**3. Implementation**

The first step in implementing any type of numerical algorithm to price American options is to assume that time can be discretized. Thus, we assume that the derivative considered may be exercised at *J* points in time. We specify the potential exercise points as *t*0 = 0 < *t*1 ≤ *t*2 ≤ ... ≤ *tJ* = *T*, with *t*0 and *T* corresponding to the current time and maturity of the option, respectively. An American option can be approximated by increasing the number of exercise points *J*, and a European option can be valued by setting *J* = 1. We assume a complete probability space (<sup>Ω</sup>, F,<sup>P</sup>) equipped with a discrete filtration F *tj<sup>J</sup> j*=0. The derivative's value depends on one or more underlying assets, which are modelled using a Markovian process, with state variables *X tj<sup>J</sup> j*=0 adapted to the filtration and with *X* (0) = *x* known. We denote by *Z tj<sup>J</sup> j*=0 an adapted payoff process for the derivative satisfying *Z tj* = *π X tj* , *tj* for a suitable function *π* (·, ·), which is assumed to be square integrable. Following, e.g., Karatzas (1988) and Duffie (1996), in the absence of arbitrage, we can specify the American option price as:

$$P\left(X\left(0\right)=\mathbf{x}\right) = \max\_{\tau\left(t\_1\right)\in\mathcal{T}\left(t\_1\right)} \mathbb{E}\left[Z\left(\tau\right)\left|X\left(0\right)\right],\tag{1}$$

where T *tj* denotes the set of all stopping times with values in {*tj*, ... , *tJ*} and where it is therefore implicitly assumed that the option cannot be exercised at time *t*0.

In the literature, the problem of calculating the American option price in Equation (1), i.e., with *J* > 1, is referred to as a discrete time optimal stopping time problem. The preferred way to solve such problems is to use the dynamic programming principle. Intuitively, this procedure can be motivated by considering the choice faced by the option holder at time *tj*: either to exercise the option immediately or to continue to hold the option until the next period. Obviously, at any time, the optimal choice will be to exercise immediately if the value of this is positive and larger than the expected payoff from holding the option until the next period and behaving optimally from there on forward. To fix notation, in the following, we let *V X tj* denote the value of the option for state variables *X* at a time *tj* prior to expiration. We define *F X tj* ≡ E[*Z τ tj*+<sup>1</sup> |*X tj* ] as the expected conditional payoff, where *τ tj*+<sup>1</sup> is the optimal stopping time. It follows that:

$$\mathcal{V}\left(X\left(t\_{\bar{j}}\right)\right) = \max\left(Z\left(t\_{\bar{j}}\right), F\left(X\left(t\_{\bar{j}}\right)\right)\right),\tag{2}$$

and it is easily seen that it is possible to derive the optimal stopping time iteratively using the following algorithm:

$$\begin{cases} \tau\left(t\_{\boldsymbol{l}}\right) = \boldsymbol{T} \\ \tau\left(t\_{\boldsymbol{l}}\right) = \boldsymbol{t}\_{\boldsymbol{l}}\mathbf{1}\_{\{\boldsymbol{Z}\left(t\_{\boldsymbol{l}}\right) \geq \boldsymbol{F}\left(\boldsymbol{X}\left(t\_{\boldsymbol{l}}\right)\right)\}} + \tau\left(t\_{\boldsymbol{k}+1}\right)\mathbf{1}\_{\{\boldsymbol{Z}\left(t\_{\boldsymbol{l}}\right) < \boldsymbol{F}\left(\boldsymbol{X}\left(t\_{\boldsymbol{l}}\right)\right)\}}, \quad 1 < \boldsymbol{j} \leq \boldsymbol{J} - 1. \end{cases} \tag{3}$$

Based on this, the value of the option in Equation (1) can be calculated as:

$$P\left(X\left(0\right)=\mathbf{x}\right) = \mathbb{E}\left[Z\left(\tau\left(t\_1\right)\right)|X\left(0\right)\right].\tag{4}$$

The backward induction theorem of Chow et al. (1971) (Theorem 3.2) provides the theoretical foundation for the algorithm in Equation (3) and establishes the optimality of the derived stopping time and the resulting price estimate in Equation (4).

### *3.1. Simulation and Regression Methods*

The idea behind using simulation for option pricing is quite simple and involves estimating expected values and therefore option prices by an average of a number of random draws. However, when the option is American, one needs to determine simultaneously the optimal early exercise strategy, and this complicates matters. In particular, it is generally not possible to implement the exact algorithm in Equation (3) because the conditional expectations are unknown, and therefore, the price estimate in Equation (4) is infeasible. Instead, an approximate algorithm is needed. Because conditional expectations can be represented as a countable linear combination of basis functions, we may write *F X tj* = ∑∞*<sup>l</sup>*=<sup>0</sup> *φl X tj cl tj*, where {*φl* (·)}<sup>∞</sup>*l*=<sup>0</sup> form a basis.<sup>4</sup> In order to make this operational, we further assume that it is possible to approximate well the conditional expectation function by using the first *L* + 1 terms such that *F X tj* ≈ *FL X tj* = ∑*Ll*=<sup>0</sup> *φl X tj cl tj* and that we can obtain an estimate of this function by:

$$\left(\hat{F}\_{L}^{N}\left(X\left(t\_{\hat{\jmath}}\right)\right)\right) = \sum\_{l=0}^{L} \phi\_{l}\left(X\left(t\_{\hat{\jmath}}\right)\right)\hat{c}\_{l}^{N}\left(t\_{\hat{\jmath}}\right),\tag{5}$$

where *c*ˆ*Nl tj* are approximated or estimated using *N* ≥ *L* simulated paths. Based on the estimate in Equation (5), we can derive an estimate of the optimal stopping time from:

$$\begin{cases} \hat{\mathbf{t}}\_{L}^{N}\left(\mathbf{t}\_{l}\right) = T\\ \hat{\mathbf{t}}\_{L}^{N}\left(\mathbf{t}\_{\hat{\mathbf{t}}}\right) = t\_{\hat{\mathbf{t}}}\mathbf{1}\_{\{Z\_{\hat{\mathbf{t}}}\left(\hat{\mathbf{t}}\_{\hat{\mathbf{t}}}\right) \geq \hat{\mathbf{t}}\_{L}^{N}\left(X\left(\mathbf{t}\_{\hat{\mathbf{t}}}\right)\right)\}} + \hat{\mathbf{t}}\_{L}^{N}\mathbf{1}\_{\{Z\_{\hat{\mathbf{t}}}\left(\hat{\mathbf{t}}\_{\hat{\mathbf{t}}}\right) \leq \hat{\mathbf{t}}\_{L}^{N}\left(X\left(\mathbf{t}\_{\hat{\mathbf{t}}}\right)\right)\}}, \quad 1 < j \leq J - 1. \end{cases} \tag{6}$$

From the algorithm in Equation (6), a natural estimate of the option value in Equation (4) is given by:

$$\mathcal{P}\_L^N\left(X\left(0\right)=x\right) = \frac{1}{N} \sum\_{n=1}^N Z\left(n, \mathfrak{f}\_L^N\left(1, n\right)\right),\tag{7}$$

where *Z n*, *τ*ˆ*NL* (1, *n*) is the payoff from exercising the option at the optimal stopping time *τ*ˆ*NL* (1, *n*) determined for path *n* according to Equation (6).

### *3.2. Implementation of the LSMC Method*

When implementing the method outlined above, one has to choose at least two things: how to generate the data, the simulated state variables, and how to approximate the value function, that

<sup>4</sup> This is justified when approximating elements of the *L*<sup>2</sup> space of square-integrable functions relative to some measure. Since *L*<sup>2</sup> is a Hilbert space, it has a countable orthonormal basis (see, e.g., Royden 1988).

is how to estimate the parameters in the approximation. The key contribution of Longstaff and Schwartz (2001) is to sugges<sup>t</sup> that the coefficients in the approximation of the continuation value, *c*<sup>ˆ</sup>*<sup>N</sup> l tj* , can be estimated in a simple cross-sectional ordinary linear (OLS) regression, where the independent variable is the discounted path-wise future payoff and the dependent variables are functions of the current state variables. In this paper, we propose to merge the LSMC method with PCS and use the symmetric method when pricing call options. Thus, instead of simulating paths from a dynamic model with a risk-free rate of *r* and dividend yield of *d*, we simulate from the same dynamic model, but with a risk-free rate of *d* and dividend yield of *r*, and instead of pricing the option as a call option with a strike price of *K* and a current value of the underlying asset of *S*, we price the option as if it had been a put option with a strike price of *S* and a current value of the underlying asset of *K*. 5 These changes are simple to make and involve no extra computational complexity or changes to the numerical procedure. In fact, for consistency, it is important to note that we use the exact same numerical procedures for simulating the paths and to implement the cross-sectional regression.<sup>6</sup>

There are two very intuitive reasons why using the symmetric method to price call options may work better than when pricing the call option using the regular method. First, as explained above, in simulation-based methods, the option price, an expectation under the risk neutral measure, is approximated by the average of a number of random realizations of future payoffs, obtained from simulated values of the appropriate state variables. This mean obviously behaves better, and the estimator will have a smaller variance when the possible realizations are bounded, as they are in the case of the payoff of a put option, than when they are unbounded, as they are in the case of the payoff of a call option. Our numerical results for the benchmark options in Section 2 indeed showed that the standard deviation of the estimates is always lower when using the symmetric method than when using the regular method.<sup>7</sup>

Second, it is easier to approximate the continuation value when this is a bounded function on a bounded interval than when this is an unbounded function on an unbounded interval. In particular, theoretically, it is straightforward to design a robust approximation scheme for the continuation value of a put option using only the simulated paths that are in the money.<sup>8</sup> For call options, on the other hand, no general theoretical results exist to justify that this is in fact feasible, and though numerical schemes are available and polynomial families that have nice properties can be used, approximating the continuation value is likely much more complicated. A further complication with the continuation value of a call option is that this is bounded below by the exercise value for large values of the underlying asset and thus asymptotically linear in the stock value. It is obviously difficult to approximate a function with these characteristics.
