Vanilla options

Option strategies

The easiest way to find the payoff of a strategy that combines standard calls/puts is to decompose the payoffs as shown below.

Greeks

As showed in the Black-Scholes section,

\[\theta + r S \Delta + \frac{1}{2}S^2\sigma^2 \Gamma = rC\]

Delta

The delta is:

• The sensitivity w.r.t. the underlying move of 1 unit. E.g. Microsoft trades at \$410 and the ATM call is at \$4.7 with a delta of 50%. If the price moves to \$411, the call price becomes 4.7 + 0.5 = \$5.2.

• The estimated probability that the underlying ends up ITM at maturity.

• The quantity of underlying to short to be delta-hedged.

• 50% when ATM. In other words, when buying a call ATM, we earn only half of the underlying’s variation.

When a position is delta-hedged, it can still be profitable on other aspects such as theta (e.g. by selling options and collecting the time decay as time passes) or vega.

\[\Delta_{call} = \mathcal{N}(d_1)\] \[\Delta_{put} = \mathcal{N}(d_1)-1\]

$d_1 = \frac{\ln\frac{S_t}{K}+\left(r+\frac{\sigma^2}{2}\right)\left(T-t\right)}{\sigma\sqrt{T-t}}$

Gamma

The gamma is:

• The sensitivity of the delta w.r.t. the underlying move

Gamma positive: when the underlying increases a lot, the delta increases even more (so the option becomes even more sensitive to the underlying). \

Example: when an investor buys a BRC, he is gamma negative (short put). It means that when the underlying decreases (and goes close to the barrier), the delta increases so the option becomes more sensitive to the underlying. Since he is also delta positive, the option would strongly decrease once close to the barrier. That’s why considering selling a structured product when the underlying arrives close to the barrier is not a good strategy.