Hybrid tree-finite difference methods for the Heston and ... · The hybrid tree-ﬁnite di↵erence...

IntroductionThe hybrid tree-finite di↵erence approachThe hybrid tree-finite di↵erence approach

Convergence resultsNumerical results

Generalization to other models

Hybrid tree-finite di↵erence methods for theHeston and Bates model with stochastic interest

rate.

Antonino Zanette

DIES, University of Udine

From joint works with:

Maya Briani* and Lucia Caramellino**

*IAC-CNR, Rome**Dept. of Mathematics, University of Roma-Tor Vergata

1 / 64




Introduction

We propose a mixed tree-finite di↵erence method in order toapproximate the Heston model.

We prove the convergence by embedding the procedure in abivariate Markov chain.

Numerical results show reliability and the e�ciency of thealgorithm.

We show how to generalize the procedure to theHeston-Hull-White model, the Heston-Hull-White2D model,and the Bates model, .

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Previous tree and finite di↵erence literature

Tree methods for the Heston model:

H. Nieuwenhuis, M. Vellekoop (2009): A tree-based methodto price American Options in the Heston Model. The Journal ofComputational Finance.

Finite di↵erences for the 2D Heston PDE:

R. Zvan, P. Forsyth, K. Vetzal (1998): A penalty method forAmerican options with stochastic volatility. J. Comp. Appl. Math.

S. Ikonen, J. Toivanen (2009): Operator splitting methods forpricing American options under stochastic volatility, Numer. Math.

K.J. in’t Hout, S. Foulon (2010): ADI finite di↵erenceschemes for option pricing in the Heston model with correlation.Int.J. Numer. Anal. Mod.

C. Chiarella, B.Kang, G.H. Meyer (2012): The evaluationof barrier option prices under stochastic volatility. Computers andMathematics with Applications. 2034-2048.

T. Haentjens, K.J. in’t Hout (2012): Alternating directionimplicit finite di↵erence schemes for the Heston-Hull-White partialdi↵erential equation. J. Comp. Finan.

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Heston Hull-White model

Finite di↵erences for the Heston Hull-White model:

T. Haentjens, K.J. in’t Hout (2012): Alternating directionimplicit finite di↵erence schemes for the Heston-Hull-White partialdi↵erential equation. .J. Comp. Finan.

Fourier cosine methods:

F.Fang, C.W. Oosterlee (2011): A Fourier-based valuationmethod for Bermudan and barrier options under Heston’s modelSIAM J. Fin. Math..

A.L. Grzelak, C.W. Oosterlee (2011): On the Hestonmodel with stochastic interest rates. SIAM J. Fin. Math..

A.L. Grzelak, C.W. Oosterlee (2012): On theCross-currency with stochastic volatility and stochastic interest.Applied Mathematical Finance.

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References

1 Appolloni E., Caramellino, L., Zanette, A. (2015):A robust tree method for pricing American options with CIRstochastic interest rate. IMA Journal of ManagementMathematics, (2015) Vol.26, Issue No. 4, 377-401.

2 Briani, M., Caramellino, L., Zanette, A. (2015): Ahybrid approach for the implementation of the Heston model.IMA Journal of Management Mathematics, DOI10.1093/imaman/dpv032

3 Briani, M., Caramellino, L., Zanette, A. (2016): Ahybrid tree/finite-di↵erence approach for Heston-Hull-Whitetype models. to appear in The Journal of ComputationalFinance, March 2018.

4 Briani, M., Caramellino, L., Zanette, A. (2016): Ahybrid approach for the implementation of the Bates modelwith stochastic interest rate.in: arXiv:1603:07225

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The hybrid tree-finite di↵erence approach

The hybrid tree-finite di↵erence approach : Main idea

Backward induction algorithm that works following a finitedi↵erence PDE method in the direction of the share process andfollowing a tree method in the direction of the other randomsources (volatility, interest rate and possibly dividend rate).

6 / 64




The Heston model

Under the risk neutral measure, the pair (S ,V ) of the share priceand the volatility process solves the SDE

dSt

St

= (r � �)dt +p

Vt

dZS

(t), S(0) = S0 > 0

dVt

= (✓ � V (t))dt + �p

Vt

dZV

(t), V (0) = V0 > 0

where:r and � are the risk free interest rate and the continuousdividend rate respectively,� is the volatility of the volatility, is the reversion speed,✓ is the long run variance,ZS

and Zr

are correlated Brownian motions:

dhZS

,ZV

it

= ⇢dt, |⇢| < 1.

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Hybrid Tree-Finite Di↵erence for the Heston model

The hybrid procedure :

a binomial tree for the volatility V ;

a a trasformation which keeps the di↵usion processes S and Vuncorrelated;

a finite di↵erence approach in the S-direction.

8 / 64




The robust tree method for V

Take N “large” and h = T/N. For n = 0, 1, . . . ,N, consider thelattice used in Appolloni, Caramellino, Zanette (2014)

Vh

n

= {vn,k}k=0,1,...,n with

vn,k =

⇣

p

V0 +�

2(2k � n)

ph⌘2

1pV0+�

2 (2k�n)ph>0

We define the multiple jumps

kh

d

(n, k) = max{k⇤ : 0 k⇤ k and vn,k + µ

V

(vn,k)h � v

n+1,k⇤},kh

u

(n, k) = min{k⇤ : k + 1 k⇤ n + 1 and vn,k + µ

V

(vn,k)h v

n+1,k⇤}

in which µV

denotes the drift coe�cient of V .

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0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 5 10 15 20 25 30

latticeV(n,k)

V(n+1,Kd)V(n+1,Ku)

Figure: Standard jumps and multiple jumps for the discreteapproximation of the process V .

10 / 64





Starting from the node (n, k), the discrete process can reach theup-jump node (n + 1, kh

u

(n, k)) or the down-jump node(n + 1, kh

d

(n, k)) with transition probability:

up-jump: phk

h

u

(n,k) = 0 _µV

(vn,k)h + v

n,k � vn+1,kh

d

(n,k)

vn+1,kh

u

(n,k) � vn+1,kh

d

(n,k)^ 1,

down-jump: phk

h

d

(n,k) = 1� phk

h

u

(n,k).

Multiple jumps & jump probabilities are set in order to match thefirst local moment of the tree and of the process V up to order 1w.r.t. h. As a consequence, as h ! 0 one gets weak convergenceon the path space.Remark. In order to obtain the convergence, we do not need torequire the Feller condition 2✓ � �2.

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The transformed process Y

We consider the di↵usion pair (Y ,V ), where

Yt

= log St

� ⇢

�Vt

.

Set ⇢ =p

1� ⇢2 and (W ,Z ) standard Brownian motion in R2.Then,

dYt

=⇣

r � � � 1

2Vt

� ⇢

�(✓ � V

t

)⌘

dt + ⇢p

Vt

dZt

, (1)

dVt

= (✓ � Vt

)dt + �p

Vt

dWt

, (2)

with Y0 = log S0 � ⇢�V0. We set

µY

(v) = r � � � 1

2v � ⇢

�v

(✓ � v) and µV

(v) = (✓ � v).

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The approximation of Y

Let V h = (V h

n

)n=0,...,N denote the tree process approximating V

and set V h

t

= V h

bt/hc, t 2 [0,T ], the associated piecewise constantand cadlad approximating path.

In order to approximate Y , we construct a Markov chain from thefinite di↵erence method.

We start from the Euler scheme: Y h

0 = Y0 and fort 2 (nh, (n + 1)h], n = 0, . . . ,N, set

Y h

t

= Y h

nh

+ µY

(V h

nh

)(t � nh) + ⇢q

V h

nh

(Zt

� Znh

),

Z being independent of the noise driving V h.

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The approximation of the pair (Y ,V )

Let f be a suitable function depending on both variables (y , v).Then,

E(f (Y(n+1)h,V(n+1)h) | Ynh

= y ,Vnh

= v)

' E(f (Y h

(n+1)h,Vh

(n+1)h) | Yh

nh

= y ,V h

nh

= v)

= E(uh(nh, y ; v ,V h

(n+1)h) | Vh

nh

= v)

where

uh(nh, y ; v , z) = E(f (Y h

(n+1)h, z) | Yh

nh

= y ,V h

nh

= v).

anduh(nh, y ; v , z) = uh(s, y ; v , z)|

s=nh

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The approximation of the pair (Y ,V )

Main fact: (s, y) 7! uh(s, y ; v , z) solves the PDE

@s

uh + µY

(v)@y

uh +1

2⇢2v@2

yy

uh = 0, y 2 R, nh < s < (n + 1)h,

uh((n + 1)h, y ; v , z) = f (y , z), y 2 R.(3)

) simple problem: one dimensional, constant coe�cients.

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The finite di↵erence scheme

The solution uh(s, y ; v , z) of the PDE problem (3) can be easilynumerically found by using a finite di↵erence method.

In practice we consider the finite grid Yh = {yj

}j2J

M

h

with equallyspaced points

yj

= Y0 + j�yh

, j 2 JM

h

= {�Mh

, . . . ,Mh

}.

The approximation of uh(nh, y ; v , z) is done by adding to (3)suitable boundary conditions - we use Neumann type conditions(but others can be chosen).

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The behavior of the solution of problem (3) changes with respectthe magnitude of the rate between the di↵usion coe�cient (⇢2v/2)and the advection term (µ

Y

(v)).

We fix a small real threshold ✏h

> 0. Then:

• case v > ✏h

: we use an implicit scheme

un+1j

� unj

h+ µ

Y

(v)unj+1 � un

j�1

2�y+

1

2⇢2 v

unj+1 � 2un

j

+ unj�1

�y2= 0,

with boundary conditions:

un�M�1 = un�M+1, unM+1 = un

M�1;

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• case v < ✏h

: we use an explicit scheme:⇤ if µ

Y

(v) � 0:

un+1j

� unj

h+ µ

Y

(v)un+1j+1 � un+1

j

�y+

1

2⇢2 v

un+1j+1 � 2un+1

j

+ un+1j�1

�y2= 0;

⇤ if µY

(v) < 0:

un+1j

� unj

h+ µ

Y

(v)un+1j

� un+1j�1

�y+

1

2⇢2 v

un+1j+1 � 2un+1

j

+ un+1j�1

�y2= 0;

here, the boundary conditions are

un+1�M�1 = un+1

�M+1, un+1M+1 = un+1

M�1.

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2-dimensional Markov chain

We obtain a finite-dimensional stochastic matrix ⇧h(v) that givesthe discrete solution {un

i

}i2J

M

of (3) at time nh in terms of thesolution {un+1

i

}i2J

M

at time (n + 1)h:

un = ⇧h(v)un+1.

By resuming, we get

E�

f (Y(n+1)h,V(n+1)h) | Ynh

= yi

,Vnh

= v�

'X

j2JM

h

⇧h(v)i,jE

�

f (yj

,V h

(n+1)h) | V h

nh

= v)�

, i 2 JM

h

.

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We take h small and we call X h = (X h

n

)n=0,1,...,N the 2-dimensional

Markov chain with transition probability law

µh(yj

, vn+1,k⇤ | y

i

, vn,k) =

8

>

<

>

:

⇧h(vn,k)ij ph

k

h

u

(n,k) if k⇤ = kh

u

(n, k)

⇧h(vn,k)ij ph

k

h

d

(n,k)if k⇤ = kh

d

(n, k)

0 otherwise,

for every (yi

, vn,k) 2 Yh ⇥ Vh

n

and (yj

, vn+1,k⇤) 2 Yh ⇥ Vh

n+1.

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Then, for n = 0, 1, . . . ,N � 1, yi

2 Yh and vn,k 2 Vh

n

we get

E(f (Y(n+1)h,V(n+1)h) | Ynh

= yi

,Vnh

= vn,k)

'X

k

⇤,j

⇧h(vn,k)i ,jp

h

k

⇤ f (yj

, vn+1,k⇤),

the above sum running on k⇤ 2 {khu

(n, k), khd

(n, k)} and j 2 JM

h

.

) we have constructed an approximation of (Y ,V ) at timestn

= nh, n = 0, 1, . . . ,N, through the Markov chain (X h

n

)n=0,1,...,N .

Before studying the convergence, we briefly discuss how to use theabove procedure to numerically price American options.

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American option pricing

Consider an American option with maturity T and payo↵ function

(Yt

,Vt

) = �(eYt

� ⇢�Vt ), t 2 [0,T ].

By considering the discrete dynamic programming principle and byusing the approximation of (Y ,V ) through the Markov chain X h

n

,n = 0, 1, . . . ,N, we approximate the price as follows:

for n = 0, 1, . . . ,N, we define Ph

(nh, y , v) for (y , v) 2 Yh ⇥ Vh

n

by

8

>

>

>

>

<

>

>

>

>

:

Ph

(T , yi

, vN,k) = (yi , vN,k) i 2 J

M

h

and vN,k 2 Vh

n

Ph

(nh, yi

, vn,k) = max

n

(yi

, vn,k), e

�rh⇥

⇥P

k

⇤,j ⇧h(v

n,k)i,j Ph

�

(n + 1)h, yj

, vn+1,k⇤

�

phk

⇤

o

,

i 2 JM

h

and vn,k 2 Vh

n

.

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Convergence conditions

We set up the dependence on the time step h for thespace-step �y , the number M giving the points of the gridYM

and the threshold ✏ that allows us to use the explicit orthe implicit finite di↵erences method.

�y ⌘ �yh

= cy

hp, M ⌘ Mh

= cM

h�q, ✏ ⌘ ✏h

= c✏hp (4)

where cM

> 0 and the constants cy

, c✏, p, q > 0 are chosen asfollows

p < 1, q > p,2c

y

⇢2�

�

r � � �⇢

�v

✓�

� < c✏, or

p = 1, q > p,2c

y

⇢2�

�

r � � �⇢

�v

✓�

� < c✏ <⇣1

2�

1

c

y

�

�

r � � �⇢

�v

✓�

�

⌘

c

2y

⇢2,

(5)

The constraint in (5) can be really satisfied, for example bychoosing c

y

> 4|r � � � ⇢�v

✓|.23 / 64




Convergence theorem

The algorithm is actually given by approximating in law thedi↵usion pair X = (Y ,V ) with the Markov chainX h = (Y h, V h).We set X h = (Y h,V h) as the piecewise constant and cadlaginterpolation in time of X h, that is

X h

t

= X h

n

, t 2 [nh, (n + 1)h), n = 0, 1, . . . ,N � 1. (6)

We set D([0,T ];R2) the space of the R2-valued and cadlagfunctions on the interval [0,T ], that we assume to be endowedwith the Skorohod topology. Our main result is the following:

Theorem

Suppose that (4) and (5) hold. Then as h ! 0, the sequence{X h}

h

= {(Y h,V h)}h

weakly converges in the space D([0,T ];R2) to thedi↵usion process X = (Y ,V ) solution to (1)-(2).

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Convergence of the options price

Let us consider pricing European options with payo↵ functionf : D([0,T ];R) ! R+.

The transformed payo↵-function g(y , v) = f (ey+⇢� v ),

(y , v) 2 D([0,T ];R2)

The associated option prices on the continuous and thediscrete model as seen at time 0 are given by

PEu

= E�

g(Y ,V )�

and Ph

Eu

= E�

g(Y h,V h)�

,

respectively, g denoting the discounted payo↵, i.e. g = e�rTg

25 / 64




Convergence of the European and American Put optionsprice

The weak convergence in Theorem 24 ensures the convergencePh

Eu

! PEu

of the European price when the discountedpayo↵-function fulfills the following requests: (y , v) 7! g(y , v)is continuous and there exists a > 0 and h⇤ > 0 such that

suph<h⇤

E�

|g(Y h,V h)|1+a

�

< 1.

As for American style options, even for simple payo↵s thingsare more di�cult because of the presence of optimal stoppingtimes. However, due to the results in Amin and Khanna 94,we can deduce the convergence of the prices for suitablepayo↵s.European and American put options can be numericallyevaluated by means of the approximating algorithm 26 / 64




Numerical results: vanilla options

We compare the performance of the hybrid tree-finitedi↵erence algorithm with the tree method of Vellekoop andNieuwenhuis. We consider as benchmark the Carr Madan FFTmethod in the European case and the Monte CarloLongsta↵-Schwartz metohd in the American case.Parameters: S0 = 100, K = 100, T = 1, r = log(1.1), � = 0,V0 = 0.1, ✓ = 0.1, = 2 and ⇢ = �0.5.In order to study the numerical robustness of the algorithmswe choose di↵erent values for �: 0.04, 0.5, 1.For � = 1 the Feller condition 2✓ � �2 is not satisfied.HTFD1: fixed number of time steps N

t

= 100 and varyingnumber of space steps N

S

= 50, 100, 200, 400;HTFD2: number of time steps equal to the number of spacesteps: N

t

= NS

= 50, 100, 200, 400.27 / 64




Numerical results: European options

NS

VN HTFD1 HTFD2 CF� =0.04 50 8.040982 7.934492 7.911034

100 8.021780 7.970437 7.970437200 8.003938 7.978890 7.983188 7.994716400 7.984248 7.980984 7.990825

� =0.50 50 8.148234 7.758954 7.746533100 7.727191 7.804520 7.804520200 7.813599 7.816749 7.821404 7.8318540400 7.910909 7.818596 7.827805

� =1.00 50 6.586889 7.214303 7.247748100 7.114225 7.225292 7.225292200 7.964052 7.228235 7.229139 7.2313083400 6.639931 7.224356 7.233742

Table: Prices of European put options. � = 0.04, 0.5, 1. S0 = 100,K = 100, T = 1, r = log(1.1), � = 0, V0 = 0.1, ✓ = 0.1, = 2,⇢ = �0.5.

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Numerical results: American options

NS

VN HTFD1 HTFD2 MC-LS� =0.04 50 9.100312 8.966651 8.932445

100 9.086233 9.016732 9.016732200 9.073722 9.028866 9.042581 9.074102400 9.063396 9.031881 9.054538

� =0.50 50 9.150887 8.763369 8.731867100 8.892206 8.841776 8.841776200 8.981855 8.862606 8.878530 8.904514400 9.058313 8.866911 8.892583

� =1.00 50 8.588392 8.185052 8.206052100 9.020989 8.263395 8.263395200 9.251595 8.281755 8.290371 8.277985400 9.102788 8.283214 8.304415

Table: Prices of American put options. � = 0.04, 0.5, 1. S0 = 100,K = 100, T = 1, r = log(1.1), � = 0, V0 = 0.1, ✓ = 0.1, = 2,⇢ = �0.5.

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Numerical results: computational time

NS

VN HTFD1 HTDF250 0.11 0.02 0.007100 0.42 0.04 0.040200 1.73 0.08 0.380400 7.06 0.16 3.040

Table: Computational times (in seconds) for European put optionsfor � = 0.5.

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American vanilla options: comparison with finite di↵erencemethods

N

S

HTFD1 HTFD2 ZFV IT-PSOR OO CPS0 = 8 50 2.077001 2.076301

100 2.077402 2.077402200 2.077510 2.077904 2.0784 2.0783 2.0790 2.0733400 2.077540 2.078141800 2.077548 2.078255

S0 = 9 50 1.330748 1.329763100 1.332058 1.332058200 1.332351 1.332923 1.3337 1.3335 1.3340 1.3290400 1.332424 1.333295800 1.332445 1.333469

S0 = 10 50 0.792953 0.791768100 0.794123 0.794123200 0.794515 0.795156 0.7961 0.7958 0.7960 0.7992400 0.794605 0.795589800 0.794632 0.795790

S0 = 11 50 0.445496 0.444646100 0.446940 0.446940200 0.447200 0.447661 0.4483 0.4481 0.4490 0.4536400 0.447273 0.447983800 0.447286 0.448131

S0 = 12 50 0.241466 0.241199100 0.242226 0.242226200 0.242360 0.242546 0.2428 0.2427 0.2430 0.2502400 0.242393 0.242678800 0.242404 0.242743

Table: Prices of American put options. S0 = 8, 9, 10, 11, 12, K = 10,T = 0.25, r = 0.1, � = 0, V0 = 0.25, ✓ = 0.16, = 5, ⇢ = 0.1,� = 0.9.

31 / 64




Numerical results: Convergence behaviour

In order to study the convergence behaviour of HTFD2 weconsider the following convergence ratio proposed in D’Halluin etal. :

ratio =P

N

2� P

N

4

PN

� PN

2

, (7)

where PN

denotes here the approximated price obtained withN = N

t

= NS

number of time steps.

N S0 = 8 S0 = 9 S0 = 10 S0 = 11 S0 = 12200 2.194914 2.653543 2.280589 3.181503 3.204384400 2.115892 2.322611 2.380855 2.237965 2.423995800 2.074822 2.140352 2.165029 2.178548 2.044881

The table suggests that the convergence ratio for HTDF2 islinear, as it is expected to be because of the tree contribution(whose error typically behaves linearly).

32 / 64




Numerical results: barrier options

We study continuously monitored barrier options and we compareour hybrid tree-finite di↵erence algorithm with the numericalresults of the method of lines MOL of Chiarella et al.

We consider European and American up-and-out call options withthe following set of parameters: K = 100, T = 0.5, r = 0.03,� = 0.05, V0 = 0.1, ✓ = 0.1, = 2, ⇢ = �0.5.The up barrier is H = 130. We choose di↵erent values for S0:S0 = 80, 100, 120.

33 / 64




Numerical results: European barrier options

NS

HTFD1 HTFD2 MOLS0 =80 50 0.913861 0.875374

100 0.893484 0.893484200 0.895127 0.900893 0.9029400 0.897820 0.902770

S0 =100 50 2.635396 2.583568100 2.606249 2.606249200 2.597363 2.591857 2.5908400 2.603679 2.594134

S0 =120 50 1.417225 1.438429100 1.485704 1.485704200 1.500692 1.482193 1.4782400 1.504755 1.486212

Table: Prices of European call up-and-out options. Up barrier isH = 130. K = 100, T = 0.5, r = 0.03, � = 0.05, V0 = 0.1, ✓ = 0.1, = 2, ⇢ = �0.5.

34 / 64




Numerical results: American barrier options

NS

HTFD1 HTFD2 MOLS0 =80 50 1.199802 1.285959

100 1.369914 1.369914200 1.400823 1.396628 1.4012400 1.400710 1.401111

S0 =100 50 8.274116 8.269779100 8.286667 8.286667200 8.284054 8.294226 8.3003400 8.283815 8.296745

S0 =120 50 21.943742 21.884228100 21.820015 21.820015200 21.785274 21.815989 21.8216400 21.779648 21.804518

Table: Prices of American call up-and-out options. Up barrier isH = 130. K = 100, T = 0.5, r = 0.03, � = x0.05, V0 = 0.1,✓ = 0.1, = 2, ⇢ = �0.5.

35 / 64




Numerical results: computational time

NS

HTFD1 HTDF250 0.007 0.017100 0.132 0.132200 0.284 1.079400 0.535 8.901

Table: Computational times (in seconds) for European Barrieroptions.

36 / 64




Conclusion

The numerical results show that our method is very stable androbust, on the contrary tree methods fail when �

r

increases.

The proposed method is e�cient and reliable.

37 / 64




Heston-Hull-White model

The model is a Heston model with a stochastic interest rate r :under the risk neutral measure, the model is

dSt

St

= (rt

� �)dt +p

Vt

dZt

, S0 > 0,

dVt

= V

(✓V

� Vt

)dt + �V

p

Vt

dW 1t

, V0 > 0

drt

= r

(✓r

(t)� rt

)dt + �r

dW 2t

, r0 > 0.

where Z , W 1 and W 2 are Brownian motions with correlations

dhZ ,W1it = ⇢1 dt, dhZ ,W2it = ⇢2 dt, dhW 1,W 2it

= 0.

Here, r follows a generalized OU process: ✓r

is a (deterministic)function determined by the market values of the zero-couponbonds.

38 / 64





The hybrid procedure:

a 2-dimensional binomial tree for the pair (V , r);


39 / 64




Transformation

By passing to the logarithm Y = ln S in the first component and takinginto account the above mentioned correlations, we reduce to thedynamics

dYt

= (rt

� ⌘ � 1

2Vt

)dt +p

Vt

�

⇢1dW1t

+ ⇢2dW2t

+ ⇢3dW3t

�

, Y0 = ln S0 2 R,

dVt

= V

(✓V

� Vt

)dt + �V

p

Vt

dW 1t

, V0 > 0,

drt

= r

(✓r

(t)� rt

)dt + �r

dW 2t

, r0 > 0,

where W = (W 1,W 2,W 3) is a standard Brownian motion in R3 and thecorrelation parameter ⇢3 is given by

⇢3 =q

1� ⇢21 � ⇢22, (⇢1, ⇢2) 2 B1(0),

B1(0) denoting the open ball in R2 centered in 0 and with radius 1.

40 / 64




The process r can be written in the following way:

rt

= �r

Xt

+ 't

(8)

where

Xt

= �r

Z

t

0Xs

ds + W 2t

and 't

= r0e�

r

t + r

Z

t

0✓r

(s)e�r

(t�s)ds.

(9)So, we can consider the triple (Y ,V ,X ), whose dynamics is given by

dYt

= µY

(Vt

,Xt

, t)dt +pVt

�

⇢1dW 1t

+ ⇢2dW 2t

+ ⇢3dW 3t

�

, Y0 = ln S0 2 R,dV

t

= µV

(Vt

)dt + �V

pVt

dW 1t

, V0 > 0,

dXt

= µX

(Xt

)dt + dW 2t

, X0 = 0,(10)

where

µY

(v , x , t) = �r

x + 't

� ⌘ � 1

2v , (11)

µV

(v) = V

(✓V

� v), (12)

µX

(x) = �r

x . (13) 41 / 64




The tree for X

For n = 0, 1, . . . ,N, consider the lattice for the process X

X h

n

= {xn,j}j=0,1,...,n with x

n,j = (2j � n)ph (14)

We define the multiple jumps

kh

d

(n, k) = max{k⇤ : 0 k⇤ k and xn,k + µ

X

(xn,k)h � x

n+1,k⇤},kh

u

(n, k) = min{k⇤ : k + 1 k⇤ n + 1 and xn,k + µ

X

(xn,k)h x

n+1,k⇤}

The transition probabilities are defined as follows

pX ,hu

(n, j) = 0_µX

(xn,j)h + x

n,j � xn+1,jh

d

(n,j)

xn+1,jh

u

(n,j) � xn+1,jh

d

(n,j)^1 and pX ,h

d

(n, j) = 1�pX ,hu

(n, j)

(15)respectively.

42 / 64




The tree for the pair (V ,X )

For n = 0, 1, . . . ,N, consider the lattice

Vh

n

⇥ X h

n

= {(vn,k , xn,j)}k,j=0,1,...,n. (16)

Starting from the node (n, k , j), which corresponds to the position(v

n,k , xn,j) 2 Vh

n

⇥ X h

n

, we define the four possible jump by setting thefour nodes at time n + 1

(n + 1, kh

u

(n, k), jhu

(n, j)) with probability phuu

(n, k , j) = pV ,hu

(n, k)pX ,hu

(n, j),

(n + 1, kh

u

(n, k), jhd

(n, j)) with probability phud

(n, k , j) = pV ,hu

(n, k)pX ,hd

(n, j),

(n + 1, kh

d

(n, k), jhu

(n, j)) with probability phdu

(n, k , j) = pV ,hd

(n, k)pX ,hu

(n, j),

(n + 1, kh

d

(n, k), jhd

(n, j)) with probability phdd

(n, k , j) = pV ,hd

(n, k)pX ,hd

(n, j),(17)

One might include correlations between any two of the Brownian motionsdriving the processes V , X . The jump probabilities are no more of aproduct-type but they solve a linear system of equations that mustinclude the matching of the local cross-moments up to order one in h.

43 / 64




We go now back to (10), that is

dYt

= µY

(Vt

,Xt

, t)dt +pVt

�

⇢1dW 1t

+ ⇢2dW 2t

+ ⇢3dW 3t

�

, Y0 = ln S0,

dVt

= µV

(Vt

)dt + �V

pVt

dW 1t

, V0 > 0,

dXt

= µX

(Xt

)dt + dW 2t

, X0 = 0,

By isolatingpVt

dW 1t

in the second line and dW 2t

in the third one, weobtain

dYt

=⇢1�V

dVt

+ ⇢2p

Vt

dXt

+ µ(Vt

,Xt

, t)dt + ⇢3p

Vt

dW 3t

(18)

with

µ(v , x , t) = µY

(v , x , t)� ⇢1

�V

µV

(v)� ⇢2pv µ

X

(x)

= �r

x + 't

� ⌘ � 12 v � ⇢1

�V

V

(✓V

� v) + ⇢2r

xpv .

(19)

The main point is that the noise W 3 is independent of the processes Vand X .

44 / 64




The hybrid Monte Carlo algorithm

The approximation we have set-up for the Heston-Hull-Whiteprocesses can be used to construct a Monte Carlo algorithm;we simulate a continuous process in space (the component Y )starting from a discrete process in space (the 2-dimensionaltree for (V ,X )).

Euler Scheme We set Y h

0 = Y0 and for t 2 [nh, (n + 1)h] withn = 0, 1, . . . ,N � 1 then

Y h

n+1 = Y h

n

+⇢1�V

(V h

n+1�V h

n

)+⇢2

q

V h

n

(X h

n+1�X h

n

)+µh+⇢3

q

hV h

n

�n+1,

(20)where µ is defined in (19) and �1, . . . ,�N

denote i.i.d. standardnormal r.v.’s, independent of the noise driving the chain (V , X ).One let the pair (V,X) evolve on the tree and simulate the processY at time nh by using (7.1)

45 / 64




The approximating scheme for the triple (Y ,V ,X )

Y

h

t

= Y

h

nh

+⇢1�V

(V h

t

�V

h

nh

)+⇢2

q

V

h

nh

(X h

t

�X

h

nh

)+µ(X h

nh

, V h

nh

, nh)(t�nh)+⇢3

q

V

h

nh

(W 3t

�W

3nh

).

(21)

If we set

Z h

t

= Y h

t

� ⇢1�V

(V h

t

�V h

nt

)�⇢2

q

V h

nh

(X h

t

�Xnt

), t 2 [nh, (n+1)h] (22)

then we have

dZ h

t

= µ(X h

nh

, V h

nh

, nh)dt + ⇢3

q

V h

nh

dW 3t

, t 2 (nh, (n + 1)h],

Z h

nh

= Y h

nh

(23)

46 / 64




Take now a function f : we are interested in approximating

E(f (Y(n+1)h) | Ynh

= y ,Vnh

= v ,Xnh

= x).

By using our scheme and the process Zh in (22), we approximateit with

E(f (Y h

(n+1)h) | Yh

nh

= y , V h

nh

= v , X h

nh

= x)

= E(f (Z h

(n+1)h +⇢1�V

(V h

(n+1)h � V

h

nh

) + ⇢2

q

V

h

nh

(X h

(n+1)h � X

h

nh

)) | Z h

nh

= y , V h

nh

= v , X h

nh

= x).

47 / 64




PDE Problem

Since (V h, X h) is independent of the Brownian noise W 3 driving Z h in(22), we can write

E(f (Y h

(n+1)h) | Yh

nh

= y , V h

nh

= v , X h

nh

= x)

= E⇣

f

⇣

⇢1�V

(V h

(n+1)h � v) + ⇢2pv(X h

(n+1)h � x); y , v , x⌘

�

�

�

V

h

nh

= v , X h

nh

= x

⌘

,

(24)

in which

f

(⇠; y , v , x) = E(f (Z h

(n+1)h + ⇠) | Z h

nh

= y , V h

nh

= v , X h

nh

= x). (25)

48 / 64




Now, in order to compute the above quantity f

(⇠), consider a genericfunction g and set

u(s, z ; v , x) = E(g(Z h

(n+1)h) | Z h

s

= z , V h

s

= v , X h

s

= x), s 2 [nh, (n+1)h].

By (23) and the Feynmac-Kac representation formula we can state that,for every fixed x 2 R and v � 0, the function (s, z) 7! u(s, z ; v , x) is thesolution to(

@s

u + µ(v , x , s)@z

u + 12⇢

23v@

2z

u = 0, s 2 [nh, (n + 1)h), z 2 R,u((n + 1)h, z ; v , x) = g(z),

(26)µ being given in (19). In order to solve the above PDE problem, we use afinite di↵erence approach.

49 / 64




The scheme on the Y -component

We get the approximation

f

�

⇠; yi

, vn,k , xn,j

�

'X

`2JM

⇧h

i`(vn,k , xn,j)f�

y` + ⇠�

, i 2 JM

.

Therefore, the expectation is computed on the approximating treefor (V ,X ) by means of the above approximation:

E(f (Y h

(n+1)h) | Y h

nh

= y

i

, V h

nh

= v

n,k , Xh

nh

= x

n,j ) 'X

a,b2{d,u}

X

`2JM

⇧h

i`(vn,k , xn,j )Tn,k,j f (`, a, b)ph

ab

(n, k, j)

(27)

where

T

n,k,j f (`, a, b) = f⇣

y` +⇢1�V

(vn+1,k

a

(n,k) � v) + ⇢2pv(x

n+1,jb

(n,j) � x)⌘

50 / 64




The algorithm for the pricing of American options

Finally, we can summarize the backward induction giving ourapproximating algorithm as follows. For n = 0, 1, . . . ,N, we definePh

(nh, y , v , x) for (y , v , x) 2 Dh

n,M as

8

>

>

>

>

>

>

<

>

>

>

>

>

>

:

P

h

(T , yi

, vN,k , xN,j ) = (yi ) for (y

i

, vN,k , xN,j ) 2 D

N,M and as n = N � 1, . . . , 0:

P

h

(nh, yi

, vn,k , xn,j ) = max

n

(yi

), e�(�r

x

n,j+'nh

)h⇥

⇥X

a,b2{d,u}

X

`2JM

⇧h

i`(vn,k , xn,j )ph

ab

(n, k, j)Tn,k,jPh

(`, a, b)o

,

for (yi

, vn,k , xn,j ) 2 Dh

n,M .

(28)

51 / 64




Heston-Hull-White2D model

dSt

St

= (rt

� ⌘t

)dt +p

Vt

dZt

,

dVt

= V

(✓V

� Vt

)dt + �V

p

Vt

dW 1t

,

drt

= r

(✓r

(t)� rt

)dt + �r

dW 2t

,

d⌘t

= ⌘(✓⌘(t)� ⌘t

)dt + �⌘dW3t

,

with initial data S0,V0, r0, ⌘0 > 0, where Z , W 1, W 2 and W 3 aresuitable and possibly correlated Brownian motions. Note that the process⌘ evolves as a generalized OU process: ✓⌘ is a deterministic function ofthe time. We consider non null correlations between the Brownianmotions driving the pairs (S ,V ), (S , r) and (S , ⌘), that is

dhZ ,W 1it

= ⇢1 dt, dhZ ,W 2it

= ⇢2 dt, dhZ ,W 3it

= ⇢3 dt.

52 / 64





The hybrid procedure:

a 3-dimensional binomial tree for the triple (V , r , ⌘);


53 / 64




Numerical results

We consider

HTFD1 refers to the (fixed) number of time steps Nt

= 50 andvarying number of space steps N

S

= 50, 100, 150, 200;

HTFD2 refers to Nt

= NS

= 50, 100, 150, 200.

HMC1 and AMC1 refer to 50 000 iterations,

HMC2 and AMC2 refer to 200 000 iterations.

The benchmark value B-AMC is obtained using the Alfonsi Monte Carlomethod AMC with a huge number of Monte Carlo simulations (1 millioniterations) and N

t

= 300 discretization time steps.In the American case, in absence of reliable numerical methods, weconsider the Longsta↵-Schwartz algorithm MC-LS with 20 exercise dates.

54 / 64




Numerical results : European Options

⇢SV

= �0.5 N

S

HTFD1 HTFD2 B-AMC HMC1 HMC2 AMC1 AMC2⇢Sr

= �0.5 50 11.202744 11.202744 11.34±0.04 11.30±0.16 11.32±0.08 11.34±0.16 11.37±0.08100 11.319814 11.331040 11.41±0.16 11.38±0.08 11.31±0.16 11.36±0.08150 11.340665 11.349902 11.36±0.16 11.36±0.08 11.35±0.16 11.38±0.08200 11.346972 11.355772 11.34±0.16 11.37±0.08 11.44±0.16 11.39±0.08

⇢Sr

= 0 50 12.526779 12.526779 12.77±0.04 12.66±0.18 12.69±0.09 12.68±0.18 12.79±0.09100 12.720651 12.705772 12.74±0.18 12.79±0.09 12.63±0.18 12.78±0.09150 12.754610 12.749526 12.74±0.18 12.79±0.09 12.68±0.18 12.81±0.09200 12.760365 12.766836 12.74±0.18 12.80±0.09 12.75±0.18 12.79±0.09

⇢Sr

= 0.5 50 13.853193 13.853193 14.04±0.04 13.88±0.19 13.92±0.10 13.97±0.20 14.05±0.10100 14.011537 14.013063 13.91±0.19 14.01±0.10 13.89±0.19 14.06±0.10150 14.031598 14.038361 13.94±0.19 14.07±0.10 13.92±0.20 14.08±0.10200 14.038235 14.045612 13.99±0.19 14.07±0.10 13.90±0.19 14.06±0.10

Table: Prices of European call options. S0 = 100, K = 100, T = 1,r0 = 0.04,

r

= 1, �r

= 0.2, ⌘ = 0.03, V0 = 0.1, ✓V

= 0.1, V

= 2,�V

= 0.3, ⇢Sr

= �0.5, 0, 0.5, ⇢SV

= �0.5.

55 / 64




Numerical results : American Options

⇢SV

= �0.5 N

S

HTFD1 HTFD2 MC-LS⇢Sr

= �0.5 50 12.090433 12.090433 12.18±0.01100 12.205014 12.212884150 12.224432 12.231392200 12.230288 12.237054

⇢Sr

= 0 50 12.912708 12.912708 13.14±0.01100 13.119121 13.101073150 13.156492 13.149182200 13.162893 13.168602

⇢Sr

= 0.5 50 13.944266 13.944266 14.15±0.01100 14.125059 14.122918150 14.146240 14.152060200 14.153288 14.160288

Table: Prices of American call options. S0 = 100, K = 100, T = 1,r0 = 0.04,

r

= 1, �r

= 0.2, ⌘ = 0.03, V0 = 0.1, ✓V

= 0.1, V

= 2,�V

= 0.3, ⇢Sr

= �0.5, 0, 0.5, ⇢SV

= �0.5.

56 / 64




NS

HTFD1 HTDF2 B-AMC HMC1 HMC2 AMC1 AMC50 0.41 0.41 223.67 0.77 3.05 2.16 7.48100 0.84 11.33 1.59 6.11 4.00 14.61150 1.37 49.99 2.33 9.13 5.87 21.64200 1.87 213.06 3.11 12.73 7.61 28.85

Table: Computational times (in seconds) for European call options.

57 / 64




Numerical results : European Options

⇢SV

= �0.5,⇢S⌘ = �0.5

N

S

HTFD1 HTFD2 B-AMC HMC1 HMC2 AMC1 AM2

⇢Sr

= �0.5 30 13.470572 13.470572 13.79 ± 0.04 13.82±0.20 13.74±0.10 13.83±0.20 13.79±0.1050 13.688842 13.671173 13.96±0.20 13.81±0.10 13.88±0.20 13.80±0.10100 13.790205 13.781519 14.00±0.20 13.80±0.10 13.68±0.20 13.73±0.10

⇢Sr

= 0 30 14.736242 14.736242 15.04 ± 0.05 15.10±0.22 14.99±0.11 14.95±0.22 15.03±0.1150 14.958094 14.946029 15.23±0.22 15.04±0.11 14.98±0.22 15.01±0.11100 15.019204 15.032709 15.23±0.22 15.04±0.11 14.80±0.21 14.97±0.11

⇢Sr

= 0.5 30 15.805046 15.805046 16.19 ± 0.03 15.21±0.22 15.04±0.11 16.04±0.23 16.17±0.1250 16.052315 16.032043 16.13±0.23 16.06±0.11 16.09±0.23 16.13±0.12100 16.155354 16.145308 16.33±0.23 16.10±0.11 15.93±0.23 16.12±0.12

Table: Prices of European call options. S0 = 100, K = 100, T = 1,r0 = 0.04,

r

= 1, �r

= 0.2, ⌘0 = 0.03, ⌘ = 1, �⌘ = 0.2, V0 = 0.1,✓V

= 0.1, V

= 2, �V

= 0.3, ⇢Sr

= �0.5, 0, 0.5, ⇢SV

= �0.5,⇢S⌘ = �0.5.

58 / 64




Numerical results : American Options

⇢SV

= �0.5,⇢S⌘ = �0.5

N

S

HTFD1 HTFD2 MC-LS

⇢Sr

= �0.5 30 14.057963 14.057963 14.37 ± 0.0150 14.290597 14.263254100 14.400377 14.381552

⇢Sr

= 0 30 14.989844 14.989844 15.32 ± 0.0150 15.253011 15.229151100 15.320569 15.331744

⇢Sr

= 0.5 30 15.826696 15.826696 16.28 ± 0.0250 16.146080 16.111559100 16.270439 16.248656

Table: Prices of American call options. S0 = 100, K = 100, T = 1,r0 = 0.04,

r

= 1, �r

= 0.2, ⌘0 = 0.03, ⌘ = 1, �⌘ = 0.2, V0 = 0.1,✓V

= 0.1, V

= 2, �V

= 0.3, ⇢Sr

= �0.5, 0, 0.5, ⇢SV

= �0.5,⇢S⌘ = �0.5.

59 / 64




NS

HTFD1 HTDF2 B-AMC HMC1 HMC2 AMC1 AMC230 2.22 2.22 284.84 0.60 2.61 1.79 6.0350 4.15 24.56 1.14 4.19 2.73 9.58100 7.95 998.1 2.02 8.06 5.05 18.70

Table: Computational times (in seconds) for European call options.

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Bates model

The Bates model di↵ers from the Heston model in the presence ofjumps in the equation for the share price S :

dSt

St

= (r � �)dt +p

Vt

dZS

(t) + dNt

, S(0) = S0 > 0

dVt

= (✓ � Vt

)dt + �p

Vt

dZV

(t), V (0) = V0 > 0.

Here, Nt

is a compound Poisson process independent of thecorrelated Brownian motion (Z

S

,ZV

) with intensity � andindependent jumps J1, J2, . . . whose common law is

log(1 + J) ⇠ N⇣

log(1 + �)� 1

2↵2,↵2

⌘

.

So, the PDE problem becomes a PIDE problem.61 / 64




@s

uh(s, y ; v , gt+h

) + L(v)uh(s, y ; v , gt+h

) = 0 y 2 R, s 2 (t, t + h),

uh(t + h, y ; v , gt+h

) = f (y , gt+h

) y 2 R,(29)

with

L(v)u = µY

(v)@y

u+1

2⇢2v@2

yy

u+

Z +1

�1[u(y + x , s)� u(y)] p(x)dx .

62 / 64




To solve the PIDE perform the following steps:

Localisation. We choose a spatial bounded computational domain⌦

l

= [�l , l ], which implies that we must choose some artificialboundary conditions.

Truncation of large jumps. This step corresponds to truncating theintegration domain in the integral part.

Discretisation. The derivatives of the solution are replaced by finitedi↵erences, and the integral terms are approximated using thetrapezoidal rule. Then the problem is solved by using anexplicit-implicit scheme (see Cont and Voltchkova, Briani et al).

63 / 64




Conclusion

We have introduced a new hybrid tree-finite di↵erence methodand a new Monte Carlo method for numerically pricingoptions in a stochastic volatility with jumps framework withstochastic interest rates.

The numerical comparisons show that both methods providegood approximation of the option prices with e�cient timecomputations.

64 / 64

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