A Full Asymptotic Series of European Call Option Prices in the SABR Model with Beta = 1

We develop two new pricing formulae for European options. The purpose of these formulae is to better understand the impact of each term of the model, as well as improve the speed of the calculations. We consider the SABR model (with 1 β= ) of stochastic volatility, which we analyze by tools from Malliavin Calculus. We follow the approach of Alòs et al. (2006) who showed that under stochastic volatility framework, the option prices can be written as the sum of the classic Hull-White (1987) term and a correction due to correlation. We derive the Hull-White term, by using the conditional density of the average volatility, and write it as a two-dimensional integral. For the correction part, we use two different approaches. Both approaches rely on the pairing of the exponential formula developed by Jin, Peng, and Schellhorn (2016) with analytical calculations. The first approach, which we call “Dyson series on the return’s idiosyncratic noise” yields a complete series expansion but necessitates the calculation of a 7-dimensional integral. Two of these dimensions come from the use of Yor’s (1992) formula for the joint density of a Brownian motion and the time-integral of geometric Brownian motion. The second approach, which we call “Dyson series on the common noise” necessitates the calculation of only a one-dimensional integral, but the formula is more complex. This research consisted of both analytical derivations and numerical calculations. The latter show that our formulae are in general more exact, yet more time-consuming to calculate, than the first order expansion of Hagan


Introduction
European options are traditionally priced and hedged by Black-Scholes [1] (1973) model, one of the natural extensions of the Black-Scholes model to make volatility stochastic. The simplest stochastic volatility models assume that the volatility and the noise driving stock prices are uncorrelated. Moreover, the Hull-White formula [2] (1987) establishes that the European option price is the expectation of the Black-Scholes option pricing formula with a time-dependent volatility. An important success of this model is that it calculates European prices which implied volatilities smile. The development of local volatility models by Dupire and Derman (1994) was a major development in handling smiles and skews. However its predictions contradict empirical findings. Thus the SABR (stochastic alpha beta rho) model, a stochastic volatility model in which the asset price is correlated with its volatility was derived by Hagan et al. [3] (2002) to resolve this problem. Alòs [4] (2006) extended the classical Hull-White formula to the correlated case by means of Malliavin calculus. The new generalization decomposes option prices as the sum of the same derivative price if there was no correlation and a correction due by correlation. Another popular model is the Heston (1993) model. In that model, the volatility is mean-reverting. The general asymptotic method presented by Fouque, Papanicolau and Sircar (2000) [5] can be used to analyze Heston's model. For more information on stochastic volatility models, we refer the reader to Gatheral [6] (2006).
Nevertheless, there are still terms of conditional expectation of functions of non-adapted processes in the new generalization of Hull-White formula. Jin, Peng and Schellhorn [7] (2016) showed that under certain smoothness conditions, a Brownian martingale can be represented via an exponential formula when evaluated at a fixed time. It is a powerful tool similar to Clark-Ocone formula that allows us to work with the conditional expectation of a random variable instead of the random variable itself.
The main goal of this research was to obtain an option pricing formula for the special case of the SABR model with 1 β = . We used two different approaches.

Preliminaries on Malliavin Calculus
The following section briefly reviews some basic facts of Malliavin Calculus required along the paper. For a complete exposition we refer to Nualart [9] (1995) and Øksendal [10]  In Section 2.4, we will enlarge our probability space to consider two standard Brownian motions.

Malliavin Derivative
0, u u t t T = ∈ , be a measurable  -adapted stochastic process such that and its Skorohod integral coincides with the It integral We will need the following results on the Malliavin derivative.

Exponential Formula
A Brownian motion martingale can be represented via an exponential formula Given ω ∈ Ω , a freezing operator t ω is defined as: The freezing operator t ω is a mapping from Ω to Ω . The following equations show some properties of the freezing operator: We denote the Malliavin derivative of order l of F at time t by l t D F , as a    (14) where the sum is over all n-tuples of non-negative integers ( ) , , Definition 2.14 Exponential Bell polynomials. The partial or incomplete exponential Bell polynomials are a triangular array of polynomials given by ( ) where the sum is taken over all sequences 1 2 1 , , , n k j j j − +  non-negative integers such that these two conditions are satisfied: is called the n th complete exponential Bell polynomials.

Extension to Two Brownian Motions
In what follows, we work with two independent Brownian motions { } 0 The freezing operators t W ω and t Z ω follow the same definition 2.9 as the one dimension case. However, each random variable are depend on the the path of single Brownian motion indicated by its subscript.

Preliminaries on Option Pricing
Throughout this paper we shall operate in the context of a complete financial market. Options are an example of a broader class of assets called contingent claims. We will study European call option pricing under stochastic framework.
The aim of this section is to review the basic objects, ideas and results of the classical Black-Scholes theory, stochastic volatility models of derivative pricing [11].
Definition 3.1 1) A contingent claim is any asset whose future payoff is contingent on the outcome of some uncertain event.
2) A European call option is a contract that gives its holder the right, but not the obligation, to buy one unit of an underlying asset for predetermined strike price K on the maturity date T.

The Black-Scholes Theory
The Black-Scholes model is widely used for the dynamics of a financial market containing derivative investment instruments. From the Black-Scholes equation, one can deduce the Black-Scholes formula, which gives a theoretical estimate of the price of European-style options. The Black-Scholes model with constant volatility under risk-neutral probability measure is that the stock price t S satisfies the following stochastic differential equation: where r and σ are constants. For reasons of convenience, we make the change of variable in the following sections, let ln t t X S = denote the logarithm of stock price, then the price t V of an European call option with payoff ( ) at time t for this model with constant volatility σ , current stock price e x , maturity time T and interest rate r, satisfy the risk-neutral pricing formula [12]: 2π is the standard normal cumulative distribution function. The derivation consists of finding a self-financing investment strategy, that replicates the call option payoff structure and assume that one continuously adjusts the replicating portfolio over time.

Stochastic Volatility Models
That it might make sense to model volatility as a random variable should be clear to the most casual observer of equity markets. Nevertheless, given the suc-   1  1  1  ln  ln  24  1920   1  1  2 3  1  ,  24 4 24 For the case of at-the money options, i.e. when 0 S K = , this formula reduces In the special case 1 β = , the SABR implied volatility formula reduces to

Exponential Functions of Brownian Motion
Marc Yor's discovery (1992) of an integral formula for joint density of the distribution of a Brownian motion and the integral of exponential Brownian motion taken over a finite time interval has been computed in the case 2 σ = . Proposition 3.2 Marc Yor's formula. Applying Brownian motion rescaling [13], this joint density of ( ) e e e sinh sin d , , 0. 2π By Lyasoff [14], (32) is equivalent to the following: x y x y t

Hull and White Formula and Extension
The no-arbitrage price at time t using the risk-neutral theory for any derivatives with terminal time T and payoff function ( ) h x is given by the risk-neutral formula below:

Hull-White Formula: Uncorrelated Volatility
Under the assumption that the volatility t σ is uncorrelated with the asset price driven by another Brownian motion t Z , i.e. when 0 ρ = , the pricing formula (35) can be further simplified. By conditioning on the path of the volatility process and using the iterated conditioning property, the European call option price is given by The inner expectation is the Black-Scholes computation with a time-dependent volatility. Since t σ is a Markov process, we can apply the Black-Scholes formula, and obtain: is the root-mean-square time future average volatility.

Hull-White Formula: Correlated Volatility
In general, the situation is more complicated when volatility is correlated with

A Generalization of Hull-White Formula
The Notice that formula (44) does not reduce the dimensionality of the problem but identifies the impact of correlation. When 0 ρ = , it is the same as (38).

Application of Marc Yor's Formula
Throughout this and next section we denote by One straightforward application of (47) is using the conditional density of

Application of Exponential Formula
Note that the volatility process in (26) is the differential notation for

Option Pricing Formula for SABR Model
Remark Equation (57) shows that s X ω is a function depends only on two random variables: ,  (3.2) in Section 3, with properly parameters.
Remark , x y s X represent a real-valued function of (s, x, y) which mimic the definition of s X ω but replace ,

First Order Approximation Pricing Formula for SABR Model
One obvious drawback of formula (66) is that the option price is a 7-dimensional integral when the volatility is correlated with underlying asset, which could be computationally expensive, even for the first order approximation. In this section, we reverse the order of the two major steps that have been used in previous section by first using the conditional probability density to solve one Brownian motion, then apply Exponential formula to the remaining.

Numerical Approximation
In Tables 1-3  The average calculation speed for each methods are listed in Table 4.

Conclusion
We derived that the European call option price for SABR model with 1 β = in two different approaches by means of Malliavin Calculus. The full Dyson series expansion is a high dimension integration with its integrand to be an infinite sum of asymptotic series. The second approach uses similar method as previous one but with different order; it yields to a first order approximation by time integral for the correction part of option price. A big advantage of the latter is that the integrand is analytic function. Besides, some partial results can be further extended to fractional Brownian motion case, which will be our future work.

− = =
, and by Lemma 2.15, Therefore, substitute M and N in (92) and combining like terms, we have ( )  Step 2: Calculation of ( )