Introduction:
An option gives the holder the right, but not the obligation, to buy or sell a specific quantity of a particular asset at a fixed price on or before a specified future date. The option is purchased by the holder and sold by the writer.
A call option gives the holder the right, but not the obligation, to buy the underlying item at a specified price. For instance, a call option on shares in ABC plc could give its holder the right to purchase 2,000 shares in ABC plc at a fixed price on or before a specified expiry date for that option.
A put option gives the holder the right, but not the obligation, to sell an underlying item at a specified price. For instance, a put option on shares in XYZ plc may give its holder the right to sell 1,000 shares in XYZ plc at a fixed price on or before a specified expiry date for the option.
An option is considered in-the-money when the exercise price of the option is more favorable for the holder than the current market price. It goes without saying that an option will only be exercised if it is in-the-money.
The price of a call option is determined by several major factors:
The factors that affect options pricing include:
- The price of the underlying security and the exercise price. Essentially, it is the difference between the strike price and the underlying market price that matters.
- The time to go to expiry. The longer the remaining period to expiry, the greater the probability that the underlying instrument will rise in value.
- Interest rates.
- Type of option. Whether it is a European option or an American option?
The Black-Scholes model:
The Black-Scholes model was introduced by Myron Scholes and Fisher Black in 1973 as a mathematical model for pricing stock options. The model has also been adapted for European-style currency options. It assumes that exchange rate changes are log-normally distributed (Anthony, 1989).
The option price can be calculated using the following formula:
Option Price = PsN(d1) – Xe(-rt)N(d2)
Where:
- P is the current price of the underlying asset
- s is the standard deviation of the asset’s returns
- N() is the cumulative distribution function of a standard normal distribution
- X is the exercise price of the option
- r is the risk-free interest rate
- t is time to maturity in years
The variables d1 and d2 are given by:
d1 = [ln(P/X) + (r + s^2/2)t] / (s sqrt(t))
d2 = d1 – s sqrt(t)
The equation is as follows: d1 = ln(PsX) + rT + 0.5ơ√T
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The formula for calculating d2 is: d2 = d1 – ơ √T
To calculate the option price, we must first determine the values of d1 and d2. To calculate d2, we need to know the value of d1. The formula for calculating the value of d1 includes ln(Ps/X), which represents the natural logarithm of the current market price (Ps) divided by the exercise price (X).
Ơ represents the standard deviation of returns for the underlying instrument, expressed as a percentage. Once values for d1 and d2 have been calculated, the final step is to determine the option price.
The second part of the call option pricing formula is Xe(-rt)N(d2). The value of N(d2) is calculated similarly to N(d1). Xe(-rt) is the exercise price for the call option multiplied by the constant value (e) raised to the power (-rT). The purpose of this adjustment is to reduce the exercise price X to its present value.
The basic form of the Black-Scholes model has been explained above. Traders widely apply it in option markets to estimate European option values. However, the five factors of the Black Scholes model, in its basic form, have several limitations.
Firstly, this assumption does not take into consideration the time value of dividends since it assumes that no dividend payments are paid on the underlying instrument. Additionally, this model only applies to European options as American options can be exercised before the exercise price.
Secondly, it is assumed that the risk-free interest rate is known and remains constant throughout the option’s life. However, a decrease in the risk-free rate will lower the value of the call option since the money saved by buying a call option instead of purchasing an underlying security decreases as well. If an option is purchased, cash saved can be invested at the risk-free rate. A decline in this rate makes buying a call option less appealing and lowers its price.
The accurate estimation of the standard deviation of returns for the underlying security is crucial, as it cannot be assumed to remain constant throughout the option’s lifespan. It should be noted that the standard deviation will vary depending on the period over which it is calculated. Additionally, it is important to recognize that this model is highly sensitive to changes in the value of standard deviation. Lastly, it assumes that there are no transaction costs or tax effects associated with buying or selling options in relation to the underlying security.
Option pricing theory has advanced significantly since 1972, when Black and Scholes published their groundbreaking paper on a model for valuing dividend-protected European options. The duo employed a replicating portfolio” consisting of the underlying asset and the risk-free asset with identical cash flows to the option being valued (Damodaran). Although their derivation is mathematically complex, there is a more straightforward binomial model for valuing options that employs the same reasoning.
The binomial and Black-Scholes models are option valuation models. The Binomial model involves determining the value of options using a tree-like format. The value of the option is determined by the expiration time period of the option and volatility. On the other hand, for the Black-Scholes model, the value of options is determined by obtaining a derivative that helps to get discount rates for options.
The binomial pricing model is a mathematical model used in finance to determine the value of an option. It involves calculating the probability of different possible outcomes and assigning a value to each outcome. The model is based on the assumption that the price of an underlying asset will either go up or down over time, and that these changes are independent of each other.
The binomial pricing model was introduced by Ross, Cox, and Rubinstein in 1979. It provides a numerical method for valuing options. The model is based on a simple formulation for the asset price process in which the asset can move to one of two possible prices during any time period.
This model breaks down the option into many potential outcomes during the time period of the option. These steps form a tree-like format where the model assumes that the value of the option will either rise or go down. This value is calculated and determined by the expiration time and volatility. Finally, at the end of the option tree, its final possible value is determined because it’s equal to its intrinsic value.
In a multi-period binomial process, valuation must proceed iteratively. This means starting with the last time period and moving backwards in time until the current point. At each step, portfolios replicating the option are created and valued to provide values for that time period. The final output from the binomial option pricing model is a statement of the option’s value in terms of the replicating portfolio, composed of shares (option delta) of the underlying asset and risk-free borrowing/lending. Option delta is a measure of how much an option price changes given a unit change in security price.
One of the strengths of the binomial model is that it provides insight into the determinants of option value. The value of an option is not determined by the expected price of the asset but rather by its current price, which is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position that requires no investment, involves no risk, and delivers positive returns.
For example, if the portfolio that replicates a call costs more than what it does in the market, an investor could buy a call and sell the replicating portfolio to guarantee profit from their difference. The cash flows on these two positions will offset each other leading to no cash flows in subsequent periods.
The option value also increases as time to expiration extends, as price movement increases and with interest rate increases.
Although the binomial model provides an intuitive understanding of the factors that determine option value, it demands a large number of inputs concerning expected future prices at each node. As we shorten time periods in the binomial model, we can make one of two assumptions about asset prices. We can assume that price changes become smaller as periods get shorter. This results in price changes becoming small as time periods approach zero, leading to a continuous price process.
Conclusion:
An option is an asset that has payoffs contingent on the value of an underlying asset. A call option provides the holder with the right to buy the underlying asset at a fixed price, while a put option provides the holder with the right to sell at a fixed price before expiration. The value of an option is determined by six variables: the current value of the underlying asset, variance in this value, strike price, life of the option, riskless interest rate, and expected dividends on the asset. This is illustrated in both Binomial and Black-Scholes models which value options by creating replicating portfolios composed of underlying assets and riskless lending or borrowing. These models can be used to evaluate assets that have option-like characteristics.
The binomial method is very useful when the only variable of interest is the price today, and nothing significant happens in intermediate dates such as dividends, splits, changes in volatility or interest rates, or early exercise. (Benninga and Wiener, 1997).
However, the Black-Scholes model is widely used by traders in option markets to estimate European option values. The model is more suitable for calculating the value of European options because it assumes that no dividend payments are made on the underlying instrument. Therefore, it does not take into consideration the time value of dividends and cannot be applied to American options as these can be exercised before the exercise price.
References:
Damodaran, A. (2001). Choosing the Right Valuation Model. Working Paper.
S. Beninga and Z. Weiner (1997) introduced the Binomial Option Pricing Model in Mathematica.
Education and Research Journal, Volume 6, Issue 4, Pages 11-14
Ross, Cox, and Rubinstein (1979) presented a simplified approach to the option pricing model.
Financial Economics Journal, Issue 7, Pages 229-263.
Hull, J. (1997). Options, Futures, and Other Derivatives. Prentice Hall Publishers: New York.
