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2.1.3 Plain vanilla options

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To avoid margin payments every day and allow investors who are not members of a clearing house to use derivatives, financial institutions created a new type of security they called options. As the name suggests, an option confers the right and no obligation for the holder (the buyer) to exchange an underlying asset (e.g. a share) for a fixed price some time in the future. Of course, the writer (the seller) enters an obligation towards the holder, but the writer is generally a large financial institution who is also a member of a clearing house.

In their most basic form (or ``flavor''), financial derivatives are commonly called vanilla:33a plain vanilla call (alternatively put) option confers its holder the right to buy (alt. sell) the underlying for a fixed amount of cash $ K$ called exercise or strike price. Depending on whether the market value of the underlying $ S$ is higher or lower than the strike price $ K$ when the option reaches the expiry date $ T$ , the option holder can choose to either exercise the option and buy (alt. sell) the underlying for a price $ K$ , or let the option expire worthless.

The terminal payoff $ \Lambda (S)=V(S,T)$ plotted in (2.1.3#fig.1) for all the possible realizations of the underlying spot price $ S$ is similar to the forward contract (2.1.2#fig.1), except that with no obligation, the option expires worthless and can never become negative.

Figure 2.1.3#fig.1: Terminal payoff diagrams $ \Lambda (S)=V(S,T)$ showing the value of plain vanilla call (left) and put options (right) as a function of possible realizations of the underlying share price $ S$ at the expiry time $ T$ .
\includegraphics[width=6cm]{figs/payCall.eps}        \includegraphics[width=6cm]{figs/payPut.eps}

A vanilla call, which carries the right to buy the underlying for a price $ K$ , has a finite value only if the underlying is more expensive on the market; the risk-free profit that can be made by exercising the call option (spending $ -K$ to buy the underlying and immediately sell it for a higher price $ S$ ) is given by the difference $ S-K$ if this is positive and zero otherwise. Similarly, a put option has a finite value provided that its holder can sell the underlying to the writer for a price $ K$ that is higher than the spot price on the market $ -S$ . Mathematically,

$\displaystyle \Lambda_\mathrm{call}=\mathrm{max}(S-K,0), \qquad\qquad\qquad\qquad \Lambda_\mathrm{put}=\mathrm{max}(K-S,0).$ (2.1.3#eq.1)

Because of the fluctuations in the underlying spot price $ S(t)$ , the value of an option $ V(S(t),t)$ before it expires is generally different from the terminal payoff. By definition, the intrinsic value of an option at a time $ t<T$ is defined from the terminal payoff as if the option would expire now with the current price of the underlying $ V(S(t),T)$ . Moreover, call and put options are said to be out-of-the-money if they have no intrinsic value and in-the-money if they have a large intrinsic value. If $ S\approx K$ , they are at-the-money and that is where their spot price is generally quoted in the press. For example, take one of the two Marconi call options quoted on Feb 23, 2002 by the Financial Times and reproduced in (2.1.3#tab.1).

Table 2.1.3#tab.1: Options traded in London and quoted on Feb 23, 2002 in the press
Option Strike Calls Puts
(*stock price) - May Aug Nov May Aug Nov
Hilton 200 17.5 23 26 5.5 9.5 13.5
(*215 1/2) 220 6.5 13 16 15.5 20 24
AstraZeneca 3500 167.5 267.5 336 129 197 249
(*3519) 3600 116.5 216 284 179.5 245.5 295.5
Marconi 15 4.5 6 7 3.5 4.5 5
(*16 3/4) 20 3 4.5 5.5 7 8 8.5

An investor who speculates on a solid rebound could buy 100 Marconi shares for GBP 1675; alternatively, he could buy 100 call (options are usually traded in units of 100) for GBP 3 each, giving him the right to buy the shares later in May for a total of GBP 2000. If the stock prices double until May (the precise expiration date is on the Saturday immediately following the third Friday of the expiration month), the net benefit from exercising the options to buy 100 shares for 20 and immediately sell them for 33 1/2 will be GBP 3350-2000=1350, a larger return on investment (1350/300=4.5) than the doubling that would have been achieved by using shares alone. If the price of the share remains below 20, however, the holder of calls with a strike at 20 will however never exercise his rights and will eventually loose all the investment made when buying the options, i.e. GBP 300.

This shows how speculators can use options to achieve larger gains for a higher risk, using an effect called gearing. Just the opposite can be achieved with hedging, where the negative correlation between an asset and its derivatives is exploited in the form of an insurance reducing the investment risk at the expense of for a lower expected return. To show an extreme case of hedging, imagine a portfolio that is long one asset, long one put and short one call with the same strike price $ K$ and expiry time $ T$ . This combination corresponds to what is called the put-call parity relation

$\displaystyle \Pi(T) = S(T)+\Lambda_\mathrm{put}-\Lambda_\mathrm{call} = S(T) + \mathrm{max}(K-S(T),0) -\mathrm{max}(S(T)-K,0) = K, \hspace{5mm}\forall S$ (2.1.3#eq.2)

and shows that the risk from the uncertain evolution of a spot price $ S(t)$ can be eliminated completely in favor of a guaranteed payoff $ K$ . Hedging is particularly important for companies that work with expensive raw materials such as gold: the right combination of options allows them to secure their activity without having to take the financial risk from volatile markets.

In general, the right combination of assets (e.g. shares) and derivatives (e.g. call or put options) can be used to expose a portfolio to any level and type of risk chosen by the investor and reap the benefit from the payoff that reflects the investor's opinion. The plots in (2.1.3#fig.2) show only at the option expiry how each term (or option series, i.e. options having the same strike price and expiry date) contributes to the put-call parity relation (2.1.3#eq.2) and cancels the investment risk.

Figure 2.1.3#fig.2: Terminal payoff diagrams illustrating the put-call parity relation.

More complicated payoffs can be obtained by combining vanilla options from the same class (i.e. same type, but different strike price and expiry dates, exercise 2.05-2.07) or even with hybrid underlyings that have only partly correlated prices. For example, combining the right amount of put options on the NASDAQ top 100 index (a symbol called QQQ) with shares from IBM, it is in principle possible to make a profit if IBM shares fall, but less than the rest of the technology market. However, remember that individual investors who are not member of a clearing house are only permitted to write covered options, where every short position such as the call ( $ -\Lambda_\mathrm{call}$ ) in the put-call parity relation has to appear in a combination with a long position in the underlying ($ +S$ ).

Finally, note that different exercise styles do affect the price of an option $ V(S,t)$ before it expires $ t<T$ : in chapter 4, we will first study the European style where the options can be exercised only on the expiry date and later in chapter 6, we will extend the models to deal with the American style where the options can be exercised anytime up to the expiry date.

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