9.2 The Trinomial Model for American Options

The American option price can only be determined numerically. Similar to the European options, the binomial model after Cox-Ross-Rubinstein can be used. In this section we introduce a less complex but numerically efficient approach based on trinomial trees, see Dewynne et al. (1993). It is related to the classical numerical procedures for solving partial differential equations, which are also used to solve the Black-Scholes differential equations.

The trinomial model (see Section 4.2) follows the procedure of the binomial model whereby the price at each time point $ t_j = j \Delta t, j=0,...,n$ can change to three instead of two directions with $ \Delta t = T/n$, see Figure 8.4. The value $ S_j^k$ at time $ t_j$ can reach to the values $ u_1 \cdot S_j^k, u_2 \cdot S_j^k, u_3 \cdot S_j^k$ at $ t_{j+1}$, where $ u_i > 0, i=1, 2, 3,$ are the suitable parameters of the model. The probability with which the price moves from $ S_j^k$ to $ u_i \cdot S_j^k$ is represented as $ p_i, i=1, 2, 3.$ The price process $ S_j, j=0, ..., n$ in discrete time is also a trinomial process, i.e. the logarithms of the price $ Z_j = \ln
S_j, j=0, ..., n$ is an ordinary trinomial process with possible increments $ \ln u_i, i=1, 2, 3.$

Fig.: Possible price movements in the trinomial model.
\includegraphics[width=1\defpicwidth]{stock.ps}

As in the binomial model three conditions must be fulfilled: The sum of the probabilities $ p_i, i=1, 2, 3,$ is one, the expectation and variance of the logarithms increments $ Z_j$ must be the same as those of the logarithms of the geometric Brownian motion over the time interval $ \Delta t$. From these conditions we get three equations:

$\displaystyle p_1+p_2+p_3$ $\displaystyle =$ $\displaystyle 1,$ (9.17)
$\displaystyle p_1\ln u_1+p_2\ln u_2+p_3\ln u_3$ $\displaystyle =$ $\displaystyle (b - \frac{1}{2}\sigma^2 )\Delta t,$ (9.18)
$\displaystyle p_1(\ln u_1)^2+p_2(\ln u_2)^2+p_3(\ln u_3)^2$ $\displaystyle =$ $\displaystyle \sigma ^2\Delta t +
(b - \frac{1}{2}\sigma^2)^2\Delta t^2.$ (9.19)

In the last equation we use $ \mathop{\text{\rm\sf E}}( Z_j^2) = var(Z_j) +(\mathop{\text{\rm\sf E}}Z_j)^2$. Since there are 6 unknown parameters in the trinomial model, we have the freedom to introduce three extra conditions in order to identify a unique and possibly simple solution of the equation system. To be able to construct a clear price tree, we require the recombination property

$\displaystyle u_1u_3=u_2^2 .$

From this, the number of possible prices at time $ t_n$ is reduced from maximal $ 3^n$ to $ 2n+1$ and consequently the memory spaces and computation time are saved. To determine the parameters of the model we still need two more conditions. We discuss two approaches of which one comes from binomial models while the other from the numerical results of partial differential equations.

a.) The first approach requires that a time step of $ \Delta t$ in the trinomial model corresponds to two time steps in the binomial model: $ u_1$ represents two upwards increments, $ u_3$ two downwards increments and $ u_2$ one upward and one downward increment (or reversed). The binomial model fulfills the recombination condition $ d = 1/u$. Since now the length of the time step is $ \Delta t/2$, it holds following Section 7.1

$\displaystyle u=e^{\sigma \sqrt{\Delta t/2}}$

and the probability with which the price moves upwards in the binomial model is:

$\displaystyle p = \frac{1}{2} + \frac{1}{2}(b -\frac{1}{2}\sigma^2)\frac{\sqrt{\Delta t/2}}{\sigma}.$

Then we get the conditions for the parameters of the trinomial model
$\displaystyle u_1$ $\displaystyle =$ $\displaystyle u^2,\, u_2=1,\, u_3=u^{-2},$  
$\displaystyle p_1$ $\displaystyle =$ $\displaystyle p^2,\, p_2=2p(1-p),\, p_3=(1-p)^2.$  

With these parameters, the trinomial model performs the same as the corresponding binomial model for the European option, requiring however only the half of the time step. It converges therefore more quickly than the binomial model against the Black-Scholes solution.

Example 9.1   Given the parameters from Table 7.1, the trinomial model delivers a price of 30,769 for a European call option after $ n=50$ steps. This corresponds exactly the value the binomial model reaches after $ n=100$ steps, see Table 7.2.

American options differ from the European in that the options can be exercised at any time $ t^*\ ,0<t^* \leq T $. Consequently the value of a call falls back to the intrinsic value if it is early exercised:

$\displaystyle C(S,t^*)=\max \{ 0,S(t^*)-K\}.$

Mathematically we have solved the free boundary problem, which is only numerically possible.

$ V_j^k$ denotes the option value at time $ t_j$ if the spot price of stock is $ S_j = S_j^k$. As in the binomial model on European options we use $ V_j^k$ to denote the discounted expectation that is calculated from the prices attainable in the next time step, $ V_{j+1}^{k+1},\ V_{j+1}^k$ and $ V_{j+1}^{k-1}$. Different from the European options, the expectation of American options may not fall under the intrinsic value. The recursion of American call price is thus:

$\displaystyle C_j^k=\max \{ S_j^k-K \, , \, e^{-r\Delta t}
[p_1C_{j+1}^{k+1}+p_2C_{j+1}^k +p_3C_{j+1}^{k-1}]\}.$

Example 9.2   Table 8.3 gives the parameters and the value of an American call option determined with steps $ n=50$ in a trinomial model. It coincides with Theorem 8.1 giving the same value 30.769 as a European option because the underlying stock issues no dividend during the running time.

The American put is on the other hand more valuable than the European. With the parameters from Table 8.3 one gets $ P_{eur}=6.05140$ and $ P_{am}=6.21159$.


Table 8.3: The value of an American call option.
spot stock price $ S_t$ 230.00
exercise price $ K$ 210.00
time to maturity $ \tau$ 0.50
volatility $ \sigma$ 0.25
interest rate $ r$ 0.04545
dividend no
steps 50
option type American call
option price 30.769


b.) In the second approach the trinomial parameters $ p_i$ and $ u_i$ are certainly decided through additional conditions. Here a certain upwards trend is shown in the whole price tree since we replace the condition $ u_2=1$ by

$\displaystyle u_2 = u \stackrel{\mathrm{def}}{=}e^{(b-\frac{1}{2}\sigma^2)\Delta t}.$

Furthermore we assume $ p_1 = p_3$ and receive therefore together with the four above-mentioned conditions:

$\displaystyle p_1=p_3=p,\ p_2=1-2p,$   with $\displaystyle \; p=\frac{\Delta t}{T h^2},$      
       
$\displaystyle u_1=ue^{\sigma h\sqrt{T/2}},\ u_2=u,\ u_3=ue^{-\sigma
h\sqrt{T/2}},$      

Where $ h$ is another free parameter. The $ p_i$ and $ u_i$ fulfill the equation system (8.17) - (8.19) exactly so that $ p_1, p_2, p_3$ are not trivial probabilities, i.e they must be between 0 and 1 and $ 0<p<1/2$. That means $ h$ must fulfill the following condition:

$\displaystyle h > \sqrt{\frac{2\Delta t}{T}}.$ (9.20)

We consider now a European option. Here the trinomial model delivers the following recursion for the possible option value dependent on the probabilities $ p_i$ and the change rates $ u_i$:

$\displaystyle V_j^k=e^{-r\Delta t}\left( \frac{\Delta t}{T h^2}V_{j+1}^{k+1} +(1-2\frac{\Delta t}{T h^2})V_{j+1}^k+ \frac{\Delta t}{T h^2}V_{j+1}^{k-1}\right).$ (9.21)

We consider $ \Delta t=-(T -t_{j+1})+(T -t_j)$ for all $ j=0,...,n-1$ and we put $ h^* = \Delta t/T$ as well as

$\displaystyle Z_j^k = V_j^ke^{-r(T -t_j)}, \; Z_{j+1}^k = V_{j+1}^ke^{-r(T -t_{j+1})}.$

The recursion (8.21) for the option value $ V_j^k$ becomes then

$\displaystyle \frac{Z_j^k-Z_{j+1}^k}{h^* }\ =\ \frac{Z_{j+1}^{k+1}-2Z_{j+1}^k + Z_{j+1}^{k-1}}{h^2}.$ (9.22)

This is the explicit difference approximation of the parabolic differential equation (6.15), see Samaskij (1984). The condition (8.20) corresponds to the well-known stability requirement for explicit difference operations. Compared to the previously discussed approach, the probabilities $ p_i$ in the variant of the trinomial models and the calculation in (8.21) are not dependent on the volatility. The recursion (8.21) depends only on the starting condition, i.e. $ S_n =
S_T$ at the exercise moment as an observation depends on $ \sigma$ from the geometric Brownian motion.