Review Direct Integration Discretely observed options Summary QUADRATURE. Dr P. V. Johnson. School of Mathematics
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1 QUADRATURE Dr P.V.Johnson School of Mathematics 2011
2 OUTLINE Review 1 REVIEW Story so far... Today s lecture
3 OUTLINE Review 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION
4 OUTLINE Review 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options
5 OUTLINE Review 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
6 OUTLINE Review Story so far... Today s lecture 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
7 REVIEW Review Story so far... Today s lecture We have examined three different numerical methods: Monte Carlo Lattice/tree methods Finite Difference Each method has its advantages and disadvantages.
8 REVIEW Review Story so far... Today s lecture We have examined three different numerical methods: Monte Carlo Lattice/tree methods Finite Difference Each method has its advantages and disadvantages. Finite difference is the most appropriate for low dimensional problems..
9 REVIEW Review Story so far... Today s lecture We have examined three different numerical methods: Monte Carlo Lattice/tree methods Finite Difference Each method has its advantages and disadvantages. Finite difference is the most appropriate for low dimensional problems.. Monte Carloshould only be usedwhennothing elsewill work!!
10 OUTLINE Review Story so far... Today s lecture 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
11 QUADRATURE Review Story so far... Today s lecture Here we examine a method using direct integration rather than Monte Carlo integration The method is averypowerfultoolto price options with discrete barriers discrete exercise dates path dependent features
12 QUADRATURE Review Story so far... Today s lecture Here we examine a method using direct integration rather than Monte Carlo integration The method is averypowerfultoolto price options with discrete barriers discrete exercise dates path dependent features The method is veryfast, and after removing non-linear errors very accurate
13 OUTLINE Review 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
14 EXPECTING AN INTEGRAL First recall that the value of an option(with constant interest rates) may be written: V(S t,t) = e r(t t) E Q t [V(S T,T)]
15 EXPECTING AN INTEGRAL First recall that the value of an option(with constant interest rates) may be written: V(S t,t) = e r(t t) E Q t [V(S T,T)] Assume S admits a probability density function F(S t,s T,T t), then wecan write E Q t [S T] = 0 F(S t,s T,T t)sds
16 EXPECTING AN INTEGRAL First recall that the value of an option(with constant interest rates) may be written: V(S t,t) = e r(t t) E Q t [V(S T,T)] Assume S admits a probability density function F(S t,s T,T t), then wecan write E Q t [S T] = 0 F(S t,s T,T t)sds and thereforewritethe valueof the optionas V(S t,t) = e r(t t) F(S t,s T,T t)v(s,t)ds 0
17 MONTE CARLO EXAMPLE f(s) A frequency distribution of S Tafter a Monte Carlo simulation The probability distribution function V(S,T) The payoff for a call option S T
18 PDES TO PROBABILITIES We can show that the Black-Scholes equation V t σ2 S 2 2 V S 2 +(r δ)s V rv = 0, (1) S can be used to generate the function probability density functionf(s t,s T,T t)
19 PDES TO PROBABILITIES We can show that the Black-Scholes equation V t σ2 S 2 2 V S 2 +(r δ)s V rv = 0, (1) S can be used to generate the function probability density functionf(s t,s T,T t) Since the Black-Scholesequationcan beshown to be a Fokker-Planck equation, the derivation of the probability density function is simple and not shown here.
20 THE VALUE OF THE OPTION Now It iseasierifwe make the substitutions x = log(s t /X), (2) y = log(s t+ t /X). (3)
21 THE VALUE OF THE OPTION Now It iseasierifwe make the substitutions x = log(s t /X), (2) y = log(s t+ t /X). (3) then wemay write V(x,t) = A(x) B(x,y)V(y,t+ t)dy, (4)
22 THE VALUE OF THE OPTION Now It iseasierifwe make the substitutions then wemay write V(x,t) = A(x) x = log(s t /X), (2) y = log(s t+ t /X). (3) B(x,y)V(y,t+ t)dy, (4) where the function A(x) contains the discounting term and other terms not involving y and B(x, y) can be thought to represent the probability density function
23 The functionais givenby and B by A(x) = 1 2σ 2 π t e 1 2 kx 1 8 σ2 k 2 t r t, (5) B(x,y) = e (x y)2 2σ 2 t +1 2 ky (6) where k = 2(r D c) σ 2 1. (7)
24 A SIMPLIFIED FORMULA Thenthe valueofan optionattime t withstock price x = log(s t /X) canbe written V(x,t) = A(x) f(x,y)dy, (8)
25 A SIMPLIFIED FORMULA Thenthe valueofan optionattime t withstock price x = log(s t /X) canbe written V(x,t) = A(x) f(x,y)dy, (8) So to valuethe option wemust numericallyevaluatean integral The solution is exact for European options
26 A SIMPLIFIED FORMULA Thenthe valueofan optionattime t withstock price x = log(s t /X) canbe written V(x,t) = A(x) f(x,y)dy, (8) So to valuethe option wemust numericallyevaluatean integral The solution is exact for European options INTUITION Onlyvalid foreuropean at all time points Can however slice complex option up into time intervals over which it isvalid tointegrate the option
27 OUTLINE Review 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
28 EXAMPLE - VANILLA CALL OPTION In the transformed variables the integrand becomes f(x,y) = B(x,y) Xmax(e y 1,0). (9) Notice here that derivitives of payoff are discontinuous
29 EXAMPLE - VANILLA CALL OPTION In the transformed variables the integrand becomes f(x,y) = B(x,y) Xmax(e y 1,0). (9) Notice here that derivitives of payoff are discontinuous Split integral up into continuous regions i.e f(x,y) = 0 fory < 0 f(x,y) = B(x,y) X(e y 1) fory 0 Choose avalueof y max to approximate Note that the regiony < 0makes no contributionto the option value
30 EXAMPLE - VANILLA CALL OPTION Now we must integrate ymax V(x,t) A(x) B(x,y) X(e y 1)dy 0
31 EXAMPLE - VANILLA CALL OPTION Now we must integrate ymax V(x,t) A(x) B(x,y) X(e y 1)dy 0 Splitthe regioninto N points and performasimpsons integration ( V(x, t) A(x) f(x,0)+4f(x, 1 2 δy)+ ( N + 1 2f(x,iδy)+4f(x,(i+ 1 ) )+f(x,n 2 )δy) + δy) i=1
32 EXAMPLE Review
33 OUTLINE Review A moving barrier Bermudan Options 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
34 DISCRETE BARRIER Review A moving barrier Bermudan Options Any discretely observed option can be easily handled by the quadrature method Assume adown-and-outcalloption suchthat attime t 1, V = 0if S < B the option is European otherwise.
35 DISCRETE BARRIER Review A moving barrier Bermudan Options Any discretely observed option can be easily handled by the quadrature method Assume adown-and-outcalloption suchthat attime t 1, V = 0if S < B the option is European otherwise. Quadrature method: Integrate tofind the value of the option at t 1 forn points in the region S B. Assume V = 0 otherwise. Use those N points tointegrate andfindthe value at t = 0 This can be extended to any number of observations.
36 QUAD IN ACTION Review A moving barrier Bermudan Options
37 OUTLINE Review A moving barrier Bermudan Options 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
38 A BERMUDAN PUT OPTION A moving barrier Bermudan Options A Bermudan option has discrete exercise dates Assume abermudanputoption withexerciseatt 1
39 A BERMUDAN PUT OPTION A moving barrier Bermudan Options A Bermudan option has discrete exercise dates Assume abermudanputoption withexerciseatt 1 Quadrature method: Find the point x 0 at which V(x 0,t 1 ) = 0 Solve forn points in the region (x 0, ) Then split the integral into two regions f(x,y) = B(x,y) X(e y 1) fory x 0 f(x,y) = B(x,y) V(x,t 1 ) fory > x 0 Integrate the two regions seperately and add together for option value ymax V(x, 0) =A(x) B(x,y)V(x,t 1 )dy +A(x) x 0 x0 y min B(x,y)X(1 e y )dy
40 OUTLINE Review Overview 1 REVIEW Story so far... Today s lecture 2 DIRECT INTEGRATION 3 DISCRETELY OBSERVED OPTIONS A moving barrier Bermudan Options 4 SUMMARY Overview
41 Overview For a vanilla European option the method is comparable to the Black-Scholes formula The method is especially fast and accurate for discretely observed options
42 Overview For a vanilla European option the method is comparable to the Black-Scholes formula The method is especially fast and accurate for discretely observed options Must split integrals into continuous regions to remove non-linearity errors - by placing nodes precisely on the discontinuities
43 Overview For a vanilla European option the method is comparable to the Black-Scholes formula The method is especially fast and accurate for discretely observed options Must split integrals into continuous regions to remove non-linearity errors - by placing nodes precisely on the discontinuities Can be extended to multiple Brownian motions and mean-reverting processes See notes for convergence rates and computation times
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