Derivatives Pricing under Beta Stochastic Volatility Model ... · The general stochastic volatility...

Applied Mathematical Sciences, Vol. 12, 2018, no. 17, 825 - 840HIKARI Ltd, www.m-hikari.com

https://doi.org/10.12988/ams.2018.8574

Derivatives Pricing under Beta Stochastic

Volatility Model Using ADI Schemes

Abdelkamel Alj and Abdelghani Benjouad

Department of EconomicsOMEGA, FSJES, Moulay Ismail University

B.P. 3102 Toulal, Meknes, Morocco

Copyright c© 2018 Abdelkamel Alj and Abdelghani Benjouad. This article is distributed

under the Creative Commons Attribution License, which permits unrestricted use, distribu-

tion, and reproduction in any medium, provided the original work is properly cited.

Abstract

We consider Alternating Direction Implicit finite difference schemesfor the numerical solution of initial boundary value problems for convec-tion diffusion equations type with mixed derivatives and non-constantcoefficients, as they arise from a beta stochastic volatility model in op-tion pricing. We present various numerical examples with realistic datasets from the literature, where we consider European call options. Thetested numerical methods provide an accurate and fast way of calculat-ing option prices under a beta stochastic volatility model.

Mathematics Subject Classification: 35K51, 65M06, 91G80Keywords: Stochastic volatility model, Option pricing, Convection-diffusion

equations, ADI finite difference schemes

1 Introduction

Stochastic volatility models present one approach to solving one of the short-comings of the Black-Scholes model. Some well known examples of stochasticvolatility models are introduced by Hull and White [6], Heston [5], Hagan et al[4] and Lipton [13]. These models, which are widely used among practitioners,assume that also the volatility follows a stochastic process. For some models,vanilla options can be priced in an analytic way. However, for more exotic op-tions an analytic formula is not available and other methods need to be used

826 Abdelkamel Alj and Abdelghani Benjouad

to approximate the prices.

In this paper, we will look at option pricing in a more general setting thanin the Black-Scholes framework. We will especially look at the Beta Stochas-tic Volatility (BSV) model suggested by Karasinski and Sepp [10]. The BSVmodel is introduced to overcome the disadvantages of other existing models:Bardhan-Karasinski [1], Langnau [12] and Sepp [15]. The model parameterscan be naturally interpreted and easily implied from historical and currentmarket data. The model dynamics incorporating the local volatility with cal-ibration that can be implemented using conventional partial-differential equa-tions (PDE) methods. The PDE approach makes use of the fact that the priceof an option in a stochastic volatility model can be represented by a two di-mensional convection diffusion PDE. The finite difference method (FDM) is aproven numerical procedure to obtain accurate approximations to the relevantPDE. For the numerical discretization of the BSV–PDE, we investigate threeAlternating Direction Implicitan (ADI) schemes. An ADI scheme proposedby Douglas & Rachford [3], an ADI scheme introduced by Craig & Sneyd [2],and modified by In’t Hout & Welfert [9], and an ADI scheme introduced byHundsdorfer & Verwer [7].

This paper is organized as follows. First, we present the model dynamicsand its discretization by Euler scheme illustrated by simulations. Then, wederive the BSV–PDE with initial and boundary conditions for European calloptions. Finally, we present the ADI schemes to obtain the European callprice by solving the BSV–PDE along a two-dimensional grid representing thestock price and the volatility. We show how to construct uniform and non-uniform grids for the discretization of the stock price and the volatility, andpresent formulas for finite difference approximations to the derivatives in theBSV model. The methods can easily be modified to allow for the pricing ofEuropean puts which requires a reformulation of the boundary conditions. Inmany cases, however, it is simpler to use put-call parity to obtain the put price.

2 Beta Stochastic Volatility Model

The BSV model was first introduced in 2012 by Karasinski & Sepp [10]. One ofthe important features of this model is that its key parameter β has a naturalinterpretation as the rate of change in the short term ATM (at-the-money)volatility given change in the stock price. The remaining two parameters- the idiosyncratic volatility of volatility and the mean- reversion rate havelesser impact on forward skews and also can be estimated from historical datawithout the need to apply time-expensive and obscure non-linear fits.

Derivatives pricing under beta stochastic volatility model 827

2.1 Model Dynamics

The BSV model is expressed on terms of the spot price St and the stochasticvolatility vt:

dSt

St

= µtdt+ vtdW1t , S0 = S, (1)

dvt = κ(ν − vt)dt+ βvtdW1t + εvtdW

2t , v0 = v, (2)

dW 1t dW

2t = 0,

where:β is the rate of change in the volatility;ε > 0 is idiosyncratic volatility of volatility;κ > 0 is the mean-reversion speed;ν > 0 is the mean of the volatility;µt is the risk-neutral drift;W 1

t ,W2t are two uncorrelated Brownians.

2.2 Euler Scheme for the BSV Model

We assume that the stock price St and its volatility vt are driven by the stochas-tic process (1) and (2). We simulate St and vt over the time interval [0, T ],which we assume to be discretized as 0 = t1 < t2 < . . . < tm = T , wherethe time increments are equally spaced with width dt. To discretize the twoprocess in the stochastic differential equations (SDE) (1) and (2), the simplestway is to use Euler discretization.

Discretization of vt: The SDE for vt in (2) in integral form is

vt+dt = vt +

∫ t+dt

t

κ(ν − vz)dz +

∫ t+dt

t

βvzdW1z +

∫ t+dt

t

εvzdW2z

The Euler discretization approximates the integrals using the left-point rule

∫ t+dt

t

κ(ν − vz)dz ≈ κ(ν − vt)dt∫ t+dt

t

βvzdW1z ≈ βvt(W

1t+dt −W 1

t ) = βvt√dtZ1∫ t+dt

t

εvzdW2z ≈ εvt(W

2t+dt −W 2

t ) = εvt√dtZ2

where Z1 and Z2 are randoms draws from the standard normal distributionN (0, 1). The right hand side involves (ν − vt) rather than (ν − vt+dt) since at


time t we dont know the value of vt+dt. This leaves us with

vt+dt = vt + κ(ν − vt)dt+ βvt√dtZ1 + εvt

√dtZ2 (3)

Discretization of St: In the same way, the SDE for St in (1) is written inintegral form as

St+dt = St +

∫ t+dt

t

µzSzdz +

∫ t+dt

t

SzvzdW1z

The Euler discretization approximates the integrals using the left-point rule∫ t+dt

t

µzSzdz ≈ µtStdt∫ t+dt

t

SzvzdW1z ≈ Stvt(W

1t+dt −W 1

t ) = Stvt√dtZ1

We end up withSt+dt = St + µtStdt+ Stvt

√dtZ1 (4)

Process for (St, vt): Start with the initial values S0 for the stock price andv0 for the volatility. Given a value for vt at time t, we first obtain vt+dt from(3) and we obtain St+dt from (4).

2.3 Derivation of a BSV–PDE

We consider the following general stochastic process{dSt = µSdt+ σSdW

1t

dvt = µvdt+ σvdW1t + σεdW

2t

(5)

Using Feynman-Kac [13] one can derive that the price of an option or derivativeis the solution of a PDE. The general stochastic volatility process (5) is used.It is assumed that the price of an option or a derivative u with underlyingasset S with volatility v is a function of the price of this underlying asset, thevolatility and time: u = u(S, v, t). Thanks to the Ito’s formula, we obtain

du =∂u

∂tdt+

∂u

∂sdSt+

∂u

∂vdvt+

1

2

∂2u

∂s2d < S >t +

∂2u

∂s∂vd < S, v >t +

1

2

∂2u

∂v2d < v >t

where: d < S >t= σ2Sdt, d < v >t= (σ2

v + σ2ε)dt, d < S, v >t= σSσvdt.

Thus, we have

du =∂u

∂tdt+

∂u

∂s(µSdt+ σSdW

1t ) +

∂u

∂v(µvdt+ σvdW

1t + σεdW

2t )

+1

2σ2S

∂2u

∂s2dt+ σSσv

∂2u

∂s∂vdt+

1

2(σ2

v + σ2ε)∂2u

∂v2dt


Figure 1: Simulations of BSV Model dynamics by Euler Scheme with :S0 = 100, v0 = 1, µ = 0.02, κ = 1.1, ν = 0.04, β = 0.74, ε = 0.86, dt = 0.001


is still

du = L(u)dt+(σS∂u

∂s+ σv

∂u

∂v

)dW 1

t + σε∂u

∂vdW 2

t (6)

where

L =1

2(σ2

v + σ2ε)∂2

∂v2+ σSσv

∂2

∂s∂v+

1

2σ2S

∂2

∂s2+ µv

∂

∂v+ µS

∂

∂s+∂

∂t

We consider the self-financing portfolio π consisting of one derivative u , −δ1shares of the underlying stock S and −δ2 units of another derivative φ. Thedynamics of the self-financing portfolio will be:

dπ = du− δ1dS − δ2dφ (7)

According to (5) and (6), this yields

dπ =

(L(u)dt+

(σS∂u

∂s+ σv

∂u

∂v

)dW 1

t + σε∂u

∂vdW 2

t

)− δ1

(µSdt+ σSdW

1t

)− δ2

(L(φ)dt+

(σS∂φ

∂s+ σv

∂φ

∂v

)dW 1

t + σε∂φ

∂vdW 2

t

)Rearranging terms results in:

dπ =

(L(u)− δ1µS − δ2L(φ)

)dt

+

(σS∂u

∂s+ σv

∂u

∂v− δ1σS − δ2

(σS∂φ

∂s+ σv

∂φ

∂v

))dW 1

t

+ σε

(∂u

∂v− δ2

∂φ

∂v

)dW 2

t

(8)

For π to be risk-free self-financing portfolio, δ1 and δ2 must satisfy the followingsystem:

σS∂u

∂s+ σv

∂u

∂v− δ1σS − δ2

(σS∂φ

∂s+ σv

∂φ

∂v

)= 0

∂u

∂v− δ2

∂φ

∂v= 0

this yields

δ1 =∂u

∂s−(∂u

∂v/∂φ

∂v

)∂φ

∂s

δ2 =∂u

∂v/∂φ

∂v


For the absence of arbitrage, portfolio returns must be equal to the following

dπ = rπdt = r[u− δ1S − δ2φ]dt

Assuming that µS = rS and using (8), the last equation can be rewritten as

L(u)− ru∂u

∂v

=L(φ)− rφ

∂φ

∂vThe left side member is only a function of u while the right side member isa function of φ only. This forces each of the two members to be equal toa function f(S, v, t) of the independent variables S, v and t. Choosing thisfunction equal to zero concludes this derivation

∂u

∂t+

1

2σ2S

∂2u

∂s2+

1

2(σ2

v + σ2ε)∂2u

∂v2+ µS

∂u

∂s+ µv

∂u

∂v+ σSσv

∂2u

∂s∂v− ru = 0

Within the framework of BSV model, we have

σS = Stvt, σv = βvt, σε = εvt

µS = rSt, µv = κ(ν − vt)Resulting in a more elegant form of the general PDE:

∂u

∂t+

1

2s2v2

∂2u

∂s2+

1

2(β2+ε2)v2

∂2u

∂v2+rs

∂u

∂s+κ(ν−v)

∂u

∂v+βSv2

∂2u

∂s∂v−ru = 0 (9)

for 0 ≤ t ≤ T, s > 0, v > 0.

2.4 Boundary conditions

The equation (9) has several solutions corresponding to all derivatives thatmay have s as underlying asset. Each solution is characterized by the boundaryconditions. For the PDE in Equation (10), we use the boundary conditionsfor a European call explained by Heston [5] and by In’T Hout and Foulon [8],among others.

Let u(s, v, t) denote the price of a European option. To obtain an approxi-mate solution of problem (9), we change the direction of time to integrate theequation in the direction of increasing t. It has been the parabolic PDE

∂u

∂t=

1

2s2v2

∂2u

∂s2+

1

2(β2+ε2)v2

∂2u

∂v2+rs

∂u

∂s+κ(ν−v)

∂u

∂v+βsv2

∂2u

∂s∂v−ru (10)

Initial condition for a European call option:

u(s, v, 0) = max(0, s−K)

where K is the strike of the option.Boundary conditions for European call option, with (0 ≤ t ≤ T ):

u(0, v, t) = 0, lims→∞

∂u

∂s(s, v, t) = e−rt, lim

v→∞u(s, v, t) = se−rt


3 Numerical discretization of the BSV–PDE

Finite difference methods are techniques to find a numerical approximation tothe PDE. To implement finite differences, we first need a discretization grid forthe two state variables (the stock price and the volatility), and a discretizationgrid for the maturity. These grids can have equally or unequally spaced incre-ments. Second, we need discrete approximations to the continuous derivativesthat appear in the PDE. Finally, we need a finite difference methodology tosolve the PDE. We use the same techniques as in [14, Chapter 10].

3.1 Mesh definitions

3.1.1 Uniform grid

Uniform grids are those that have equally spaced increments for the two statevariables. These grids have two advantages. On the one hand, they are easyto construct, on the other hand, since the increments are equal, the finitedifference approximations to the derivatives in the PDE take on a simple form.To discretize the computational domain [0, smax]×[0, vmax]×[0, T ], we partitionthe domain in space using a mesh

• si = i ∗ ds, i = 0, ...,ms, ds = (smax − smin)/ms

• vi = j ∗ dv, j = 0, ...,mv, dv = (vmax − vmin)/mv

and in time using a mesh

• tn = n ∗ dt, n = 0, ..., nt, dt = T/nt

3.1.2 Non Uniform grid

Following the work of Kluge [11] and In’T Hout & Foulon [8], we apply anon-uniform grid that is finer around the strike price K and around the spotvolatility v0 = 0. Their grid of size ms + 1 for the stock price is

si = K + c sinh(ξi), i = 0, . . . ,ms

with

• c > 0 is a constant that controls the fraction of the mesh points si thatlie around s = K

• ξ0 < ξ1 < · · · < ξms are equidistant points given by

ξi = sinh−1(−Kc) + i∆ξ


Figure 2: Spacial grid for K = 100, ms = 20, mv = 20, c = K/10 andd = vmax/50

where

∆ξ =1

ms

(sinh−1

(smax −Kc

)− sinh−1

(−Kc

))The grid of size mv + 1 for the volatility is

vj = d sinh(ηj), j = 0, . . . ,mv

with

• d > 0 is a constant that controls the fraction of the mesh points vj thatlie around v = 0

• η0 < η1 < · · · < ηmv are equidistant points given by

ηj = j∆η

where

∆η =1

mv

sinh−1(vmax/d)

3.2 Derivatives approximation

Starting from the following approximations, calling uni,j the numerical approx-imation of the exact solution at the grid point (i, j, n):


First-order derivatives

∂u

∂s(si, vj, tn) =

uni+1,j − uni−1,jsi+1 − si−1

,∂u

∂v(si, vj, tn) =

uni,j+1 − uni,j−1vj+1 − vj−1

when uniform grids are used, the denominators are replaced by 2ds and 2dv,respectively.

Second-order derivatives

We consider the central differences for second-order derivatives as presented in[11, 8].

∂2u

∂s2(si, vj, tn) =

uni−1,j(si − si−1)(si+1 − si−1)

−2uni,j

(si − si−1)(si+1 − si)

+uni+1,j

(si+1 − si)(si+1 − si−1)

∂2u

∂v2(si, vj, tn) =

uni,j−1(vj − vj−1)(vj+1 − vj−1)

−2uni,j

(vj − vj−1)(vj+1 − vj)

+uni,j+1

(vj+1 − vj)(vj+1 − vj−1)

When uniform grids are used, the denominators are replaced by ds2 and dv2,respectively.

Mixed derivative

Finally, the mixed derivative of an interior point appears in [11] as

∂2u

∂s∂v(si, vj, tn) = a−1,−1u

ni−1,j−1 + a−1,0u

ni−1,j + a−1,1u

ni−1,j+1

+ a0,−1uni,j−1 + a0,0u

ni,j + a0,1u

ni,j+1

+ a1,−1uni+1,j−1 + a1,0u

ni+1,j + a1,1u

ni+1,j+1


where the coefficients are

a−1,−1 =∆si+1

∆si(∆si + ∆si+1)× ∆vj+1

∆vj(∆vj + ∆vj+1),

a−1,0 =−∆si+1

∆si(∆si + ∆si+1)× ∆vj+1 −∆vj

∆vj∆vj+1

,

a−1,1 =−∆si+1

∆si(∆si + ∆si+1)× ∆vj

∆vj+1(∆vj + ∆vj+1),

a0,−1 =∆si+1 −∆si

∆si∆si+1

× −∆vj+1

∆vj(∆vj + ∆vj+1),

a0,0 =∆si+1 −∆si

∆si∆si+1

× ∆vj+1 −∆vj∆vj∆vj+1

,

a0,1 =∆si+1 −∆si

∆si∆si+1

× ∆vj∆vj+1(∆vj + ∆vj+1)

,

a1,−1 =∆si

∆si(∆si + ∆si+1)× −∆vj+1

∆vj(∆vj + ∆vj+1),

a1,0 =∆si

∆si+1(∆si + ∆si+1)× ∆vj+1 −∆vj

∆vj∆vj+1

,

a1,1 =∆si

∆si+1(∆si + ∆si+1)× ∆vj

∆vj+1(∆vj + ∆vj+1).

In these coefficients, the increments are ∆si = si − si−1 and ∆vj = vj − vj−1.When the grid is uniform, the mixed derivative reduces to the much simplerform:

∂2u

∂s∂v(si, vj, tn) =

uni+1,j+1 + uni−1,j−1 − uni−1,j+1 − uni+1,j−1

4dsdv

3.3 ADI schemes

ADI schemes have the advantage of being stable and showing good convergencefor a small number of time points.After having discretized the PDE (10) in space and using U(t) = U(s, v, t) ascompact notation, we obtain a large system of ODEs of the form:

U ′(t) = LU(t)

U represents, for each t > 0 , the solution to the finite difference scheme atthe grid point (s, v), i.e., the approximation to the exact solution u(s, v, t).The idea behind ADI schemes is that the components of L are treated sepa-rately, so that certain components are treated explicitly, and others implicitly.


3.3.1 ADI Matrices

We decompose L into three matrices A0 , A1 and A2, so that

L = A0 + A1 + A2

where

A0 = βsv2∂2u

∂s∂v

A1 = rs∂u

∂s+

1

2s2v2

∂2u

∂s2− 1

2ru

A2 = κ(ν − v)∂u

∂v+

1

2(β2 + ε2)v2

∂2u

∂v2− 1

2ru

The system of equations can be solved using various ADI schemes that all workiteratively, by updating a given U(t− 1) to a new value U(t). All the schemesrequire an initial value U0. For the call option, this initial value is exactly thepayoff of this option. Denote by I the identity matrix. The schemes coveredby [8] and [14] are the following.

Douglas-Rachford (DR) Scheme [3]

This is the simplest ADI scheme under consideration. Given U(t − 1) , weupdate to U(t) using the following steps.

Step 1. Y0 = [I + dtL]U(t− 1)Step 2. Yk = [I − θdtAk]−1[Y0 − θdtAkU(t− 1)] for k = 1, 2Step 3. Set U(t) = Y2 .

In’t Hout - Welfert (IW) Scheme

This scheme has recently been introduced by In’t Hout & Welfert [9] to obtainmore freedom in the choice of the parameter θ as compared to the second-orderCraig-Sneyd Scheme [2].

Step 1. Y0 = [I + dtL]U(t− 1)Step 2. Yk = [I − θdtAk]−1[Yk−1 − θdtAkU(t− 1)], for k = 1, 2Step 3. Y0 = Y0 + θdt[A0Y2 − A0U(t− 1)]Step 4. Y0 = Y0 + (1

2− θ)dt[LY2 − LU(t− 1)]

Step 5. Yk = [I − θdtAk]−1[Yk−1 − θdtAkU(t− 1)], for k = 1, 2Step 6. Set U(t) = Y2 .


Hundsdorfer-Verwer (HV) Scheme

The HV scheme has been introduced by Hundsdorfer & Verwer [7]. Thisscheme is a different extensions to the DR scheme.

Step 1. Y0 = [I + dtL]U(t− 1)Step 2. Yk = [I − θdtAk]−1[Yk−1 − θdtAkU(t− 1)], for k = 1, 2Step 3. Y0 = Y0 + 1

2dt[LY2 − LU(t− 1)]

Step 4. Yk = [I − θdtAk]−1[Yk−1 − θdtAkY2], for k = 1, 2Step 5. Set U(t) = Y2 .

θ–schemes

The parameter θ controls the type of weighing being implemented.

θ = 0 produces the fully explicit scheme,θ = 1/2 produces the Crank-Nicolson scheme,θ = 1 produces the fully implicit scheme.

An ADI scheme is specified by the scheme itself and the value of the parameterθ.

4 Numerical experiments

To illustrate and check the model performances, we will present the results ofthe simulations given by the ADI schemes. We use a European Call optionwhose the exact price under Heston model using the closed-form solution, is4.5806. The parameters used in our numerical experiments areK = 100; r = 0.025;T = 0.15 years;κ = 1.5; ν = 0.04; β = −0.2; ε = 0.03;

Figure 3 display European call option price functions u in the three ADISchemes (DR, IW, HV) under uniform and non-uniform grids on the domain(s, v) ∈ [0, 200]× [0, 0.5].


Figure 3: European call option price functions u in the three ADI Schemesunder Uniform and NonUniform Grids

Using nt = 20 time steps and a uniform grid with ms = mv = 40. With theIn’t Hout - Welfert method the prices under the explicit, implicit, and Crank-Nicolson schemes are 4.5949, 4.5606, and 4.5613, respectively, all of which areaccurate for pennies. With a non-uniform grid reduced to nt = 10 time stepsand ms = mv = 20, the results are comparable in accuracy and require much


less computational time. The complete results are in Tables 1 for the uniformgrid and in Table 2 for the non-uniform grid.

Table 1: ADI Prices Under a Uniform Grid, Size 40× 40× 20

ADI Explicit Error Implicit Error Crank-Nicol ErrorDR 4.5955 0.0149 4.5368 0.0437 4.5611 0.0195IW 4.5949 0.0144 4.5606 0.0200 4.5613 0.0192HV 4.5949 0.0144 4.5606 0.0199 4.5613 0.0192

Table 2: ADI Prices Under a NonUniform Grid, Size 20× 20× 10

ADI Explicit Error Implicit Error Crank-Nicol ErrorDR 4.6121 0.0316 4.5644 0.0162 4.5886 0.0080IW 4.6039 0.0233 4.5877 0.0072 4.5884 0.0079HV 4.6037 0.0231 4.5878 0.0072 4.5884 0.0078

5 Conclusions

In this paper, we have derived a PDE from the Beta Stochastic Volatilitymodel, the resolution of which gives the price of a European option as a func-tion of the time, the price of the underlying asset and the volatility. We havepresented some of the finite difference methods that are commonly used to ob-tain European prices. A comparative numerical results was presented followingthree Alternating Direction Implicit schemes, especially: Douglas - Rachford,In’t Hout - Welfert and Hundsdorfer - Verwer, which have the advantage ofbeing stable and showing good convergence for a small number of time points.

References

[1] I. Bardhan, P. Karasinski P, Random Volatility Model Implied by S&POption Prices, Goldman Sachs internal paper, 1993.

[2] I. J. D. Craig, A. D. Sneyd, An alternating-direction implicit schemefor parabolic equations with mixed derivatives, Comp. Math. Appl., 16(1988), 341 - 350. https://doi.org/10.1016/0898-1221(88)90150-2

[3] J. Douglas, H. H. Rachford, On the numerical solution of heat conductionproblems in two and three space variables, Trans. Amer. Math. Soc., 82(1956), 421 - 439. https://doi.org/10.1090/s0002-9947-1956-0084194-4


[4] P.S. Hagan, D. Kumar, A.S. Lesniewski, D.E. Woodward, Managing SmileRisk, Willmot Magazine, 1 (2002), 84 - 108.

[5] S. L. Heston, A closed-form solution for options with stochastic volatilitywith applications to bond and currency options, The Review of FinancialStudies, 6 (1993), no. 2, 327 - 343.https://doi.org/10.1093/rfs/6.2.327

[6] J. Hull, A. White, The pricing of options on assets white stochastic volatil-ities, Journal of Finance, 42 (1987), 281 - 300.https://doi.org/10.2307/2328253

[7] W. Hundsdorfer, J. G. Verwer, Numerical Solution of Time-DependentAdvection-Diffusion-Reaction Equations, Springer, Berlin, 2003.https://doi.org/10.1007/978-3-662-09017-6

[8] K. J. In’t Hout, S. Foulon, ADI finite difference schemes for option pricingin the Heston model with correlation, Int. J. Numer. Anal. Model., 7(2010), no. 2, 303 - 320.

[9] K. J. In’t Hout, B. D. Welfert, Unconditional stability of second-orderADI schemes applied to multi-dimensional diffusion equations with mixedderivative terms, Appl. Num. Math., 59 (2009), 677 - 692.https://doi.org/10.1016/j.apnum.2008.03.016

[10] P. Karasinski, A. Sepp, The beta stochastic volatility model, Risk Maga-zine, 25 (2012), 66 - 71.

[11] T. Kluge, Pricing Derivatives in Stochastic Volatility Models Using theFinite Difference Method, Ph.D. Thesis, Technical University, Chemnitz,Germany, 2002.

[12] A, Langnau, Introduction to the Stochastic Implied Beta Model, MerrillLynch internal paper, 2004.

[13] A. Lipton, The vol smile problem, Risk Magazine, (2002), 81 - 85.

[14] F. D. Rouah, The Heston Model and Its Extensions in Matlab and C,Wiley, 2013. https://doi.org/10.1002/9781118656471

[15] A. Sepp, Efficient numerical PDE methods to solve calibration and pricingproblems in local stochastic volatility models, Conference Global Deriva-tives Trading & Risk Management, Paris, 2011.

Received: May 30, 2018; Published: July 14, 2018

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