Derivative Valuation

American Options: A comparative analysis of the Cox Ross Rubinstein (CRR) and Longstaff & Schwartz (LS) valuation methods.

Paul McAteer, MSc, MBA

pcm353@stern.nyu.edu

1.0 Introduction ¶

This technical note compares the prices obtained from the two most common methods employed for the valuation of American-Style options. We implement the binomial "tree-based" technique of Cox Ross Rubinstein¹ (CRR) and the least-squares Monte Carlo method of Longstaff & Schwartz ² (LS). The design, implementation and output of both are validated by comparing our results to those produced in the academic literature.

¹Cox, J., S. Ross, and M. Rubinstein (1979): “Option Pricing: A Simplified Approach.” Journal of Financial Economics, Volume 7, Issue 3, pp. 229–263.
²Longstaff, Francis & Schwartz, Eduardo. (2001). Valuing American Options by Simulation: A Simple Least-Squares Approach. Review of Financial Studies. 14. 113-47.

2.1 LS Method: Model Theory and Design ¶

The valuation of an American-style option using the Longstaff Shwarz approach requires the Monte Carlo simulation of $N$ realizations of the asset path from now to expiration using time steps of $δt$ . If the random walk is lognormal then we will simulate

S_{j + 1} = S_{j} \exp ((r - \frac{1}{2} σ^{2}) δ t + σ \sqrt{δ} t ϕ)

$S_{j+1}= S_{j}\exp\ (( r-\frac{1}{2}\ \sigma^2)\ \ \delta t+\sigma\sqrt\delta t\ \phi)$

Where $\phi$ is a Normally distributed random variable, $r$ is the risk free rate, $\sigma$ is volatility and and $S_{j}$ is the stock price after $j$ time steps.

For the purposes of exposition, we will follow the example of the Longman Shwartz paper which demonstrates the algorithm over three time steps and 8 simulated price paths, considering a 3-year American put option on a non-dividend-paying stock that can be exercised at the end of year 1, the end of year 2, and the end of year 3. The risk-free rate is 6% per annum (continuously compounded). The current stock price is 1.00 and the strike price is 1.10. In the implementation phase, the timesteps and simulations will obviously be significantly greater. Having obtained the 8 simulations of the asset path, we calculate the intrinsic value for each simulation at each time step. The present value of the average of all of the payoffs at expiration is the value of a European option. See below for the Stock Price Matrix and the Intrinsic Value Matrix.

From the stock price matrix, there are five paths where the option is in the money at time $t+2$ . These are paths 1, 3, 4, 6, and 7. Obviously at the final exercise date, $t+3$ , the optimal exercise strategy for an American option is to exercise the option if it is in the money (ITM). Prior to the final date however, at $t+2$ , the optimal strategy is to compare the immediate exercise value $X-S$ with the expected cash flows from continuing $E[V|S]$ , and then exercise if immediate exercise is more valuable, i.e. where $X-S > E[V|S]$ . Conversely, one would continue to hold where the continuation value is more valuable, i.e. where $E[V|S] > X-S$

The conditional expectation function is obtained at each time step by regressing the present value of subsequent realized cash flows from continuation at t+2 on the simulated stock price at the time step t+2 when the continuation/exercise decision is made. That is we assume an approximate relationship of the form:

E [V | S] = \hat{V} = a + b S + c S^{2}

$E[V|S] =\ \hat{V}= \ a\ +bS\ +\ cS^{2}$

Where $S$ is the stock price at the 2-year point, $\hat{V}$ is the expected value of continuing discounted back to the two year time point, based on the expected subsequent prices generated in the Monte Carlo simulation process. For this example, we visualize below the present value of subsequent realized cash flows and the conditional expectation of the discounted value of subsequent realized cash flows.

Having obtained a vector of continuation values for t+2, we now compare these with a vector of immediate exercise values at t+2 and determine optimal exercise behaviour:

We use this to construct a matrix composed of the cashflows which would result from optimal exercise behaviour, conditional on not exercising prior to t+2. Note that if there is a positive entry in the time t+2 column it means that we should have exercised there (if not earlier) and so the cashflows for later times are set to zero.

Proceeding recursively, we next examine whether the option should be exercised at time t+1. From the stock price matrix, there are again five paths where the option is in the money at time 1. These are paths 1, 4, 6, 7, and 8.