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. P ayo ® ( ca ll ) = max ( S - K, 0 ) P ayo ® ( pu t ) = 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 : V V = $ 0 : 633

Option pricing methods

Brownian Motion

Brownian Motion - example d S ( t ) S ( t ) = ¹ ( t ) d t + ¾ ( t ) d W ( t )

Mathematical model Asset price process: Option value equation: Numeraire: N ( t ) = exp ( Z t 0 r ( ¿ ) 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 Q N S ( t ) N ( t ) ; V ( t ) N ( t ) S ( t ) V ( t ) N ( t ) = E ( V ( v ) N ( v ) ) ; 8 0 < t < v < 1 V ( 0 ) Q N N ( t ) = exp ( R t 0 r ( ¿ ) d ¿ )

Expectation method II

Multivariate quadrature Product of univariate quadrature. Monte Carlo methods. Quasi Monte Carlo methods. Sparse grids.

Univariate quadrature – Trapezoidal rule I f = R 1 ¡ 1 f ( x ) d x ¼ Q f = P n k = 1 w k f ( x k ) R b a f ( x ) d x ¼ ( b ¡ a ) f ( a + b 2 )

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 h n = 2 ¡ n Á n ; i ( x ) = Á ( x ¡ x n ; i h n ) Á ( x ) = ( 1 ¡ j x j x 2 [ ¡ 1 ; 1 ] 0 o t h erw i se x n ; i = i h n ; 1 · i < 2 n ; i o dd

Hierarchical basis II

Hierarchical basis III

Hierarchical quadrature Z 1 ¡ 1 f ( x ) d x ¼ n X l = 1 X i 2 I c l ; i Z 1 ¡ 1 Á l ; i ( x ) d x

Full grid

Cost/Gain Gain: Costs: 2 ¡ 2 j l j 1 2 j l j 1 ¡ d

Sparse grid

Comparison – 3D

Smolyak I

Smolyak II

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 !