Binomial Model

14Dec 0

Last Updated on 2024-12-14 by Admin

The Binomial Option Pricing Method is a widely used and flexible approach to value options, including both call and put options. It is particularly useful because it can handle a variety of scenarios, such as American options (which can be exercised before expiration), dividends, and other features that more sophisticated models like Black-Scholes may not address. The binomial model approximates the price of an option by breaking down the time to expiration into multiple small intervals (steps), creating a binomial tree to represent all the possible price movements of the underlying asset during that time.

In this detailed explanation, we’ll break down the binomial method for both call options and put options and show how they are handled step-by-step.

Variables

To understand how the binomial method works for both call and put options, we first need to define the key parameters used in the model:

S_₀: Initial price of the underlying asset (e.g., stock).
K : Strike price of the option.
T : Time to expiration (typically in years).
r : Risk-free interest rate (annualised, compounded continuously).
σ : Volatility of the underlying asset (annualised).
N : Number of time steps (periods) until the option expires.
u : Up factor, representing the percentage increase in the asset price in each step.
d : Down factor, representing the percentage decrease in the asset price in each step.
p : Risk-neutral probability of the price moving up in each period.
Call Option (C) : The value of the call option.
Put Option (P) : The value of the put option.

Method

Step 1: Define Parameters and Build the Binomial Tree

In the binomial model, the price of the underlying asset can either move up or down in each period. To determine the up and down factors, we use the following formulas:

Up Factor (u) =

$$𝑒^{𝜎\sqrt{Δ𝑡}}$$

where Δt is the length of each period typically;

$$Δt=\frac{T}{N}$$

$Down\;Factor\;(d) =\frac{1}{𝑢}$

(since the down factor is the inverse of the up factor).

Next, we calculate the risk-neutral probability (p) that the price will go up in a given period:

$$p=\frac{𝑒^{𝑟Δ𝑡}−𝑑}{𝑢−𝑑}$$

Where r is the risk-free interest rate, and Δt is the length of each time period.

The risk-neutral probability is used to price the option as if the expected return of the underlying asset were the risk-free rate.

Step 2: Build the Binomial Tree for Asset Price Evolution

At each step, the price of the asset can either go up by a factor u or down by a factor d. Starting from the initial price S_₀, we create a tree of possible prices. For example:

At time 0 (the initial node), the price is S_₀.
After the first time step (period 1), the price can be:
- S_₀ * u (if the price goes up), or
- S_₀ * d (if the price goes down).
At time 2, the price can be:
- S_₀ * u² (if the price goes up twice),
- S_₀ * ud (if the price goes up once and down once),
- S_₀ * d² (if the price goes down twice).

This tree structure continues for N steps, generating all possible future asset prices.

Step 3: Calculate Option Payoffs at Expiration (Terminal Nodes)

At expiration (the final time step N), the payoff of the option depends on whether it is a call or a put:

Call Option Payoff at time T:

$$C_T=max(S_T−K,0)$$

where S_T is the stock price at expiration and K is the strike price.

- If S_T > K, the call option is in the money and the payoff is the difference $S_T – K$
  
  .
- If S_T ≤ K, the call option is out of the money and the payoff is 0.

Put Option Payoff at time T:

$$P_T=max(K−S_T,0)$$

- If S_T < K, the put option is in the money and the payoff is the difference $K – S_T$
  
  .
- If S_T ≥ K, the put option is out of the money and the payoff is 0.

Step 4: Work Backwards to Calculate the Option Value at Earlier Nodes

After calculating the payoffs at the terminal nodes (expiration), we move backwards through the tree to calculate the option’s value at each earlier node. At each node, the value of the option is the discounted expected value of the option at the next time step, considering the probabilities of the price moving up or down:

For each node at time t: $C_t = e^{-r \Delta t} \left[ p \cdot C_{\text{up}} + (1 – p) \cdot C_{\text{down}} \right]$

where:
- C_t is the option value at time t,
- C_up is the option value at the next node (if the price moves up),
- C_down is the option value at the next node (if the price moves down),
- p is the risk-neutral probability of the price moving up,
- e^{-r \Delta t} is the discount factor to account for the time value of money.

This process is repeated at each node in the tree, moving backward in time until we reach the initial time step (time 0). The value at time 0 represents the option’s fair price.

Example 1: Call Option

Let’s go through an example of calculating a call option price using the binomial model.

Given Parameters:

S_₀ = 100 (initial stock price),
K = 105 (strike price),
T = 1 year (time to expiration),
r = 5% (risk-free interest rate),
σ = 20% (volatility),
N = 2 time steps.

Step 1: Calculate u, d, and p.

Δt = $\frac{T}{N} = \frac{1}{2}$

years.
u = $e^{\sigma \sqrt{\Delta t}} = e^{0.20 \times \sqrt{0.5}} ≈ 1.151$

,
d = $\frac{1}{u} ≈ 0.869$

,
p = $\frac{e^{r \Delta t} – d}{u – d} = \frac{e^{0.05 \times 0.5} – 0.869}{1.151 – 0.869} ≈ 0.577$

.

Step 2: Construct the Binomial Tree.

Starting with S_₀ = 100:

After one period, the price can be:
- S_₀ * u = 100 * 1.151 = 115.1 (up),
- S_₀ * d = 100 * 0.869 = 86.9 (down).
After two periods, the price can be:
- S_₀ * u² = 100 * 1.151² = 132.3 (up-up),
- S_₀ * ud = 100 * 1.151 * 0.869 = 100 (up-down or down-up),
- S_₀ * d² = 100 * 0.869² = 75.5 (down-down).

Step 3: Calculate Payoffs at Expiration.

At expiration (T = 1 year):

For S = 132.3: Payoff = max(132.3 – 105, 0) = 27.3,
For S = 100: Payoff = max(100 – 105, 0) = 0,
For S = 75.5: Payoff = max(75.5 – 105, 0) = 0.

Step 4: Work Backwards.

Now, calculate the option’s value at each earlier node.

At t = 0.5 (after the first period):

For S = 115.1: $C_{\text{up}} = e^{-0.05 \times 0.5} [0.577 \times 27.3 + (1 – 0.577) \times 0] ≈ 13.4$

,
For S = 86.9: $C_{\text{down}} = e^{-0.05 \times 0.5} [0.577 \times 0 + (1 – 0.577) \times 0] = 0$

.

At t = 0 (the initial time):

$C_0 = e^{-0.05 \times 0.5} [0.577 \times 13.4 + (1 – 0.577) \times 0] ≈ 7.5$

.

So, the value of the call option today is $7.50.

Example 2: Put Option

Using the same parameters as in the previous example, let’s calculate the price of a put option.

Step 1: Calculate u, d, and p.

These parameters are the same as before:

u ≈ 1.151,
d ≈ 0.869,
p ≈ 0.577.

Step 2: Construct the Binomial Tree.

The same binomial tree applies for the asset price evolution as we did for the call option.

Step 3: Calculate Payoffs at Expiration.

For the put option, the payoff is calculated as;

$P_T = \max(K – S_T, 0)$

For S = 132.3: Payoff = max(105 – 132.3, 0) = 0,
For S = 100: Payoff = max(105 – 100, 0) = 5,
For S = 75.5: Payoff = max(105 – 75.5, 0) = 29.5.

Step 4: Work Backwards.

At t = 0.5 (after the first period):

For S = 115.1: $P_{\text{up}} = e^{-0.05 \times 0.5} [0.577 \times 0 + (1 – 0.577) \times 5] ≈ 2.1$

,
For S = 86.9: $P_{\text{down}} = e^{-0.05 \times 0.5} [0.577 \times 29.5 + (1 – 0.577) \times 0] ≈ 13.6$

.

At t = 0 (the initial time):

$P_0 = e^{-0.05 \times 0.5} [0.577 \times 2.1 + (1 – 0.577) \times 13.6] ≈ 7.5$

.

So, the value of the put option today is $7.50.

Conclusion

The Binomial Option Pricing Method is a powerful and flexible model for pricing both call and put options, especially when dealing with American-style options or options with features not easily modeled by the Black-Scholes formula. By discretizing time into small intervals and using a binomial tree to model the possible movements of the underlying asset, we can calculate the fair value of options by working backwards from expiration. The method is intuitive, but its accuracy improves with a larger number of time steps (N) and can accommodate a wide range of market conditions.

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