1. American Options using Monte Carlo
Universita degli Studi di Verona May Matt Davison

2. American Put Option
An American Put option is a contract which gives its holder the right to sell a stock for $K at any time between now and maturity time T.
For instance, consider an American Put with K = $12 and T = 0.5 years (6 months) written on a stock with price today (t=0) of S= $10.
You could take $2 from this contract today: Buy stock for $10, sell it for $12.
Or you could wait. Which should you do?

3. When to exercise?
Instead of taking the $2 = $12-$10 today, you could wait and perhaps take more later.
Which should you do?
Should you take the $2 now?
Should you wait until later?
Let’s see how the first month works

4. S(1 month) = $9.95

5. S(2 months) = $8.45

6. S(3 months) = $8.08

7. S(4 months) = $8.70

8. S(5 months) = $8.98

9. S(6 months) = $10.11

10. What is best?
If you had perfect foresight, the best would be to exercise at t = 0.22, when S = $7.76, for proceeds of $12-$7.76 = $4.24
Sometimes the stock price does nothing but rise from the beginning, and it would have been best to take the $2 you were offered at the beginning.
What do we mean by “best”

11. The Best decision
We define the best decision as the one that leads to the best average outcome
The best decision does not always, or even typically, lead to the best possible outcome.
But nor does it lead to the worst possible outcome.
But if you do something over and over again it will be best.

12. The best decision
At every point we compare the value of exercising immediately (taking St-K) and continuing to hold the option (for value of V(St,t;K,T)
We do whichever is higher value
If it is PDEs we have:

13. PDE for put option P(S,t) AND early exercise boundary θ(t)
δP/δt + ½ σ2S2δ2P/δS2 + rSδP/δS – rP = 0
P(S,T) = max(S-K,0)
P(∞,t) = 0
P(θ,t) = K- θ
δP/δS(θ,t) = -1
Note “extra” smooth pasting boundary condition which allows determination of θ(t).

14. How can we price this with Monte Carlo?
Begin by assuming Bermudan, rather than American, exercise.
That is to say, over the period T, we can exercise N times.
For now assume they are equally spaced T1 = T/N, T2 = 2T/N, … Tk = kT/N,… TN = T.
The value of this Bermudan option will be less than the otherwise similar American option.

15. Need a stock price model
Use the introductory stock price model is
dS = rSdt + σSdW
Where Z is a draw from a standard normal random variable
St = S0exp[(r- ½ σ2)t]exp[σ√tZ]
Note we price in the risk-neutral measure.
Exact solution here means we can simulate prices just at the exercise times Tk .

16. Work backwards
We simulate price paths using our exact solution or using one of our numerical SDE methods.
At time TN = T our exercise decision is easily made. As the option is expiring, we take either K-ST or 0, whichever is better.

17. At the previous exercise period TN-1?
Here in principal it is also easy.
Either exercise right away and take the exercise value P(SN-1) = K-SN-1
Or wait until the next (and final) period.
We don’t know what we’ll get that period.
But we can construct a hedged portfolio to guarantee us the “hold value”: H(SN-1) = exp[-r(T-(N-1)T/N]EQ(K-ST|ST-1)
Or H(SN-1) = exp[-rT/N] EQ(K-ST|ST-1)

18. Optimal exercise at TN-1 and TN-2
Then, at each stock price SN-1
Take Max[P(SN-1),H(SN-1)]
The recursion is now obvious:
At time TN-2 the hold value is: H(SN-2) = exp-rT/N EQ{Max[P(SN-1),H(SN-1)]|SN-2}
And we take Max[P(SN-2),H(SN-2)]

19. Dynamic Programming So the recursion is: V(Sk) = max(P(Sk),H(Sk))
P(Sk) = payoff function, eg. K-Sk
H(Sk) = exp-rT/NEQ{Max[P(Sk+1),H(Sk+1)]|Sk} k < N
H(SN) = 0
And work from N to 1
This is Bellman’s principle of dynamic programming and it works only because the stock prices {S1,S2,….SN} form a Markov process.

20. Solving this with Monte Carlo
In principle this is easy. Just compute the required expectations
EQ{Max[P(Sk+1),H(Sk+1)]|Sk}
using Monte Carlo.
Let’s consider how this might work.
We’ll see if we do it Naively we are doomed to failure.

21. Naïve Monte Carlo Bermudan pricing
Suppose we want the value of the option at a single point S0.
Because Monte Carlo is slow to converge, we need to simulate M values of S1 to get a good average of the hold value.
For each of these M values of S1, we need M values of S2, etc.

22. Simulations are expensive…
The total simulation load is M at step T1 M2 at step T2
M3 at step T3
MN at the final time TN
Total load is M + M2 +… MN = M(MN-1)/M-1).
If N = 5 and M = 1000, this is about 1015 simulations -- really crazy.

23. and wasteful ….
Most of the points simulated (1015 are on the last surface)
Arguably you can get away without that last surface as the H(SN-1) is the European put formula.
So that’s now only 1012 simulations.
Of which the overwhelming majority are on the second last surface. Still wasteful.

24. And inaccurate! The Monte Carlo simulation is inaccurate
We are taking max(P,H) at each time step.
And we are likely wrong about a bunch of these choices.
This generates error which ramps up over time.
It is for these reasons that early versions of Hull’s book on quantitative finance said that Monte Carlo method’s couldn’t be used to solve American Options problems.

25. Ways around this
One way is to simulate M stock prices at each of the N time periods (total cost M*N, a very manageable 5000 with our example numbers)
Then figure out what the probability of going from each of the M stock prices at time k to each of the other stock prices at time k+1
For GBM this probability problem is solvable.
Then work out the various hold values using these probabilities.

26. This is the stochastic mesh idea
Due to Broadie and Glasserman.
It is no longer expensive and wasteful, but it is still inaccurate, for the reasons explained above.
Even this inaccuracy can be turned to advantage, because it biases the method.
Depending on our choices we can bias it high or low and so get bounds for option prices.

27. Least Squares Monte Carlo
Probably the most popular American options pricing model today.
This was developed by Longstaffe and Schwartz
This also works by simulating M different stock prices on each of N exercise dates.
But it links them together as paths. On each path the option is exercised either 1 or 0 times, according to a regression based rule.
This is used to get the average price.
I will explain it using exactly their example.

28. LSMC Example
Consider an option with three exercise times after an initial time. So 4 times in total
We can generate prices at each of these times.
The prices are as indicated in the following table.
The option is a Put with strike $1.10
The time value of money gives a discount factor from time k+1 to time k of

29. Stock Price Paths Path T=0 T=1 T=2 T=3 1 1.00 1.09 1.08 1.34 2 1.16
1.26
1.54 3
1.22
1.07
1.03 4
0.93
0.97
0.92 5
1.11
1.56
1.52 6
0.76
0.77
0.90 7
0.84
1.01 8
0.88

30. Cash flow matrix at time 3
Path
T=1
T=2
T=3 1 -
0.00 2 3
0.07 4
0.18 5 6
0.20 7
0.09 8

31. Time 2 decision
If the put is out of the money at time 2 there is no decision to make; hold for some uncertain future non-negative value, or exercise now for a certain current loss.
The option is only in the money at 5 of the 8 simulated time 2 stock prices. (paths 1,3,4,6,7; not 1,2 or 8)

32. Data to regress at time 2 1 1.08 0.00x 0.94176 2 - 3 1.07
Path
X (T =2 stock prices)
Y (PV of T = 3 put payouts) 1
1.08
0.00x 2 - 3
1.07
0.07 x 4
0.97
0.18 X 5 6
0.77
0.20 X 7
0.84
0.09 X 8

33. Regression step We regress on the basis functions {1,X,X2}
(We would never pick these basis functions in real life, this is just for simplicity)
With this the resulting conditional expectation is E[Y|X] = X X2
We can now compare exercise or hold as per the next table:

34. Time 2 exercise decision
Path
Exercise (1./1-S2)
Hold (from regression) 1
0.02
0.0369 2 - 3
0.03
0.4461 4
0.13
0.1176 5 6
0.33
0.1520 7
0.26
0.1565 8

35. Cash flows
We use the above matrix to decide the cash flows (which come only from exercise, not from holding) at time 2.
If we exercise at time 2 we no longer get cash flows (if any) at time 3, since the option has already been exercise.
Put that together and the next table shows the time 2 and 3 cash flows

36. Cash flow matrix at time 2
Path
Time 1
Time 2
Time 3 1 --
.00 2 3
.07 4
.13 5 6
.33 7
.26 8

37. Time 1
Now we look at the in the money paths at time 1: paths 1,4,6,7 and 8
Along each of these paths we write down the PV of the cash flow on the path: discounted one period if received at time 2, discounted 2 periods if received at time 3.
[Our example has only one path with time 3 cash flows; this is path 3 which isn’t in the money at time 1, so we don’t see that]

38. Regression data for time 1
Path X y 1
1.09
.00 x 2 -- 3 4
0.93
0.13 X 5 6
0.76
0.33 X 7
0.92
0.26 X 8
0.88
0.00 X

39. Time 1 regression results
Again regress on the basis functions {1,X,X2}
Get E[Y|X] = – 3.335X X2
Use this to make exercise decisions, see next table

40. Optimal Exercise decision at time 1
Path
Exercise
Hold 1
0.01
0.0139 2 -- 3 4
0.17
0.1092 5 6
0.34
0.2866 7
0.18
0.1175 8
0.22
0.1533

41. Cash flows
This optimal rule induces the following optimal stopping rule table

42. Stopping Rule
Path
Time 1
Time 2
Time 3 1 2 3 4 5 6 7 8

43. Optimal cash flow matrix
From this stopping rule we can get the following optimal cash flow matrix

44. Full Cash flow matrix Path T=1 T=2 T=3 1 .00 2 3 .07 4 .17 5 6 .34 7
.18 8
.22

45. Final pricing step
Just discount all the cash flows in the matrix back to time 0 (time 3 x ( )3, time 2 x ( )2, time 1 x ( ).
Add them up, divide by the number of paths.

46. Important fact
This is not any more accurate or faster than using Finite Difference if the number of dimensions is small (< 3, maybe < 4).
In fact, are serious biases in the results.
It may be easier to code and to explain however.
And for complicated problems, very crucial.

47. References
For Monte Carlo Methods: Monte Carlo Simulation & Finance, Don L McLeish, Wiley Finance, 2005 For Stochastic DEs: D. Higham, An Algorithmic Intro to Numerical Simulation of SDEs, SIAM Review 32, , 2001 For LSMC: F. Longstaff & E.S. Schwartz, Valuing American Options by Simulation: A Simple Least Squares Approach. Review of Financial Studies For all three: Monte Carlo Methods in Financial Engineering, Paul Glasserman, Springer Verlag 2000