By: Tracy Porter
Copyright 2007
When one considers that the evolution of man’s mental reasoning has spanned many millennium, with leaps and bounds of intellectual progress occurring within the last 10,000 years, the science of finance is a relatively new concept. Quite often the figures associated with this field are not tangible because quite often people cannot actually see the money they have been given the responsibility for accounting for. Perhaps the fields of accounting and finance became a discipline in their own right when, in the mid-1400’s, Frater Luca Bartolomes Pacioli, a polymath and friend of Leonardo da Vinci, developed the double entry accounting method still taught in schools and widely in industry today (Bryce, 2002). Considering the fact that mankind has been using Pacioli’s accounting method for nearly 600 years, derivatives are a relatively new concept, with futures contracts not actually being introduced until as late as the early 1970’s (University of Leicester, 2001). Therefore, in the grand scheme of things, the science of finance is relatively new: perhaps not actually becoming a discipline in its own right until Franco Modigliani and Merton Miller’s work entitled, “The Cost of Capital, Corporation Finance, and the Theory of Investment”, which appeared in the June 1958 edition of the American Economic Review (Poulsen, 2006).
Pricing of Equity Index Futures
Futures are a type of derivative security not dissimilar from the forward, but allows the purchaser a little more flexibility. A derivative security is an asset whose value depends on the value of some other asset, such as stock in a company, oil, grain, or cattle. The value of the derivative is derived from the value of the underlying asset, and it is for that reason that derivative securities, such as futures, can be used by hedgers, speculators, and arbitrageurs for trading purposes (Cuthbertson and Nitzshe, 2001).
In the science of finance, arbitrage is the practice of taking advantage of price differentials between two or more markets: matching deals are struck to capitalize on the imbalance in the market, and the profit being the difference between the two market prices. A very practical example of arbitrage would be an entrepreneur purchasing products and services from a country where wages and the standard of living are comparatively low, such as Mexico, Romania, China, and India, and selling those products and services in a country where the wages and cost of living are high.
For instance, many engineering firms outsource their back-office functions to India because the Indians are well educated and speak English. For argument’s sake, a client would like an estimate on a study and is willing to pay a senior engineer £125 an hour for 120 hours of work, which means that he is willing to pay £15,000 for the study. When taking into account the salary, payroll burden, office space, and telecommunications, and engineer working in the UK can very well cost £75 an hour with the total cost of his work being £9,000, so the firm will make a profit of £6,000 for the study. An engineer working in India with the same qualifications and level of experience, however, would only cost the company £9 an hour, with a total cost to the company of £1,080. The engineering firm, therefore, would stand to profit £13,920. Because the firm will stand to make £7,920 more profit if he outsources the work to India, this opportunity is an arbitrageur’s dream. The situation that will arise, however, as it did in Japan after World War II and South Korea after the Korean War, is that as the Indian workers make their own deals and become more confident in their ability to provide superior products and services, they will demand higher salaries. As a result, the profits that companies will make by outsourcing to India will continue to decrease until it will become no longer economically viable to outsource to that country. When it is no longer profitable to outsource to India then the arbitrageurs, who are in this case businesses, will begin looking for new ways to limit costs and maximize profits.
The above example is symptomatic of Eugine Fama’s Efficient Market Hypothesis. Arbitrage is possible only when there is an inequality in the market, as in the case of Indian workers being paid substantially less than their British counterparts. As the arbitragers or businesses seek to take advantage of this inequality, the cost of work in India is likely to increase and the cost of work in the UK is likely to decrease. Over time, the inequalities in price will likely level out so that equilibrium will be established, or restored. As a result, arbitrage will no longer be viable and investors or businesses will only earn a competitive expected return on their capital employed, thereby making the market efficient (University of Leicester, 2007).
The futures contract is a standardized document that was first introduced in the 1970’s and is popular because it provides a hedging mechanism for those organizations that would not be eligible to purchase other hedging instruments, such as the forward (University of Leicester, 2001). The future, like the forward, involves the future exchange of an item, such as a commodity, currency, or stock, at an agreed upon rate and maturity date. These trades take place on an organised exchange, such as the Chicago Mercantile Exchange (CME), and the contracts are revalued, or marked to market, every day. When an individual buys or sells a futures contract on an item, such as cocoa, he is trading the legal right to the terms of the contract, not the item itself, which in this instance would be cocoa (Cuthbertson & Nitzsche, 2001).
An individual who purchases a futures contract is required to post a daily margin, which is based upon the closing settlement price of each day’s trading. This margin is a cast deposit by the buy or seller of the futures contract and is a guarantee that he will fulfill the contract. Each day the exchange decides upon a settlement price for each futures contract, a process known as marking to market. If the value of the account has risen it is credited as a gain and the surplus can be withdrawn as cash; if it has fallen then it has incurred a loss (University of Leicester, 2001).
It goes without saying that most people trade in futures contracts because they want to make a profit, and that is where arbitrage comes in because the primary motive of the arbitrageur is to make a profit (Markose, 2006).
Arbitrage plays a very important role in determine the price of futures. Arbitrage, it should be noted, provides a strong link, or correlation, between the spot price and future price of an item (Cuthbertson and Nitzsche, 2001).
One formula that arbitrageurs use in the case of covered interest arbitrage so that riskless profits can be made unless the rate is determined by (Cuthbertson and Nitzsche, 2001):-
Formula 1
S = the spot exchange rate
F = the futures exchange rate
Rd = the domestic interest rate
Rf = the foreign interest rate
For example, as at November 2007 the Bank of England exchange rate is 5.75%, the US Federal Reserve exchange rate is 4.75%, and a share in a hypothetical firm is £100. Therefore:-
One way to create riskless arbitrage is to create a synthetic forward. Using the same data that has been presented above:-
Share price S = £100In order to create a synthetic future the arbitrageur borrows £100 today and purchases the share, which he can deliver in a year. At the same time, the arbitrager sells a futures contract for £105.75, to be completed in a year’s time. The cost of the synthetic future in one year is £105.75, which is calculated as £100 * (1 + .0575). After one year the arbitrager sells the futures contract and delivers one share, thereby making a riskless profit of £0.
Bank of England interest rate R = 5.75
Quoted future price F = £105.75
Time of maturity T = 1 year
If the future had sold for £110 then the arbitrageur would have made a riskless profit of £4.25. If, however, he had miscalculated and sold the future for £105, he would have made a loss of £.75.
The above example illustrates, therefore, that if the future is the same price as the synthetic future, the arbitrage profit can be made, and this premise can be summed up by the following formula (Cuthbertson and Nitzshe, 2001):-
Formula 2
Although options were sole over the counter in the US in the 1960’s, no one knew how to correctly price them. It was this pricing dilemma, therefore, that brought together the superior intellects of the economists Myron S Scholes and Fischer Black, who consulted the applied mathematician, Robert Merton, about devising a formula that would determine the appropriate value for derivatives, namely options. The three men’s collaboration produced a paper that combined the option and the underlying asset to yield a riskfree portfolio. This paper, entitled, “The Pricing of Options and Corporate Liabilities,” was published in the May/June 1973 edition of the Journal of Political Economy, and is the basis for the resulting formula, aptly name the Black-Scholes Model (Cuthbertson and Nitzsche, 2001).
A call option gives the owner the right to buy a specific quantity of shares in a specific company at a fixed rate, while a put option gives the owner the right so sell a specific quantity of shares in a specific company at a fixed rate. The Black-Scholes formula for the value of a call can be written as (Lumby and Jones, 2003):-
Formula 3
(d1) = (ln(S/X) + (Rf * T)/σ * T) + (.5 * σ * squarerootT)
and
(d2) = (d1) – σ * squarerootT
S = Current market price of the shares
X = Options exercise price
Rf = Riskfree rate of interest
T = Time, in years, until the option expires
Σσ = Volatility (as measured by standard deviation) of the share price
N = Cumulative area under the normal curve
When the Black-Scholes Model was first developed, assumptions were made to enable it to be used as a tool to price options. According to Fischer Black, these assumptions were (Chew, 2001):-
1. The stock’s volatility is known and doesn’t change over the life of the option.
2. The stock’s price changes smoothly.
3. The short-term interest rate never changes.
4. Anyone can borrow or lend as much as he wants at a single rate.
5. An investor who sells the stock or the option short will have the use of all the proceeds of the sale and receive any returns from investing those proceeds.
6. There are no trading costs for either the stock or the option.
7. An investor’s trades do not affect the taxes he pays.
8. The stock pays no dividends.
9. An investor can exercise the option only at expiration.
10. There are no takeovers or other events that can end the option’s life early.
Even taking the above mentioned assumptions into consideration, one is nevertheless left with several factors that will determine the price of the option. These assumptions are (Lumby and Jones, 2003):-
1. The exercise price.
2. The current market share price.
3. The time to expiry of the option.
4. The volatility of the share price.
5. The time value of money or the riskfree investment.
6. Whether the option is European or American.
7. Whether dividends are paid on the shares.
The Exercise Price
A key aspect of an option is being able to determine its price, or the option premium. The option premium consists of two elements, which are its intrinsic value and its time value.
The intrinsic value of an option, or the profit that would be made if it is exercised today, is expressed as (Lumby and Jones, 2003):-
Formula 4
Formula 5
center> Exercise price + Premium paid on option = Breakeven market price
A further examination of Formula 4 will reveal that all other factors remaining the same, if the share price increases the intrinsic value of the call will also increase. In the same line of thought, if the share price decreases then the intrinsic value of the call will also decrease. Therefore, the share price can be said to be directly proportional to the intrinsic value of the call.
Time to Expiry of the Option
Because options are contracted to be exercised at a certain date in the future, the time to expiry of the option is an important determinant in the market vale of an option, and is illustrated by the following formula (Lumby and Jones, 2003):-
Formula 6
Volatility of the Share Price
Under normal circumstances, investors do not like the idea of taking risks with their money, but the option is designed to protect investors from adverse movements in the stock market while at the same time allowing them to take advantage of favourable movements. For options, therefore, risk can bee seen as a good thing. A decline in a stock’s price can be seen as increasing volatility, while an increase in the stock’s price is seen as a decrease in volatility (Chew, 2001). Volatility, however, is a determinant of time value, so if the volatility increases, the time value will also increase. Looking at formula 6, therefore, will show that all other factors being the same, an increase in volatility will lead to an increase in time value, which will result in an increase in the market value of the option. Some people could argue that the volatility makes no difference to the market value of the option because, referring to formula 4, a decrease in the share price will decrease the intrinsic value of the option as well as increasing volatility. Therefore, although an increase in volatility (↑) will cause the time value to increase, it will also cause the intrinsic value of the option to decrease, so the overall effect that this movement has on the market value of the option may very well be negligible.
Time Value of Money
Whilst the formula for the intrinsic value of a call in formula 4 illustrates how the share price and exercise price affect it, if one is going to analyse how the riskfree rate of interest, of the time value of money, determines the price of an option, the formula for the intrinsic value of a call will need to be expanded to include the time value determinants (Lumby and Jones, 2003):-
Formula 7
Formula 7 illustrates that if the riskfree interest rate increases then the present value of the future exercise price will decrease, which will have the effect of increasing the intrinsic value of the call. It can be said, therefore, that the riskfree interest rate is directly proportional to the intrinsic value of a call.
American or European?
The type of option is also a determining factor in its price because American options tend to be slightly more valuable than European ones. The reason for this is because European options can only be exercised on their expiry date, whilst American options can be exercised at any time up to their expiry date. In practice, however, options are normally never exercised before their expiry date because of the gamble associated with the time value.
Referring back to formula 6, it can be shown that options are more valuable if they are sold unexercised because the seller would receive the market value of the option. The exercised option, however, would only be worth its intrinsic value (Lumby and Jones, 2003).
Dividend Payments
Whether or not dividend will be paid on the shares will affect the value of the option. If dividends are paid on shares, there is the possibility that American options will be exercised before the expiry date so that the owner can gain access to the shares and consequently the dividend payments associated with those shares. Therefore, if dividends are paid on shares then the market value of a call is reduced. The market value of a put, however, is increased because the paying of dividends means it is less likely that this option will be exercised before the expiry date (Chew , 2001).
Seven determinants of option pricing have been discussed, and these determinants are the framework by which options pricing can be achieved. While the above mentioned determinants are very significant to help the investor to determine the optimum price for an option, according to Fischer Black, there are other factors that are also highly significant when determining an option’s price. These factors are temporary jumps in stock prices, borrowing penalties, short-selling penalties, trading costs, taxation, and corporate takeovers. While these determinants will not be discussed in detail, it is important to realize that there are other determinants in addition to the seven mentioned above that will have an effect on an option’s price (Chew, 2001).
Put-Call Parity Theorem
There are four basic financial securities, for which all investments can be constructed, and these securities are: (1) shares, (2) riskfree bonds, (3) call options, and (4) put options. A fundamental relationship exists between these four securities, which is written as :-
Formula 8
S = Value of share
P = Value of put
B = Value of bond
C = Value of call
When speculating with options, however, the bond can be interchanged with the exercise price (X), so the formula can be rewritten as:-
Formula 9
A particular traded share option has the following available information:-
Value of call = 39p
Present value of exercise price =793p
Value of share = 762p
Value of put = 80p
The question regarding the given data is whether to buy or sell a put on the share in question. In order to answer that question, formula 9 needs to be rearranged as follows:-
Formula 10
It is also worth mentioning that the intrinsic value of the put is calculated as (Lumby and Jones, 2003):-
Formula 11
Therefore, it makes no difference, which model an individual chooses to use, in this particular exercise an individual would make a gain if he chose to sell the put at 80p.
References
Brealey, R. et al. (2006). Corporate Finance: International Edition, McGraw-Hill: New York, New York
Bryce, R. (2002). Pipe Dreams: Greed, Ego and the Death of Enron, Public Affairs: Oxford, England
Chew, D. (2001). Corporate Finance: Where Theory Meets Practice, McGraw Hill: New York, USA
Cuthbertson, K. and Nitzsche, D. (2001). Financial Engineering: Derivatives and Risk Management, John Wiley & Sons Ltd: Chichester, England
Lumby, S. and Jones, C. (2003). Corporate Finance Theory and Practice, Seventh Edition, Thomson: London, England
Markose, S (2006). Stock Market Crashes: On Hedging and Arbitrage in Stock Index Futures
URL: http//www.essex.ac.uk//ccfea/prospectivestudents/presessional/2006MarkoseEC907N10FTSE-100%20Futures.doc
[31 Oct 07]
Poulson, A. (2006). Corporate Debt
URL: http://www.econlib.org/library/enc/corporatedebt.html
[26 August 2006]
University of Leicester (2007). Module 4 MSc in Finance MN7032D Corporate Finance, Learning Resources: Cheltenham, England
University of Leicester (2001). MSc in Finance: 2506 International Finance, Learning Resources: Cheltenham. England.
