Introduction

Understanding how to calculate the value of options is fundamental in the world of finance. Binomial models provide a versatile framework for pricing options, offering insights into both European and American options. In this chapter, we will explore the one-step and two-step binomial models, which serve as the building blocks for more complex option pricing models. We will dive into the calculations and concepts behind pricing European and American call and put options, equipping you with essential tools for option valuation.

One-Step Binomial Model

The one-step binomial model is a simple framework used to estimate the value of financial options, especially European options. In this model, it is assumed that the underlying asset’s price can take one of two possible values: it can either go up by a certain factor or down by another factor over a fixed time period.

Assume that the initial stock price is $S_0$. Then, the stock price can take values $S_U$ or $S_D$ in the next step. The stock price tree for one-step binomial model is given below:

    graph LR
    S0[S0] --> Su[Su] 
    S0 --> Sd[Sd] 
  

Here, $S_U = S_0 U$ and $S_D = S_0D = \left(\frac{S_0}{U}\right)$, where $U$ and $D$ are sizes of the upward and downward movements. The size and probability of these movements are calculated as per below formula.

• Size of Movement: The sizes of the upward and downward movements are defined as functions of the volatility and the length of the steps in the binomial model:

$$\begin{aligned} & U=e^{\sigma \sqrt{t}} \\ & D=e^{-\sigma \sqrt{t}}=\frac{1}{e^{\sigma \sqrt{t}}}=\frac{1}{U} \end{aligned} $$

Where:

• $U$ = size of the up-move factor
• $D$ = size of the down-move factor
• $\sigma$ = annual volatility of the underlying asset’s returns
• $t$ = the length of the step in the binomial model

Based on above movement sizes, $S_U$ and $S_D$ are calculated as below:

$$\begin{aligned} & S_U= S_0 U \\ & S_D=S_0 D = \frac{S_0}{U} \end{aligned} $$

• Probability of Movement: The risk-neutral probabilities of upward and downward movements are then calculated as:

$$\begin{aligned} & \pi_{u}=\frac{(e^{rt}-D)}{(U-D)} \\ & \pi_{d}=(1-\pi_{d}) \end{aligned}$$

Where:

• $\pi_{u}$ = probability of an up move
• $\pi_{u}$ = probability of a down move
• $r$ = continuously compounded annual risk-free rate

• Pricing Options: To price the options, we need to find the expected price of the option at maturity and then discount it back to its present value. To illustrate, let’s consider a European call option. The call option tree for one-step binomial model is given below::

    graph LR
    C0[C0] --> CU[CU] 
    C0 --> CD[CD] 
  

Here, $C_U$ = $max(0,S_U-K)$ and $C_D$ = $max(0,S_D-K)$.

• The expected value of the option in one year is given by: $$(C_U \times \pi_u+C_D \times \pi_d)$$

• The present value of the option’s expected value is given by: $$C_0 = \frac{(C_U \times \pi_u+C_D \times \pi_d)}{e^{r \times 1}}$$

The corresponding put option can be valued using the put-call parity formula given below:

$$P_0 + S_0 = C_0 + Ke^{-rt}$$

Two-Step Binomial Model

The two-step binomial model is an extension of the one-step model. It allows for a more granular estimation of option values by introducing an additional intermediate time step. In this model, the asset’s price can move up, stay constant, or move down in two time steps.

The stock price tree for two-step binomial model is given below:

    graph LR
    S0[C0] --> SU[SU] 
    S0 --> SD[SD] 
    SU --> SUU[SUU]
    SU --> SUD[SUD or SDU] 
    SD --> SUD[SUD or SDU]
    SD --> SDD[SDD]
  

Step 1: First, we calculate the size and probability of upward and downward movements.

Step 2: Then, we calculate the stock price at each node.

Step 3: In this step, we begin to value the option by starting at the end of the tree and working backwards. This process is referred to as backward induction (or rolling back through the tree).

American Options

American options can be exercised at any time before expiration. To value American options, we need to calculate below at each node:

• Option’s worth if it is exercised at the node
• Option’ worth if it is not exercised

The value at the node is the greater of these two.