Pricing American Options Using Least Square Monte Carlo

Definition An American put (call) option is a type of derivatives that gives its holder the right, but not the obligation, to sell (buy) the underlying security at anytime until the maturity date \(T\), and for a pre-determined exercise price \(K\).

An American option is priced at a higher value than a similar European option (with the same exercise price and maturity date), since the American option gives its holder the right to exercise the option at any time during the life of the option and not only at maturity \(T\). It therefore gives its holder the ability to exercise the option at any time prior and including the maturity date. The additional features of American options makes it an important problem, but more difficult to solve that, so far, there is no exact analytical solution to price an American option. Therefore, numerical approximations are used to price such options. G. Barone-Adesi and R.E. Whaley (1987) have derived an analytical approximation to price American options, but it does not give accurate results when the option has a very short or very long maturity date. They modeled the premium of the early exercise of the American option as a diffusion process. Binomial methods and finite difference methods are also used to price such options, but it becomes impractical when there are multiple stochastic variables such as volatility and interest rates. The partial differential equation of an American option follows the following inequality $$ \frac{\partial P}{\partial t} + \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + r S \frac{\partial P}{\partial S} - r P \leq 0. \label{PDE} $$

F.A. Longstaff and E.S. Schwartz (2001) have introduced a simple, but efficient Monte Carlo approach that is based on simulating paths. The method uses least square method to estimate the present value of the conditional expected pay-off and is called least squares Monte Carlo method (LSM). For every generated path, at each time step, the pay-off from immediate exercise is compared with the present value of the pay-off from continuation. The option holder has two options at each time step - that is - to exercise the option immediately or to continue holding it. If the immediate pay-off is higher than the expected pay-off from continuation, the option is then exercised. At maturity, all in-the-money paths that are left will be exercised. Prior to the maturity date, a quadratic least squares approximation is used by regressing the present value of the expected pay-off at time \(t_i+1\) on the security price at time \(t_{i}\), which is for a put option as follows $$\mathrm{E} [\mathrm{Y}|S_{t_{i}}]=e^{-r\Delta t}\mathrm{E} \left[ \mathrm{max} (K-S_{t_{i+1}},0)\right]=a_0+a_1 S_{t_{i}} +a_2 S_{t_{i}}^2 $$ Only the paths that are in-the-money at time \(t_i\) (i.e. \(K-S^{[k]}_{t_{i}}>0\)) are considered in the regression to allow for a better prediction in the region of interest. The fitted value of the regression is unbiased estimate of the conditional expectation function, but this is not covered in this notes. The least squares method is used rather than the actual path values because the option holder does not know the when the option price has a maximum pay-off, but can estimate it. Chebyshev and Laguerre polynomials are considered as alternatives to the least squares fitted model.

Numerical example Our aim is to find minimum time \(\tau \in (0,T]\) such that \(\mathrm{E} [ e^{-r\tau} \mathrm{max} (K-S_{\tau},0)]\) is maximised. A numerical example is covered here to price an American put option with exercise price \(K=11\), risk-free monthly interest rate \(r=0.01\), maturity \(T=\) 3 months. Table 1 is an example of a sample of eight paths that are generated for the security price with an initial price \(S_{t_{0}}=10\).

Table 1

Path	\(S_{t_{0}}\)	\(S_{t_{1}}\)	\(S_{t_{2}}\)	\(S_{t_{3}}\)
1	10.0000	9.3758	11.9631	11.0691
2	10.0000	11.2394	10.9945	13.7906
3	10.0000	9.5320	10.2757	19.0931
4	10.0000	11.0030	12.0361	9.1672
5	10.0000	5.7184	6.0453	6.4382
6	10.0000	11.5133	9.8262	7.2811
7	10.0000	15.8629	15.1192	21.1174
8	10.0000	13.1436	13.0515	18.6872

In order to determine the optimal strategy to exercise the option for each of the paths, we start by treating the American option as a European option by setting the exercise time for all the paths to maturity \(t_{3}\) as shown in Table 2.

Table 2

Path	Exercise time
1	\(t_{3}\)
2	\(t_{3}\)
3	\(t_{3}\)
4	\(t_{3}\)
5	\(t_{3}\)
6	\(t_{3}\)
7	\(t_{3}\)
8	\(t_{3}\)

If all paths are left until maturity \(t_{3}\), then the pay-off will be \(\mathrm{max} (K-S^{[k]}_{t_{3}},0)\). Now, let us consider \(t_{2}\), which is one time step before maturity, and use the least squares approach to compute the coefficients of the following quadratic model $$\mathrm{Y}=e^{-0.01}\mathrm{E} \left[ \mathrm{max} (K-S^{[k]}_{t_3},0)\right]=a_0+a_1 S^{[k]}_{t_2} +a_2 \left(S^{[k]}_{t_2} \right)^2, \label{LS1} $$ for all \(k\) such that \(K-S^{[k]}_{t_{2}}>0\). Obviously, the optimal decision that is made for paths that are out-of-money is to leave them until maturity, not to receive a zero pay-off when \(K-S^{[k]}_{2} \leq 0\). If the option is in-the-money, the option holder needs to decide whether to exercise the option immediately or to continue with the option and exercise it at maturity. The coefficients, \(a_i\), can be determined using regression on the in-money paths shown in Table 3.

Table 3

Path	\(S_{t_{2}}|K-S_{t_{2}}>0\)	\(\mathrm{E} [\mathrm{Y}]\)
1
2	10.9945	\(e^{-0.01} 0=0\)
3	10.2757	\(e^{-0.01} 0=0\)
4
5	6.0453	\(e^{-0.01}4.56=4.51\)
6	9.8262	\(e^{-0.01}3.72=3.68\)
7
8

The resulted coefficients are \(a_0=-17.4434\), \(a_1=6.1141\) and \(a_2=-0.4158\). The least squares prediction model is then used to predict the expected pay-off of continuation and the predicted values are compared to the pay-off resulted from immediate exercise of the option as shown in Table 4.

Table 4

Path	\(K-S_{t_{i}}\)	\(\mathrm{max} (\hat{\mathrm{Y}},0)\)
1
2	0.01	0
3	0.72	1.48
4
5	4.96	4.32
6	1.17	2.49
7
8

By comparing the two choices in Table 4, it is optimal to exercise paths 2 and 5 at time \(t_2\), but to hold the other paths until maturity. The optimal exercise strategy proposed in Table 2 will then be updated and replaced by the strategy in Table 5.

Table 5

Path	Exercise time
1	\(t_{3}\)
2	\(t_{2}\)
3	\(t_{3}\)
4	\(t_{3}\)
5	\(t_{2}\)
6	\(t_{3}\)
7	\(t_{3}\)
8	\(t_{3}\)

We now aim to decide the optimal strategy at time \(t=1\) after identifying the strategy at time \(t=2\). The coefficients of the quadratic model are estimated using the in-the-money paths at time \(t=1\) as shown in Table 6.

Table 6

Path	\(S_{t_1}|K-S_{t_{1}}>0\)	\(\mathrm{E} [\mathrm{Y}]\)
1	9.3758	\(e^{-0.01} 0\)
2
3	9.5320	\(e^{-0.01}0.72=0.71\)
4
5	5.7184	\(e^{-0.01}4.95=4.9\)
6
7
8

The resulted coefficients are \(a_0=95.3\), \(a_1=-24.6333\) and \(a_2=1.5432\). The least squares prediction model is then used to predict the expected pay-off of continuation. The predicted discounted pay-off is then compared with the pay-off resulted of immediate exercise of the option at time \(t=1\) as shown in Table 7.

Table 7

Path	\(K-S_{t_{i}}\)	\(\mathrm{max} (\hat{\mathrm{Y}},0)\)
1	1.62	0
2
3	1.47	0.71
4
5	5.28	4.9
6
7
8

By comparing the two choices in Table 7, it is optimal to exercise paths 1, 3 and 5 at time \(t_1\), but to continue holding the other paths. The optimal exercise strategy in Table 5 has then to be updated and replaced by the strategy in Table 8.

Table 8

Path	Exercise time
1	\(t_{1}\)
2	\(t_{2}\)
3	\(t_{1}\)
4	\(t_{3}\)
5	\(t_{1}\)
6	\(t_{3}\)
7	\(t_{3}\)
8	\(t_{3}\)

The pay-offs of the strategy in Table 8 are then discounted back to time \(0\) as shown in Table 9.

Table 9

Path	Discounted factor	pay-off
1	\(e^{-0.01}\)	1.62
2	\(e^{-0.02}\)	0.01
3	\(e^{-0.01}\)	1.47
4	\(e^{-0.03}\)	1.83
5	\(e^{-0.01}\)	5.28
6	\(e^{-0.03}\)	3.72
7	\(e^{-0.03}\)	0
8	\(e^{-0.03}\)	0

The option price is then calculated by averaging the discounted pay-off over all paths, which gives 1.71. Note that a European option based on the same sample paths is priced at 1.26. There is no point of considering the opportunity of exercising the option at time 0 as the pay-off of 1 is less than the option value.

The sample paths in the listed example with an exercise price of 11

Eight sample paths in another sample with an exercise price of 11 using 7 time steps

---

References

1. Longstaff, Francis A., and Eduardo S. Schwartz. Valuing American options by simulation: A simple least-squares approach. Review of Financial studies 14, no. 1 (2001): 113-147.
   
2. Wilmott P, Howison S and Dewunne J, The Mathematics of Financial Derivatives: A Student Introduction, (1995).