On a New Parametrization Class of Solvable Diffusion Models and

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On a New Parametrization Class of Solvable Diffusion Models and Transition Probability Kernels? Sebastian Tudor1,† , Rupak Chatterjee1,2 and Igor Tydniouk3 1

2

Financial Engineering Division, Stevens Institute of Technology, Hoboken, NJ 07030, USA Center for Quantum Science and Engineering, Stevens Institute of Technology, Hoboken, NJ 3 Citigroup Global Markets Incorporated, New York, NY 10013, USA (This version: January 2018)

We present in this paper a novel parametrization class of analytically tractable local volatility diffusion models used to price and hedge financial derivatives. A complete theoretical framework for computing the local volatility and the transition probability density is provided along with efficient model calibration for vanilla option prices, and reliable extrapolation procedures for producing the volatility surface. Our approach is based on the spectral analysis of the pricing operator in the time invariant case, along with some specific properties of the class that we propose. In order to show the advantages of our model, numerical examples with market data are analyzed. Finally, some extensions of our parametrization class are discussed in the conclusions. Keywords: Local Volatility, Transition Probability, Spectral Decomposition, Robust Model Calibration JEL Classification: C65 and G13

1.

Introduction

The constant–volatility Black–Scholes–Merton model dSt = St (rdt + σdWt ), see Black and Scholes (1973), Merton (1973b), Merton (1973a), is still among the most used models for option pricing in financial practice. However, more recent evidence suggests that a constant volatility model is not adequate Rubinstein (1994) and the pricing formula has significant biases, see e.g. Rubinstein (1985), Dupire et al. (1994). Indeed, numerically inverting the Black-Scholes formula on real data sets supports the notion of asymmetry with option prices (volatility skew), as well as a dependence on time to expiration (volatility term structure). The challenge is to accurately and efficiently model this volatility smile and understand the dynamics of the volatility process. The quantitative finance literature has considered local volatility (LV) and stochastic volatility (SV) models as means of explaining the observed market smile. Local volatility models assume that the volatility σ(S, t) is a deterministic function of the underlying price S and of the time t > 0. Assuming that we know European option prices C(K, T ) for a continuous set of strikes K and expiries T , Dupire, Derman, and Kani, see Dupire et al. (1994) and Derman et al. (1996), provided

? The

first author was supported by the Hanlon Financial Systems Laboratory, Hoboken, NJ 07030, USA. The second author was partially supported by the CME Group Foundation. † Corresponding author. Email: [email protected]

1

Electronic copy available at: https://ssrn.com/abstract=3108105

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the formula that links C(K, T ) and local volatility in the scalar case: v u ∂C u u ∂T σ(K, T ) = u . t1 ∂2C K2 2 ∂K 2

(1)

The Dupire formula (1) guaranties the existence of the unique stochastic process St (implied process) that “re-prices” vanilla options for the given sets of strikes and expiries, and can be interpreted as “some kind of average over all possible instantaneous stochastic correlations (an effective correlation)” Gatheral (2011). Nevertheless, one can identify a few issues with Dupire’s local volatility formula: (i) in reality, the strike price and expiry dates are discrete, and therefore, in order to compute (1) we need numerical methods, e.g., interpolation, finite difference approximations, that are known to possess stability issues; (ii) there is no direct connection to the transition probability density (TPD) of the underlying stochastic process; (iii) interpolation and extrapolation of local volatilities is very subjective; (iv) price and time dimensions are considered together and cannot be treated separately; (v) it cannot be easily extended to the multivariate case. Stochastic volatility models assume that the volatility itself is a stochastic process Hull and White (1987). In this case, the market becomes incomplete since it is not possible to hedge and trade the volatility with a single underlying asset. For local and stochastic volatility models, the resulting partial differential equation becomes complicated and only a few exact solutions are known Heston (1993), Albanese et al. (2001). A good presentation of solvable models is given in Linetsky (2004), where the authors used spectral decomposition techniques to obtain transition probability densities. More recently, solvable diffusion models with linear and mean–reverting drifts were analyzed and promising numerical results were obtained Campolieti and Makarov (2017), Campolieti and Makarov (2012). Perhaps the most important problem that needs to be addressed here is how to compute the price of a given financial instrument, and to characterize and/or restore the transition probability density p(t, x|x0 ) from market prices (also known as the inverse problem of option pricing). The pricing problem for exotic options, such as barriers, cliquets, etc., is rather difficult, since it involves complicated PDEs and complex pricing algorithms, see Fusai and Roncoroni (2007), Clewlow and Strickland (1996), Seydel (2012), Rouah (2013) for a good presentation of analytics, numerical techniques and algorithms. Model calibration is also an important problem, both from numerical and theoretical point of view, since it involves nonlinear and/or non-smooth optimization techniques. 1.1.

Solution Methods

In order to tackle the pricing problem, the calibration problem, and the inverse problem for a diffusion process, we present here a novel parametrization class, based on the spectral analysis of the infinitesimal generator Øksendal (2003) in the time homogeneous case. The technical tools used here span wide topics of mathematical physics: spectral decomposition and spectral analysis of operators Phillips and Hille (1957), partial differential equations Strauss (1992), and methods of operator identities Sakhnovich (2012). These tools have been widely used in the quantitative financial literature, see e.g. Henry-Labord`ere (2005), Henry-Labord`ere (2007), Henry-Labord`ere (2008), Linetsky (2004), and several extensions and generalizations can be made. We propose in this paper a new local volatility model, that can be easily extended and generalized. Our modeling approach relies on a special class of functions, namely solutions of some nonlinear integrable differential equations Tydniouk (2017). The method of operator identities Sakhnovich (2012), not used in finance so far, proved to be a powerful tool in many areas, e.g. interpolation problems, spectral analysis, and inverse spectral problems among others. As it turns out, by using these tools we can address important and still unsolved problems in quantitative finance, e.g. interpolation of the volatility surface, reliable extrapolation of the volatility surface beyond calibration Electronic copy available at: https://ssrn.com/abstract=3108105

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intervals, a rigorous definition of the local correlation for multidimensional diffusions, etc.

1.2.

Contributions

Our goal is to provide a complete theoretical framework for modeling the local volatility function σ(S, t) that will address certain issues with Dupire’s formula, and to recover the transition probability density p(t, x|x0 ) of the process St in the scalar case. Furthermore, it will become clear in the following that our model has several advantages over the existing ones: (i) the non-arbitrage necessary condition can be imposed as p(t, x|x0 ) ≥ 0; (ii) the model is largely solvable and numerically efficient solutions to recover the TPD are available; (iii) efficient algorithms for computing the prices of options; (iv) an easy to implement, reliable, and elegant method for model calibration, due to the specific class of function that our model relies on. Numerical and theoretical examples will show the advantages of our model. It will become clear in the sequel that the local correlation function is a natural extension under our parametrization class.

1.3.

Outline of the paper

The paper is organized as follows. We begin with some preliminaries, where we briefly introduce the mathematical concepts, and present the market model and the pricing problem in the most general setup. Kolmogorov equations and the spectral decomposition theorem, as well as a brief discussion on one dimensional time–homogeneous diffusions will also be presented in the preliminaries. In Section 3 we present our new model for derivative pricing, and show an explicit solution for the simplest case. Calibration to option prices is presented in Section 4, and a simple example is analyzed in detail. Section 5 gives a comprehensive analysis of our model using market data. The paper ends with several conclusions and a proposal for future extensions and research. All proofs are deferred to an Appendix. Some useful technical results for nonlinear integrable differential equations and methods of operator identities are also given in the Appendix.

2.

Preliminaries

We begin with basic notations and definitions. Let R, R+ , and Rn denote the real numbers, the non-negative real numbers, and the n−dimensional real space, respectively. Here H denotes a real separable Hilbert space equipped with an inner product hx, yi ∈ R. The Borel σ−algebra on S ⊂ Rn will be denoted with B(S). We write B(S) to denote the Banach space of real–valued, bounded and Borel measurable functions on S with the supremum norm. We say that a linear operator D : Dom(D) ⊂ H → H is densely defined if the subset Dom(D) is dense in H. The adjoint operator is the uniquely defined linear operator D† satisfying hDx, yi = hx, D† yi. We call the operator D symmetric if ∀x, y ∈ Dom(D) we have hDx, yi = hx, Dyi. If in addition D† = D, D is called self–adjoint or Hermitian. We say that a number λ ∈ C is a regular value of D if the operator D − λI has a bounded inverse, i.e. Ker (D − λI) = {0}. Any λ that is not an regular value is called a singular value. The set of all regular values is called the resolvent of D, and the set of all singular values is called the spectrum of D and is denoted with Spec(D). It can be shown that the spectrum of a self–adjoint operator is a non-empty closed subset of R. The norm of D is equal to the absolute maximum eigenvalue, i.e. kDk = supλ∈Spec(D) |λ|. A stochastic process Xt (ω) is said to be Markovian if for any 0 ≤ s ≤ t and any f ∈ B(Rn ) we have E[f (Xt )|Fs ] = E[f (Xt )|Xs ]. The transition probability density of a continuous Markov process is denoted with p(x, t|y, s), and the process is time–homogeneous if p(x, t|y, s) = p(x, t − s|y, 0) for any 0 ≤ s ≤ t. It can be shown that all joint finite–dimensional probability densities of a Markov process can be determined in terms of p(x, t|y, s). Moreover, it is well–known that the TPDs are not arbitrary and satisfy the Chapman–Kolmogorov equation, see Cox and Miller (1977).

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2.1.

The Market Model

e) be a probability space, and let Ft , t ≥ 0 be a filtration on F, satisfying the usual Let (Ω, F, P conditions. Throughout the paper we will assume without loss of generality that the interest rate is zero. We also assume that there exist a probability measure P under which the market model is characterized by an Itˆ o diffusion process {Xt }t≥0 whose regular state space is some open interval S = (e1 , e2 ) ⊂ R: dXt = b(Xt , t)dt + σ(Xt , t)dWtP , X0 = x0 ,

(2)

where WtP is a P–Wiener process, x0 ∈ R is the initial condition, b : R × R+ → R and σ : R × R+ → R are measurable functions that satisfy the standard assumptions under which the SDE (2) has a unique t–continuous solution Xt (ω), see e.g. Øksendal (2003). All expectations are taken with respect to the risk–neutral measure P. Consider now a cash flow or a payoff 1 f : R → R contingent on the state of the process at some time T > 0, i.e. f (XT ). The pricing problem is to find a price for these contracts. Formally, the price at time 0 < t < T is an operator Pt : M → R+ which maps the space of all measurable functions M on a measurable space (Ω, F, P) to a non– negative value. The fair price at time t > 0 for a contract with payoff f (XT ) at time T can be represented as the risk–neutral expectation: i h V (x, t) = EP f (XT ) Ft (3) h i   = EP f (XT ) Xt = x = Pt (f ) (x). The second equality follows from the Markov Property of Xt . Several approaches are available for computing the fair price in (3), see Karatzas and Shreve (2012), Shreve (2004), Fusai and Roncoroni (2007), Clewlow and Strickland (1996). We will focus here on the PDE approach. The Feynman–Kac Theorem Øksendal (2003) shows that V (x, t) satisfies a parabolic PDE of the form ∂V (x, t) + DV (x, t) = 0, t > 0, x ∈ R, ∂t

(4)

with terminal condition V (x, T ) = f (x). The second–order differential operator D, also known as the infinitesimal generator of the Itˆ o diffusion (2), is given by D := b(x, t)

∂ 1 ∂2 + σ 2 (x, t) 2 . ∂x 2 ∂x

(5)

We recall now the link between the valuation of an European option, and the backward Kolmogorov equation. Let C(x, t, T ) be the fair value of an European option at time t > 0 maturing at T > t, with strike price K > 0. Then we have C(x, t, T ) = EP [max(XT − K, 0) Xt = x]. By definition of conditional expectation, C(x, t, T ) depends on the conditional transition probability density p(T, z|t, x) by: Z C(x, t, T ) = f (z)p(T, z|t, x)dz. (6) R

1 In

the most general setting, f can be a functional, and can depend continuously and/or discreetly on time.

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With the above and equation (4), one can obtain the backward Kolmogorov equation. We shall assume further that the solutions are time homogeneous, i.e. p(T, z|t, x) ≡ p(τ, z|x) only depends on τ := T − t and not on t and T separately. Then the backward Kolmogorov equation can be written as:  ∂p(τ, z|x)   = Dp(τ, z|x), ∂τ   p(0, z|x) = δ(z − x).

(7)

where the operator D is given in (5). The initial value problem (7) will be the main equation ∂ used in our analysis. Similarly, one can write ∂τ C(x, τ ) = DC(x, τ ), with the initial condition C(x, τ = 0) = f (x). Furthermore, C(x, t, T ) ≡ C(x, τ ) depends on τ = T − t and not on t and T separately, and we have C(x, τ ) = Pτ f (x).

2.2.

Spectral Decomposition

Let H = L2 (R, µ) be the Hilbert space of real–valued, square integrable functions on S ⊂ R with R the inner product h φ, ψi := S φ(x)ψ(x)µ(dx), ∀φ, ψ ∈ H. Then it can be shown that the operator D : Dom(D) ⊂ H → H given in (5) is a symmetric, densely defined operator on H. In addition, one can prove that D is unbounded and self–adjoint, see McKean (1956), Langer and Schenk (1990). Furthermore, the pricing semigroup {Pt , t ≥ 0} from (3) restricted to the space B(S) ∩ H extends uniquely to a strongly continuous semigroup of self-adjoint contractions in H. In this setup, one can apply the Spectral Decomposition Theorem to obtain the spectral representation as in Linetsky (2004): Z V (x, t) = Pt f (x) =

f (z)p(t, z|x)dz S

Z p(t, z|x) = Spec(D)

e−λt

2 X

ui (x, λ)uj (z, λ)ρij (dλ),

(8)

i,j=1

with t ≥ 0, x, z ∈ S. Here, the functions ui (x, λ), i = 1, 2 are eigenvalues of the operator D, and thus can be written as the solutions of the initial value problem: (Dui )(x, λ) = λui (x, λ),

u1 (x0 , λ) = 1, u01 (x0 , λ) = 0,

u2 (x0 , λ) = 0, u02 (x0 , λ) = 1,

(9)

(10)

where the derivative is taken with respect to x, x0 ∈ S is arbitrary, and λ ≥ 0. Here, ρ : R → R2×2 is a Borel measurable function called the spectral matrix, such that ρ11 ≥ 0, ρ22 ≥ 0, ρ212 ≤ ρ11 ρ22 . Note that the matrix ρ is semi–positive definite and symmetric. Furthermore, for f ∈ H, the

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spectral expansion for the value function can be written in the form: Z V (x, t) = Pt f (x) =

2 X

e−λt

Spec(D)

ui (x, λ)Gj (λ)ρij (dλ),

(11)

i,j=1

Z where Gj (λ) =

f (z)uj (z, λ)dz. S

3.

A new parametrization class of Local Volatility Models

Consider one dimensional Itˆ o processes. In addition to the spectral decomposition, we will apply a general transformation on the space variable, called the Liouville transformation, see Olver (2014). Let us consider the following time–homogeneous diffusion process with drift: dXt = b(Xt )dt + σ(Xt )dWt , X0 = x0

(12)

Note that we have assumed for simplicity that the functions b(x, t) and σ(x, t) are separable1 . Furthermore, if we consider an asset with price Xt whose forward is given by Ft := R T Xt exp t r(Xs )ds , then the SDE (12) becomes even simpler in the forward measure: Ft = σ(Ft )dWtF . 3.1.

(13)

The Liouville transformation

We need at this point some facts from the Sturm–Liouville theory, see Titchmarsh (2013), Dunford and Schwartz (1963), Levitan and Sargsian (1975). Consider a Sturm–Liouville (SL) second order ordinary differential equation with λ ∈ R and x ∈ S: 1 Au(x) := − σ 2 (x)u00 (x) − b(x)u0 (x) = λu(x). 2

(14)

The Sturm–Liouville operator A is the negative of the diffusion infinitesimal generator, i.e. A := −D. While the infinitesimal generator D is non-positive, the SL operator A is non-negative. Recall that by using the so–called Liouville transformation, any second order differential operator can be reduced to the the canonical form. Equation (14) can be written as 1 2 σ (x)u00 (x) + b(x)u0 (x) + λu(x) = 0. (15) 2 o nR x dt u(x). Then equation (15) takes the form Consider now the transformation ϕ(x) = exp 0 σb(t) 2 (t) ϕ00 (x) + I(x)ϕ(x) = 0, with I(x) = σ14 (2σ 2 λ − σ 2 b0 + 2bσσ 0 − b2 ), where we omitted the argument x for brevity. For x0 ∈ S ⊂ R arbitrary, define the Liouville Transformation by √ Z s(x) = 2

x

x0

1 The

dz , σ(z)

(16)

time–dependent separable process dXt = b(Xt )µ2 (t)dt + σ(Xt )µ(t)dWt is equivalent to the process (12) under the change R of local time τ = 0t µ2 (s)ds.

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and denote v(s) := u (x(s)). Then it can be checked that the function v(s) in the transformed coordinates satisfies the Sturm–Liouville equation −v 00 (s) + Q(s)v(s) = λv(s)

(17)

The potential function Q(s) has, in the transformed coordinates, the form 1 Q(s) = 2

(

σ 0 (s) σ(s)

0

1 − 2



σ 0 (s) σ(s)

2

− σ 2 (s)

"

b(s) σ 2 (s)

0

1 − 2



b(s) σ 2 (s)

2 #) .

(18)

Remark 1 Please note that, by abuse of notation, we denoted both functions σ(x) and σ e(s) := σ(x(s)) (in the original and transformed space variables, respectively) with σ. Any possible confusion will be clarified a priori.  Remark 2 In particular, if we assume that b(x) = 0, ∀x ∈ S, then the potential function Q(s) takes the form "    # 1 σ 0 (s) 0 1 σ 0 (s) 2 Q(s) = − . (19) 2 σ(s) 2 σ(s) Note that the equation (19) is a differential Riccati equation with respect to the function ψ(s) = σ 0 (s) d  σ(s) = ds ln(σ(s)). 3.2.

The Parametrization Class

Our approach is mainly based on the parametrization of potentials Q(s) by a special class of functions analyzed in detail in Tydniouk (2017). Consider the determinants: 

cosh(χ1 ) ω1 sinh(χ1 ) · · · ω1N −1 cosh(χ1 )



     cosh(χ2 ) ω2 sinh(χ2 ) · · · ω N −1 cosh(χ2 )  δ1 (s) = det  2 ,   .. .. .. ..   . . . . N −1 cosh(χN ) ωN sinh(χN ) · · · ωN cosh(χN ) (20) 

sinh(χ1 ) ω1 cosh(χ1 ) · · · ω1N −1 sinh(χ1 )



     sinh(χ2 ) ω2 cosh(χ2 ) · · · ω N −1 sinh(χ2 )  δ2 (s) = det  2 ,   .. .. .. . .   . . . . N −1 sinh(χN ) ωN cosh(χN ) · · · ωN sinh(χN ) when N ∈ N? is odd (for N even, the last column of δ1 and δ2 is in terms of sinh(·) and cosh(·), respectively, since sinh(·) and cosh(·) alternate column–wise), and N 1 Y ωk − αi χk (s) := ωk s − ln , k = 1, 2, . . . , N. 2 ωk + αi

(21)

i=1

The set of numbers (parameters) Ω = {ω1 , ω2 , . . . , ωN } and A = {α1 , α2 , . . . , αN } are such that α1 < α2 < · · · < αN < ω1 < · · · < ωN . We consider the following class of potential functions, with

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δ1 , δ2 given by (20):

Q(s) = 2

d2 ln |δ1 (s)|. ds2

(22)

For the class of potentials (19), the function σ(s) admits the representation (for more details, see the Appendix B for a brief overview on singular solutions of non–linear integrable differential equations, NIDE):  σ(s) =

δ1 (s) δ2 (s)

2 .

(23)

Remark 3 Notice from Appendix B that ln(σ(s)) = ξ(s) is chosen to be a solution of a nonlinear integrable differential equation (NIDE), namely the sinh–Gordon equation. This is actually the central idea of this paper: this specific parametrization for Q(s) and σ(s) gives a general theoretical framework for modeling the LV and the TPD, it allows interpolation/extrapolation of the volatility skew (smile) in a reliable way, it solves the pricing problem, and it solves the model calibration problem via the method of inverse interpolation problem Tydniouk (2017), thus avoiding numerical optimization. Moreover, for this parametrization class, the solution of the Sturm–Liouville equation can be obtained in explicit form. Remark 4 Elements of the spectral matrix function ρ(λ) are in this case rational functions with constant coefficients depending on the parameters Ω and A. Remark 5 One can observe that in this case, the functions of interest, namely the potential Q(s), the volatility function σ(s), and the transition probability density P (t, s|s0 ), are parameterized by the same set of parameters Ω and A. Remark 6 We consider in this paper s ∈ R+ or s ∈ R, i.e. the problem on half or full real axis, with the potential Q(s) satisfying the relations: Z

∞

−∞

(1 + |s|) Q(k) (s) ds < ∞, k = 1, 2, . . .

Moreover, in this case the spectrum Spec(D) consists of both discrete and continuous parts. Remark 7 Consider the eigenvalue problem Dφλ = λφλ , where λ ∈ Spec(D) are the eigenvalues, and φλ are the eigenvectors, assumed to be normalized functions (distributions), i.e., R Spec(D) φλ (s)φλ (s0 )dλ = δ(s − s0 ). Then the unique solution of the Kolmogorov equation (7) is given by: P (t, s|s0 ) =

R

Spec(D) e

−λt φ

λ (s)φλ (s0 )dλ

(24) =

P

k

e−λk t φ

λk (s)φλk (s0 )

+

R Spec/Specd (D)

e−λt φ

λ (s)φλ (s0 )dλ,

where we have split the spectrum into a discrete part {λk }k∈I = Specd (D) and a continuous part Spec/Specd (D). Here, P (t, s|s0 ) with s = s(x) is a space–transform of the transition probability density p(t, x | x0 ), see Henry-Labord`ere (2007). Remark 8 The transforms of the TPD P (t, s|s0 ) and of the option price V (s0 , t) can be obtained

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by using the solutions of SL equation (17): Z

2 X

e−λt

P (t, s|s0 ) = Spec(D)

vi (s, λ)vj (s0 , λ)ρij (λ)dλ,

(25)

i,j=1

Z f (s)P (t, s|s0 )ds.

V (s0 , t) =

(26)

S

Remark 9 In risk management, see Chatterjee (2014), the Greeks are vital tools. The Greeks measure the sensitivity of the option prices to a small change in a given underlying parameter, on which the value of the instrument is dependent. Component risks can be easily calculated in our framework, and the portfolio rebalanced accordingly to achieve a desired exposure: ∂ ∆(s0 , t) := V (s0 , t) = ∂s0

R

Γ(s0 , t) :=

∂2 V (s0 , t) = ∂s20

R

ν(s0 , t) :=

∆(s0 , t) ∂ V (s0 , t) = 0 . ∂σ σ (s0 )

S

S

f (s)

∂ P (t, s|s0 )ds ∂s0

f (s)

∂2 P (t, s|s0 )ds ∂s20

(27)

We picked for illustration purposes Delta, Gamma, and Vega, respectively. Notice that we can write Vega (sensitivity to volatility) as a function of Delta. Remark 10 Note that the functions of interest σ(s), P (t, s|s0 ), V (s0 , t), and the Greeks (27) are given in transformed coordinates. To obtain the formulae in terms of the initial coordinate x, we still need to invert the SL transformation (16). But this can be obtained by using the following explicit map between p(t, x|x0 ) and its transform P (t, s|s0 ), with s = s(x), s0 = s(x0 ): P (t, s|s0 ) = κeη(x,x0 ) p(t, x|x0 ),

(28)

where κ is a constant factor that ensures the initial condition limt→0 P (t, s|s0 ) = δ(s − s0 ), and η(x, x0 ) is given by: 1 η(x, x0 ) = − ln 2



σ(x) σ(x0 )



Z

x

+ x0

b(z) dz. σ 2 (z)

(29)

Notice that this mapping can be checked by direct substitution, since we recover the Schr¨odinger ∂2P type PDE in transformed coordinates, ∂P ∂t = ∂s2 + Q(s)P. For a detailed proof of this mapping using supersymmetric methods see Henry-Labord`ere (2008) and Henry-Labord`ere (2007).  3.3.

An explicit solution

We will present here in more detail our proposed model for the simplest parametrization class, namely for N = 1. From (20) we have: δ1 (s) = cosh(χ(s)), δ2 (s) = sinh(χ(s)), 1 ω − α χ(s) = ωs − ln . 2 ω + α

(30)

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Furthermore, by using (22) and (23) the local volatility function σ(s) and the potential Q(s) in transformed coordinates are given by: σ(s) = σ1 coth2 (χ(s)), Q(s) = −

(31)

2ω 2 , cosh2 (χ(s))

where σ1 ∈ R+ . Having the volatility function in transformed coordinates, we can obtain the inverse mapping x = x(s) by using (16). Notice first that (16) implies: dx σ(s) = √ ds 2

⇒

1 x(s) = √ 2

Z

s

σ(s0 )ds0 .

(32)

s0

Therefore:  1 σ1  x(s) = √ s − s0 − (coth χ(s) − coth χ(s0 )) . ω 2

(33)

Notice that equation (31) gives an explicit formula for the local volatility σ(s) in terms of s. To obtain the formula in terms of the initial coordinate x, we need to invert the SL transformation (16). The following results provides a method for obtaining an implicit equation for σ(x) when N = 1. The proof is deferred to the Appendix. Proposition 3.1 Consider N = 1. Let σ1 ∈ R+ be a constant and let x0 ∈ S be arbitrary. Then the implicit equation for the local volatility function σ(x) in the original coordinates is given by: s

r σ(x) √ 1 + σ(x) 1 2ω σ1 r − ln (x − x0 ). =± σ1 2 σ1 σ(x) 1 − σ1

(34)

Remark 11 Graphical representations for σ(x) in original coordinates are given in Figure 1, with different values of the parameters ω, σ1 , x0 . Note from the plots that our LV function has even in the simplest case N = 1 the volatility skew. Furthermore, the LV function can be easily adjusted to obtain any desired asymptotes via model calibration. We have proposed a novel model for the local volatility function, and simplified the option pricing problem and interpolation of volatility to solving a Schr¨odinger–Euler PDE in original coordinates and a Sturm–Liouville eigenvalue problem in transformed coordinates: −v 00 (s) + Q(s)v(s) = λv(s),

(35)

where Q(s) is given in (31). Two cases can be further analyzed: s ∈ [0, ∞), and s ∈ R. On the half–line, i.e. when s ∈ [0, ∞), the spectral function ρ(λ) is scalar valued. On the full line, s ∈ R, the spectral function becomes a diagonal 2 × 2 rational matrix function, with the same entries on the diagonal, see Sakhnovich (2012) for a comprehensive proof. Since most practical cases involve non–negative states, we shall restrict our analysis to the half–line. The following result gives the complete solution of the pricing and interpolation/extrapolation problems. √ R x du Proposition 3.2 Let N = 1 and assume that, in the transformed space s(x) = 2 σ(u) with x(s) in (33). Q(s) and σ(s) are as in (31). ConsiderR the mapping (28), with η(x, x0 ) given in (29). t Define the discount factor from 0 to t by D0t := e− 0 rs ds . Then then following statements hold:

newParametrizationClass

5

LV function in original coordinates with ω = 0.12, σ 1 = 0.1, x 0 = 5, α = 0.1

4.5

4

3.5

σ(x)

3

2.5

2

1.5

1

0.5

0 4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

9.5

x

5

LV function in original coordinates with ω = -0.12, σ 1 = 0.5, x 0 = 8, α = 0.1 "+" in the implicit equation "-" in the implicit equation

4.5

4

3.5

3

σ(x)

January 23, 2018

2.5

2

1.5

1

0.5 -2

0

2

4

6

8

10

12

14

16

18

x

Figure 1. The LV function for N = 1 in original coordinates for different values of the parameters ω, α, σ1 , x0 . We can pick either sign in (34), so that we obtain the reflected function σ(x) with respect the line x = x0 .

January 23, 2018

newParametrizationClass

(i) The spectrum is Λ = {−ω 2 , −α2 , 0} ∪ (0, ∞) and the spectral function is ω 2 − α2 ρ(λ) = √ . π λ(α2 + λ)

(36)

(ii) The explicit solutions of the SL problem (35) are v1 (s, λ) = √

v2 (s, λ) = √

w1 (s, λ) 2 ω ) (ω cosh(sω)

λ (λ +

− α sinh(sω))

,

w2 (s, λ) 2 ω ) (ω cosh(sω)

, − α sinh(sω)) √ √  √ 

w1 (s, λ) = sinh(sω) λ α2 + ω 2 cos λs + α ω 2 − λ sin λs + √   √  √
λs − 2α λ cos λs ω cosh(sω) λ − α2 sin λ (λ +

√ 
w2 (s, λ) = ω α2 + λ sin λs cosh(sω)− √   √  √

λs − λ ω 2 − α2 cos λs sinh(sω) α λ + ω 2 sin (iii) The unique TPD in transformed space is P (t, s|s0 ) =

R

e−λt v1 (s, λ)v2 (s0 , λ)ρ(λ)dλ,

λ∈Λ

=2

P5

i=1 Res[Ψ, pi ]

+2

R∞ 0

Ψ(s0 , s, λ, t)λdλ,

2

with Ψ(s0 , s, λ, t) := e−λ t v1 (s, λ2 )v2 (s0 , λ2 )ρ(λ2 ), and pi ∈ {±iω, ±iα, 0} are the poles of Ψ. (iv) The unique TPD in the original coordinates and the price of an European option with payoff f (XT ) = max{XT − K, 0} := (XT − K)+ are given by 1 −η(x,x0 ) e P (t, s(x)|s(x0 )) κ r C(K, t|x0 ) σ(K)σ(x0 ) = (x0 − K)+ + × D0t 2 R 1 − e−λt × v1 (s(K), λ)v2 (s(x0 ), λ)ρ(λ) dλ. λ λ∈Λ p(t, x|x0 ) =

(v) The Delta of the European call option is given by: s ∆(K, t|x0 ) 2ω σ(K)
= I{x0 >K} − × D0t σ(x0 ) sinh 2χ(s0 ) R 1 − e−λt × v1 (s(K), λ)v2 (s(x0 ), λ)ρ(λ) dλ λ λ∈Λ s +

σ(K) R ∂v2 1 − e−λt v1 (s(K), λ) (s0 , λ)ρ(λ) dλ, σ(x0 ) λ∈Λ ∂s λ

January 23, 2018

newParametrizationClass

where s0 = s(x0 ), and the partial derivative solution in part (ii).

∂v2 (s, λ) can be evaluated by using the explicit ∂s

Remark 12 The TPD for N = 1 and synthetic parameters is presented in Figure 2. The dynamics of p(t, x|x0 ) in time is also presented, with t1 and t2 fixed. In order to calculate the TPD surface, we used Proposition 3.2 part (iv). The residuals were calculated explicitly using Mathematica, and the integration over (0, ∞) was computed via the Gauss–Laguerre quadrature. Further, it can be checked numerically (by finite differences) that this function satisfies the corresponding PDE in original coordinates.

4.

Calibration to option prices

We have developed a procedure for obtaining a numerically efficient solution for the TPD and for pricing financial derivatives. One important issue remains to be addressed: how to determine the parameters Ω, A. We present in this section a novel calibration method for our model, which is based on Theorems B.2 and B.4 in the Appendix B, see also Tydniouk (2017). Therefore, by using our proposed parametrization class, we can calibrate without any optimization procedure. Let C(t, x; T, K) be the fair price at time t of an European Call Option with maturity T and strike K. The inverse problem of option pricing is to restore the TPD p(t, x|x0 ), i.e., to build the implied diffusion process for some fixed T and set of strikes {xk , k = 1, 2, . . . , M }. Lemma 4.1 The following PDE holds: 1 ∂ 2 F (x, t) ∂F (x, t) = σ 2 (x, t) , ∂t 2 ∂x2

(37)

with terminal condition F (xk , T ) = vk , k = 1, 2, . . . , M , where vk is the forward price of an European Option with maturity T > 0 and strike xk , k = 1, . . . , M . Note that we have 2N parameters to estimate from option prices, A = {α1 , . . . , αN }, and Ω = {ω1 , . . . , ωN } and so we need 2N equations. As we will see, we need the derivatives at some point x0 , which we will pick to be At–The–Money. The main idea of our calibration method is to use the results in Theorem B.3. We will consider for illustration purposes the case N = 1 in the following subsection. 4.1.

Calibration for a simple model

Consider N = 1. Define now, in transformed coordinates, the following function, known as a Pseudo–Exponential potential (see Appendix B): b0 (s) :=

σ(s) ˙ , σ(s)

(38)

where σ(s) ˙ denotes the derivative with respect to s. From Theorem B.3, we can recover our parameters by using the polynomial D(s, λ), since its roots gives the set Ω. For N = 1, the model has two parameters, namely α, ω. We have that (see Example B1 in the Appendix):  D(λ) = (−λ + a1 )(λ + a1 ) − b20    

1 − R1 σ(s0 ) 2ωs0 α=ω , R1 := e 1 + R1 ω + a1

(39)

newParametrizationClass

Transition Probability Density

0.7 0.6

p(t,x|x0 )

0.5 0.4 0.3 0.2 0.1 0 6 5 6

4

5

3

4 2

3 2

1

x

0

1

x0

0

Transition Probability Density

0.5

0.4

p(t,x|x0 )

January 23, 2018

0.3

0.2

0.1

0 1

6 5

2 4

3 3

4 2

5

x0

6

1

x

Figure 2. The transition probability density p(t, x|x0 ) for N = 1 and variable time. We considered the parameters to be α = 0.2856, ω = 0.4285 and we picked t1 = 0.04 years (top) and t2 = 0.8 years (bottom).

January 23, 2018

newParametrizationClass

with b0 (s) =

2 σ(s) ˙ b0 (s) σ ¨ (s)σ(s) − (σ(s)) ˙ , a1 (s) = 0 = . σ(s) b0 (s) σ(s)σ(s) ˙

Therefore, in order to find the parameters α, ω, we need to find numerically b0 and a1 . Thus, the problem reduces to finding σ(s), σ(s) ˙ and σ ¨ (s) numerically, for some arbitrary s. But this can be ∂ done using (37). Denote Θ(x, t) := ∂t C(x, t). By taking two derivatives with respect to x we get:  1 2   Θ(x, t) = 2 σ (x)Cxx (x, t)    Θx (x, t) = σ(x)σ 0 (x)Cxx (x, t) + 12 σ 2 (x)Cxxx (x, t)      Θxx (x, t) = (σ(x)σ 0 (x))0 Cxx (x, t) + 2σ(x)σ 0 (x)Cxxx (x, t) + 12 σ 2 (x)Cxxxx (x, t)

(40)

Notice now that σ(x)σ 0 (x) = σ(x)

dσ(s) ds √ = 2σ(s), ˙ ds dx

√ d √ ds d d 2¨ σ (s) (σ(x)σ 0 (x)) = 2 (σ(s)) ˙ = 2 (σ(s)) ˙ = . dx dx dx ds σ(x) Using the equations above in (40) at t = T, x = x0 (fixed maturity and at the money, ATM), we obtain:  1 2   θ 0 = 2 σ w2    √  θ1 = 2σw ˙ 2 + 12 σ 2 w3 (41)     √ σ   θ = 2¨ w2 + 2σw ˙ 3 + 12 σ 2 w4 , 2 σ where we used s0 = s(x0 ), σ(x0 ) = σ(s0 ) 1 , and ∂ i C(x, t) wi := , ∂xi x=x0 ,t=T

∂ j Θ(x, t) θj = . ∂xj x=x0 ,t=T

Solve the system of equations (41) for σ, σ, ˙ σ ¨ , and recover the parameters α, ω as described above, see equations (39). Notice that we used the derivatives ATM for C(x, t) and Θ(x, t). In order to obtain these numbers from market data (European option prices), we use Taylor series expansions. The following problem needs to be addressed: given the set of strikes {xk , k = 1, 2, . . . , M } and the European option prices vk := C(xk , T ), k = 1, 2, . . . , M , find the first 4 derivatives wi , i = 1, . . . , 4 at (x0 , T ). The Taylor Series expansion of C(x, t) at point x0 is: C(x, t) = C(x0 , t) + C 0 (x0 , t)(x − x0 ) +

1 00 C (x0 , t)(x − x0 )2 + 2!

1 1 + C 000 (x0 , t)(x − x0 )3 + C (4) (x0 , t)(x − x0 )4 + O((x − x0 )5 ), 3! 4! 1 Since our volatility model is invariant to shift, we can pick any constant σ to enforce σ(x ) = σ(s ). Thus we have an extra 1 0 0 degree of freedom for calibration.

January 23, 2018

newParametrizationClass

where the derivative is taken with respect to x. At x = xk , k = 1, 2, . . . , M , and t = T we have 1 that vk = w0 + w1 (xk − x0 ) + 21 w2 (xk − x0 )2 + 61 w3 (xk − x0 )3 + 24 w4 (xk − x0 )4 + O((xk − x0 )5 ), and so the system of equations becomes:  v1 = w0 + w1 (x1 − x0 ) + 21 w2 (x1 − x0 )2 + 16 w3 (x1 − x0 )3 +          v2 = w0 + w1 (x2 − x0 ) + 12 w2 (x2 − x0 )2 + 16 w3 (x2 − x0 )3 +

1 24 w4 (x1

− x0 )4

1 24 w4 (x2

− x0 )4

  v3 = w0 + w1 (x3 − x0 ) + 12 w2 (x3 − x0 )2 + 16 w3 (x3 − x0 )3 +        v4 = w0 + w1 (x4 − x0 ) + 12 w2 (x4 − x0 )2 + 16 w3 (x4 − x0 )3 +

1 24 w4 (x3

− x0 )4

1 24 w4 (x4

− x0 )4

(42)

We can solve the system above for w1 , w2 , w3 , w4 . Similarly, in order to obtain θ0 , θ1 , θ2 , we write v −v − tk = θ0 + θ1 (xk − x0 ) + 21 θ2 (xk − x0 )2 + O((xk − x0 )3 ), where tk := k∆t k , k = 1, . . . , M . Here, vk− denotes the option chain of yesterday (t − ∆t), and ∆t is the sampling time. Thus we have:   t1 = θ0 + θ1 (x1 − x0 ) + 21 θ2 (x1 − x0 )2 

(43)

t2 = θ0 + θ1 (x2 − x0 ) + 21 θ2 (x2 − x0 )2

Remark 13 Note that for the case N = 1, we only need four European option prices, since the model is fairly simple. For greater values of N , more prices and derivatives will be needed. Thus our model can be easily calibrated to any desired accuracy, the only restriction being the available market prices. Remark 14 Notice that one source of error in our calibration procedure is the Taylor series approximation, which is used to compute derivatives at (x0 , T ). Several alternative approaches are available: (i) one can consider a larger number of terms, giving rise to an overdetermined system of linear equations, for which approximate solutions can be found via ordinary least squares; (ii) one can fit a polynomial to market data; then, the problem of computing p0 (x0 ) becomes trivial.

5.

Numerical results

We test now the consistency of our model using market data. We consider options on the Dow Jones Industrial Average (DJI) for two consecutive days, October 3 and October 4, 2016, and we pick “today” to be October 4 2016. The price of European calls with different strikes and fixed 45 = 0.1786 years from maturity (we picked here options expiring on November 18 2016, i.e. T = 252 “today”) are plotted in Figure 3. Let N = 1 for our parametrization class and consider the calibration method described in Section 4.1. Equation (42) can be written in matrix form as: 

x1 − x0 21 (x1 − x0 )2 16 (x1 − x0 )3

   v 1 − w0 1 2 1 3   v2 − w0   x2 − x0 2 (x2 − x0 ) 6 (x2 − x0 )  =  v 3 − w0    x3 − x0 1 (x3 − x0 )2 1 (x3 − x0 )3  2 6 v 4 − w0  

x4 − x0 21 (x4 − x0 )2 16 (x4 − x0 )3

1 24 (x1

− x0 )4



   w − x0   1    w2  ,   1 4  w3 24 (x3 − x0 )  w4  1 24 (x2

1 24 (x4

)4 

− x0 )4

where x0 = 182.068 is the price today, {xk }4k=1 are strike prices in the interval [0.98x0 , 1.02x0 ], {vk }4k=1 are option prices corresponding to the selected strikes, and w0 = 3.4250 is the op-

January 23, 2018

newParametrizationClass

tion price at the money, in this case for the strike K = 182 (one can check that 182 − 1 < x0 −3 ). But this is a linear system of equations and can be easily solved to obtain w = 10  | 0.0152, 0.3524, −4.3022, −7.5302 . By using yesterday’s price, we compute the first variation in time of the option price {tk }4k=1 , v −v −

defined as tk = k∆t k , where vk− denotes the option chain of yesterday, i.e. at time (t − ∆t), and ∆t = 1/252 is the sampling time. The first variation in time of the option prices is also plotted in Figure 3. To find the coefficients θ1,2 we need to solve the following linear system, see (43): 





t1 − θ0 = t2 − θ0

x1 − x0 21 (x1 − x0 )2 x2 − x0 12 (x2 − x0 )



   θ1 , θ2 2

where θ0 = −113.4 is the first variation in time of the option | price at the money, in this case for K = 182. We have the solution θ = −12.1987 −11.7976 . Solve now (41) for σ, σ, ˙ σ ¨ , and then use equation (39) to obtain the parameters to be ω = 0.4285, α = 0.2856. Further, set x0 = S0 = 182.0680, and we pick σ1 = 8, since limx→∞ σ(x) = σ1 . In Figure 4, we present our local volatility function versus the Dupire local volatility, obtained numerically via (1) and finite difference approximations. Table 1 presents a pointwise comparison between Dupire local volatility and our local volatility model, given in Proposition 3.1. Note that interpolation and extrapolation of the local volatility function becomes a natural extension under our complete theoretical framework. Moreover, we have implemented numerically the formula for the price C(K, T ) in Proposition 3.2 part iv, and the results are given in Figure 4. The residuals were calculated explicitly using Mathematica, and the integral over (0, ∞) was computed in Matlab using the Gauss–Laguerre quadrature. The functions s(x) and σ(x) were computed graphically via the implicit form. The TPD in original coordinates p(t, x|x0 ) was calculated for t1 = 0.04 years and t2 = 0.8 years, see Figure 2. Please note that, even in this simple case with N = 1, we recover, with good precision, the prices for (plain) vanilla options. In Table 1, we show, for the given set of strikes, the market price C M versus C(K, T ), and we calculate the absolute and relative errors. Please note that the replication, while far from being perfect, shows that our model is indeed consistent, even for the simplest parametrization class, having N = 1. Future extensions will investigate more general parametrization classes, e.g. with N = 2, 3. The RMSE was computed as: v u 22  2 u1 X CjM (K, T ) − Cj (K, T ) = 0.220021. RM SE = t 22 j=1

Furthermore, we computed for illustrative purposes two Greeks, namely Delta and Gamma. The functions ∆(K) and Γ(K) are presented graphically in Figure 5. We also show in Table 1, for the given set of strikes, the computed Greeks via Proposition 3.2. The integral was computed using the Gauss–Laguerre quadrature.

6.

Conclusions and future work

We proposed in this paper a new parametrization class of solvable local volatility diffusion models, mainly used for pricing and hedging financial derivatives. We obtained the transition probability density and the local volatility function for the simplest parametrization class and showed the easiness to calibrate and price plain (vanilla) options. Several theoretical extensions and applications for our parametrization class are possible, thus

January 23, 2018

newParametrizationClass

Strike K

Market price C M

Price via Prop. 3.2

Absolute Error

Relative Error

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192

12.0000 11.1250 10.3250 9.4750 8.6250 7.8000 7.0000 6.2000 5.4500 4.7500 4.0750 3.4250 2.7750 2.2150 1.7100 1.2700 0.9000 0.6050 0.3850 0.2450 0.1450 0.0950

12.5973 11.6056 10.6252 9.6690 8.7322 7.8201 6.9395 6.0986 5.2822 4.5242 3.8099 3.1628 2.5666 2.0434 1.5875 1.2036 0.8931 0.6434 0.4511 0.3030 0.1998 0.1274

0.5973 0.4806 0.3002 0.1940 0.1072 0.0201 0.0605 0.1014 0.1678 0.2258 0.2651 0.2622 0.2084 0.1716 0.1225 0.0664 0.0069 0.0384 0.0661 0.0580 0.0548 0.0324

0.0498 0.0432 0.0291 0.0205 0.0124 0.0026 0.0086 0.0164 0.0308 0.0475 0.0650 0.0765 0.0751 0.0775 0.0717 0.0523 0.0077 0.0635 0.1716 0.2367 0.3782 0.3413

Strike K

Dupire LV (%)

LV via (34) (%)

Delta

Gamma

171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192

17.9432 17.5861 17.4771 16.9420 16.4473 16.1867 15.7133 15.1021 14.6740 14.2469 13.8241 13.2304 12.5798 11.9863 11.3966 10.8341 10.2821 9.7581 9.2811 8.8797 8.6500 8.5781

20.6100 19.4600 18.3800 17.3800 16.4500 15.5900 14.8000 14.0800 13.4100 12.8100 12.2600 11.7700 11.3200 10.9200 10.5600 10.2400 9.9600 9.7100 9.4900 9.2900 9.1200 8.9700

0.8154 0.8059 0.7942 0.7800 0.7626 0.7417 0.7168 0.6876 0.6528 0.6135 0.5685 0.5194 0.4652 0.4085 0.3501 0.2925 0.2384 0.1884 0.1444 0.1060 0.0759 0.0524

0.0199 0.0217 0.0238 0.0262 0.0288 0.0318 0.0351 0.0386 0.0423 0.0459 0.0492 0.0520 0.0539 0.0546 0.0539 0.0515 0.0478 0.0427 0.0368 0.0303 0.0241 0.0183

Table 1. Comparison between our model and market data for maturity T = 0.1786 years. The absolute and relative errors are reasonably small even in the simplest case with N = 1.

newParametrizationClass

Option Prices for Dow Jones Industrial Average

15

Prices today with S 0 = 182.068 Prices yesterday with S 0 = 183.0815

Call Option price C

10

5

0 165

170

175

180

185

190

195

Strike price K First variation of option price C in time

0

-20

-40

-60

-80

Θ(K)

January 23, 2018

-100

-120

-140

-160

-180

-200 170

175

180

185

190

195

Strike price K

Figure 3. Call Options Prices and Theta (first variation in time) of DJI with ∆t =

1 252

newParametrizationClass

Local Volatility with N=1 and Dupire Local Volatility

22

LV function (ω = 0.42852, σ 1 = 8, x0 = 182.068) Dupire Local Volatility from market prices

20

Volatility (%)

18

16

14

12

10

8 170

175

180

185

190

195

200

205

Strike price K Calculated vs. Market Option Prices

14

Calculated prices today using our model Market prices today with underlying 182.068

12

10

Call Option price C

January 23, 2018

8

6

4

2

0 170

175

180

185

190

195

200

205

Strike price K

Figure 4. Comparison between our LV model, Dupire Local Volatility, and market prices. Note that we recover the prices for vanilla options. We have from calibration that ω = 0.4285, α = 0.2856, σ1 = 8, x0 = 182.068.

newParametrizationClass

Delta for the Call Option computed via our formula

0.9

0.8

0.7

0.6

∆(K)

0.5

0.4

0.3

0.2

0.1

0 170

175

180

185

190

195

200

205

Strike price K Gamma for the Call Option computed via our formula

0.06

0.05

0.04

Γ(K)

January 23, 2018

0.03

0.02

0.01

0 170

175

180

185

190

195

200

205

Strike price K

Figure 5. The Greeks (Delta and Gamma) as a function of K, calculated using our model and Proposition 3.2

January 23, 2018

newParametrizationClass

demonstrating the versatility of our approach. The first natural idea is to consider N = 2. Consistency of the dynamics of the TPD as maturity increases, adding time–dependence to our model, as well as extensions to additive subordinate diffusions Li et al. (2016) will be explored in a future work, see below for a glimpse. In a future paper, we also plan to extend this approach to n–dimensional diffusion models, and thus to introduce a concept somewhat underrepresented in the financial literature, namely the local correlation function, see Langnau (2009).

6.1.

The parametrization class for N = 2

The first natural extension is to consider N = 2. In this case, the parametrization class is given by four parameters, namely α1 , α2 , ω1 , ω2 . The functions δ1 (s) and δ2 (s) can be obtained via: 1 log 2



1 χ2 (s) = sω2 − log 2



χ1 (s) = sω1 −

 δ1 (s) = ω1 sinh  −ω2 sinh

1 log 2 

δ2 (s) = ω1 sinh  −ω2 sinh

1 log 2 



(ω1 − α1 ) (ω1 − α2 ) (α1 + ω1 ) (α2 + ω1 )

(ω2 − α1 ) (ω2 − α2 ) (α1 + ω2 ) (α2 + ω2 )

1 log 2

1 log 2







(ω2 − α1 ) (ω2 − α2 ) (α1 + ω2 ) (α2 + ω2 )







 − sω2 cosh





− sω1 cosh



(ω2 − α1 ) (ω2 − α2 ) (α1 + ω2 ) (α2 + ω2 )

(ω1 − α1 ) (ω1 − α2 ) (α1 + ω1 ) (α2 + ω1 )

(ω1 − α1 ) (ω1 − α2 ) (α1 + ω1 ) (α2 + ω1 )





1 log 2

1 log 2





,

,

(ω2 − α1 ) (ω2 − α2 ) (α1 + ω2 ) (α2 + ω2 )

(ω1 − α1 ) (ω1 − α2 ) (α1 + ω1 ) (α2 + ω1 )





 − sω2 

− sω1 ,

     (ω1 − α1 ) (ω1 − α2 ) 1 − sω2 cosh log − sω1 2 (α1 + ω1 ) (α2 + ω1 )

     (ω2 − α1 ) (ω2 − α2 ) 1 log − sω1 cosh − sω2 , 2 (α1 + ω2 ) (α2 + ω2 )

please see (20). As before, we can compute explicitly the volatility function and the potential in transformed coordinates using the equations (23), (22). In order to simplify our formulae, we introduce first some notations. Define: " √ # √ p ( λ − iα1 )( λ − iα2 ) 1 2 2 √ a2 = ln √ a1 = (λ + α1 )(λ + α2 ) 2 ( λ + iα1 )( λ + iα2 ) p b1 = (ω12 − α12 )(ω22 − α22 ) d1 =

p (ω22 − α12 )(ω22 − α22 )

  1 (ω1 + α1 )(ω1 + α2 ) b2 = ln 2 (ω1 − α1 )(ω1 − α2 )   (ω2 + α1 )(ω2 + α2 ) 1 d2 = ln 2 (ω2 − α1 )(ω2 − α2 )

Then:  σ(s) = σ1

δ1 (s) δ2 (s)

2 , Q(s) = 

  (ω12 − ω22 ) ω12 − ω22 + ω22 cosh(2b2 + 2ω1 s) + ω12 cosh(2d2 + 2ω2 s) 2 ω1 cosh(b2 + ω1 s) cosh(d2 + ω2 s) − ω2 sinh(b2 + ω1 s) sinh(d2 + ω2 s)

It can be easily checked that the Riccati equation (19) holds. Moreover, most results of Proposition 3.2 hold identically for N = 2. For (i) and (ii), new solutions for the SL problem (14) and the spectral

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function need to be calculated: the spectrum is given by Λ = {−ω12 , −ω22 , −α12 , −α22 , 0} ∪ (0, ∞), the explicit solutions of the SL problem are: v1 (s, λ) =

f1 (s, λ) , G(s, λ)

v2 (s, λ) =

f2 (s, λ) G(s, λ)

where  √ f1 (s, λ) = a1 cosh(a2 + i λs)(ω2 (λ + ω12 ) sinh(b2 + ω1 s) sinh(d2 + ω2 s) −ω1 (λ + ω22 ) cosh(b2 + ω1 s) cosh(d2 + ω2 s))  √ √ −i λ(ω12 − ω22 ) sinh(a2 + i λs) sinh(b2 + ω1 s) cosh(d2 + ω2 s) √

√ λ(ω22 − ω12 ) cosh(d2 + ω2 s) sinh(a2 − is λ) sinh(b − 2 + ω1 s) √ −i cosh(a2 − is λ)(ω1 (λ + ω22 ) cosh(b2 + ω1s) cosh(d2 + ω2 s)

f2 (s, λ) = a1

−(λ + ω12 )ω2 sinh(b2 + ω1 s) sinh(d2 + ω2 s))

 G(s, λ) = (λ + ω12 )(λ + ω22 )(ω1 cosh(b2 + ω1 s) cosh(d2 + ω2 s)  −ω2 sinh(b2 + ω1 s) sinh(d2 + ω2 s) . It was checked in Mathematica that the SL equation −v 00 (s) + Q(s)v(s) = λv(s) holds for both eigenfunctions v1 (s, λ) and v2 (s, λ) given above. Notice that the poles of the function 2 Ψ(s0√, s, λ, t) := e−λ t v1 (s, λ2 )v2 (s, λ2 )ρ(λ2 ) are in this case {±iω1 , ±iω2 , ±iα1 , ±iα2 , 0}, where i = −1. The remaining results (iii), (iv), (v) will hold identically. The volatility function σ(s) can still be inverted to obtain σ(x) in the original coordinates, via a similar trick as in Proposition 3.1. For N = 2, we can calibrate the model by using a similar reasoning as above. Given the point s0 and the numbers b0 (s0 ), b00 (s0 ), b000 (s0 ), b000 0 (s0 ), it was shown in Tydniouk (2017) how to construct the coefficients a1 , a2 , b2 of the polynomials {Dkj (λ)}k,j=1,2 . 6.2.

Extensions to additive subordinate diffusions

Additive subordinations, presented recently in Li et al. (2016), are a useful tool for constructing and analyzing time–inhomogeneous Markov processes with analytical tractability. As argued in the paper, such processes have proved to be very useful in financial applications. One can easily note that this generalization can be used in conjunction with our new model, thus making it more appealing from both theoretical and practical points of view. We make some remarks and formulate some conjectures in relation with our approach. Conjecture 1 It can be shown that, in our case, by using assumption 6.1 in Li et al. (2016), that the following holds in the transformed space (please see Remark 8): Z

ψ

P (t1 , t2 ; s|s0 ) =

e−

Rt

2 t1

ψ(λ,u)du

Spec(D)

R

− Spec(D) e

vi (s, λ)vj (s0 , λ)ρψ ij (λ)dλ,

(44)

i,j=1

V ψ (s0 , t1 , t2 ) = Ptψ1 ,t2 f (s0 ) = =

2 X

Rt

2 t1

R S

f (s)P ψ (t, s|s0 )ds

ψ(λ,u)du

2 P i,j=1

(45) Fi (λ)vj (s0 , λ)ρψ ij (λ)dλ,

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R where Fj (λ) := S f (z)vj (z, λ)dz. Moreover, the spectrum, the spectral function ρψ (λ), and the solutions to the SL problem in transformed coordinates will remain the same. Conjecture 2 We conjecture that a similar equation must hold for inverting the transformation, with s = s(x), s0 = s(x0 ): P ψ (t1 , t2 ; s|s0 ) = C2 eη

ψ

(x,x0 ) ψ

p (t1 , t2 ; x|x0 ).

The polynomial η ψ (x, x0 ) can be determined by using the generator G ψ , given in Li et al. (2016), Theorem 4.2. Moreover, we can obtain similar results as in Proposition 3.2. Furthermore, most of our results will hold in the case of subordinated diffusions with a slight modification in the R − tt2 ψ(λ,u)du −λt exponential term, namely that e will be replaced by e 1 .

Appendix A: Proofs Proposition 3.1 Proof. To avoid possible confusions, we shall denote here σ e(s) := σ(x(s)). We have with SL transform (16) that x0 (s) :=

σ 0 (x) :=

σ e(s) d x(s) = √ , ds 2

√ σ d dσ(x(s)) ds de σ (s) ds e0 (s) σ(x) = = = 2 . dx ds dx(s) ds x0 (s)ds σ e(s)

With this and (31), we have: 2ωσ1 cosh(χ) σ e(s) = σ1 coth (χ(s)), σ e (s) = − sinh3 (χ) 0

2

√ 4 2ω σ (x) = − . sinh(2χ) 0

⇒

With this, it can be easily checked that the Riccati equation (19) holds. On the other hand, by standard manipulations of σ e(s) = σ1 coth2 (χ(s)) and by using the hyperbolic identity cosh2 (χ) − sinh2 (χ) = 1, we obtain: r sinh(χ) = ±

s σ1 σ e(s) , cosh(χ) = ± σ e(s) − σ1 σ e(s) − σ1

p σ1 σ e(s) sinh(2χ) = 2 , |e σ (s) − σ1 |

⇒

and therefore, σ 0 (x) =

√ |e dσ(x) σ (s) − σ1 | =2 2 p . dx σ1 σ e(s)

But this is a differential equation with separable variables. By integration, we obtain: √ σ 2 2ω dσ = ± √ dx σ − σ1 σ1 √

⇒

√

σ−

√

r σ1 atanh

σ σ1



√ ω 2 = √ (x − x0 ), σ1

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where is the inverse function on tanh(·). It can be further calculated as atanh(x) = atanh(·) 1+x 1 2 ln 1−x , with −1 < x < 1. Finally, we obtain (34), and the proof is complete. Proposition 3.2 Proof. First notice that the following equality holds: Q(s) = −

2ω 2 2ω 2 (ω − α)(α + ω) = − (ω cosh(sω) − α sinh(sω))2 cosh2 (χ(s))

With this, it can be easily checked (either by hand or in Mathematica) that the functions in statement (ii) satisfy the SL equation (35). The spectral function was computed to be (36), and thus the spectrum had two components: a discrete part and a continuous part, Λ = {−ω 2 , −α2 , 0}∪ (0, ∞). Using the Spectral decomposition Theorem, we can write the transformed TPD and price as R −λt P (t, s|s0 ) = e v1 (s, λ)v2 (s0 , λ)ρ(λ)dλ, λ∈Λ

(A1) R

Ve (s0 , t) =

f (s)P (t, s|s0 )ds.

s∈[0,+∞)

The integral expression for P (t, s|s0 ) can be further simplified by using Cauchy Residue Theorem. √ In order to avoid branch points that appear due to λ in the denominator, we make the change 2 of variable λ ← λ2 . Denote Ψ(s0 , s, λ, t) := e−λ t v1 (s, λ2 )v2 (s0 , λ2 )ρ(λ2 ). With this, the integral becomes: P (t, s|s0 ) = 2

5 X

Z

∞

Res[Ψ, pi ] + 2

Ψ(s0 , s, λ, t)λdλ, 0

i=1

where pi ∈ {±iω, ±iα, 0} are the poles of Ψ. The TPD in the original coordinates p(t, x|x0 ) can be obtained by using the explicit mapping (28), and the functions s = s(x), s0 = s(x0 ), with x(s) as in (33). Numerical inversion can be used in this case. Further, if we assume that the interest rate is deterministic, then the forward ft satisfies a driftless Ito process, i.e., dft = σ(ft )dWt . Then the value of an European call option with strike K and maturity T is given by (τ = T − t): Z

∞

C(K, τ |xt ) = DtT

max{x − K, 0}p(τ, x|xt )dx K

Doing an integration by parts and by applying Ito’s formula on the payoff max{Xt − K, 0} we obtain dXt = I(Xt − K)dXt + 21 σ 2 (t, K)δ(Xt − K)dt, the option price C can be rewritten as C(K, t|xt ) σ 2 (K) = max{xt − K, 0} + DtT 2

τ

Z

p(t0 , K|xt )dt0 .

0

√ Using the mapping (28) with C2 = σ(x0 )/ 2 and b(x) = 0 we obtain C(K, τ |xt ) = (xt − K)+ + DtT

r

σ(K)σ(xt ) 2

Z 0

τ

P (t0 , s(K)|st )dt0 .

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Now, plug in the spectral decomposition for P (t0 , s|st ) given in (iii), and do the integration over time to obtain the expression in (iv). Note that in (iv) we denoted, by abuse of notation, the initial time t with 0 and the maturity T with t. Finally, the value of the Greek Delta ∆(K, t|x0 ) was obtained 0 (s) 4ω by taking the derivative of C(K, t|x0 ) with respect to x0 , and using that σσ(s) = − sinh(2χ(s)) . This completes the proof.

Lemma 4.1 Proof. Recall that the following forward PDE holds for the option price C(t, x; T, K):     

∂C ∂C 1 ∂2C + r(K, T )K = σ 2 (K, T )K 2 − r(K, T )C ∂T 2 ∂K 2 ∂K

(A2)

C(T, x; T, K) = (x − K)+

Note that t, x are regarded as constants, and so C(t, x; T, K) is only a function of T, K, i.e. C = RT tT ,T ) r s ds t C(K, T ). Let ftT = Xt e , y := fKtT , and consider the transformation F (y, T ) = C(yf DtT ftT . By simple manipulations, we obtain ∂F 1 2 ∂2F = σ b (y, T ) 2 , where σ b(y, T ) = σ(yftT , T ). ∂T 2 ∂y

(A3)

We denote, by abuse of notation, T, y and σ b with t, x and σ, respectively. Notice that if we assume a zero interest rate, we obtain the needed result without any additional transformation. This concludes the proof of this lemma.

Appendix B: Singular Solutions of NIDE We present in this Appendix for completeness some results from Tydniouk (2017), where the method of operator identities was successfully applied to obtain explicit solutions of some Nonlinear Integrable Differential Equations (NIDE) using the inverse spectral problem approach. We consider the following NIDEs: ∂2 φ(x, t) = 4 sinh φ(x, t), ∂x∂t ∂ ψ(x, t) ∂t ∂ ρ(x, t) ∂t

1 ∂3 ∂ 3 ψ(x, t) + |ψ(x, t)|2 ψ(x, t), 4 ∂x3 2 ∂x   √ i ∂2 2 = ρ(x, t) − |ρ(x, t)| ρ(x, t) , i = −1. 2 2 ∂x

=−

(B1)

In the literature, these equations are known as sinh–Gordon equation (SHG), the modified Korteveg–de Vries equation (MKdV), and the nonlinear Schr¨odinger equation (NSE), respectively. Explicit solutions to these NIDE can be obtained, by using the method of inverse spectral problem. We sketch the main results below. Theorem B.1 Let Ω = {ω1 , . . . , ωN } and A = {α1 , . . . , αN } be two sets of complex numbers such that ωi 6= ωk for i 6= k, Re(ωk ) > 0, and each of the sets Ω, A is symmetrical with respect to the

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real axis. Consider δ1 (x, t), δ2 (x, t) as in (20), with N 1 Y ωk − αi χk (x, t) = ωk x + Θ(ωk )t − ln , k = 1, 2, . . . , N, 2 ωk + αi

(B2)

i=1

where Θ(ωk ) = 1/ωk for SHG, Θ(ωk ) = −ωk3 for MKdV, and Θ(ωk ) = iωk2 for NSE. Then the solution ξ(x, t) of NIDE (B1) can be written as: δ1 (x, t) ξ(x, t) = 2 ln δ2 (x, t)

(B3)

in case of SHG equation and (1)

ξ(x, t) = 2

(1)

δ1 (x, t)δ2 (x, t) − δ2 (x, t)δ1 (x, t) δ1 (x, t)δ2 (x, t)

(B4)

(1)

in case of MKdV and NSE equations. Here, δi (x, t), i = 1, 2 represents determinants obtained (1) from δi (x, t), i = 1, 2, respectively, by differentiating the first column w.r.t. x. Consider now the functions {Dij (x, t, λ)}2i,j=1 as polynomials w.r.t. the spectrum λ ∈ C (for more details on the significance of these polynomials and their connection with NIDEs and dynamical systems, we refer the reader to Tydniouk (2017)):

D11 (x, t, λ) = (−1)N

N P

(−1)k ak (x, t)λN −k , a0 = 1,

k=0

D12 (x, t, λ) =

NP −1

bk (x, t)λN −1−k ,

k=0

(B5) D21 (x, t, λ) = (−1)N −1

NP −1

(−1)k bk (x, t)λN −1−k ,

k=0

D22 (x, t, λ) =

N P

ak (x, t)λN −k , a0 = 1.

k=0

The following results gives differential relations between the coefficients {ak (x, t)}N k=1 and N −1 {bk (x, t)}k=0 of the functions (B5). Theorem B.2 Let H(x, t) = [b0 (x, t) b1 (x, t) · · · bN −1 (x, t) a1 (x, t) · · · aN (x, t)]T be a N × 1 vector function on R × R+ . Then H(x, t) satisfies the following Riccati–type system of differential equations: ∂ H(x, t) = H(x, t)F H(x, t) − GH(x, t), ∂x

(B6)

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where F = [O1×N 2 0 · · · 0] ∈ R1×2N 

ON ×N

     G=   ON ×N  

ON ×N 

0 2 0 ··· 0 0 2 ···   .. .. .. . . . . . .  0 0 0 ··· 0 0 0 ···

   0    0   ∈ R2N ×2N  ..   .   2  0

and

b0 (x, t) =

 δ1 (x, t) 1 ∂  ,  ln −    8 ∂x δ2 (x, t)

for SHG equation (B7)

 (1) (1)   δ (x, t)δ2 (x, t) − δ2 (x, t)δ1 (x, t)   1 , for MKdV and NSE equations. 4δ1 (x, t)δ2 (x, t)

Remark 15 Note that we can rewrite equation (B6) in a simplified form: 

ON ×1 b1 (x, t) .. .



    1 ∂   H(x, t) = b0 (x, t)H(x, t) −  .   2 ∂x  bN −1 (x, t)  0 But this is equivalent with the following Riccati–type system of nonlinear differential equations (the derivative is taken w.r.t. x):  b00    0    b1

= 2(b0 a1 − b1 ) = 2(b0 a2 − b2 ) .. .

   b0 = 2(b0 aN −1 − bN −1 )    b0N −2 = 2b a 0 N −1 N −1

 a01    0    a1

= 2b20 = 2b0 b1 .. .

(B8)

   a0 −1 = 2b0 bN −2    aN 0 = 2b0 bN −1 N

Potentials of type (B7) are called Pseudo Exponential (PE) and therefore our solutions of NIDE can be considered as a special scalar case of PE potentials. Observe that PE potentials parametrized by 2N parameters according to Theorems B.1 and B.2. The following theorem solves the inverse problem for the solution of NIDE having PE potentials. Theorem B.3 Let b0 (x, t) be a PE potential, meromorphic on R × [0, ∞) and analytic at (x0 , t0 ). Then it is uniquely defined by b0 (x0 , t0 ),

∂ ∂ 2N −1 b0 (x0 , t0 ), . . . , 2N −1 b0 (x0 , t0 ). ∂x ∂x

Proof. See Tydniouk (2017) for the complete proof. We only describe here the procedure for recovering the sets of parameters Ω and A. Fix t = t0 . Firstly, find the roots {e ωk }N k=1 of the polynomial

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D(x0 , λ) = D11 (x0 , λ)D22 (x0 , λ) − D12 (x0 , λ)D21 (x0 , λ). Compute now the ratios D12 (x0 , ωk ) D22 (x0 , ωk ) = . D11 (x0 , ωk ) D21 (x0 , ωk )

R=

By solving this system of linear equations w.r.t. {α}N k=1 , we can recover the set A. Example B1. Let N = 1, t0 = 0 and assume that b0 (x) is a Pseudo Exponential (PE) potential for MKdV and NSE equations, see expression (B7). Then b0 (x) is given by (1)

b0 (x) =

(1)

δ1 (x)δ2 (x) − δ2 (x)δ1 (x) 2δ1 (x)δ2 (x)

cosh2 (χ) − sinh2 (χ) ω = sinh(χ) cosh(χ) sinh(2χ)   ω − α , = ωcsch 2ωy − ln ω + α =ω

(B9)

where csch(x) = 1/ sinh(x) is the Hyperbolic Cosecant function. With this, we obtain the following Riccati–type system, see (B8): b00 (x) = 2b0 (x)a1 (x),

a01 (x) = 2b20 (x).

  ω − α . It is easy to verify that these equations are satisfied with a1 (x) = −ω coth 2ωx − ln ω + α We construct now the polynomials {Dij (x, t, λ)}2i,j=1 : D11 (x, λ) = −λ + a(x), D22 (x, λ) = λ + a(x),

D12 (y, λ) = b0 (x), D21 (y, λ) = b0 (x),

and compute the roots λ1,2 of the polynomial D(x, λ) = (−λ + a1 (x))(λ + a1 (x)) − b20 (x) = a21 (x) − λ2 − b20 (x) = ω2


cosh2 (2χ) ω2 ω2 2 = − − λ cosh2 (2χ) − 1 − λ2 2 2 2 sinh (2χ) sinh (2χ) sinh (2χ)

= ω 2 − λ2 = (ω − λ)(ω + λ) to be λ1,2 = ±ω. Further, we calculate the ratios

R(x) =

1 − cosh(2χ) b0 (x) ω + a1 (x) sinh(2χ) = = = sinh(2χ) − cosh(2χ) = −e−2χ(x) 1 −ω + a1 (x) b0 (x) sinh(2χ)

ω − α , from which we can easily compute α as Therefore ln R(x) = 2χ(y) = 2ωx − ln ω + α α=ω

R − e2ωx . R + e2ωx

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 A connection between the solutions of the considered NIDE and other non-linear differential equations can be found via the Miura transformation: M [f (x)] = f 2 (x) ±

df (x) . dx

(B10)

Using this transform, solutions of MKdV equation can be converted into the solutions of the Korteweg–deVries (KdV) equation, given by: ∂u(x, t) 1 ∂ 3 u(x, t) 3 ∂u(x, t) =− + |u(x, t)| , 3 ∂t 4 ∂x 2 ∂x

(B11)

and solutions of NSE can be converted into the solutions of NSE with opposite sign by the nonlinear term. The corresponding image can be represented in the standard form: P (x, t) = −2

∂ 2 ln δ(x, t) ∂x2

(B12)

where δ ≡ δ1 when choosing ”+” in (B10), and δ ≡ δ2 when choosing ” − ”, see also (20). If δ(x, t) 6= 0, ∀(x, t) ∈ R × R+ , then P (x, t) is called the reflectionless (RL) potential representing the N −soliton solution of the corresponding non-linear equation. Example B2. Consider the solution of MKdV for N = 1: ψ(x, t) = 2ωcsch (2χ(x, t)) , where χ(x, t) = ωx − ω 3 t − 21 ln ω−α ω+α . It can be easily checked that the function P1 (x, t) = ψ 2 (x, t) +

∂ψ(x, t) = −2ω 2 sech2 (χ(x, t)) ∂x

satisfies the KdV equation (B11) and δ(x, t) = cosh χ(x, t). Similarly, it can be checked that the function P2 (x, t) = ψ 2 (x, t) −

∂ψ(x, t) = −2ω 2 csch2 (χ(x, t)) ∂x

satisfies the KdV equation (B11) with δ(x, t) = sinh χ(x, t). Please note that P1 (x, t) is RL potential representing the classical 1-soliton solution of the KdV equation. The function P2 (x, t) is singular and strictly speaking doesn’t represent RL potential, but because of the similarities between them in further considerations we shall refer to all the Miura-transformed potential as reflectionless-like (RL-like) potentials.  Combining results obtained in Theorem B.3 and properties of Miura-transformed potentials, we can solve an inverse problem for the RL-like potentials. Theorem B.4 Let a(x, t) be a RL–like potential, meromorphic on R × [0, ∞) and analytic at (x0 , t0 ). Then it is uniquely defined by q(x0 , t0 ),

∂ ∂ 2N −1 q(x0 , t0 ), . . . , 2N −1 q(x0 , t0 ). ∂x ∂x

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