# Binomial and Black and Scholes Pricing models

Introduction:

An option gives its holder the right (but not the obligation) to buy or sell a specific quantity of a specific asset at a fixed price on or before a specified future date. An option is purchased by the option holder and is sold by the option writer.

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A call option gives its holder the right but not the obligation, to buy the underlying item at the specified price. For example, a call option on share in ABC plc might give its holder the right to buy 2,000 shares in ABC plc, at a fixed price on or before a specified expiry date for the option.

A put option gives its holder the right, but not the obligation, to sell the underlying item at a specified price. For example, a put option on shres in XYZ plc might 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 would be in-the-money if the exercise price of an option is more favorable for the option holder than the current market price. Obviously, an option would only be exercised if it is in the money.

The major factors in determining the price of a call option are as follows:

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 an European option or an American option?

The Black- Scholes model:

The Black and Scholes model was introduced by Myron Scholes and Fisher Black in the year 1973 as a mathematical model for pricing stock options. This model has been adapted for European style currency options. This model assumes that exchange rate for exchange rate changes are lognormally distributed (Anthony, 1989)

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Option Price = PsN(d1) – Xe(-rt)N(d2)

Where d1 = ln(PsX)+ rT +0.5 + ơ √T

ơ √T

d2 = d1 – ơ √T

To calculate the option price we must first calculate the values of d1 and d2. To calculate d2 we need the value of d1. The formula of the value of d1 includes the item ln(PsX). This means the ‘natural log’ of Ps/X, which is the share’s current market price, divided by the exercise price.

Ơ is a measurement of the standard deviation of returns of the underlying instrument which will be given as a percentage figure. Having calculated values of d1 and d2, the final step is to work out the option price.

The second part of the call option pricing formula is Xe(-rt)N(d2). The value of N(d2) is calculated similar to N(d1). Xe(-rt) is the exercise price for the call option multiplied by the value of the constant (e) to the power (-rT). The purpose of this adjustment is to reduce the value of the exercise price X to the present value.

The basic form of the Black-Scholes model has been explained above. It is widely applied by traders in option markets to give an estimate of European option values. However, the five factors of the Black Scholes model, in its basic form, suffer from a number of limitations.

Firstly, it assumes that no dividend payments are paid on the underlying instrument, thus not taking into consideration the time value of dividends and it applies to European options only since American options can be exercised before the exercise price.

Secondly, it assumes that the risk free interest rate in known and is constant throughout the option’s life. Although a reduction in the risk free rate will reduce the value of the call option because the money saved by purchasing the call option rather than the underlying security is reduced. If an option is purchased the cash saved could be invested at the risk free rate. A reduction in the risk free rate makes purchasing the call option relatively unattractive and reduces the option price.

The standard deviation of return of the underlying security has to be accurately estimated apart from it being constant throughout the option’s life. However, standard deviation will vary depending on the period over which it is calculated. The model is extremely sensitive to the value of standard deviation. Finally, it assumes that no transaction costs or tax effects involving buying or selling the option in the underlying security.

Option pricing theory has made vast strides since 1972, when Black and Scholes published their path-breaking paper providing a model for valuing dividend-protected European options. Black and Scholes used a “replicating portfolio” –– a portfolio composed of the underlying asset and the risk-free asset that had the same cash flows as the option being valued (Damodaran). While their derivation is mathematically complicated, there is a simpler binomial model for valuing options that draws on the same logic.

The binomial and the Black and Schole models are option valuing models, the Binomial model involves determining the value of options using a tree like format whereby the value of the option is determined by the expiration time period of the option and volatility, for the Black and Schole model the value of options is determined by simply getting a derivative that helps get the discount rates of options.

Binomial pricing model:

The binomial pricing model was introduced by Ross, Cox and Rubinstein in 1979; it provides a numerical method, in which valuation of options can be undertaken. The binomial option pricing model is based upon a simple formulation for the asset price process in which the asset, in any time period, can move to one of two possible prices.

This model breaks down the option into many potential outcomes during the time period of the option, this steps form a tree like format where by the model assumes that the value of the option will rise or go down, this value is calculated and it is determined by the expiration time and volatility. Finally at the end of the tree of the option the final possible value is determined because the value is equal to the intrinsic value.

In a multi-period binomial process, the valuation has to proceed iteratively, i.e. starting with the last time period and moving backwards in time until the current point in time. The portfolios replicating the option are created at each step and valued, providing the values for the option in that time period. The final output from the binomial option pricing model is a statement of the value of the option in terms of the replicating portfolio, composed of shares (option delta) of the underlying asset and risk-free borrowing/lending. (An option delta is a measure of how much an option price changes given a unit change in the 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 by its current price. This is a direct consequence of arbitrage. If the option value deviates from the value of the replicating portfolio, investors can create an arbitrage position, i.e., one that requires no investment, involves no risk, and delivers positive returns. To illustrate, if the portfolio that replicates the call costs more than the call does in the market, an investor could buy the call, sell the replicating portfolio and guarantee the difference as a profit. The cash flows on the two positions will offset each other, leading to no cash flows in subsequent periods. The option value also increases as the time to expiration is extended, as the price movement increase, and with increases in the interest rate.

While the binomial model provides an intuitive feel for the determinants of option value, it requires a large number of inputs, in terms of expected future prices at each node. As we make time periods shorter 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 leads to price changes becoming small as time periods approach zero, leading to a continuous price process.

Conclusion:

An option is an asset with payoffs which are contingent on the value of an underlying asset. A call option provides its holder with the right to buy the underlying asset at a fixed price, whereas a put option provides its holder with the right to sell at a fixed price, any time before the expiration of the option. The value of an option is determined by six variables – the current value of the underlying asset, the variance in this value, the strike price, life of the option, the riskless interest rate and the expected dividends on the asset. This is illustrated in both the Binomial and the Black-Scholes models, which value options by creating replicating portfolios composed of the underlying asset and riskless lending or borrowing. These models can be used to value assets that have option-like characteristics.

Binomial method is very useful when the price today is the only variable of interest and nothing important happens in the intermediate dates (like dividends, splits, changes in volatility or interest rates, early exercise.) Benninga and Wiener (1997).

However, the Black-Scholes model is widely applied by traders in option markets to give an estimate of European option values since the model is more suitable for calculating the value of European option as it assumes that no dividend payments are paid on the underlying instrument, thus not taking 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,The Binomial Option Pricing Model, Mathematica in

Education and Research Journal, 6, Issue 4, page 11-14

Ross, Cox and Rubinstein (1979) “Option pricing model: A simplified approach”

Financial Economics Journal, issue 7, page 229 to 263

Hull J. (1997) Options, Future and Derivatives, Prentice hall publishers, New York