Notes, Computer Lab and Problems

1.3.1 Bibliographic remarks: For a global view of modern corporate finance, theory and practice, read Brealey et al. (2011), a renown textbook in business schools. For a deeper mathematical analysis and detailed account of derivatives see Hull (2009), Cox and Rubinstein (1985) and Neftci (2000). A more practical view of options, their history and a long list of trading strategies is in The Options Institute (1995).

A review of the Efficient Market Hypothesis is given by Fama (1991) and Park and Irwin (2007), and can be complemented with the discussion on the subject found in Campbell et al. (1997, §1.5). For an in-depth treatment of the theory of Computa-tional Complexity see Papadimitriou (1994). To complement the topics treated here see Rubinstein (2006) chronological review of the most influential research papers in the financial theory of investments. From this source one learns, for example, the historical development of the DCF models (Eqs. (1.17) and (1.20)) and gets the authorship record straight: the formulas are often attributed to Gordon and Shapiro,²³ but they were first developed in 1938 by Williams in his PhD thesis (Williams 1938).

For more on the history of derivatives and asset pricing models read Bernstein (1992,

22seehttp://www.claymath.org/millennium-problems.

23M. Gordon and E. Shapiro, “Capital equipment analysis: the require rate of profit”, Management Science 3, 102–110, 1956.

Ch. 11). It is from this source where we learned about the possibly first option con-tract in history (see following note). Finally, for your bedtime reading we recommend the financial chronicle of Paulos (2007).

1.3.2 Possibly the First Option Contract in History: Aristotle reports in his book of Politics that the Greek mathematician and philosopher Thales the Milesian, incited by their common fellow’s belief of the inutility of his maths, set out to prove them all wrong and worked on a financial scheme based on his knowledge of astronomy.

Reading the stars one winter, Thales was able to foresee that by the coming year the olive harvest would be larger than usual; so he took his savings and paid all the owners of olive presses in Chios and Miletus for the right to use their presses in the upcoming harvest-time. He was able to negotiate this reservation of presses at a low price since no one could know what the harvest would be that much time ahead, and hence no one bid against his offer. Aristotle ends this anecdote as follows: “When the harvest-time came, and many [presses] were wanted all at once and of a sudden, he let them out at any rate which he pleased, and made a quantity of money. Thus he showed the world that philosophers can easily be rich if they like, but that their ambition is of another sort”.²⁴

1.3.3 On Short Selling: Short selling a stock is a bet for the future decrease of the price. If the short seller is a well-known investment corporation, and since these short positions are communicated to the public by the market authorities, the knowledge of this news can induce the stampede from the stock by many small investors, which would indeed knock down the price. The European Commission has adopted a series of regulatory technical standards on short selling that can be read athttp://ec.europa.

eu/internal_market/securities/short_selling_en.htm. See also what the US Securities and Exchange Commission writes about short sales athttp://www.sec.gov/answers/

shortsale.htm.

1.3.4 R Lab: We can reproduce in R Fig.1.3showing the different performances for four US stocks and the Value Line Index. To compute rate of benefits or returns from a given time series^xuse^quantmod’s function^Delt(x). To express the resulting return series in percentage terms just multiply the series by 100. The function^cumsum(x) does cumulative sum (for each time t, sum of terms up to time t). After loading quantmod, write the following commands in your R console:

> symbols=c(’ˆVLIC’,’GE’,’KO’,’AAPL’,’MCD’)}

> getSymbols(symbols,src=’yahoo’,from="2012-02-01",to="2013-02-01")

> #obtain adjusted closed

> VLICad = VLIC$VLIC.Adjusted; GEad= GE$GE.Adjusted;

+ KOad=KO$KO.Adjusted; AAPLad=AAPL$AAPL.Adjusted;

+ MCDad = MCD$MCD.Adjusted

> #compute cumulative sum (cumsum) of daily returns (Delt) + #Remove first term of the series, with [-1,],

+ #since cumsum is not defined for it.

24source Aristotle, Politics, Book I §11. In collection: Encyclopedia Britannica, Great Books of the Western World, 9, p. 453 (1982). See also Bernstein (1992, Ch. 11).

1.3 Notes, Computer Lab and Problems 35

> vl = cumsum((Delt(VLICad)*100)[-1,])

> ge = cumsum((Delt(GEad)*100)[-1,])

> ko = cumsum((Delt(KOad)*100)[-1,])

> ap = cumsum((Delt(AAPLad)*100)[-1,])

> md = cumsum((Delt(MCDad)*100)[-1,])

> ###range of values for the plot

> lim = c(min(vl,ge,ko,ap,md),max(vl,ge,ko,ap,md))

> ###the plot

> plot(vl,main="",ylim=lim,xlab="dates",ylab="% benefits")

> lines(ge,col="green"); lines(ko,col="red")

> lines(ap,col="violet"); lines(md,col="yellow")

> legend(x="topleft",cex=0.4,c("VLIC","GE","KO","AAPL","MCD"), + lty=1, col=c("black","green","red","violet","yellow"))

1.3.5 Research Problem: Which market index is best? No market index has shown to be optimal for the purposes of (long term) asset allocation with highest return and lowest risk. Thus it is important to compare and statistically test which index (or measure of indexing) has historically come closer to this optimal goal. Further-more, to research new ways of computing an index that may realize these optimality requirements. This is important because investors typically perceive an index to be a neutral choice of long term risk factor exposure. Some papers on the subject: Amenc, N., F. Goltz, and V. Le Sourd, 2006, Assessing the Quality of Stock Market Indices, EDHEC Pub.; and Arnott, R.D., J. Hsu, and P. Moore, 2005, Fundamental Indexa-tion, Financial Analysts Journal 60 (2), 83–99.

1.3.6 Problem (Simple Bounds on Calls and Puts): Using no arbitrage arguments show that for options on stocks: (i) the stock’s price is an upper bound for the price of a call; and (ii) the strike price is an upper bound for the price of a put.

1.3.7 Problem (More Strategies with Options): The following is a list of well-known investment strategies obtained by different combinations of put and call options on the same underlying asset. For each one of these strategies compute the payoff function and draw the profit graph. Additionally argue about the situations where the strategy is profitable.

The strangle: Similar to the straddle (see Example 1.7). It consists on buying a put and a call with the same expiration date but different strike prices. If Kcis the strike price for the call, and Kp is the strike price for the put, then the strategy requires Kc> Kp.

The strip: This strategy consists of long positions in one call and two puts, all with the same strike price and expiration date.

The strap: This one consists of long positions in two calls and one put, all with the same strike price and expiration date.

The butterfly spread: This is made with options of the same type. Suppose we use calls and that the underlying asset is a stock. Then a butterfly spread of calls consist on short selling two calls with strike price K0close to the current stock price, and buying two calls, one with strike price K0−c and the other with strike price K0+ c, where 2c > 0 is the length of the spread chosen by the investor.

1.3.8 Problem: Use the replicating portfolio method (Example 1.11) to estimate the current value of a six-month call option on a stock with the following characteristics.

The current price ise25, the exercise price is e20, at exercise time it is estimated that the stock will be in the range ofe15 to e40, and the six-month risk-free interest rate is 5 %. Also, compute the value of the corresponding put option using the put-call parity relation.

1.3.9 Problem: In Example 1.11 we encounter the option delta (for a call), which is computed as the quotient of the spread of possible option prices over the spread of possible share prices Eq. (1.22). Denote this quantityσC, and we have seen that it gives the number of shares needed to replicate a call option. Define analogouslyσP

to be the option delta for a put, and show that for a call and a put European options on the same stock and with the same strike price,σP = σC−1. (Hint: use the put-call parity for European options.)

Chapter 2