Vanilla Option

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:

• N is the notional

• S is the underlying price, at expiry

• K is the strike price

For a put option, the payoff is:

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

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 $N=1$, $K=100$, $C_L=5$, $C_S=50$

Example

Alice and Bob enter in a Vanilla Call option contract on SOL/USD. The contract has a notional of 10 SOL, and the strike is set to the current spot price, $35. Alice deposits an option premium of 5 USDC and Bob deposits 100 USDC as collateral. The duration is set to 1 week.

After 1 week has passed, the profit & loss will depend on the final price of SOL/USD:

• if SOL/USD is at $40, the profit for Alice will be 10*(40-35) = $50. So she will claim 50 USDC from the contract (+45 USDC PnL). Bob will be able to claim the 5 USDC of premium + 50 USDC of remaining collateral (-45 USDC PnL)

• if SOL/USD is at $30, the option expires worthless. Alice will lose the option premium (-5 USDC PnL), while Bob will able to claim 5 USDC + his original collateral of 100 USDC (+5 USDC PnL)