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 Call Option Price

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 greeks are actually used in the following example. The prices are included for reference and to get a feeling of the strategy prices, but are not directly used.

The Put Option Price

The greeks of the put option are derived in analogy with the call option.

The Portfolio

Again, 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 .

The four graphs are summarized below: the delta, gamma, theta and vega are displayed.

Greeks for the simple call and the put option

When the portfolio consists of a single derivative, the greeks follow the patterns below. For the sequel we will assume that today's price is about 100. Observe that Theta and Vega peak at around 100. This implies that longing an ATM call or put will give maximum profits if the volatility increases, but the price will decline rapidly due to the time decay. For profits to be realized, this volatility increase has to take place soon. Of course the value of a call will also increase in value if the underlying price increases, while the value of a put will increase when the underlying price falls.

Greeks for the put

Spreads

Bull Spreads

Longing a bull spread offers a delta which is similar to longing a call [that is in the around-the-money area]. In contrast, both Theta and Vega are around zero. This implies that a bull spread will disentangle the price movement from the time decay and the riskiness changes. This strategy will pay off if the market turns ``bullish'', regardless of the future volatility changes. In addition this strategy allows for a degree of patience: the near zero vega imples that the speculator can afford some time before the market becomes bullish. The time decay is not going to generate losses, at least in the short run.

Bear Spreads

The bear spread exhibits the same patterns with the bull spread, with the delta being negative. Profits for the speculator who longs a bear spread will come if the market turns ``bearish''. Instead of replicating the above graphs, it is interesting to observe the patterns if the strike prices move further in-the-money and out-of-the-money. Also observe that since the spread is constructed using calls, there is an initial cashflow towards the speculator.

Butterfly Spreads

The butterfly spread exhibits a near zero delta. Therefore the value of the spread will not be affected by small changes of the underlying price. On the other hand it offers a substantial negative vega, which implies that its value will increase when the volatility decreases. A speculator that anticipates this scenario will profit by longing butterfly spreads.  For instance, if the volatility falls by 5%, the price will increase by around 0.5, which is a return of 25% for the speculator. The positive theta indicates that the value of the spread is also anticipated to increase as the maturity is approached.

Straddles, Strips, Straps and Strangles

Straddles, strips and straps are similar to the butterfly spreads, but without offering the up- and downside risk bounds. Longing a straddle will bring profits if the volatility increases. Although similar in principle, one can see that they do not offer the same stability, since both delta and gamma can take larger values. This, coupled with the significant negative value of Theta implies that the volatility has to increase significantly in the short run for the speculator to enjoy profits.

Using more puts skews the straddle towards the strip, and makes delta negative. Using more calls skews the straddle towards the strap and makes delta positive. Again, the high values of the Gamma and Theta dictate that volatility changes have to take place quickly for speculating to be successful.

A strangle expands the area where the Vega is high. The drawback is the higher price of the contract and the high Gamma and Theta values, like the contracts discussed above.