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2
Asymptotics of Implied Volatility in the Gatheral Double Stochastic Volatility Model

Gatheral’s (2008) double-mean-reverting model by is motivated by empirical dynamics of the variance of stock price. No closed-form solution for European option exists in the above model. In this chapter, we study the behavior of the implied volatility with respect to the logarithmic strike price and maturity near expiry and at-the-money. Using the method by Pagliarani and Pascucci (2017), we explicitly calculate the first few terms of the asymptotic expansion of the implied volatility within a parabolic region.

2.1. Introduction

The history of implied volatility can be traced back at least to Latané and Rendleman (1976), where it appeared under the name “implied standard deviation”, i.e. the standard deviation of asset returns, which are implied in actual European call option prices when investors price options according to the Black-Scholes model. For a recent review of different approaches to determine implied volatility, see Orlando and Taglialatela (2017). To give exact definitions, we use Pagliarani and Pascucci (2017).

In order to briefly explain our contribution to the subject, we will introduce some notations. Let d ≥ 2 be a positive integer, let T0 > 0 be a time horizon, let T ∈ (0, T₀], and let { Z_t : 0 ≤ t ≤ T } be a continuous ℝ^d-valued adapted Markov stochastic process on a probability space with a filtration . Assume that the first coordinate St of the process Z_t represents the risk-neutral price of a financial asset, and the d - 1 remaining coordinates Y_t represent stochastic factors in a market with zero interest rate and no dividends.

On one hand, we have the time t no-arbitrage price of a European call option with strike price K > 0 and maturity T is Ct,T,K = v(t,St,Yt,T,K), where

and where (t, s, y) ∈ [0,T] × (0, ∞) × ℝ^{d 1}. We change to logarithmic variables and define the option price by

where x is the time t log price of the underlying asset, k is the log strike of the option, and (t,x, y) ∈ [0, T] × ℝ × ℝ^d-1.

On the other, the Black-Scholes price in logarithmic variables is

[2.1]

and τ = T − t ∈ [0, T], x, k ∈ ℝ, is the cumulative distribution function of the standard normal random variable.

DEFINITION 2.1.- The implied volatility σ = σ(t,x, y, T, k) is the unique positive solution of the nonlinear equation

REMARK 2.1.- In the literature on option pricing, there are concepts of model implied volatility and market implied volatility. If the right-hand side of the above equation, i.e. u(t,x,y,T,k), refers to the European option price under a given model, then σ = σ(t,x,y,T,k) is called the model implied volatility. If u(t,x, y,T,k) is replaced by the observed market option price, then we have the so-called market implied volatility. Here, we work with the (model) implied volatility.

Pagliarani and Pascucci (2012) derived a fully explicit approximation for the implied volatility at any given order N ≥ 0 for the scalar case. Lorig et al. (2017) extended this result to the multidimensional case. Denote the above approximation by .

Pagliarani and Pascucci (2017) proved that under some mild conditions, the following limits exist:

where the limit is taken as (T,k) approaches (t,x) within the parabolic region

for an arbitrary positive real number λ and nonnegative integers m and q.

Moreover, Pagliarani and Pascucci (2017) established an asymptotic expansion of the implied volatility in the following form:

[2.2]

as (T,k) approaches (t,x) within .

We apply the above described theory to the double-mean-reverting model by Gatheral (2008) given by the following system of stochastic differential equations:

[2.3]

where the Wiener processes are correlated: , and where parameters κ1,κ2, θ, ξ1, ξ2, α1, α2 are the positive real numbers. Note that while S₀ is observable in the market, ν0 and are usually not observable and may be calibrated from the market data on options.

In this model, with rate κ₁ the variance νt mean reverts to a level which itself moves over time to the level θ at a (usually slower rate) κ2, hence the name double-mean-reverting. Here, parameters α1, α2 ∈ [1/2,1]. In the case of α1 = α2 = 1/2, we have the so-called double Heston model; in the case of α1 = α2 = 1, the double lognormal model; and finally, in the general case, the double CEV model (Gatheral 2008).

The DMR model can be consistently calibrated to both the SPX options and the VIX options. However, due to the lack of an explicit formula for both the European option price and the implied volatility, the calibration is usually done using time-consuming methods like the Monte Carlo simulation or the finite difference method. In this chapter, we provide an explicit solution to the implied volatility under this model.

In section 2.2, we formulate three theorems that give the asymptotic expansions of implied volatility of orders 0, 1 and 2. Detailed proof of Theorems 2.1 and 2.2 as well as a short proof of Theorem 2.3 without technicalities are given in section 2.3.

2.2. The results

Put xt = ln St.

THEOREM 2.1.- The asymptotic expansion of order 0 of the implied volatility has the form

THEOREM 2.2.- The asymptotic expansion of order 1 of the implied volatility has the form

THEOREM 2.3.- The asymptotic expansion of order 2 of the implied volatility has the form

[2.4]

2.3. Proofs

PROOF OF THEOREM 2.1.- First, we perform the change of variable χt = ln St in the system [2.3]. Using the multidimensional Itô formula, we obtain

The infinitesimal generator of this system is

with z = (x,y,z)^T. We have

From Pagliarani and Pascucci (2017), Definition 3.4, we have

[2.5]

where the terms on the right-hand side of equation [2.5] are the values of the functions given by (Pagliarani and Pascucci 2017, Equation 3.15), when , and . (Pagliarani and Pascucci 2017, Equation 3.15) is recursive, and we define the above functions for n = 0 first.

Following Lorig et al. (2017), Appendix B, put . From Lorig et al. (2017), Equation 3.2, we have

where . It follows that . Then, we have

and Theorem 2.1 follows from [2.2] and [2.5].

PROOF OF THEOREM 2.2.- Let n ≥ 1, and let h be an integer with 1 ≤ h ≤ n. The Bell polynomials are defined by Pagliarani and Pascucci (2017) in Equation E.5

where the sum is taken over all sequences { ji : 1 ≤ i ≤ n – h + 1 } of non-negative integers such that

Let u^BS (σ; τ,x,k) be the Black-Scholes price [2.1]. Pagliarani and Pascucci (2017, Equation 3.15) has the form

[2.6]

For the sake of simplicity, we have omitted the last three arguments of the function u^BS and all arguments of the functions and .

To define , consider the differential operator

where

[2.7]

and

are the terms of the Taylor expansions of the functions aij(z) and ai(z) around the point .

Following Pagliarani and Pascucci (2017), define the vector by

the matrix by

and the operator by

[2.8]

Define the set In,h by

and the operator as the differential operator acting on the z-variable and defined by (Pagliarani and Pascucci 2017, Equation D.2) as

The function in equation [2.6] is defined by (Pagliarani and Pascucci 2017, Equation D.1)

[2.9]

Here, we wrote all the arguments of the function to show that it

does not depend on y and z.

Note that a₁₁(z) = -a₁(z). It follows that

and equation [2.9] can be written as

where the operator is given by (Lorig et al. 2017, Equation 3.14) as

[2.10]

It is well-known that

The first term on the right-hand side of equation [2.6] takes the form of (Lorig et al. 2017, Equation 3.13)

It follows that there exist functions such that

(see Lorig et al. (2017), Equation 3.15). This is because the function does not depend on y and z.

From Lorig et al. (2017), Lemma 3.4, we have

[2.11]

where

and where

is the mth Hermite polynomial.

We must still calculate the expression in the third line of equation [2.6] for h ≥ 2 (it is equal to 1 when h = 1). From Lorig et al. (2017), Proposition 3.5, we have

[2.12]

where the coefficients ch,h−2q are defined recursively by

Using equations [2.7] and [2.10], we explicitly calculate:

and

where the dots denote the terms containing and ∙ The functions take the form

Equation [2.6] gives

Then,

and

As T → t and k → x, the second and third terms disappear. Calculating the derivative with respect to k, we obtain

and Theorem 2.2 follows.

PROOF OF THEOREM 2.3.-

Equation [2.6] takes the form

[2.13]

The sets I₂,h are I₂,₁ = {(2)}, I₂,₂ = {( 1, 1)}. We have a₁₁,₂(x,y,z) = 0. It follows that equation [2.10] with n = 2 includes only summation over the set I₂,₂ and takes the form

While calculating the operator using equation [2.8], we need to calculate only the coefficients of the three partial derivatives with respect to the variable x. We obtain

The following integrals are important for calculations:

The operator takes the form

Calculation of the first term on the right-hand side of equation [2.13] using equation [2.11] may be left to the reader.

Next, we calculate the left-hand side of equation [2.12] for h = 2. Using the Hermite polynomials H0(ζ) = 1, H1 (ζ) = 2ζ and H2(ζ) = 4ζ2 - 2, we obtain

Combining everything together, we obtain the formula for

[2.14]

where the ellipsis denotes the terms satisfying the following condition: the limits of the term, its first partial derivative with respect to T and its first two partial derivatives with respect to k as (T,k) approaches (t,x) within are all equal to 0.

On the right-hand side of equation [2.14], the first term, the partial derivatives with respect to T of the second, fourth and sixth terms, the first partial derivative with respect to k of the third term, and the second partial derivative with respect to k of the fifth term give nonzero contributions to the right-hand side of the asymptotic expansion [2.4].

2.4. References

Gatheral, J. (2008). Consistent modelling of SPX and VIX options. The Fifth World Congress of the Bachelier Finance Society, London.

Latané, H.A. and Rendleman Jr., R.J. (1976). Standard deviations of stock price ratios implied in option prices. J. Finance, 31(2), 369–381.

Lorig, M., Pagliarani, S., Pascucci, A. (2017). Explicit implied volatilities for multifactor local-stochastic volatility models. Math. Finance, 27(3), 926–960.

Orlando, G. and Taglialatela, G. (2017). A review on implied volatility calculation. J. Comput. Appl. Math., 320, 202–220.

Pagliarani, S. and Pascucci, A. (2012). Analytical approximation of the transition density in a local volatility model. Cent. Eur. J. Math., 10(1), 250–270.

Pagliarani, S. and Pascucci, A. (2017). The exact Taylor formula of the implied volatility. Finance Stoch., 21(3), 661–718.

Chapter written by Mohammed ALBUHAYRI, Anatoliy MALYARENKO, Sergei SILVESTROV, Ying NI, Christopher ENGSTRǑM, Finnan TEWOLDE and Jiahui ZHANG.