Share Options and Option Pricing
1 Introduction
In this chapter we will explain what share options are, and explain the Black Scholes option pricing model which you can be expected to use to calculate the value of an option.
2 Share option
A share option gives the holder the right to buy or sell a share at a fixed price on a future date. A call option gives the holder the right to buy the share, whereas a put option gives the holder the right to sell a share.
An investor wanting an option will have to pay for it, whether or not they ultimately decide to exercise it.
The share price of Madrid plc is currently $2.00.
Johnson holds a call option with an exercise price of $1.80, exercisable in 3 months time.
What will Johnson do if the share price in 3 months time is:
$2.50
$1.50
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3 Option prices
As already stated, if an investor wishes to have an option to buy or sell shares, they will have to pay for it (whether or not they ultimately choose to exercise the option).
Several factors will determine the value of the option – the most obvious being the current share price and the exercise price.
The share price of Lisbon plc is currently $2.50.
A call option is available with an exercise price of $2.00, exercisable immediately.
What will be the value of the option?
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Although the last example should be very obvious, it is unrealistic in that options are not exercisable immediately but at some date in the future.
The full list of factors that will determine the price of an option is as follows:
current share price
the exercise price
the time to expiry of the option
the risk free interest rate
the volatility of the share price
Although option prices in practice are determined by the dealers, in line with market forces, Black and Scholes developed a formula for determining the value which is very commonly used in practice.
4 The Black Scholes Option Pricing Model
The formulae that Black and Scholes developed are as follows:
4.1 Call option:
Where
4.2 Put option:
p = c – Pa + Pee-rt
Where:
Pa = the current share price
Pe = the exercise price of the option
e = the exponential constant (2.7183)
r = the annual risk free rate of interest
t = the time (in years) until expiry of the option
S = the share price volatility
N(d) = the probability that a deviation of less than d will occur in a normal distribution. (You do not need to know this – you just need to know how to find the value using normal distribution tables).
Current share price is $2.90.
Exercise price $2.60 in 6 months time.
Risk free rate of interest is 6% p.a.
Standard deviation of rate of return on share is 40%
What is the value of a call option?
What is the value of a put option?
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Current share price is $1.50
Exercise price $1.80 in 3 months.
Risk free rate of interest is 10% p.a.
Standard deviation of rate of return on share is 40%
What is the value of a call option?
What is the value of a put option?
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Important:
Although it is important that you understand what the formula is doing, you can no longer be asked to calculate using the formula in the exam.
If the BSOP model is examined in your exam then you will be provided with a ‘mini-spreadsheet’ called the “BSOP calculator”. You will just need to enter the variables (Pa, Pe, etc.) and the “BSOP calculator” will automatically calculate the values of both call and put options.
To see how it works, you must watch the ACCA video explaining it, which you can find here:
5 The use of options
One use of options is as a way of rewarding managers of a company in a way that motivates them to increase the share price.
By giving call options to the managers, it becomes very much in their interest to take decisions that increase the share price.
Very often these options are not traded options and therefore the formula in the previous section can be used to place a value on them.
Speculators also deal in options. The reason for this is that if (for example) you expect the price of a share to increase, then you could make money simply by buying shares and then selling them at the later, higher, price. As alternative, however, would be to buy call options. As the share price increases then so too will the option price.
The financial manager is not a speculator. Consider, however, the following situation – the company currently has an investment in shares in another company. They intend to sell the shares in six months time, and expect the price to increase. The are however worried in case they are wrong and the price should fall. How can they protect the company against the possible fall?
If the share price were to fall, then so too would the value of call options. In order to profit out of the fall the company will need to sell call options now (and would be able to buy them later and make a profit, should the price fall).
This hedging is known as a ‘delta hedge’. The slight problem is that the change in the option price will not be the same as the change in the share price and therefore we need to be able to calculate how many options to deal in. We will cover the arithmetic shortly, but first we need to consider the Greeks!
6 The Greeks
From day to day the price of an option will change. It will change due to changes in all the factors listed in section 3 of this chapter.
Black and Scholes also produced formulae to measure the rate of change in the options price with changes in each of the factors listed. You do not need to know the formulae, but you need to be aware of the names given to each of the measures, and they are as follows:
6.1 Delta
The rate at which the option price changes with the share price (=N(d1))
6.2 Theta
The rate at which the option price changes with the passing of time.
6.3 Vega
The rate at which the option price changes with changes in the volatility of the share
6.4 Rho
The rate at which the option price changes with changes in the risk-free interest rate
6.5 Gamma
The rate at which delta changes
Although you will not need the formulae for each of these, you may need to know about the relevance of delta. This is because in the very short term, delta enables us to predict the effect on the option price of movements in the share price. It will be equal to N(d1), and we can use it to decide how many options we need to trade in to protect ourselves against movements in the share price.
Note: you can no longer be expected to be aware of Vega or of Rho
7 The Delta Hedge
If you own shares and you are worried that the share price might fall, then sell some call options. As the share price falls, so will the value of the options. (You can buy back at a profit).
The problem is to decide how many call options we need to sell.
Current share price is $1.50
Call option exercise price is $1.80 in 3 months
Risk free interest rate is 10% p.a.
Standard deviation of rate of return on share is 40%
Martin owns 1,000 shares.
Devise a delta hedge to protect against changes in the share price.
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The problem with a delta hedge is that our answer to example 6 will only protect us in the very short term. The reason for this is that over a longer term changes in the other factors will also affect the option price. For this reason the delta hedge will have to be continuously reviewed and changes made (which is why the other Greeks are of importance to a trader in options). You will not be expected to deal with this but you can be expected to be aware of the problem (and therefore of the other Greeks).
8 ‘Styles’ of options
A European option can only be exercised at the date expiration, whereas an American option can be exercised at any time up to the date of expiration. The terms refer to the ‘style’ of option and have nothing to do with where the dealing in the options takes place.
In either case, options can be traded prior to expiration (i.e. you can buy an option and later sell the option, before the expiration date)
The Black Scholes formula applies to European options.
9 Long and short dealings
As mentioned in the last paragraph, it is perfectly possible to buy and sell options rather than simply buying, holding until expiration and then exercising.
An investor who buys an option (and later either sells it or exercises it) is said to be taking a ‘long position’ (and so buying a call option can be referred to as a ‘long call’, and similarly buying a put option can be referred to as a ‘long put’)
Less obviously, it is possible to sell an option (that you do not own) and buy it later to finish the deal. This is known as taking a ‘short position’ (leading to the terms ‘short call’ and ‘short put’)
Be aware of the terms in this paragraph, but they do not affect the arithmetic.