Options (Finance)

An option is a contract that gives the holder the right to exercise the option (=buy/sell a stock) at the strike price at or before the strike date.

• It is a type of Derivative
• The person who gives out the option contract is the writer.
• The person who gets the option contract is the holder.

Types of Options §

• Exercise timing
  • American option: anytime on or before the strike date.
  • European option: at the strike date only.
• Exercise type
  • Call option: you hold the right to sell the stock at the strike price.
  • Put option: you hold the right to buy the stock at the strike price.

Definitions §

• $S(t)$: price of underlying asset, at time $t$
• $C(t)$: price of a call option, at time $t$
• $P(t)$: price of a put option, at time $t$
• $K$: strike price of a call or put option

Intuition. (Payoff Diagram for Options)

Call Option §

Assume continuous compounding at rate $r$ between time period $[t_0,t_1]$. Assume No-Arbitrage.

• Payoff: $\text{max}[S(t_1)-K,0]$
  • $0$ is when the stock price as decreased, so you don’t decide to sell.
  • according to the law of one price, must equal to $=C(t_1)$
• Profit: $\text{max}[S(t_1)-K,0]-C(t_0)\exp [r(t_1-t_0)]$
  • First term is the payoff
  • Second term is because you borrowed money at the risk-free rate $r$ to buy the call option.
  • We don’t know what $C(t_0)$ is yet…see Binomial Option Pricing Model

thm. Put-Call Parity. let call option entered at time $t_0$ and put at time $t_0$. Then

$$\underset{\text{ Call }}{\underbrace{C(t_0)}}+\underset{\text{ Cash }}{\underbrace{K{e}^{-r(t_1-t_0)}}}=\underset{\text{ Put }}{\underbrace{P(t_0)}}+\underset{\text{ e−q(t1−t0) units of underlying }}{\underbrace{S(t_0)e^{-q(t_1-t_0)}}}$$

Intuition. You can always use a stock plus a put option to simulate an equivalent call option, and v.v.

Proof Outline. let ${\Pi }_A$ long $1$ call, have $K{e}^{-r(t_1-t_0)}$ units of cash. Let ${\Pi }_B$ long $1$ put, have $e^{-q(t_1-t_0)}$ units of underlying stock. At time $t_1$ they will have equivalent value, thus at time $t_0$ their value must be same by LOP.

thm. Put-Call-Forward Relation.

• Forward entered at $t=0$ and ends at $t=T$, with strike price $K$
• $C(t),P(t)$ are call and put entered at time $t$, with strike price $K$

$$C(t)-P(t)={\mathfrak{F}}_T(t)$$

Proof Outline.

Various Portfolios that Can Be Constructed from Options §

def. Moneyness of Option. A call option’s moneyness at time $t$ is:

• If $S_t>K$: call is in (=on) the money
• If $S_t=K$: Call is at the money
• If $S_t<K$: Call is out of the money, i.e. worthless.

Strategy	When you predict…	Profit Diagrams (Not Payoff)
Straddle	Underlying will move up/down a little
Strangle	Underlying will move up/down a lot
Bull/Bear Spread	Underlying will go up (bull) / down (bear)
Butterfly Spread	Underlying will not move much (long) / move a lot (short)