options are derivatives which gives you the *permission* to make a transaction at a particular date.

There are two main types of options:

- call: gives permission to
**buy**a security on or before the “exercise” date - puts: gives permission to
**sell**a security on or before the “exercise” date

For this article, we will define \(S_{t}\) to be the stock price at the time \(t\), \(K\) as the option’s strike price, \(C_{t}\) to be the price of the “call” option, and \(P_{t}\) to be the price of the “put” option at strike price \(K\); lastly \(T\) we define as the maturity date.

Naturally, the actual values \(C_{t}\) and \(P_{t}\) are:

\begin{equation} \begin{cases} C_{t} = Max[0, S_{T}-K] \\ P_{t} = Max[0, K-S_{T}] \\ \end{cases} \end{equation}

you either make no money from the option (market price is more optimal), or make some difference between the strike price and the market price.

The nice thing here is that little \(Max\) term. An option, unlike a futures contract, has no buying obligation: you don’t have to exercise it. The payoff is always non-negative!

NOTE!!! \(C_{t}\) at SMALL \(t\) is measured at \(Max[0,S_{*T*}, K}]\), using \(S\) of LARGE \(T\). This is because, even when—currently—the stock is trading at $60, the right to buy the stock in \(T\) months for $70 is not worthless as the price may go up.

To analyze options, we usually use the Black-Scholes Formula.

## American vs European Options

- American options are excercisable at or before the maturity date.
- European options are exrcercisable only at the maturity date.

## Analyze Options as Insurance

All insurance contracts are actually a form of an option, so why don’t we analyze it as such?

A put option—-

- Asset insured: stock
- Current asset value: \(S_{0}\)
- Term of policy: \(T\)
- Maximum coverage: \(K\)
- Deductible: \(S_0-K\)
- Insurance premium: \(P_{t}\)

A call option is covariant with a put option; so its isomorphic, and so we will deal with it later.

A few differences:

- American-style early exercise: (you can’t, for normal insurance, exercise it without something happening)
- Marketability: you can’t give normal insurance to other people
- Dividends: holding a stock pays dividends (an option’s value goes down as dividends)