Vanilla Option Contract

A vanilla option contract is the most popular of non-linear derivatives. It gives the option buyer the right, but not the obligation, to buy or sell the underlying. At expiry the payoff for the Long (option buyer) **is:

$$ \max (0, N(S-K)) $$

for a call option, where:

For a put option, the payoff is:

$$ \max (0, N(K-S)) $$

Payoff profile of the forward for $N=1$ and $K=100$

Payoff profile of the forward for $N=1$ and $K=100$

Note that the payoff above is that of an idealized option, i.e. one with infinite collateral. In practice, the Short (option seller) deposits an upfront amount of collateral, which means that the maximum profit and loss for the buyer is capped.

In particular, given $C_L$ (option premium) and $C_S$ (collateral deposited by the Short side), the final payoff for the Long side will be:

$$ \min(C_S, \max(0, N(S-K))-C_L $$

for a call option, and:

$$ \min(C_S, \max(0, N(K-S))-C_L $$

for a put option. So the Long side will lose the option premium in any case, and can make at most $C_S$.

Payoff profile of the forward for $N=1$, $K=100$, $C_L=5$, $C_S=50$

Payoff profile of the forward for $N=1$, $K=100$, $C_L=5$, $C_S=50$