American Options with Monte Carlo

American Options with Monte Carlo

Tommaso GabbrielliniSiena, 20 Maggio 2011

Black&ScholesVery basic recap

The Black&Scholes model assumes a market in which the tradable assets are:- A risky asset, whose evolution is driven by a geometric

brownian motion- the money market account, whose evolution is deterministic

Black&Scholes (2)Valuing a derivative contract

A derivative can be perfectly replicated by means of a self-financing dynamic portfolio whose value exactly matches all of the derivative flows in every state of the world. This approach shows that the values of the derivative (and of the portofolio) solves the following PDE

where the terminal condition at T is the derivative’s payoff.

Black&Scholes (3)There exists a very important result: the Feynman-Kac theorem.It mathematically states the equivalence between the solution of this PDE and an expectation value.

If f(t0,S(t0)) solves the B&S PDE, then it is also solution of

i.e. it’s the expected value of the discounted payoff in a probability measure where the evolution of the asset is

This probability measure is the Risk Neutral Measure

Black&Scholes and numerical methodsSince there exist such an equivalence, we can compute option prices by means of two numerical methods

PDE: finite difference (explicit, implicit, crank-nicholson)suitable for optimal exercise derivatives

IntegrationQuadrature methods

Monte Carlo Methodssuitable for path dependent options

European OptionLet’s recall the European Call/Put option:

It’s a derivative contract in which the holder of the option has the right to buy/sell the asset at expiry at a fixed price (the strike).

The price at time t can be computed as

where the expectation is taken in the risk neutral probability.

American Put OptionThe american version of the put option gives the holder the right to exercise it any time before the expiration date.

Will there be cases in which it is convenient to “early” exercise the option?Yes. Here’s a case.

Imagine you bought an american put and at t1 the stock drops to zero, with no chance to ever going back to a strictly positive value (like in the Black&Scholes model)

American Put Option (2)

K = 10

t1 T

The holder, at t1, wonders if it is worth exercising the option.

Which is the optimal strategy?

I have to compare the values of the possibilities

1. The option is exercised at t1, the holder gets K

2. The option is exercised later, suppose at maturity, the value is

It’s convenient to exercise at t1!

American Call OptionWhat about american call option?

Will there be cases in which it is convenient to “early” exercise the option?

Well, it depends on dividends.

Imagine you bought an american call and at t1 the stock goes so high that the probability to finish out of the money at expiry is negligible (S >> K)

American Call Option (2)No dividends

t1 T

K = 10




1. The option is exercised at t1, the holder gets S(t1) - K

2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption)

It’s better to wait!

t1 T

K = 10




1. The option is exercised at t1, the holder gets S(t1) - K

2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption)

American Call Option (3)With dividends

It might be better to exercise

With dividends things are different. As in the previous example, but now the stock pays a dividend yield q:

Bermudan Put OptionThe bermudan option is similar to an american option, except that it can be early exercised once only on a specific set of dates

t1 t2 t3 t4 t5 T

In the graphPut at strike K, maturity 6 years, and each year you can choose whether to exercise or wait.

Bermudan Put Option. A simple exampleLet’s consider a simple example:a put option which can be exercised early only once.

t1 T

Valuation of the simple exampleCan we price this product by means of a Monte Carlo?

Yes, let’s see how.

Let’s implement a MC which actually simulates, besides the evolution of the market, what an investor holding this option would do (clearly an investor who lives in the risk neutral world).

In the following example we will assume the following dataS(t) = 100, K = 100, r = 5%, s = 20%, t1 = 1y, T = 2y

Valuation of the simple example1. We simulate that 1y has passed, computing the new value of the asset and the new value of the money market account

2. At this point, I (the investor) could exercise. How do I know if it’s convenient? In case of exercise I know exactly the payoff I’m getting. In case I continue, I know that it is the same of having a European put option.

Valuation of the simple example (2)In mathematical terms I have the payoff in t1 is

So the premium of the option is the average of this discounted payoff calculated in each iteration of the monte carlo procedure

Where P(t1, T, S(t1), K) is the price of a put (which I can compute analytically!)In the jargon of american products, P is referred to the continuation value, i.e. the value of holding the option instead of early exercising it.

Some more considerationsI could have priced this product because I have an analytical pricing formula for the put. What if I didn’t have it?

Brute force solution: for each realization of S(t1) I run another Monte Carlo to price the put.

This method (called Nested Monte Carlo) is very time consuming. For this very simple case it’s time of execution grows with N2… which becomes prohibitive when you deal with more than one exercise date!

Introducing Longstaff-SchwarzA finer solution

For each realization of S(t1) I go on with the following step simulating S(T)

t0 t1 T1 100 94.08641 68.097332 100 87.59017 102.20353 100 131.1194 121.32944 100 112.1032 98.534625 100 81.33602 98.154376 100 212.0479 206.74387 100 118.9995 110.15718 100 77.46154 56.856779 100 164.9462 160.4879

10 100 91.20603 71.07494

For each path compute at time t1 the discounted payoff given the value S(T)

i.e. t0 t1 T disc .payoff

1 100 94.08641 68.09733 30.346754272 100 87.59017 102.2035 03 100 131.1194 121.3294 04 100 112.1032 98.53462 1.3939113515 100 81.33602 98.15437 1.7556168446 100 212.0479 206.7438 07 100 118.9995 110.1571 08 100 77.46154 56.85677 41.039109669 100 164.9462 160.4879 0

10 100 91.20603 71.07494 27.51436479

Introducing Longstaff-Schwarz (2)Plot the discounted payoff Pi versus Si(t1) (as an example, by means of the scatter plot in excel)

t1 disc .payoff94.08641 5.6251807287.59017 11.80459474131.1194 0112.1032 081.33602 17.75372602212.0479 0118.9995 077.46154 21.43924169164.9462 091.20603 8.365080751

Introducing Longstaff-Schwarz (3)On this plot, add the analytical price of the put as a function of Si(t1)

t1 disc .payoff put94.08641 5.62518072 8.0653329587.59017 11.80459474 11.65149926131.1194 0 0.511529034112.1032 0 2.38390643281.33602 17.75372602 15.95968568212.0479 0 0.00019481118.9995 0 1.39941790577.46154 21.43924169 19.0155111164.9462 0 0.02226524391.20603 8.365080751 9.542377625

Introducing Longstaff-Schwarz (4)The analytical price of the put is a curve which kinds of interpolate the cloud of monte carlo points.

Observation.

Today the price can be computed by means of an average on all discounted payoff (i.e. the barycentre of the cloud made of discounted payoffs)

Maybe….

The future value of an option can be seen as the problem of finding the curve that best fits the cloud of dicounted payoffs (up to the date of interest)???

Introducing Longstaff-Schwarz (5)Below there’s a curve found by means of a linear regression on a polynomial of 4° order.

Introducing Longstaff-Schwarz (6)We now have a pricing formula for the put to be used in my MC:

The formula is obviously fast:the cost of this algorithm is performing the best fit

Please note that I could have used any form for my curve (non only a polynomial). This method has the advantage that it can be solved as a linear regression, which is fast.

Longstaff-Schwarz algorithmLet’s consider now a generic bermudan optionHere’s the Longstaff-Schwarz algoritm

1. Generate the MC trajectories of the underlying up to maturity

2. Compute the payoff at maturity and discount it to the previous exercise date

3. Regress the last column as a function of the previous one, compute the continuation value for each path and calculate what you would get from exercise.

100 72.31062 81.05736 96.04066 90.91403 68.04453 66.75914

100 72.31062 81.05736 96.04066 90.91403 68.04453 31.61969

Continuation value 20.1 Exercise 100 72.31062 81.05736 96.04066 90.91403 31.95547 31.61969

Longstaff-Schwarz algorithm (2)

4. Compare those two numbers. In this particular path the payoff in case of exercise is greater than the continuation value. Exercise it and go to next step and discount the payoff.

5. As in step 3, compute the continuation value and the payoff in case of exercise

6. Now the continuation value is greater. Don’t exercise: the payoff value is replaced with the discounted adjacent number (more on this in next slide)

Continuation value 20.1 Exercise 100 72.31062 81.05736 96.04066 90.91403 30.39698




Longstaff-Schwarz algorithm (3)Theoretically we should have done this

This is correct, but it is generally less accurate because the continuation value provided by the interpolating function is accurate only in a region close to the exercise boundary. That’s why it is used the previous step.

Continuation value Exercise 100 72.31062 81.05736 96.04066 14.5


7. Ok, iterate till you get the price!

What is the Longstaff-Schwarz algorithmRecall that pricing a derivative means solving a backward partial differential equation

i.e. starting from the payoff, and proceeding backward in time, you compute at each time and for each value of S the option value.

Did I say option value? Well, I could have said continuation value

Therefore I can naturally price american/bermudan products

What is the Longstaff-Schwarz algorithm (2)Longstaff-Schwarz method is thus a way to introduce backward evaluation in a Monte Carlo approach (which is naturally forward looking)

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American Options with Monte Carlo