Trinomial Model¶

Introduction¶

The binomial model restricts the stock price to two possible outcomes at each time step: up or down. While this yields a clean, complete market in which every contingent claim can be uniquely replicated, real markets offer a richer set of possibilities. The trinomial model generalizes the binomial framework by allowing three possible price movements per period: up, middle, and down.

This seemingly minor extension has a profound structural consequence. With three possible states but only two traded assets (stock and bond), the payoff space of the market is two-dimensional while the state space is three-dimensional. As a result, not every contingent claim can be replicated, and the market becomes incomplete. The risk-neutral measure is no longer unique---there is an entire family of equivalent martingale measures, each producing a different arbitrage-free price for the same derivative.

The trinomial model thus serves a dual pedagogical purpose:

• Computational: the Boyle (1988) trinomial tree is a practical numerical method that often converges faster than the binomial tree
• Conceptual: it provides the simplest concrete example of market incompleteness, motivating the Second Fundamental Theorem of Asset Pricing

One-Period Trinomial Model¶

Setup¶

We work on a single period \(t \in \{0, \Delta t\}\) with two traded assets.

Recall (see § Binomial Model — Setup): the bank account \(B_{\Delta t} = e^{r\Delta t}\) with \(r \geq 0\) the continuously compounded risk-free rate. The novelty here is in the stock dynamics.

Risky asset (stock). Starting from \(S_0 > 0\), the stock price at time \(\Delta t\) takes one of three values:

where the multiplicative factors satisfy \(u > m > d > 0\) and the physical probabilities satisfy \(p_u, p_m, p_d > 0\) with \(p_u + p_m + p_d = 1\).

The Sample Space¶

The one-period trinomial model lives on a finite probability space \((\Omega, \mathcal{F}, \mathbb{P})\) with three states:

The filtration is \(\mathcal{F}_0 = \{\emptyset, \Omega\}\) (no information at time \(0\)) and \(\mathcal{F}_{\Delta t} = 2^\Omega\) (full information at time \(\Delta t\)). A contingent claim is any \(\mathcal{F}_{\Delta t}\)-measurable random variable, which in this finite setting is simply a vector \(H = (H_u, H_m, H_d)\) specifying the payoff in each state.

No-Arbitrage Condition¶

Recall (see § Binomial Model — No-Arbitrage Condition) the definition of arbitrage (a portfolio with \(V_0 \leq 0\), \(V_{\Delta t} \geq 0\) in every state, and \(V_{\Delta t} > 0\) in at least one state) and the shorting argument. The same argument applies here: short the stock if \(e^{r\Delta t} \geq u\), go long if \(e^{r\Delta t} \leq d\). The middle factor \(m\) never enters — only the extremes can dominate the bond.

Risk-Neutral Measures¶

The Martingale Condition¶

Recall (see § Risk-Neutral Measure): a risk-neutral measure (equivalent martingale measure) is a measure \(\mathbb{Q}\) that is equivalent to \(\mathbb{P}\) (assigns positive mass to every state) and under which the discounted stock price is a martingale, \(S_0 = e^{-r\Delta t}\mathbb{E}^{\mathbb{Q}}[S_{\Delta t}]\). Specialized to \(\Omega = \{\omega_u, \omega_m, \omega_d\}\) with masses \((q_u, q_m, q_d)\), the three defining conditions become \(q_u, q_m, q_d > 0\), \(q_u + q_m + q_d = 1\), and:

Characterizing the Set of Risk-Neutral Measures¶

We now have three unknowns \((q_u, q_m, q_d)\) subject to:

This is a system of two linear equations in three unknowns with positivity constraints. The solution set is a one-parameter family. Using the normalization constraint to eliminate \(q_m = 1 - q_u - q_d\) and substituting into the martingale equation:

We can parameterize the family by a free parameter. Let \(q_d = \lambda\) where \(\lambda > 0\). Then:

The constraints \(q_u > 0\), \(q_m > 0\), \(q_d = \lambda > 0\) restrict \(\lambda\) to an open interval \((\lambda_{\min}, \lambda_{\max})\).

Geometric Interpretation¶

The martingale condition defines a plane (actually a line, after intersecting with the probability simplex) in the simplex \(\{(q_u, q_m, q_d) : q_i \geq 0, \sum q_i = 1\}\). In the binomial model, two constraints on two unknowns yield a single point. In the trinomial model, two constraints on three unknowns yield a line segment in the interior of the simplex.

Market Incompleteness¶

Replication Failure¶

Recall (see § Replicating Portfolio) that in the binomial model any claim \(H = (H_u, H_d)\) is replicable because two unknowns \((\Delta, \beta)\) solve a \(2\times 2\) system. In the trinomial model the system becomes \(3 \times 2\):

This is a system of three equations in two unknowns \((\Delta, \beta)\). Generically, such a system has no solution.

Claims that fail this condition---such as a generic European call option---cannot be replicated. The market is incomplete.

Connection to the Second Fundamental Theorem¶

Recall (see § FTAP): First FTAP — no-arbitrage iff a risk-neutral measure exists; Second FTAP (see § Complete Markets and Uniqueness) — completeness iff that measure is unique.

In the trinomial model:

• Risk-neutral measures exist (by no-arbitrage), so the First FTAP is satisfied
• Multiple risk-neutral measures exist, so by the Second FTAP, the market is incomplete

No-Arbitrage Price Bounds¶

The Pricing Interval¶

Since the risk-neutral measure is not unique, the risk-neutral pricing formula \(V_0 = e^{-r \Delta t} \, \mathbb{E}^{\mathbb{Q}}[H]\) gives a different price for each choice of \(\mathbb{Q}\). Every such price is consistent with no-arbitrage. The set of all no-arbitrage prices forms an interval:

where \(\mathcal{Q}\) is the set of all risk-neutral measures. The infimum and supremum are taken over the one-parameter family derived above.

Super-Replication Interpretation¶

The upper bound \(\sup_{\mathbb{Q}} e^{-r \Delta t} \, \mathbb{E}^{\mathbb{Q}}[H]\) equals the cost of the cheapest portfolio that super-replicates \(H\) (i.e., \(V_{\Delta t} \geq H\) in every state). Similarly, the lower bound equals the negative of the super-replication cost of \(-H\). These bounds have direct financial meaning:

• A seller of the claim needs at most the upper bound to hedge
• A buyer would pay at most the lower bound to avoid being overcharged relative to the market

Any price within the open interval is consistent with no-arbitrage, but the market does not determine a unique fair value. Additional criteria---such as utility maximization, model calibration, or risk preferences---are needed to select a specific price.

The Boyle Parameterization¶

Motivation¶

For computational purposes, we need a specific choice of \(u\), \(m\), \(d\) (and often a specific risk-neutral measure) for the trinomial tree. Boyle (1988) proposed choosing parameters to match the first two moments (mean and variance) of the stock's log-return under the risk-neutral measure, plus an additional symmetry condition.

Construction¶

Fix the time step \(\Delta t\) and the volatility \(\sigma > 0\). Boyle's parameterization sets:

Note that \(u \, d = 1\) and \(m = 1\) (the stock is unchanged in the middle state). The risk-neutral probabilities are chosen to match the mean and variance of \(\ln(S_{\Delta t}/S_0)\):

Advantages of the Trinomial Tree¶

Compared to the Cox-Ross-Rubinstein binomial tree:

• Three branches per node produce a more refined lattice, often giving faster convergence to the continuous-time (Black-Scholes) price
• The lattice naturally recombines: an up-then-down path, a middle-then-middle path, and a down-then-up path can all reach the same node
• The extra degree of freedom allows better moment matching and more flexibility in fitting dividend yields or time-varying parameters
• Trinomial trees are especially natural for interest rate models (e.g., the Hull-White trinomial tree in Chapter 20)

Worked Example: European Call in the Trinomial Model¶

We now compute the interval of no-arbitrage prices for a European call option in a concrete one-period trinomial model.

Parameters¶

The risk-free growth factor is \(e^{r \Delta t} = e^{0.05} \approx 1.05127\).

Verify no-arbitrage: \(d = 0.8 < 1.05127 < 1.2 = u\). The condition holds.

Stock and Call Payoffs¶

Consider a European call with strike \(K = 100\):

Is the Call Replicable?¶

Check the replication condition:

Since \(1 \neq 0\), the call is not replicable. The market is incomplete for this claim.

Computing the Price Interval¶

The risk-neutral probabilities must satisfy:

Parameterize by \(q_d = \lambda > 0\). From the two equations:

Positivity constraints:

• \(q_d = \lambda > 0\): requires \(\lambda > 0\)
• \(q_u = 0.25634 + \lambda > 0\): automatically satisfied for \(\lambda > 0\)
• \(q_m = 0.74366 - 2\lambda > 0\): requires \(\lambda < 0.37183\)

So \(\lambda \in (0, \, 0.37183)\).

Call price as a function of \(\lambda\):

Bounds:

• As \(\lambda \to 0^+\): \(V_0 \to 19.0246 \times 0.25634 \approx 4.877\)
• As \(\lambda \to 0.37183^-\): \(V_0 \to 19.0246 \times (0.25634 + 0.37183) \approx 19.0246 \times 0.62817 \approx 11.949\)

Interpretation¶

The wide interval \((4.88, \, 11.95)\) reflects the fundamental pricing ambiguity in incomplete markets. In practice, a trader would narrow this interval by:

1. Adding traded instruments: if a second option is liquidly traded, its market price pins down additional constraints on \(\mathbb{Q}\), shrinking the interval (or eliminating it entirely if the market becomes complete)
2. Imposing a model: choosing a specific parameterization (e.g., Boyle's) selects one \(\mathbb{Q}\) from the family
3. Utility-based pricing: an agent's risk preferences select a unique price within the interval

Summary¶

What's Next¶

Exercises¶

Exercise 1. Consider the trinomial model with \(S_0 = 100\), \(u = 1.3\), \(m = 1.05\), \(d = 0.7\), \(r = 5\%\), and \(\Delta t = 1\). Verify the no-arbitrage condition. Then parameterize the family of risk-neutral measures by \(q_d = \lambda\) and determine the admissible range of \(\lambda\).

Exercise 2. Using the trinomial model from Exercise 1, compute the no-arbitrage price interval for a European put with strike \(K = 100\). Express \(V_0(\lambda)\) as a function of \(\lambda\) and find the supremum and infimum over the admissible range. Compare the width of the interval to that of a call with the same strike.

Exercise 3. Prove that a contingent claim \(H = (H_u, H_m, H_d)\) in the one-period trinomial model is replicable if and only if the payoff is affine in \(S_{\Delta t}\), i.e., \(H = a \cdot S_{\Delta t} + b\) for some constants \(a, b\). Show that this is equivalent to the condition:

Exercise 4. In the worked example from the text (\(S_0 = 100\), \(u = 1.2\), \(m = 1.0\), \(d = 0.8\), \(r = 5\%\)), the binomial price of the call equals the upper bound of the trinomial interval. Prove this is not a coincidence: show that for any claim with \(H_m = H_d\) (the middle and down payoffs coincide), the binomial price always equals the upper bound of the trinomial price interval.

Exercise 5. Suppose the trinomial market is "completed" by adding a traded European call option with strike \(K = 100\) and observed market price \(C_0 = 8.50\) (using the same parameters as the worked example in the text). With three assets (stock, bond, call) and three states, the risk-neutral measure becomes unique. Find the unique risk-neutral measure \((q_u, q_m, q_d)\) and use it to price a European put with strike \(K = 110\).

Exercise 6. For the Boyle (1988) parameterization with \(\sigma = 0.20\), \(r = 0.05\), and \(\Delta t = 0.25\), compute \(u\), \(m\), \(d\), \(q_u\), \(q_d\), and \(q_m\). Verify that (a) \(q_u + q_m + q_d = 1\), (b) the martingale condition \(q_u u + q_m m + q_d d = e^{r\Delta t}\) holds, and (c) \(\text{Var}^{\mathbb{Q}}[\ln(S_{\Delta t}/S_0)] \approx \sigma^2 \Delta t\).