Options pricing refers to the process of determining the fair value of options contracts (calls and puts). An option is a financial derivative that gives the holder the right, but not the obligation, to buy or sell an underlying asset at a specific price (strike price) on or before a specific expiration date.
Options can be complicated to price due to various factors such as the asset’s price, volatility, time to expiration, and the risk-free rate. To determine the fair price, several mathematical models are used, the most well-known being Black-Scholes and Binomial Trees.
🔹 1. Black-Scholes Model
The Black-Scholes model is one of the most widely used models for pricing European-style options (options that can only be exercised at expiration). It calculates the theoretical price of an option based on several key factors.
Key Factors in Black-Scholes:
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S (Stock Price): The current price of the underlying asset.
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K (Strike Price): The price at which the option holder can buy (call) or sell (put) the asset.
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T (Time to Expiration): The time remaining until the option expires, typically expressed in years.
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σ (Volatility): The standard deviation of the asset’s returns, reflecting the level of risk or uncertainty.
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r (Risk-Free Rate): The return on a risk-free asset, like a government bond.
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C (Call Option Price) and P (Put Option Price): The prices of call and put options.
Black-Scholes Formula:
For a call option, the formula is: C=S0N(d1)−Ke−rTN(d2)C = S_0 N(d_1) - K e^{-rT} N(d_2)
For a put option, the formula is: P=Ke−rTN(−d2)−S0N(−d1)P = K e^{-rT} N(-d_2) - S_0 N(-d_1)
Where:
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d1=ln(S0/K)+(r+σ22)TσTd_1 = \frac{\ln(S_0 / K) + (r + \frac{σ^2}{2}) T}{σ \sqrt{T}}
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d2=d1−σTd_2 = d_1 - σ \sqrt{T}
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N(x) is the cumulative standard normal distribution.
Limitations:
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The Black-Scholes model assumes constant volatility and risk-free rate, which is rarely the case in real markets.
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It can only be used for European options (which can only be exercised at expiration).
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It assumes no transaction costs or taxes.
🔹 2. Binomial Trees Model
The Binomial Trees model is a more flexible option pricing model, allowing for the calculation of option prices through a discrete-time framework. It can handle American options, which can be exercised before expiration, unlike European options.
Key Features of the Binomial Tree Model:
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Discretization: The model works by dividing the time to expiration into multiple small intervals.
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Up and Down Movements: At each interval, the asset price either increases or decreases by a fixed proportion (called the "up factor" and "down factor").
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Risk-Neutral Probability: The model assumes that the probability of an upward or downward movement is adjusted by the risk-free rate.
Steps in Binomial Model:
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Construct a binomial tree: The underlying asset’s price changes in discrete steps. The tree consists of nodes, where each node represents a possible price of the asset at a given point in time.
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Calculate option value at the end: At the terminal nodes (the last point in the tree), the option's value is calculated based on the payoff, which is either max(S−K,0)\max(S-K, 0) for a call option or max(K−S,0)\max(K-S, 0) for a put option.
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Work backward: Calculate the option value at each node by working backwards from the terminal nodes, using the risk-neutral probability to discount the future payoffs.
The formula for a call option is: C=e−rΔt(pCu+(1−p)Cd)C = e^{-rΔt} \left(pC_u + (1-p)C_d\right) Where:
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C_u and C_d are the values of the option in the up and down states, respectively.
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p is the risk-neutral probability of an upward move.
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Δt is the time step (i.e., the duration of each interval).
Advantages of the Binomial Model:
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Can handle American options (which can be exercised anytime before expiration).
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More flexible, as it allows for varying volatility, interest rates, and dividends over time.
Limitations:
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More computationally intensive than Black-Scholes, especially for long-term options or those with many time intervals.
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Requires more inputs to model, which can increase the complexity.
🔹 Comparison Between Black-Scholes and Binomial Trees
| Aspect | Black-Scholes | Binomial Trees |
|---|---|---|
| Type of Option | European Options | European and American Options |
| Pricing Method | Continuous (closed-form solution) | Discrete (step-by-step approach) |
| Time to Expiration | Assumes a fixed time to expiration | Can handle varying expiration times |
| Volatility | Assumed to be constant | Can handle changing volatility |
| Risk-Free Rate | Assumed to be constant | Can handle changing rates |
| Computational Complexity | Relatively simple (closed-form formula) | More complex (requires multiple steps) |
| Accuracy | Less flexible (limited to constant inputs) | More flexible (can account for varying factors) |
🔚 Summary:
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Black-Scholes is a closed-form solution used primarily for European options with constant volatility and interest rates.
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Binomial Trees offer a discrete, flexible method that can handle American options and varying inputs over time.
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Both models are used to calculate the theoretical price of options, but the choice of model depends on the type of option and the complexity of the factors involved.