.1016/j.physa.2014.03.030Get rights and content

Highlights

  • Stochastic volatility models for extremely correlated cases ρ=±1 are analyzed.

  • Stochastic volatility models at ρ=±1 are constrained Hamiltonian systems.

  • Dirac’s method of singular Hamiltonian systems is used.

  • The propagator is obtained as a path integral over the volatility alone.

  • By semi-classical arguments, the propagator is evaluated for the simplest SABR model.

Abstract

The stochastic volatility models used in the financial world are characterized, in the continuous-time case, by a set of two coupled stochastic differential equations for the underlying asset price S and volatility σ. In addition, the correlations of the two Brownian movements that drive the stochastic dynamics are measured by the correlation parameter ρ(1ρ1). This stochastic system is equivalent to the Fokker–Planck equation for the transition probability density of the random variables S and σ. Solutions for the transition probability density of the Heston stochastic volatility model (Heston, 1993) were explored in Dragulescu and Yakovenko (2002), where the fundamental quantities such as the transition density itself, depend on ρ in such a manner that these are divergent for the extreme limit ρ=±1. The same divergent behavior appears in Hagan et al. (2002), where the probability density of the SABR model was analyzed. In an option pricing context, the propagator of the bi-dimensional Black–Scholes equation was obtained in Lemmens et al. (2008) in terms of the path integrals, and in this case, the propagator diverges again for the extreme values ρ=±1. This paper shows that these similar divergent behaviors are due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems for the most extreme correlated limit ρ=±1. In this way, the stochastic dynamics of the ρ=±1 cases are different of the 1<ρ<1 case, and it cannot be obtained as a continuous limit from the ρ±1 regimen. This conclusion is achieved by considering the Fokker–Planck equation or the bi-dimensional Black–Scholes equation as a Euclidean quantum Schrödinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at ρ=±1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac’s method for constrained systems (Dirac, 1958, 1967) must be employed, and Dirac’s analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral (Henneaux and Teitelboim, 1992), in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path-integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics for both extreme limits, the probability density function is evaluated by using semi-classical arguments, in an analogous manner to that developed in Hagan et al. (2002), for the SABR model.

Introduction

The stochastic volatility models permit to generalize the Black–Scholes model to the non constant volatility case and are widely used to evaluate option prices in the financial world. A generic stochastic volatility model is characterized in continuous time by two coupled stochastic differential equations for the underlying asset price S and the volatility σ of the form  [1], [2]dS=μ(S,σ,t)dt+G(S,σ,t)dWS(t)dσ=r(S,σ,t)dt+q(S,σ,t)dWσ(t) where μ, G, r and q are arbitrary functions of S, σ and t that defines the stochastic volatility model completely. The two Brownian motions dWS and dWσ satisfy the product rule dWSdWσ=ρdt in Itô’s sense, and the parameter 1ρ1 is called the correlation parameter. Between the different plethora of models, the best known are the Heston model  [3], the Ornstein–Uhlenbeck model  [1], the CEV model  [4], [5], the SABR volatility model  [6], the GARCH model  [7], the 3/2 model, the Hull and White model  [8] and the Chen model  [9]. Other variations include incorporating a “jump diffusion” term, which gives rise to integro-differential equations  [10] for the cost of the option. Some stochastic volatility models are even capable to capture some important statistical properties of real markets, called stylized facts, such as the autocorrelation and the leverage effect  [11], [12], [13], [14]. To account these stylized facts for the real financial data, empirical analysis implies that |ρ| must be of the order of 0.5   [11], [12]. Although the extreme case ρ=±1 is not realized in the real world (from a statistical point of view), these values can be satisfied for “outliers” events in the sense of Ref.  [15] and may be very important in a financial crisis scenario. Also, from a structural mathematical perspective, it is necessary to understand the behavior of the stochastic volatility models for the full range of its parameter values, in particular the case ρ=±1.

In mathematical terms too, the stochastic system (1), (2) is equivalent to the Fokker–Planck equation for the transition probability density of the random variables S and σ, and the Heston stochastic volatility model, were explored in Ref.  [16] through the solution of the Fokker–Planck equation. There, the fundamental quantities (such as the transition density itself) depend on ρ in such a way that these quantities are divergent for the extreme limit ρ=±1. The same divergent behavior appears in Ref.  [17], where the probability density of the SABR model was analyzed.

In an option pricing context, the propagator of the bi-dimensional Black–Scholes equation in terms of the path integrals was found in Ref.  [18]. In this case, the propagator diverges again for the extremely correlated case ρ=±1. The same divergence appears in Refs.  [19], [20], where propagators for the Black–Scholes equation for different stochastic volatility models, were constructed. Thus, one can ask: What is the reason for the divergent behavior of the probability density? What is the probability density function at ρ=±1 itself? What is the propagator for the extremely correlated limit? In Ref.  [21], the propagator for the Black–Scholes equation at the extreme limit ρ=±1 was obtained for a particular class of stochastic volatility models. At the same time, the divergent behavior of the propagator was explained by the existence of constraints that appear only in the ρ=±1 case.

This paper generalizes the results obtained in Ref.  [21] for an arbitrary stochastic volatility model and shows that the divergent behavior of the transition probability density or the option price propagator is due to a universal property of the stochastic volatility models in the continuum: all of them are second class constrained systems at the extreme correlated limit.

This conclusion is achieved by considering the Fokker–Planck equation or the bi-dimensional Black–Scholes equation as a Euclidean quantum Schrödinger equation. Then, the analysis of the underlying classical mechanics of the quantum model, implies that stochastic volatility models at ρ=±1 correspond to a constrained system. To study the dynamics in an appropriate form, Dirac’s method for constrained systems  [22], [23] must be employed and Dirac’s analysis reveals that the constraints are second class. In order to obtain the transition probability density or the option price correctly, one must evaluate the propagator as a constrained Hamiltonian path-integral  [24], in a similar way to the high energy gauge theory models. In fact, for all stochastic volatility models, after integrating over momentum variables, one obtains an effective Euclidean Lagrangian path integral over the volatility alone. The role of the second class constraints is determining the underlying asset price S completely in terms of volatility, so it plays no role in the path integral. In order to examine the effect of the constraints on the dynamics of both extreme ρ=±1 cases, the probability density function is evaluated by using semi-classical arguments, in an analogous way to that developed in Ref.  [17], for the SABR models.

The paper is structured as follows: Sections  2 Stochastic volatility models and the Fokker–Planck equation, 3 Stochastic volatility models and option pricing show that the Fokker–Planck equation for the transition probability density and the Black–Scholes equation for the option price, can be written as the same Euclidean quantum Schrödinger equation for a charged particle in external electric and magnetic fields. Section  4 is devoted to the analysis of the Euclidean framework both in classical and quantum terms in a generic way. In Section  5, the application of the Euclidean framework is done to the Fokker–Planck/Black–Scholes equations in order to determine its classical behavior and to establish how the second class constraints appear in the extreme correlated limit. Section  6 is devoted to the correct implementation of the propagator as a Hamiltonian constrained path integral. In Section  7, the semi-classical approximation is used to evaluate the transition probability density, with the object to determine the role of the constraints in semi-classical terms. Finally Section  8 contains the conclusions and further perspectives of this work.

Section snippets

Stochastic volatility models and the Fokker–Planck equation

In this section, the Fokker–Planck equation for a general stochastic volatility model will be presented and the interpretation of this equation as a Euclidean quantum Schrödinger equation, will permit to find the underlying associated classical dynamic of the quantum model.

The transition probability density P=P(t,S,σ|t,S,σ) for the stochastic differential equations  (1), (2) satisfies the Fokker–Planck equation  [25]Pt=122(G2P)S2+ρ2(GqP)Sσ+122(q2P)σ2(μP)S(rP)σ. It is

Stochastic volatility models and option pricing

The stochastic volatility models can be applied to option pricing. In this case, the stochastic system (1), (2), together with the assumption of no-arbitrage and by using a self financing portfolio constructed from the underlying asset and options  [1], [2], imply the following generalized Black–Scholes equation for the option price P(S,σ,τ)Pτ+12G22PS2+ρGq2PSσ+12q22Pσ2+r(SPSP)+(pλq)Pσ=0 where the function λ=λ(S,σ,τ) is called the “market price of risk”. As mentioned in Ref.  [2],

Euclidean framework

In this section, the analysis of the Euclidean path integral and Euclidean classical dynamics associated with Eq. (9) is done in a generic way.

Euclidean classical dynamics for the Fokker–Planck/Black–Scholes equations

In this section, the classical dynamic associated to the Fokker–Planck/Black–Scholes equations is studied. For the rest of the paper it is assumed that both functions G, q in Eqs. (1), (2) satisfy  G(S,σ,t)0 and q(S,σ,t)0  in order to obtain a non-trivial coupled stochastic system of equations for the underlying price S and the volatility σ.

The transition probability density as a path integral

The transition probability density “wave function” P is given by P(S,σ,t)=K±(S,σ,t|S,σ,0)P0(S,σ)dSdσ where the propagator is K±(S,σ,t|S,σ,0)=Sσ|eHˆ±t|Sσ and Hˆ± is the Fokker–Planck/Black–Scholes Hamiltonian operator evaluated at ρ=±1.

For the probability density P, the initial condition is just Dirac’s delta  [25]P0(S,σ)=δ(SS,σσ) so the transition probability is the propagator itself P(S,σ,t)=K(S,σ,t|S,σ,0). The Euclidean path integral for the propagator of systems with

Semi-classical approximation

In order to explore the role of the constraints in the dynamic, the semi-classical method will be used to find an approximation to the transition density probability K. For this, the solutions of the classical motion equations are needed. To obtain the classical equations of motion, in general one must search for the extreme of action given in the exponent of the path integral (95)A=t0t[(σ̇Aσ)22q2+Φ]dτ subject to the constraint equation (96), that is, one must consider the extended action Ã=

Conclusions and further research

Thus, stochastic volatility models in continuous time for the extreme correlated case, are second class constrained Hamiltonian systems. This is a universal property, that is, valid for arbitrary μ, G, r and q functions in the stochastic system (1), (2). Because the emergence of the constraint for ρ=±1, its stochastic dynamic is completely different of the 1<ρ<1 case. Thus, one cannot take the limit ρ±1 from the 1<ρ<1 world, procedure that, in fact, gives divergent quantities. The correct

References (28)

  • Tim Bollerslev

    Generalized autoregressive conditional heteroskedasticity

    J. Econometrics

    (1986)
  • J. Perello et al.

    Physica A

    (2004)
  • M. Contreras et al.

    Option pricing, stochastic volatility, singular dynamics and constrained path integrals

    Physica A

    (2014)
  • Wilmott Paul

    Paul Wilmott on Quantitative Finance

    (2000)
  • Gatheral Jim

    The Volatility Surface

    (2006)
  • S. Heston

    A close form solution for option with stochastic volatility, with applications to bond and currency options

    Rev. Financ. Stud.

    (1993)
  • J. Cox

    Notes on Option Pricing I: Constant Elasticity of Diffusions. Unpublished Draft

    (1975)
  • D.C. Emanuel et al.

    Further results of the constant elasticity of variance call option pricing model

    J. Financ. Quant. Anal.

    (1984)
  • Patrick S. Hagan, Deep Kumar, Andrew S. Lesniewski, Diana E. Woodward, Managing smile risk, Wimott magazine July 2002...
  • John Hull et al.

    The pricing of options on assets with stochastic volatilities

    J. Finance

    (1987)
  • Lin Chen

    Stochastic mean and Stochastic volatility: a three-factor model of the term structure of interest rates and its application to the pricing of interest rate derivatives

    Financ. Mark. Inst. Instrum.

    (1996)
  • Rama Cont et al.

    Financial Modelling with Jump Processes

    (2004)
  • D. Delpini et al.

    Phys. Rev. E

    (2011)
  • G. Bormetti

    J. Stat. Mech.

    (2008)
  • Cited by (0)

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