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

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

3. What is a compound option
An option on an option
Can be of type:
Call on Call (CoC)
Call on Put (Caput)
Put on Put (PoC)
Put on Call (PoP)
We looked at a Call on Call compound option

4. What is a compound option
Notation
T1 maturity date of the compound option
K1 strike price of the compound option
T2 maturity date of the underlying option
K2 strike price of the underlying option

5. What is a compound option
(Call on Call) At time T1 the value is
max(C(ST1 ,K2 ,(T2 - T1)) - K1 , 0)
Where C is the regular Call-option formula
Similar for the other compound option-types.
Ex Caput at time T1
Max(P(ST1 ,K2 ,(T2 - T1)) - K1 , 0)

6. Changes to the original MC-solver
Simulate N trajectories of the underlying asset S until time T1 - obtains N # of ST1
For each ST1, a C(ST1 ,K2 ,(T2 - T1)) is calculated, using N trajectories of S

7. Quasi-Monte Carlo methods
Instead of using random numbers quasi-random 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.

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

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

10. Results
T1=0.5, T2 =1, K1= 20, K2=20, h=5e-3 N=500, sigma=0.25, r=0.1

11. Results
ERROR PLOT
Approx = – x

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