Analytical Path-Integral Pricing of Moving-Barrier Options Under Non-Gaussian Distributions
In this work we present an analytical model, based on the path-integral formalism of Statistical Mechanics, for pricing options using first-passage time problems involving both fixed and deterministically moving absorbing barriers under possible non-gaussian distributions of the underlying object. We adapt to our problem a model originally proposed to describe the formation of galaxies in the universe of De Simone et al. (2011), which uses cumulant expansions in terms of the Gaussian distribution, and we generalize it to take into acount drift and cumulants of orders higher than three. From the probability density function, we obtain an analytical pricing model, not only for vanilla options (thus removing the need of volatility smile inherent to the Black & Scholes (1973) model), but also for fixed or deterministically moving barrier options. Market prices of vanilla options are used to calibrate the model, and barrier option pricing arising from the model is compared to the price resulted from the relative entropy model.
Keywords: non-gaussian distribution; stochastic processes; first-passage time; moving barrier, Black and Scholes model; cumulant expansion; path integral; Breeden-Litzenberger theorem; relative entropy.
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In Stochastic Processes, the first passage time , defined as the time a system takes to cross a barrier for the first time - usually associated to survival analysis - appears in several branches of science: from Biology, in cell transport phenomena, to Economics, in credit default events; Sociology, in group decisions; Physics, in Statistical Mechanics, Optics, Solid State; Chemistry, in reactions and corrosion; and Cosmology. In this latter case, to form a galaxy, the concentration of mass needs to reach a critical value, which can be seen as a barrier, not necessarily fixed, and possibly subject to a stochastic process.
The problem of finding the probability distribution of first passage time was first studied by Schrödinger (1915) in the context of a Brownian motion in a physical medium, and it was shown to be given by the Inverse Normal distribution. In Statistics, the distribution was first obtained by Wald (1947) in likelihood-ratio tests. In stochastic calculus, the problem can be studied in terms of the transition probability distribution between states emerging from boundary conditions imposed to the Fokker-Planck equation (Gardiner (2004), Risken (1989)), from which the cumulative probability distribution that the first passage time occurs after a given instant , that is, , is derived.
The study of first passage time depends on the distribution of the underlying process that is assumed. In the case of galaxy formation, Maggiore & Riotto (2010a) and Maggiore & Riotto (2010b) discusses the treatment of Gaussian distributions, while Maggiore & Riotto (2010c) and De Simone et al. (2011) treat non-Gaussian diffusion, the latter including the case of moving barriers. Usually, the non-Gaussian approach is developed in terms of expansions based on a benchmark distribution, which is commonly taken as the Gaussian one.
In Finance, the first passage time problem may arise in derivative contracts that establish deactivation or activation conditions, upon the passage of a time dependent variable through a barrier :
The most frequently used distribution in Finance is the lognormal distribution for prices, in the context of the Black & Scholes (1973) model hypothesis. A single barrier knock-up-and-out european call option (KUO european call) is then a contract that enables its holder to buy a certain asset , the underlying asset, at maturity date , paying the contract strike price , as long as the underlying asset does not cross a contractual barrier level . Mathematically, the payoff of such contract is
and its value at any time under the Black & Scholes (1973) hypothesis has closed-form analytical solution, as shown in Shreve (2004). However, although the lognormal distribution requires a single parameter of volatility, evidence is that vanilla (non-barrier) options of different strikes demand different implied volatilities, given their market prices. This structure of volatility, dependent on the strike values, is called volatility smile, and through time it generates the volatility surface. An additional issue is related to barrier clauses in the contract: should we select the volatility according to the strike, to the barrier, or both? Furthermore, in the special case of moving barrier options, Kunitomo & Ikeda (1992) provide analytical solution to the pricing problem, but in a Black-Scholes setup of one volatility.
Several approaches have been proposed in order to explain the volatility smile and, subsequently, to allow exotic options’ pricing, such as barrier options. Among them we may cite the local volatility model (Dupire (1994)), the stochastic volatility model (Heston (1993)), the jump model (Merton (1976)), the relative entropy model (Avellaneda et al. (2001)); and also non-Gaussian distribution models based on expansions, an example of which is the Edgeworth expansion (Rubinstein (1998) and Balieiro & Rosenfeld (2004)). While some of them pose difficulties related to the need of a complete set of market prices to build the volatility surface, others that rely on numerical or simulation implementations demand careful attention regarding the behaviour of the process surounding the barrier region.
In this paper we adapt the non-Gaussian model of galaxy formation, based on cumulant expansion, developed by De Simone et al. (2011), to the pricing of both fixed and deterministically moving up-and-out barrier options. By doing so, in the limiting case of infinite barrier values, we also obtain a non-Gaussian vanilla pricing model. Our adaptation consists on introducing a drift term in the expansion and also extending it to an arbitrary number of cumulants. In addition, we derive the martingale condition for risk-neutral pricing. The methodology employs the path integral formalism of Statistical Mechanics (Risken (1989)), and results in closed-form expressions for vanillas, fixed and deterministically moving barrier options. The development also takes into account the behaviour of the expansion in the neighbourhood of the barrier.
The model parameters, which are the cumulants, are calibrated with vanilla options and are afterwards used to price barrier options. As long as data for barrier options almost always refer to market-to-model quotes, we compare our results to the ones delivered by the relative entropy model (Avellaneda et al. (2001)).
With respect to the organization of the paper, we begin by describing the theoretical framework, which encompasses the development of Maggiore & Riotto (2010a) to De Simone et al. (2011). We first present the general formulation of cumulant expansion in terms of path integrals and recover the result for the Gaussian fluctuation and fixed barrier. Then we generalize to the case of non-Gaussian fluctuation and moving barrier. Next, we describe the calibration procedure, followed by the barrier options pricing. Finally, we present our conclusions. We collect in the appendices some results used in the text.
2 Cumulant expansion and the path integral formalism
In this section, we present the cumulant expansion, connecting it to the path integral formalism. Let be the stochastic variable whose distribution we wish to model. In our case, , where is the volatility parameter, and and are the underlying values at and , respectively. A path begins at , with , and evolves until the final instant of time , where . We assume the time discretization , with . A price path is a collection , such that . If there is no absorbing barrier, . The probability density in the space of trajectories can be described by the expected value of a product of Dirac delta functions:
which follows from
which is the probability density. aaaSee Risken (1989), section 2.4. In terms of , the probability that the variable assumes the value at instant , from , at , in trajectories that never exceed , is given by:
And the probability that the path remains in the region , for all instants lower than , is:
This equation represents the sum over all possible paths, thus representing the path integral that computes the probability function. We will express it in terms of the cumulants of the distribution. The characteristic function is the Fourier transform of the distribution bbbSee Risken (1989), section 2.3.:
The joint moment function is defined by
The joint moments are the coefficients of the Taylor expansion of the characteristic function:
The joint cumulants of a distribution are related to the characteristic function by
can be expressed in terms of the cumulants. To see this we use the following representation of the Dirac delta function
Substituting in (2),
We define the integration measure
This is the expansion of the probability of a given path in terms of the joint cumulants.
Keeping only terms with is equal to or , that is, (we will justify this shortly) results in:
Had we considered other values of , , , different from zero and one, we would have included generalized moments, beyond the usual covariance between two variables. For instance, the covariance between the fourth power of a variable and the cube of another variable , in the case of and , etc, and they contribute at higher orders. In our case, where we seek to calibrate market data, it will be enough to consider just the usual moments (variance, kurtosis, etc) and, thus, we will not consider covariances and its generalizations in the combinations of the several orders of the variables. In this notation, for example, in , when , we have the cumulant linked to the variance; in , when , the cumulant related to asymmetry, etc.
Using this expansion in (2.3) one obtains
This equation is the path integral representation of the probability distribution in termos of the cumulants.
3 Gaussian Fluctuations
In this Section we show that our formalism reproduces the well-known formulae for the price of barrier options for Gaussian fluctuations as a sanity check. In the case of Gaussian fluctuations, the cumulants are zero, except those satisfying cccRisken (1989), section 2.3.3.:
Besides, , where is the covariance between and .
Consider the case of Markovian processes, where just the previous state of the variable influences the present state:
We will denote and the Gaussian probability and probability density under the Markov hypothesis, that is, when the particle executes a Markovian Gaussian Brownian motion. In a stationary stochastic process, the moments are constant along time, and their values only depend on the least instante between periods. If the variable is standard Gaussian, as in a Wiener process,
The probability density (3.2) becomes:
To illustrate, consider one variable . Then,
In the case of Gaussian Markovian variables, with and , where is the drift:
Therefore we can write
In the presence of a fixed barrier, the probability density in the case of and up absorbing barrier , to be used in the call KUO pricing, under the Black-Scholes assumptions, is (Shreve (2004)):
4 Analytical expansion for non-Gaussian distributions with moving barrier in the path-integral formalism
In this Section, the non-Gaussian distribution with absorbing moving barrier is obtained from the path integral formulation. As in the work of De Simone et al. (2011), we present two alternative approaches in the expansion: (i) first, the hypothesis of Sheth & Tormen (2002), which states that instants are insignificant compared to in derivatives higher than the first order and (ii) second, barrier moves slowly. The latter we call “adiabatic barriers”.
The accomplishment of this task involves expanding the non-Gaussian distribution with moving barrier, , in an expression of the form:
where, in each approach (Sheth-Tormen and adiabatic barriers), assumes different formats, both involving the Gaussian distribution with fixed barrier (3.13), plus terms regarding moving barriers.
4.1 The Sheth-Tormen approach
Consider the expansion in cumulants (2.15), in the case of a barrier that moves according to a deterministic rule :
with given by (2.14)
Next, we assume that the barrier does not change significantly and expand in a Taylor series around . Therefore,
Redefining the variables , :
Since is a dummy variable, we will use the notation again. We work with the expansion until the 5th order, generalizing it later.
The first line of this equation is the Gaussian term. Applying the Taylor expansion to the exponential term of the non-Gaussian part (2nd and 3rd lines of (4.8)), one can write:
The summation term involving the barrier can also be expanded in Taylor series. We also consider up to second order:
(4.9) can be rewritten as:
Using as given by (B.22), we can decompose as
where the gaussian markovian piece was already given in Eq.(3.13).
The remainder of this Section is devoted to the computation of the different terms in Equation (4.12).
Defining , we note that