Options

Futures and forwards share a very important characteristic: when the delivery date arrives, the delivery must take place. The agreement is binding for both parties: the party with the short position has to deliver the goods, and the party with the long position has to pay the agreed price. Options give the party with the long position one extra degree of freedom: she can exercise the contracts if she wants to do so; whereas the short party have to meet the delivery if they are asked to do so. This makes options a very attractive way of hedging an investment, since they can be used as to enforce lower bounds on the financial losses. In addition, options offer a very high degree of gearing or leverage, which makes them attractive for speculative purposes too. The main characteristics of a plain vanilla option contract are the following:

• The maturity $\tau$: The time in the future, up to which the contract is valid;

• The strike or exercise price $X$: The delivery price. Remember that the long party will assess whether or not this price is better than the current market price. If so, then the option will be exercised. If not the option will be left to expire worthless;

• Call or put: The call option gives the long party the right to buy the underlying security at the strike price from the short party. The put option gives the long party the right to sell the underlying security at the strike price to the short party. The short party has to obey the long party's will;

• American or European: The American option gives the right to the long party to exercise the contract at any time they wish, up to the maturity date. If the option is European, it can be exercised on the maturity date only; and

• Details concerning the delivery.

Apart from the plain vanilla contracts which are American or European, a lot of other exotic options have appeared recently, mostly as OTC contracts. These include Asian options, digital options, lookback options, etc. Traders have been rather imaginative when it comes to designing new derivative securities.

Unlike the forward or futures contracts, and because of the payoff asymmetries, the initial value of an option, say $C$, is not equal to zero. Apart from the above characteristics, the option price is generally affected by:

• The volatility [or uncertainty] of the underlying asset, from today up to the maturity date. In fact, it is common in the literature for option markets to be described as markets where volatility is traded;

• The level of the interest rates, in fact the whole term structure, and the stochastic behavior of them if they are unknown;

• The dividends, coupon payments, costs of storage, and other cash flows that are possible before the maturity date; and

• Commissions and the way the margin is marked.

As an example of the payoff asymmetries, the profits from a long European call option position look typically like the ones given in figure 1.3. A great deal of time will be dedicated discussing how option contracts are priced.

Figure 1.3: European call option payoffs