Binomial pricing models

What Is Binomial Pricing Models?

Binomial pricing models are a class of financial modeling techniques used to value financial derivatives, most commonly options. Belonging to the broader category of derivatives pricing models, these models simplify the potential price movements of an underlying asset into a discrete "binomial tree" where, at each step, the asset's price can only move up or down. This framework allows for the calculation of an option's fair value by working backward from expiration, incorporating the concept of risk-neutral probability to eliminate potential arbitrage opportunities.

History and Origin

The seminal work on the binomial pricing model was published in 1979 by John Cox, Stephen Ross, and Mark Rubinstein in their paper titled "Option pricing: A simplified approach".⁷ Their contribution, often referred to as the Cox-Ross-Rubinstein (CRR) model, provided a more intuitive and computationally accessible alternative to the complex partial differential equations used in the Black-Scholes model, which had been introduced a few years prior. The CRR model demonstrated that the complex valuation of options could be simplified into a sequence of binary choices, making the underlying economic intuition of option pricing more transparent and understandable.

Key Takeaways

Binomial pricing models value options by simulating discrete upward or downward movements in the underlying asset's price over time.
They construct a "binomial tree" of possible future prices, calculating the option's value at each node by working backward from expiration.
Unlike continuous-time models, binomial models are particularly well-suited for valuing American-style options, which can be exercised at any time before expiration.
The model assumes a risk-neutral world where expected returns are the risk-free rate, simplifying the valuation process.
Accuracy generally increases with the number of time steps used in the model, as it better approximates continuous price movements.

Formula and Calculation

The core of the binomial pricing model involves calculating the probability of an upward or downward movement in the underlying asset's price, and then using these probabilities to determine the option's value at each node of the binomial tree. The model works backward from the option's expiration date.

For a single-period model, the value of a call option at time t=0 (current time) can be calculated as:

$C_0 = e^{-r\Delta t} [p C_u + (1-p) C_d]$

Where:

(C_0) = Current value of the call option
(e) = The base of the natural logarithm (approximately 2.71828)
(r) = The risk-free interest rate (continuously compounded)
(\Delta t) = Length of one time step (e.g., time to expiration divided by number of steps)
(p) = Risk-neutral probability of an upward movement
(C_u) = Value of the call option if the underlying asset price moves up
(C_d) = Value of the call option if the underlying asset price moves down

The risk-neutral probability (p) is calculated as:
$p = \frac{e^{r\Delta t} - d}{u - d}$

And the up (u) and down (d) factors are calculated based on the volatility ((\sigma)) of the underlying asset:
$u = e^{\sigma \sqrt{\Delta t}}$
$d = e^{-\sigma \sqrt{\Delta t}} = \frac{1}{u}$

For a put option, the formula is similar, replacing (C_u) and (C_d) with (P_u) and (P_d), which are the values of the put option in the up and down states, respectively. The option value at expiration (final nodes) is simply its intrinsic value: for a call option, (\text{max}(0, S_T - K)), and for a put option, (\text{max}(0, K - S_T)), where (S_T) is the asset price at expiration and (K) is the exercise price.

Interpreting the Binomial Pricing Models

Interpreting the output of binomial pricing models involves understanding the concept of a "tree" of possible outcomes. Each node in the tree represents a potential price for the underlying asset at a specific point in time. The model's primary output is the fair present value of the option, derived by discounting the expected future payoffs back to today.

A key interpretation is that the binomial model, by working backward, implicitly considers the optimal exercise strategy for American options at each decision point. For a call option, if its intrinsic value (current stock price minus strike price) at an intermediate node is greater than its calculated value from holding it to the next step, the model assumes immediate exercise. This dynamic valuation feature is a significant advantage over models that only consider exercise at expiration, such as the basic Black-Scholes model for European options. The resulting option price represents the theoretical price at which no riskless profit can be made.

Hypothetical Example

Consider valuing a 6-month call option on a stock with a current price of $100 and an exercise price of $105. Assume a risk-free rate of 5% per annum and a stock volatility of 20% per annum. Let's use two time steps, each representing three months ((\Delta t) = 0.25 years).

Calculate up (u) and down (d) factors:
(u = e^{0.20 \sqrt{0.25}} = e^{0.20 \times 0.5} = e^{0.1} \approx 1.1052)
(d = e^{-0.1} \approx 0.9048)
Calculate risk-neutral probability (p):
(p = \frac{e^{{0.05 \times 0.25} - 0.9048}{1.1052 - 0.9048} = \frac{e}{0.0125} - 0.9048}{0.2004} = \frac{1.01257 - 0.9048}{0.2004} \approx 0.5377)
(1-p \approx 0.4623)
Construct the binomial tree for the stock price:
- Initial: $100
- After 3 months:
  - Up (Su): $100 * 1.1052 = $110.52
  - Down (Sd): $100 * 0.9048 = $90.48
- After 6 months:
  - Up-Up (Suu): $110.52 * 1.1052 = $122.10
  - Up-Down (Sud): $110.52 * 0.9048 = $100.00
  - Down-Up (Sdu): $90.48 * 1.1052 = $100.00
  - Down-Down (Sdd): $90.48 * 0.9048 = $81.87
Calculate option values at expiration (t=6 months):
- Cuu = max(0, 122.10 - 105) = $17.10
- Cud = max(0, 100.00 - 105) = $0.00
- Cdu = max(0, 100.00 - 105) = $0.00
- Cdd = max(0, 81.87 - 105) = $0.00
Work backward to calculate option values at t=3 months:
- Option value at Up node (Cu):
  (C_u = e^{-0.05 \times 0.25} [0.5377 \times 17.10 + 0.4623 \times 0.00])
  (C_u = e^{-0.0125} [9.2007 + 0] \approx 0.98757 \times 9.2007 \approx $9.08)
  (For an American option, check early exercise: max(intrinsic value, calculated value) = max(110.52-105, 9.08) = max(5.52, 9.08) = 9.08)
- Option value at Down node (Cd):
  (C_d = e^{-0.05 \times 0.25} [0.5377 \times 0.00 + 0.4623 \times 0.00])
  (C_d = e^{-0.0125} = $0.00)
  (For an American option, check early exercise: max(intrinsic value, calculated value) = max(90.48-105, 0) = max(-14.52, 0) = 0)
Work backward to calculate option value at t=0 (C0):
(C_0 = e^{-0.05 \times 0.25} [0.5377 \times 9.08 + 0.4623 \times 0.00])
(C_0 = e^{-0.0125} [4.8829] \approx 0.98757 \times 4.8829 \approx $4.82)

The theoretical price of the call option today, using this two-step binomial model, is approximately $4.82. This example illustrates the step-by-step process, taking into account the time value of money by discounting back through each period.

Practical Applications

Binomial pricing models are widely used in financial markets, particularly for complex derivatives and scenarios where continuous-time models fall short. A primary application is the valuation of American-style options, which grant the holder the right to exercise at any time before expiration. The step-by-step nature of the binomial tree naturally accommodates decisions about early exercise, a feature not readily handled by the basic Black-Scholes formula.

Beyond vanilla call options and put options, binomial models are adaptable for pricing more complex instruments, including exotic options with path-dependent payoffs, such as barrier options or Bermudan options. They can also be extended to value real options within corporate finance, where strategic business decisions (e.g., the option to expand, abandon, or defer a project) are treated as options on an underlying asset (the project itself). Researchers have explored applying generalized binomial models to contexts where "real-world" probabilities, rather than strict risk-neutral probabilities, are more intuitive for practitioners, such as in valuing corporate bonds or inferring probabilities of success in real-option projects.⁶

Limitations and Criticisms

Despite their intuitive appeal and flexibility, binomial pricing models have several limitations. One significant critique is that the model assumes discrete price movements, meaning the underlying asset can only move to one of two predefined prices in each time step.⁵ In reality, asset prices move continuously. While increasing the number of time steps enhances accuracy by better approximating continuous movement and can even converge to the Black-Scholes result, it also increases computational intensity, especially for long-dated options or those with many steps.⁴,³

Another limitation is the assumption of constant volatility over the life of the option.² In practice, market volatility is rarely constant and tends to fluctuate over time, which can lead to inaccuracies in the model's output. Furthermore, like many theoretical models, the binomial pricing model typically assumes frictionless markets, meaning no transaction costs or taxes, and the ability to borrow and lend at a single risk-free rate. These assumptions do not perfectly reflect real-world market conditions.

Binomial Pricing Models vs. Black-Scholes Model

The binomial pricing models and the Black-Scholes model are two foundational approaches to option pricing, but they differ in their underlying methodology and applicability. The Black-Scholes model, developed in the early 1970s, is a continuous-time model that provides a closed-form analytical solution for valuing European options. It assumes that the price of the underlying asset follows a geometric Brownian motion and that trading is continuous.

In contrast, the binomial pricing models (specifically the Cox-Ross-Rubinstein model) are discrete-time models that approximate asset price movements through a series of upward or downward steps, forming a "tree" structure. This discrete approach makes them highly flexible for valuing options that can be exercised at multiple points in time, such as American options, which the standard Black-Scholes model cannot directly handle. While the Black-Scholes model is computationally faster for simple European options, the binomial model's step-by-step nature allows for easier adaptation to exotic options with complex features like early exercise provisions or path-dependent payoffs. Theoretically, as the number of time steps in a binomial model approaches infinity, its results converge to those of the Black-Scholes model.¹

FAQs

What type of options are best priced with a binomial model?

Binomial pricing models are particularly well-suited for valuing American-style options because their discrete-time framework allows for checking the optimal decision to exercise the option at each step before its expiration. This flexibility also makes them useful for exotic options with path-dependent features.

How does the number of steps affect the accuracy of the binomial model?

Increasing the number of steps in a binomial pricing model generally improves its accuracy. With more steps, the discrete price movements better approximate the continuous movement of an underlying asset in the real world. However, this also increases the computational complexity.

Can the binomial model be used for assets other than stocks?

Yes, the binomial pricing model is versatile and can be adapted to value options on various underlying assets beyond stocks, including currencies, commodities, and even real assets in the context of financial modeling for strategic business decisions. The core principles of discounting expected future payoffs remain applicable.

What is the primary advantage of the binomial pricing model over Black-Scholes?

The primary advantage of the binomial pricing model is its ability to handle options that can be exercised at multiple points in time, such as American options. The step-by-step tree structure allows for the incorporation of early exercise decisions, which the basic Black-Scholes model, being a closed-form solution for European options, cannot directly account for.

Is the risk-neutral probability a real-world probability?

No, the risk-neutral probability used in binomial pricing models is a theoretical construct, not a reflection of actual probabilities in the real world. It is derived under the assumption that investors are indifferent to risk, and all assets yield the risk-free rate of return. This assumption simplifies the valuation process by eliminating the need to estimate actual risk preferences.