Options

Motivation: why this matters in quant finance

Options are the first place where finance becomes genuinely nonlinear. A share of stock pays one dollar more when the stock price rises by one dollar. A call option may pay nothing for a long time and then suddenly begin behaving like the stock itself. That kink is why options are used for hedging, speculation, volatility trading, and risk transfer.

The standard pricing formula

V_0 = e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)^+]

only makes sense once the payoff is understood as a random variable. The same object feeds put-call parity, risk-neutral valuation, the Black-Scholes formula, and the Greeks. Without a clean options vocabulary, every later derivatives note becomes notation soup.

The informal idea

An option is a contract that gives one side a choice. A call gives the holder the right, but not the obligation, to buy the underlying at a fixed strike

K

. A put gives the right to sell at

K

. The choice is valuable because the holder exercises only in favourable states.

At expiry $T$ , the European payoffs are:

C_T=(S_T-K)^+, \qquad P_T=(K-S_T)^+.

The notation $(x)^+=\max(x,0)$ encodes limited downside for the holder. A call holder loses at most the premium paid; a call writer can lose much more if the stock rallies.

Formal definitions

An option contract specifies an underlying asset, strike

K

, expiry

T

, exercise style, and payoff rule.

Contract	Right	European expiry payoff
Call	Buy the underlying at $K$	$(S_T-K)^+$
Put	Sell the underlying at $K$	$(K-S_T)^+$

A European option can be exercised only at expiry. An American option can be exercised at any time up to expiry. A Bermudan option can be exercised on a specified set of dates.

The time- $0$ price is not the expected payoff under the historical measure. In an arbitrage-free complete model, the price is the discounted expectation under the risk-neutral measure:

V_0=e^{-rT}\mathbb{E}^{\mathbb{Q}}[\text{payoff}].

Key properties

Nonlinear payoff

The call payoff has a kink at

K

. Below

K

it is zero; above

K

it has slope one. This kink is why option portfolios have convexity and why hedging requires the Greeks, not just a static share position.

Moneyness

A call is in the money when

S>K

, at the money when

S \approx K

, and out of the money when

S<K

. For puts the inequalities reverse. Moneyness is a state description, not a statement that the trade is profitable after premium.

Intrinsic and time value

For a call, intrinsic value is $(S_0-K)^+$ . The market price is usually larger because there is time for the underlying to move favourably before expiry. The difference is time value.

No-arbitrage bounds

For a non-dividend stock,

0 \le C_0 \le S_0, \qquad 0 \le P_0 \le Ke^{-rT}.

A European call also satisfies $C_0 \ge S_0-Ke^{-rT}$ . Bounds are model-free sanity checks before any Black-Scholes calculation.

Worked examples

Example 1: call and put payoffs

Let $K=100$ . If $S_T=120$ , the call pays $20$ and the put pays $0$ . If $S_T=80$ , the call pays $0$ and the put pays $20$ . The payoff diagram is two straight lines meeting at the strike.

Example 2: a protective put

A stock plus a put produces payoff

S_T+(K-S_T)^+=\max(S_T,K).

The put turns a risky stock position into a position with a floor at $K$ . This is portfolio insurance: upside remains, downside is capped, but the premium must be paid up front.

Example 3: call-put spread identity

A long call and short put with the same strike has payoff

(S_T-K)^+-(K-S_T)^+=S_T-K.

Discounting the fixed strike leg gives put-call parity:

C_0-P_0=S_0-Ke^{-rT}

for a non-dividend stock.

Common confusions and pitfalls

"A call is a bet that the stock will rise." It is more precise to say a call is exposure to upside convexity. A stock can rise and the call can still lose money if the rise is too small relative to the premium and time decay.

"In the money means profitable." In-the-money only compares

S_T

with

K

. Profit compares payoff with the premium paid and financing costs.

"American options are always much more valuable." Early exercise matters most for puts and dividend-paying calls. A non-dividend American call should not be exercised early, so its value matches the European call.

"The payoff and the price are the same object." The payoff is the cash flow at expiry. The price is today's no-arbitrage value of that random future cash flow.

Where this goes next

Put-Call Parity: The model-free identity linking calls, puts, stock, and cash.
Risk-Neutral Valuation: Turns option payoffs into present values under the pricing measure.
Black-Scholes Formula: Prices European options when the stock follows geometric Brownian motion.
Delta: Measures the local stock exposure of an option price.