Cox ross rubinstein model

Cox-Ross-Rubinstein Model: Definition, Formula, Example, and FAQs

What Is the Cox-Ross-Rubinstein Model?

The Cox-Ross-Rubinstein (CRR) model is a discrete-time model used in financial derivatives pricing to value options. It is a foundational tool within quantitative finance, providing a simplified, step-by-step approach to understanding how option prices are determined. The Cox-Ross-Rubinstein model constructs a binomial tree to represent the potential movements of an underlying asset over a specified period. At each node of the tree, the asset's price can only move to one of two possible future prices: up or down. This probabilistic framework allows for the calculation of the option's value at each step, working backward from the expiration date to the present. The model's intuitive structure makes it particularly useful for demonstrating complex option pricing concepts, including the valuation of American options which can be exercised before maturity.

History and Origin

The Cox-Ross-Rubinstein model, often simply referred to as the binomial option pricing model, was introduced by John C. Cox, Stephen A. Ross, and Mark Rubinstein in their seminal 1979 paper, "Option Pricing: A Simplified Approach."⁸ This paper provided a more accessible and intuitive alternative to the continuously compounded models that preceded it, most notably the Black-Scholes model. The authors designed the Cox-Ross-Rubinstein model to be a discrete-time framework, which made the complex mechanics of option valuation easier to grasp for academics and practitioners alike. Its development was a significant step forward in financial modeling, offering a clear path to valuing not only European options, which can only be exercised at expiration, but also American options, which allow for early exercise.

Key Takeaways

• The Cox-Ross-Rubinstein model is a discrete-time framework for valuing options, simplifying price movements into up or down steps.
• It constructs a binomial tree to map out possible future prices of the underlying asset.
• The model calculates the option value by working backward from the expiration date, incorporating risk-neutral probability.
• It is particularly useful for pricing American options due to its ability to evaluate early exercise opportunities.
• As the number of time steps increases, the results of the Cox-Ross-Rubinstein model converge with those of continuous-time models like the Black-Scholes model.

Formula and Calculation

The Cox-Ross-Rubinstein model calculates the value of an option by constructing a binomial tree representing the possible price paths of the underlying asset. The key parameters for building the tree and calculating the option's value are the up (u) and down (d) factors, and the risk-neutral probability (p).

The up and down factors are typically derived from the volatility of the underlying asset and the time step duration ((\Delta t)):
$u = e^{\sigma \sqrt{\Delta t}}$
$d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u}$
where:

• (u) = up factor
• (d) = down factor
• (\sigma) = volatility of the underlying asset
• (\Delta t) = length of a single time step

The risk-neutral probability (p) of an upward movement is given by:
$p = \frac{e^{r \Delta t} - d}{u - d}$
where:

• (r) = risk-free interest rate

To value a call option or put option using the CRR model, one starts at the final nodes of the tree (at expiration) and calculates the option's intrinsic value at each node:
For a call option: (\max(S_T - K, 0))
For a put option: (\max(K - S_T, 0))
where:

• (S_T) = underlying asset price at expiration
• (K) = strike price

Then, working backward through the tree, the option's value at each earlier node is calculated as the discounted expected value of its future values:
$C_t = e^{-r \Delta t} [p C_{u} + (1-p) C_{d}]$
where:

• (C_t) = option value at time (t)
• (C_u) = option value if the underlying moves up
• (C_d) = option value if the underlying moves down

For American options, at each node during the backward calculation, the model compares the calculated discounted expected value to the immediate exercise value and chooses the greater of the two.

Interpreting the Cox-Ross-Rubinstein Model

The Cox-Ross-Rubinstein model provides an interpretable path-dependent valuation of options by illustrating how the option's value evolves with the underlying asset price. The construction of the binomial tree allows users to visualize all possible price states the asset could take, making the valuation process transparent. Each node in the tree represents a specific possible price for the asset at a given point in time value.

Interpreting the model involves understanding that the option's price at any node is derived from the expected future values of the option, discounted back to the present using the risk-free rate, under the assumption of a risk-neutral world. This framework ensures that there are no arbitrage opportunities. By varying the number of steps, users can see how the discrete model approaches the continuous-time behavior of real markets, providing a more refined price estimate. The ability to evaluate intermediate nodes is especially crucial for American-style options, where the decision to exercise early depends on the option's intrinsic value versus its remaining time value.

Hypothetical Example

Consider valuing a 1-year European call option on a non-dividend-paying stock with a current price of $100 and a strike price of $100. Assume the annual volatility ((\sigma)) is 20% and the risk-free rate ((r)) is 5%. We will use a two-step binomial tree, so (\Delta t = 0.5) years.

First, calculate the up and down factors:
$u = e^{0.20 \sqrt{0.5}} \approx e^{0.1414} \approx 1.152$
$d = e^{-0.20 \sqrt{0.5}} \approx e^{-0.1414} \approx 0.868$

Next, calculate the risk-neutral probability of an upward movement:
$p = \frac{e^{0.05 \times 0.5} - 0.868}{1.152 - 0.868} = \frac{e^{0.025} - 0.868}{0.284} = \frac{1.0253 - 0.868}{0.284} = \frac{0.1573}{0.284} \approx 0.5539$
Thus, (1-p \approx 0.4461).

Now, construct the binomial tree for the stock price:

• Initial Stock Price ((S_0)): $100
• After 1st step (0.5 years):
  • (S_{uu}) (up-up): (100 \times 1.152 \times 1.152 = 132.71)
  • (S_{ud}) (up-down, or (S_{du}) down-up): (100 \times 1.152 \times 0.868 = 100)
  • (S_{dd}) (down-down): (100 \times 0.868 \times 0.868 = 75.34)

Finally, calculate the option value at each node by working backward:
At expiration (1 year), the call option values are (\max(S_T - K, 0)):

• (C_{uu} = \max(132.71 - 100, 0) = 32.71)
• (C_{ud} = \max(100 - 100, 0) = 0)
• (C_{dd} = \max(75.34 - 100, 0) = 0)

Now, move back to the middle nodes (0.5 years):

• Value at up node ((C_u)): (e^{-0.05 \times 0.5} [0.5539 \times C_{uu} + 0.4461 \times C_{ud}])
  (C_u = e^{-0.025} [0.5539 \times 32.71 + 0.4461 \times 0] \approx 0.9753 \times [18.106 + 0] \approx 17.66)
• Value at down node ((C_d)): (e^{-0.05 \times 0.5} [0.5539 \times C_{ud} + 0.4461 \times C_{dd}])
  (C_d = e^{-0.025} [0.5539 \times 0 + 0.4461 \times 0] \approx 0)

Finally, the option value at time 0:

• (C_0 = e^{-0.05 \times 0.5} [0.5539 \times C_u + 0.4461 \times C_d])
  (C_0 = e^{-0.025} [0.5539 \times 17.66 + 0.4461 \times 0] \approx 0.9753 \times [9.784] \approx 9.54)

The estimated price of the European call option using the two-step Cox-Ross-Rubinstein model is approximately $9.54.

Practical Applications

The Cox-Ross-Rubinstein model is widely used in finance for its flexibility and intuitive approach to valuing derivative securities. Its primary application lies in option pricing, particularly for instruments where early exercise is a possibility, such as American options. Unlike some continuous-time models, the CRR framework naturally accommodates features like dividends, making it suitable for valuing options on dividend-paying stocks.

Beyond standard options, the model can be adapted to price more complex financial instruments with embedded options, such as convertible bonds or callable bonds, by modifying the payoff structure at each node. Financial professionals also use the Cox-Ross-Rubinstein model for hedging strategies, as it provides implied delta values at each node, which can inform dynamic portfolio adjustments. Investors interested in learning more about the basics of options and their associated risks can consult resources provided by regulatory bodies.⁷

Limitations and Criticisms

While the Cox-Ross-Rubinstein model offers a valuable and intuitive framework for option pricing, it is not without limitations. A primary criticism is its discrete-time nature, which assumes that the price of the underlying asset can only move in distinct "up" or "down" steps at specific intervals.⁶ In reality, asset prices fluctuate continuously. While increasing the number of steps in the binomial tree can approximate continuous movement more closely, this significantly increases computational complexity.

Another key assumption of the Cox-Ross-Rubinstein model is constant volatility and a constant risk-free rate over the life of the option.⁵ In practice, market volatility can change rapidly, and interest rates are subject to shifts. The model also assumes a perfectly efficient market with no transaction costs, taxes, or other frictions, which do not reflect real-world trading conditions.⁴ Furthermore, some studies have identified issues when the model is used to value options on assets paying continuous dividends, leading to potential inconsistencies in valuation.³ Despite these criticisms, the model remains a robust educational and practical tool, particularly for its ability to illuminate the mechanics of option valuation.

Cox-Ross-Rubinstein Model vs. Black-Scholes Model

The Cox-Ross-Rubinstein (CRR) model and the Black-Scholes model are both fundamental in derivative securities valuation, but they differ in their methodological approach.

The Black-Scholes model, developed in 1973, provides a closed-form analytical solution for pricing European options, assuming a continuous stock price path and constant volatility, among other assumptions.² It is highly efficient for European options due to its direct formula. Conversely, the Cox-Ross-Rubinstein model, while more computationally intensive for a large number of steps, offers a more flexible and transparent framework, especially for options that can be exercised before their expiration date. Over time, as the number of discrete steps in the Cox-Ross-Rubinstein model increases, its results will converge with those obtained from the Black-Scholes model, demonstrating their underlying theoretical connection.¹

FAQs

What types of options can the Cox-Ross-Rubinstein model value?

The Cox-Ross-Rubinstein model can value both European and American options. Its step-by-step, backward calculation from the expiration date makes it particularly well-suited for American options, as it can evaluate the optimal time for early exercise at each node of the binomial tree.

How does the Cox-Ross-Rubinstein model handle dividends?

The Cox-Ross-Rubinstein model can incorporate dividends by adjusting the underlying asset's price at the specific nodes where a dividend is paid or by reducing the stock price by the present value of expected future dividends. This flexibility allows for a more accurate valuation of options on dividend-paying stocks.

Is the Cox-Ross-Rubinstein model more accurate with more steps?

Yes, increasing the number of steps in the Cox-Ross-Rubinstein model generally leads to greater accuracy. As the number of time steps approaches infinity, the discrete binomial tree model converges to the continuous-time models like Black-Scholes, providing a more precise approximation of the option pricing. However, this also increases the computational effort required.

What is the concept of "risk-neutral probability" in the CRR model?

Risk-neutral probability is a theoretical concept used in the Cox-Ross-Rubinstein model (and other option pricing models) that allows for the valuation of options without needing to estimate investors' risk preferences. It assumes that expected returns for all assets, including the underlying stock and the option, are equal to the risk-free rate. This simplifies the calculation because future cash flows can be discounted at the risk-free rate.

Can the Cox-Ross-Rubinstein model be used for other financial instruments?

While primarily known for option pricing, the underlying principles of the Cox-Ross-Rubinstein model—modeling asset price movements as a series of discrete up or down steps—can be adapted to value other derivative securities or financial instruments with embedded options, where the payoff depends on future paths of an underlying asset.