208 Mathematics for Finance
9.3 Speculating with Derivatives...............................
9.3.1 Tools .............................................
Options can be used as building blocks to design sophisticated investment in-
struments. We shall consider an investor with specific views on the future be-
haviour of stock prices and willing to take risks. Our task will be to design a
portfolio of securities with a prescribed payoff profile that would satisfy this
kind of investor.
Suppose that the investor expects the stock price to rise and wants to gamble
on that. One simple way is to buy a call option. An option with strike priceX′
close to the current stock price is considerably cheaper than the stock itself,
creating a risky leverage position, as will be seen in the case study to follow. The
premium may be reduced by selling a call option with strike priceX′′>X′.In
this way we can build a so-calledbull spreadwith payoff shown in Figure 9.2.
This strategy will bring a good return if stock price increases are moderate.
Figure 9.2 Bull spreadUsing put options with strike pricesX′<X′′, selling the former and pur-
chasing the latter, we can construct abear spreadwith positive payoff for low
future stock prices, see Figure 9.3. This may be employed by an investor who
expects a moderate decline in the stock price.
Figure 9.3 Bear spreadAn investor who believes that the stock price will stay unaltered or change
insignificantly may choose abutterfly. It is constructed from three call options
with strike pricesX′<X′′<X′′′.Two calls are bought, one with strikeX′
and one with strikeX′′′, and two calls with strike priceX′′are sold. Figure 9.4