The Cox-Ross-Rubinstein (CRR) option pricing model is a widely used binomial method for valuing options. Developed by John Cox, Stephen Ross, and Mark Rubinstein in 1979, the CRR model provides a discrete-time framework that approximates the continuous Black-Scholes model. This method is particularly valuable for pricing American-style options, which can be exercised before expiration. The binomial approach offers a flexible and intuitive way of modeling the evolution of asset prices over time, making it one of the most practical tools for traders and financial analysts.
Overview of the Binomial Model
The CRR model relies on a binomial tree to model potential movements in the price of an underlying asset. At each time step, the asset price can either move up or down by a specific factor. This approach provides a simple yet effective way to capture the stochastic nature of asset prices.
The key assumptions of the CRR model include:
- The underlying asset follows a multiplicative binomial process.
- The risk-free rate remains constant throughout the option’s life.
- No arbitrage opportunities exist in the market.
- The option can be exercised at discrete points in time.
The model uses the following variables:
- S: Current price of the underlying asset.
- K: Strike price of the option.
- T: Time to maturity.
- r: Risk-free interest rate.
- σ: Volatility of the underlying asset.
- n: Number of time steps in the binomial tree.
The Binomial Tree Structure
The CRR model builds a binomial tree where each node represents a potential price of the underlying asset at a given point in time. The tree is constructed using the following formulas for the up and down movements:
- Upward movement factor: u=eσΔtu = e^{\sigma\sqrt{\Delta t}}
- Downward movement factor: d=1ud = \frac{1}{u}
- Risk-neutral probability: p=erΔt−du−dp = \frac{e^{r\Delta t} – d}{u – d}
Where:
- Δt=Tn\Delta t = \frac{T}{n} represents the time step.
- uu and dd define the possible movements in asset prices.
- pp is the probability of an upward move under the risk-neutral measure.
At each time step, the price of the asset evolves according to: Si+1,j=Si,j⋅uS_{i+1, j} = S_{i,j} \cdot u Si+1,j+1=Si,j⋅dS_{i+1, j+1} = S_{i,j} \cdot d
Option Valuation Using Backward Induction
Once the binomial tree is constructed, option values are determined through backward induction. The process involves the following steps:
- Calculate terminal payoffs: At the final time step TT, the option’s value is calculated based on its intrinsic value.
- For a call option: Cn=max(Sn−K,0)C_n = \max(S_n – K, 0)
- For a put option: Pn=max(K−Sn,0)P_n = \max(K – S_n, 0)
- Work backwards through the tree: At each earlier node, the option value is computed as the discounted expected value of the option at the next time step: Vi=e−rΔt[pVi+1,j+(1−p)Vi+1,j+1]V_i = e^{-r\Delta t} [ pV_{i+1, j} + (1 – p)V_{i+1, j+1} ] where ViV_i represents the option value at each node.
- Early exercise for American options: If the option is American-style, at each node, we compare the computed value with the intrinsic value and take the maximum: Vi=max(Vi,Si−K)V_i = \max(V_i, S_i – K) for calls and Vi=max(Vi,K−Si)V_i = \max(V_i, K – S_i) for puts.
Advantages of the CRR Model
- Flexibility: Unlike the Black-Scholes model, which assumes continuous trading, the CRR model can accommodate different time steps and changing market conditions.
- American Option Pricing: The binomial model allows for the valuation of American options, which can be exercised early.
- Transparency and Intuition: The step-by-step construction of the tree makes it easy to understand and implement.
- Incorporation of Various Factors: The model can be extended to include dividends, varying interest rates, and stochastic volatility.
Limitations of the CRR Model
- Computational Complexity: As the number of time steps increases, the number of calculations grows exponentially, making the model computationally expensive for large nn.
- Assumption of Constant Volatility: The model assumes that volatility remains constant, which may not hold in real markets.
- Risk-Neutral Assumption: The reliance on risk-neutral valuation might oversimplify market behavior.
Comparison with the Black-Scholes Model
While the Black-Scholes model provides a closed-form solution for European options, the CRR model uses a discrete-time approach. The key differences between the two models include:
- Applicability: The CRR model is suitable for both European and American options, whereas the Black-Scholes model applies only to European options.
- Computational Approach: The CRR model uses a step-by-step binomial tree, while Black-Scholes relies on a continuous differential equation.
- Flexibility: The binomial model allows for variations in volatility, dividends, and exercise policies, making it more adaptable.
Practical Implementation
To implement the CRR model in practice, financial analysts often use programming languages like Python, MATLAB, or R. Below is a Python-based implementation:
import numpy as np
def binomial_option_pricing(S, K, T, r, sigma, n, option_type="call", american=False):
dt = T / n
u = np.exp(sigma * np.sqrt(dt))
d = 1 / u
p = (np.exp(r * dt) - d) / (u - d)
# Price tree
prices = np.zeros((n+1, n+1))
for i in range(n+1):
for j in range(i+1):
prices[j, i] = S * (u ** (i-j)) * (d ** j)
# Option values at maturity
values = np.zeros((n+1, n+1))
for j in range(n+1):
if option_type == "call":
values[j, n] = max(0, prices[j, n] - K)
else:
values[j, n] = max(0, K - prices[j, n])
# Backward induction
for i in range(n-1, -1, -1):
for j in range(i+1):
values[j, i] = np.exp(-r * dt) * (p * values[j, i+1] + (1-p) * values[j+1, i+1])
if american:
if option_type == "call":
values[j, i] = max(values[j, i], prices[j, i] - K)
else:
values[j, i] = max(values[j, i], K - prices[j, i])
return values[0, 0]
# Example usage
S, K, T, r, sigma, n = 100, 100, 1, 0.05, 0.2, 100
print("European Call Option Price:", binomial_option_pricing(S, K, T, r, sigma, n, "call"))
print("American Put Option Price:", binomial_option_pricing(S, K, T, r, sigma, n, "put", american=True))
Conclusion
The Cox-Ross-Rubinstein binomial model is a powerful and flexible method for option pricing. Its ability to handle American options, incorporate dividends, and adapt to changing market conditions makes it an essential tool for financial professionals. Despite its computational intensity, advances in computing have made the CRR model an indispensable part of modern option pricing methodologies.
