Option pricing with sparse grid quadrature

JASS 2007

Marcin Salaterski

Overview

• Option– Definition– Pricing

• Quadrature– Multivariate– Univariate– Sparse grids

• Hierarchical basis• Smolyak

Option

• Agreement in which the buyer has the right to buy (call) or sell (put) an asset at a set price on or before a future date.

• Value determined by an underlying asset.

Payo®(call) = max(S-K, 0)Payo®(put) = max(K-S, 0)

Long put (Bought „selling” option)

Short call (Sold „buying” option)

Option pricing example I

Option pricing example II

• Construct a riskless, self-financing portfolio.– Start with no money.– Take a loan at a compound interest rate.– Buy underlying assets and sell an option.– After some time sell assets and repay option.– Repay loan.– Finish with no money.

Option pricing example III

$22£ ±¡ $1 = $18£ ±¡ $0

± = 0:25

$18£ 0:25 = $4:50

$4:50£ e¡ :12£ 0:25 = $4:367

$20£ 0:25 = $4:367+V

V = $0:633

Option pricing methods

Brownian Motion

Brownian Motion - example

dS(t)S(t) = ¹ (t)dt+¾(t)dW(t)

Mathematical model

• Asset price process:

• Option value equation:

• Numeraire:N(t) = exp(

Z t

0r(¿)d¿)

V(T) = max(S(T) ¡ K ;0)

\begin{eqnarray}dS(t) &=& \mu^{P}S(t)dt + \sigma S(t)dW^{P}(t) \nonumber\end{eqnarray}

Expectation method I

• Choose appropriate Numeraire

.

• Calculate drift ,so that are martingales, i.e. .

• Find the distribution of under measure.

• Calculate .

¹ QN S(t)N (t) ;

V (t)N (t)

S(t)

V (t)N (t) = E ( V (v)N (v) );80< t < v < 1

V(0)

QN

N(t) = exp(Rt0r(¿)d¿)

Expectation method II

V(0) = N(0)EQN(V(T)N (T)

)

V(0) = exp(¡ rT)EQN(max(exp(logS(T)) ¡ K ;0))

V(0) = exp(¡ rT)Z 1

¡ 1max(exp(y) ¡ K ;0)

1

¾pTÁ(y ¡ ¹

¾pT)dy

Á(x) =1

p2¼exp(

¡ x2

2)

(2)

Multivariate quadrature

• Product of univariate quadrature.

• Monte Carlo methods.

• Quasi Monte Carlo methods.

• Sparse grids.

Univariate quadrature – Trapezoidal rule

If =R1¡ 1f (x)dx ¼Qf =

P nk=1

wkf (xk)Rbaf (x)dx ¼(b¡ a)f (a+b2 )

Univariate quadrature methods

• Newton-Cotes – even point distance, hierarchical

• Clenshaw-Curtis – Chebyshev polynomials, hierarchical

• Gauss – polynomials, not hierarchical

Quadrature by Archimedes

Hierarchical basis I

• Basis function

• Distance between points

• Grid points

• Local basis functions

hn = 2¡ n

Án;i (x) = Á( x¡ xn ;ihn

)

Á(x) =

(1¡ jxj x 2 [¡ 1;1]0 otherwise

xn;i = ihn ;1 · i < 2n ; i odd

Hierarchical basis II

Hierarchical basis III

Hierarchical quadratureZ 1

¡ 1f (x)dx ¼

nX

l=1

X

i2 I

cl;i

Z 1

¡ 1Ál;i (x)dx

Full grid

Cost/Gain

• Gain:

• Costs:

2¡ 2jl j1

2jl j1¡ d

Sparse grid

Comparison – 3D

0

200000

400000

600000

800000

1000000

1200000

Costs n=10

Full Grid

Sparse Grid

Smolyak I

Smolyak II

Q(d)l f =

X

kik· l+d¡ 1

(¢ (1)i1

­ ::: ­ ¢ (1)id)f

¢ (1)i = Q(1)

i ¡ Q(1)i ¡ 1

Qaf =n1X

i=1

w1;i f (x1;i )

Qbf =n2X

i=1

w2;i f (x2;i )

(Qa ­ Qb)f =n1X

j =1

w1;j (n2X

i=1

w2;i f (x1;j ;x2;i ))

Smolyak III

Comparison

Literature

• On the numerical pricing of financial derivatives based on sparse grid quadrature – Michael Griebel, Numerical Methods in Finance, An Amamef Conference INRIA, 1. February, 2006

• Slides to lecture Scientific Computing 2 – Prof. Bungartz, TUM

• An Introduction to Computational Finance Without Agonizing Pain - Peter Forsyth• Mathematical Finance – Christian Fries, not published

yet• PDE methods for Pricing Derivative Securities - Diane

Wilcox

Thank you !