Week 10

Question 1

Compute option price from expectation

Answer 1

Question 2

Steps for Monte Carlow option pricing

Answer 2

1) discretise time

2) discretise SDE

3) use LLN

Question 3

Advantages + disadvantages of Monte Carlo option pricing

Answer 3

Adv)
1) easy conceptually
2) very easy to implement (typically)
3) easily adapted to path dependent options

Dis)
1) very difficult to adapt to American options
2) only get value of option at Time 0 and for one particular stock price

Question 4

If an option pays Ψ(S_T) in the BS model, then at time t and stock price s, the option is worth ?

Answer 4

Question 5

Steps for finite difference method for options pricing

Answer 5

Discretise time and space (stock price)

Approximate derivatives in PDE

Rearrange

Solve backwards in time + space

Question 6

Discretise time and space for finite difference method

Answer 6

Question 7

Approximate derivatives in BS PDE by finite difference

Answer 7

Question 8

Rearrange BS PDE at end of finite difference

Answer 8

And for m = M (the edge point) we assume linearity so can work out from points M-1 and M-2

Question 9

Advantages + dis for FD methods for options pricing

Answer 9

Adv)
1) we get prices for all values of s, t
2) can be easily adapted to American and barrier options
3) non random res

Dis)
1) difficult to implement
2) curse