Monte Carlo methods to price compound options Group

Monte Carlo methods to price compound options Group C Alexander Sundin Henrik Näkne

Agenda • • • What is a compound option Changes to the original MC-solver Quasi-Monte Carlo methods Halton points Results Summary

What is a compound option • An option on an option • Can be of type: – Call on Call (Co. C) – Call on Put (Caput) – Put on Put (Po. C) – Put on Call (Po. P) • We looked at a Call on Call compound option

What is a compound option • Notation – T 1 maturity date of the compound option – K 1 strike price of the compound option – T 2 maturity date of the underlying option – K 2 strike price of the underlying option

What is a compound option • (Call on Call) At time T 1 the value is max(C(ST 1 , K 2 , (T 2 - T 1)) - K 1 , 0) Where C is the regular Call-option formula Similar for the other compound option-types. Ex Caput at time T 1 Max(P(ST 1 , K 2 , (T 2 - T 1)) - K 1 , 0)

Changes to the original MC-solver • Simulate N trajectories of the underlying asset S until time T 1 - obtains N # of ST 1 • For each ST 1, a C(ST 1 , K 2 , (T 2 - T 1)) is calculated, using N trajectories of S

Quasi-Monte Carlo methods • Instead of using random numbers quasirandom number can be used. • Examples of these are Sobol and Halton points. • Because the quasi-random numbers fill out space more evenly the quasi-MC solver should converge faster.

Halton points Halton point fills the space more evenly than randn()

Halton points An example of trajectories simulated in 50 time steps with the different methods.

Results T 1=0. 5, T 2 =1, K 1= 20, K 2=20, h=5 e-3 N=500, sigma=0. 25, r=0. 1

Results • ERROR PLOT Approx = 0. 371 – 0. 2423 x

Summary • For the usual MC convergence of sqrt(N) N^2 trajectories have to be made. • Heavy calculation-wise. • Careful using quasi random….