option pricing with sparse grid quadrature jass 2007 marcin salaterski
TRANSCRIPT
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 !