1. Numerical Methods for derivatives pricing

2. 2 American Monte Carlo It’s difficult to price an option with early exercise using Monte Carlo But some options require Monte Carlo to price oPath-dependent options oOptions on baskets oOptions priced using models with many factors DO NOT DISTRIBUTE

3. 3 Derivatives Options with early exercise are called American if you can exercise at any time, or Bermudan if you can exercise only at certain times Hence a Monte Carlo scheme that allows you to price options with early exercise is called American Monte Carlo, or AMC for short DO NOT DISTRIBUTE

4. 4 What’s the problem? But Monte Carlo doesn’t use backward induction So the problem is: How can we compute the continuation value? DO NOT DISTRIBUTE

5. 5 Exercise value The continuation value is the value of the option if we continue to hold it As such it is the risk-neutral expectation of the payoff of the option So we need to be able to calculate conditional expectations in Monte Carlo DO NOT DISTRIBUTE

6. 6 Tree and Exercise values In a tree the conditional expectation comes for free For example the conditional expectation at the circled point here is the value calculated on the subtree that starts from the point DO NOT DISTRIBUTE

7. 7 Path dependent in a tree BTW, that’s why it is difficult to price path-dependent options with trees: this conditional valued does not depend on how we get there (e.g. on which one of the dashed lines) DO NOT DISTRIBUTE

8. 8 Optimal Exercise in Monte Carlo There is no sub-Monte Carlo starting from the circled point DO NOT DISTRIBUTE

9. 9 Conditional Expectation by Regression The first American Monte Carlo algorithm was devised by Longstaff-Schwartz It used linear regression DO NOT DISTRIBUTE

10. 10 Trees and Finite Difference Methods Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

11. 11 Binomial Trees SDE for stock in the BS model, risk-neutral measure dS t =rS t dt+  S t dW t Approximation of this process: after a small time step the stock can go either up or down Sd Su S

12. 12 Binomial Trees Take u and d such that u*d = 1 After two steps the tree recombines: Su S Su 2 Sd 2 S Sd

13. 13 Binomial Trees After seven steps

14. 14 Binomial Trees In the risk neutral measure all discounted tradable assets are martingales That means that S = (p u *Su+p d *Sd)e -rdt If we add the condition p u +p d =1, we get a 2x2 system that can be solved Sd Su S pupu pdpd

15. 15 Binomial Trees The solution is The sizes of the up move and the down move are chosen to ensure that the volatility of the stock is 

16. 16 Binomial Trees Option pricing is done by backward induction Start at the last time step: the price of the option is known, it’s simply the payoff Then proceed backward: at each node of the tree the price of the option is the calculated from the price of the option at the two nodes branching from it V2V2 V1V1 V pupu pdpd V = exp(-r*dt)(p u V 1 +p d V 2 ) V2V2 pdpd V2V2 pdpd V2V2 pdpd V2V2 pdpd V2V2 pdpd V2V2 V pdpd V2V2 V pdpd V2V2 V1V1 V pdpd V2V2 V1V1 V pdpd V2V2 V1V1 V pdpd V2V2 V1V1 V pdpd V2V2 V1V1 V pdpd V2V2 V1V1 V pdpd V2V2

17. 17 Binomial Trees A binomial tree is implemented in the spreadsheet “Trees”

18. 18 Main Points Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

19. 19 Trinomial Trees In a trinomial tree there are 3 nodes branching from each node

20. 20 Trinomial Trees The simplest way to create a trinomial tree is to concatenate two time steps of a binomial tree Su S Su 2 Sd 2 S Sd Sd 2 Su 2 S S

21. 21 Trinomial Trees Graph of the price given by the trinomial tree vs the true price

22. 22 Binomial vs Trinomial The price given by the trinomial tree does not exhibit the zig-zag pattern Don’t read too much into this: the trinomial tree simply picks every second price given by the binomial tree Because this trinomial tree with N time steps gives the same price as a binomial with 2N time steps

23. 23 Binomial vs Trinomial The real advantage is speed: oA binomial tree with 2N steps takes four times longer to run than one with N steps oBut the same result can be obtained from a trinomial tree with N steps, which takes only 33% longer to run

24. 24 Main Points Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

25. 25 Explicit Finite Difference Schemes The Black-Scholes equation Change the variable S = exp(x*S 0 ) New equation:

26. 26 Explicit Finite Difference Schemes Discretize the variables t 0 =0, t 1, …, t N =T ; t i+1 -t i =dt x -M, x -M+1, …, x 0 =0, x 1, … x M ; x i+1 -x i = dx Approximate the partial derivatives in our PDE:

27. 27 Explicit Finite Difference Schemes Use these points for the time derivative Use these points for the space derivative

28. 28 Explicit Finite Difference Schemes This allows us to calculate V(t i,x j ) using only the values of V at the next time-step t i+1 :

29. 29 Explicit Finite Difference Schemes In other words this scheme produces a trinomial tree The node (i,j) is calculated in terms of the three nodes branching from it:

30. 30 Explicit Finite Difference Schemes Graph of the price given by the trinomial tree vs the true price (same parameters as for the trees)

31. 31 Explicit Finite Difference Schemes Sometimes the explicit schemes are unstable For example take a look at the s/s “Finite differences”, tab “Explicit – Instability” We show the graph of the calculated value of the option at successive time steps At maturity (t=2) the value is simply the payoff, so no error

32. 32 Explicit Finite Difference Schemes As the time to maturity increases from 0 to 0.3 we start to see some zig-zag:

33. 33 Explicit Finite Difference Schemes As the time to maturity increases from 0.3 to 0.6 the instability becomes huge

34. 34 Main Points Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

35. 35 Implicit & Crank-Nicolson The implicit and CN schemes were designed to fix the instability problem of the explicit scheme They are slightly more complicated to implement: they require to solve a linear system when we do the backward induction But are stable And CN converges faster

36. 36 Implicit & Crank-Nicolson Use these points for the time derivative For implicit, use these points for the space derivative

37. 37 Implicit & Crank-Nicolson Use these points for the time derivative For CN, use all these points for the space derivative

38. 38 Implicit & Crank-Nicolson To be able to run the implicit and CN schemes in the s/s “Finite differences”, you need to load the addin “matrix.xla” For that you should save “matrix.xla” and “Funcostumize.dll” in the same folder The function that solves a tri-diagonal system in this addin is called “SysLin3”

39. 39 Implicit & Crank-Nicolson Graph of the option price calculated with an implicit scheme vs the true price

40. 40 Implicit & Crank-Nicolson Graph of the option price calculated with Crank-Nicolson vs the true price

41. 41 Main Points Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

42. 42 Barrier Options Path-dependent options are difficult to implement in a finite difference scheme But barrier options are only mildly path-dependent It’s very simple to modify a scheme to price a knock-out barrier option: simply set the option price to zero for all the nodes beyond the barrier To price a knock-in you use the KI-KO parity

43. 43 Barrier Options Path-dependent options are difficult to implement in a finite difference scheme But barrier options are only mildly path-dependent It’s very simple to modify a scheme to price a knock-out barrier option: simply set the option price to zero for all the nodes beyond the barrier To price a knock-in you use the KI-KO parity

44. 44 Barrier Options However, this straightforward implementation runs into trouble:

45. 45 Barrier Options Cranc-Nicolson is not much better:

46. 46 Barrier Options The reason is that the barrier option is discontinuous To fix this you need to make sure the barrier is among the grid points of your tree or FD scheme In the next slide we show an implicit scheme where we the barrier is indeed in the grid The improvement is clear (there is still some error due to too few points being used)

47. 47 Barrier Options

48. 48 Main Points Binomial Trees Trinomial Trees Explicit Finite Difference Schemes Implicit & Crank-Nicolson Barrier Options American Put

49. 49 American Put The American put does not have an analytical formula You need a numerical scheme to calculate its value, such as trees or FD With trees or FD it is almost trivial to modify the scheme to allow for early exercise You calculate the continuation value

50. 50 American Put You do the backward induction in the following way: At the last time step the option is equal to the payoff At each other timestep you calculate the continuation value, as you do for the European option And then you replace this with the larger of the continuation value and the immediate exercise

51. 51 American Put Of course, a scheme that is faster for European options has all the chances to be faster for American options as well In the next slide you can see how the binomial and trinomial trees converge with an increasing number of steps

52. 52 American Put

53. 53 Trees and FD vs Monte Carlo Trees and FD advantages: oFast convergence oVery easy to implement American optionality Trees and FD disadvantages: oSlow in higher dimensions. Impractical for dim > 4 oDifficult to use for path-dependent options

54. 54 Trees and FD vs Monte Carlo Monte Carlo advantages: oVery easy to implement oWorks well in higher dimensions oPerfect for path-dependent options Monte Carlo disadvantages: oSlow convergence oDifficult to implement American optionality