Preliminaries

First, we need to clear up the memory and set the global variables: the risk free rate, the asset volatility and the initial price.

The Stock Price

The discounted payoffs are straigntforward. You just discount the possible future price and subtract today's value.

The Call Option Price

The pricing of the call option is a bit more complicated. When the maturity is reached, on expiration, the price of the option is just its payoffs. For all times before expiration, one needs a pricing formula. For simplicity we make the assumption that the Black-Scholes pricing formula is correct. If this is not the case, one can substitute any other more accurate formula.

The Put Option Price

Exactly as the call option.

The Portfolio

Here a portfolio consists of vectors. The first element of the vector denotes the number of contracts bought/sold, the second the type of the contract, and the third its characteristics. E.g. in the portfolio

the second columns indicates a position where we sell one call option with maturity years and strike price .

Stock--Option Combination

Here we investigate the various combination of option with the underlying stock.

Covered Call

If one is a call option short, she can cover her position by buying the underlying stock. In this case, if the option is exercised she can just deliver the stock she already owns.

Covered Put

On the other hand, if one owns the stock and wants to cover her position, she can long a put option. In that case, if the stock price drops below the strike price she will just exercise the option to unload it at a more favorable price.

Spreads

Spreads consist of options alone. This makes them very cheap to engage into. They can be very useful for speculative purposes, due to their high gearing, and can be viewed as a bet on the future movements of the market. Of course their hedging value is also significant.

Bull Spreads

A bull spread will give us profits if the market ends up ``bullish''. It consists of a combination of a long low strike and a short high strike call, or put. Because part of the shorting will finance the longing, the initial investment is very low, or even negative. The drawback is that the positive payoffs are not unlimited as in the straight options case.

Bear Spreads

A bear spread will give us profits if the market ends up ``bearish'', i.e. if it drops. It consists of a combination of a short low strike and a long high strike call, or put. It is just the opposite of a bull spread.

Butterfly Spreads

A Butterfly spread is a combination of four calls [or four puts]. It can also be seen as a combination of a bull and a bear spread. The matrix below shows the structure: Short one high and one low strike calls, and long two medium strike ones. If the strikes are symmetric, you get payoffs as the ones presented below. In general, a butterfly spread will pay off if the market shows some kind of stagnation, neither bullish nor bearish. Again, the shortings finance part of the longings and the initial investment is small.

Calendar Spreads

Calendar spreads use options with different maturities to create nonlinear patterns similar to the butterfly spreads. Some examples are given below.

Other Combinations

The spreads above used only calls or puts. Here we present some of the strategies that involve combinations of calls and puts.

Straddles

By longing a call and a put, one can create a straddle. The payoffs are similar to the shorting of a butterfly spread, but without the limit of the profits that the spread provides.

Strips

Using more puts skews the straddle towards the strip, and makes the losses lighter. The drawback is that the profits are also smaller if the market ends up bullish.

Straps

Using more calls skews the straddle towards the strap. The losses are again lighter compared to the straddle. The drawback is that the profits are also smaller if the market ends up bearish.

Strangles

By setting the strike prices further apart, one creates the strangle. The losses are smaller than the straddle, but the market has to become more volatile for the strategy to score profits.