Pricing Theory for Range Accrual Options Based on the Trinomial Tree Model

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Abstract

The Range Accrual Option is a strongly path-dependent derivative whose final payoff depends on the number of days the underlying asset price remains within a specified range during a predefined observation period. This paper aims to explain how to use the trinomial tree numerical method, combined with risk-neutral pricing theory, to price such products. Starting from the formal definition of the product's payoff, we will detail the construction process of the trinomial tree and focus on describing the backward induction pricing algorithm that incorporates state variables (accumulated interest days).

1. Product Structure and Mathematical Model

1.1 Payoff Function

Let the product's life be $[0, T]$, with $M$ discrete observation dates $\{t_1, t_2, ..., t_M\}$ defined within this interval. Consider an asset price path $S(t)$.

At each observation date $t_m$, define an indicator function:

$$\mathbb{I}_m(S(t_m)) = \begin{cases} 1 & \text{if } L \leq S(t_m) \leq H \\ 0 & \text{otherwise} \end{cases}$$

where $L$ and $H$ are the lower and upper bounds of the range, respectively.

The product's maturity payoff $V_T$ is the principal $N$ plus accumulated interest:

$$V_T = N \cdot \left( 1 + R \cdot \frac{A}{M} \right)$$

where $R$ is the annualized coupon rate, and $A = \sum_{m=1}^{M} \mathbb{I}_m(S(t_m))$ is the number of effective interest-accruing days.

Under the risk-neutral measure $\mathbb{Q}$, the fair value $V_0$ of the product at time $t=0$ is the present value of its expected payoff:

$$V_0 = e^{-rT} \cdot \mathbb{E}^{\mathbb{Q}} \left[ V_T \right] = e^{-rT} N \cdot \left( 1 + \frac{R}{M} \mathbb{E}^{\mathbb{Q}} \left[ \sum_{m=1}^{M} \mathbb{I}_m(S(t_m)) \right] \right)$$

1.2 Pricing Challenges
The core difficulty lies in computing the expectation $\mathbb{E}^{\mathbb{Q}}[A]$. Since the indicator function $\mathbb{I}_m$ depends on the entire path $S(t)$, this problem generally lacks an analytical solution. The trinomial tree method approximates this solution by discretizing the asset price process and time, and computing the expectation over all possible paths.

2. Trinomial Tree Construction

2.1 Process Discretization
Assume the underlying asset price $S$ follows a geometric Brownian motion under the risk-neutral measure:

$$dS = (r - q) S dt + \sigma S dW^{\mathbb{Q}}$$

where $r$ is the risk-free rate, $q$ is the continuous dividend yield, and $\sigma$ is the volatility.

Divide the time interval $[0, T]$ into $N$ equal subintervals, $\Delta t = T/N$. Construct a trinomial tree such that each node at time $t_i = i \Delta t$ corresponds to an asset price $S_{i,j}$. Here, $i$ is the time step $(i = 0, 1, ..., N)$, and $j$ is the price level index.

2.2 Price Movements and Probabilities
From node $(i, j)$, the price can move in three possible ways in the next time step:

1. Up Move: $S_{i+1, k+1} = S_{i,j} \cdot u$
2. Middle Move: $S_{i+1, k} = S_{i,j} \cdot m$ (typically set $m=1$)
3. Down Move: $S_{i+1, k-1} = S_{i,j} \cdot d$

The parameters $u, m, d$ and their corresponding risk-neutral probabilities $p_u, p_m, p_d$ must match the first two moments (expectation and variance) of the risk-neutral process.

A common setting is:

$$u = e^{\lambda \sigma \sqrt{\Delta t}}, \quad d = e^{-\lambda \sigma \sqrt{\Delta t}} = \frac{1}{u}, \quad m=1$$

where $\lambda$ is a scaling parameter, often set to $\sqrt{3}$ to optimize convergence speed.

The probabilities are determined by solving the following system of equations:

$$\begin{aligned} &p_u + p_m + p_d = 1 \\ &p_u (u - 1) + p_m (m - 1) + p_d (d - 1) = (r - q) \Delta t \\ &p_u (u - 1)^2 + p_m (m - 1)^2 + p_d (d - 1)^2 = \sigma^2 \Delta t + [(r - q) \Delta t]^2 \end{aligned}$$

3. Pricing Algorithm: Backward Induction

The key to pricing lies in handling path dependency. We introduce a state variable $a$ at each tree node $(i, j)$, which represents the number of effective interest-accruing days along the price path from inception to time $t_i$. Thus, the value at each node is a function: $V_{i,j}(a)$.

3.1 Initialization (Maturity Date $i = N$)
At maturity, the accumulated days $a$ are known for all paths. For each node $(N, j)$ and each possible state $a$, its value is the deterministic payoff:

$$V_{N, j}(a) = N \cdot \left( 1 + R \cdot \frac{a}{M} \right)$$

3.2 Backward Induction (For $i = N-1, N-2, ..., 0$)
For each time step $i$, each price node $j$, and each state $a$:

1. Calculate the current node's asset price: $S_{i,j} = S_0 \cdot u^{j}$ (the index mapping needs adjustment based on the tree's geometric structure).

2. Determine if the current step accrues interest: Check if time $t_i$ belongs to the preset observation dates $\{t_1, t_2, ..., t_M\}$.

   • If it is an observation date, calculate the indicator function value $\mathbb{I}_{i,j}$:

     $$\mathbb{I}_{i,j} = \begin{cases} 1 & \text{if } L \leq S_{i,j} \leq H \\ 0 & \text{otherwise} \end{cases}$$

     The contribution of the current step to the accumulated days is $\mathbb{I}_{i,j}$. The new state variable is $a_{\text{new}} = a + \mathbb{I}_{i,j}$.

   • If it is not an observation date, the accumulated days remain unchanged, $a_{\text{new}} = a$.

3. Calculate the continuation value: The value at the current node is the risk-neutral expected discounted value of its three successor nodes. The state variable for the successor nodes is $a_{\text{new}}$.

   $$V_{i,j}(a) = e^{-r \Delta t} \left[ p_u \cdot V_{i+1,j_u}(a_{\text{new}}) + p_m \cdot V_{i+1,j_m}(a_{\text{new}}) + p_d \cdot V_{i+1,j_d}(a_{\text{new}}) \right]$$

   where $j_u$, $j_m$, $j_d$ are the indices of the nodes reached from $(i,j)$ via up, middle, and down moves, respectively.

3.3 Root Node Value
At the root node $(0, 0)$, the initial accumulated days are $a=0$. Therefore, the theoretical price of the product is:

$$V_0 = V_{0,0}(0)$$

4. Numerical Implementation Considerations

1. Discretization of the State Variable: The state variable $a$ (accumulated days) is inherently discrete, with possible values ranging from $0$ to $M$ (total observation days). This greatly simplifies the problem, as $a$ has only $M+1$ possible values, avoiding the "curse of dimensionality". At each node $(i,j)$, we only need to store $M+1$ values $V_{i,j}(a)$ for $a=0,1,...,M$.

2. Computational Complexity: The complexity of this algorithm is $O(N \times J \times M)$, where $N$ is the number of time steps, $J$ is the number of price nodes (approximately $2N+1$), and $M$ is the number of observation days. Although more complex than pricing a vanilla option, it is significantly more efficient than Monte Carlo simulation (which requires a vast number of paths to achieve comparable accuracy).

3. Alignment of Tree and Observation Dates: For accurate calculation, it should be ensured that each preset observation date $t_m$ exactly corresponds to a time node $t_i$ in the tree. This typically means the number of time steps $N$ needs to be greater than or equal to the number of observation days $M$, and $\Delta t$ should be set to $T/N$.