Financial Options: Calls and Puts

Two contracts, four positions

A call option gives the holder the right (not obligation) to buy the underlying asset at a fixed strike price $K$ on or before expiration $T$.

A put option gives the holder the right to sell the underlying at strike $K$ on or before $T$.

For each contract you can be long (the buyer/holder, who paid the premium) or short (the seller/writer, who collected the premium). Four positions total.

This lesson assumes European options (exercisable only at $T$). American options (exercisable any time before $T$) follow the same structure with one extra wrinkle.

Payoff diagrams

Let $S_T$ be the stock price at expiration. Payoff at expiration:

Position	Payoff
Long call	$\max(S_T - K, 0)$
Short call	$-\max(S_T - K, 0)$
Long put	$\max(K - S_T, 0)$
Short put	$-\max(K - S_T, 0)$

Sketch on a $(S_T, \text{payoff})$ axis:

• Long call: zero for $S_T < K$, then upward 45° from $K$.
• Long put: downward 45° from $K$ for $S_T < K$, zero for $S_T > K$.
• Short positions are mirror images across the x-axis.

Profit diagram = payoff diagram shifted by the premium paid (or received). A long call with premium $2 and strike $50 breaks even at $S_T = 52$.

Put-call parity

For European options on a non-dividend-paying stock:

$$C + \frac{K}{(1+r_f)^T} = P + S_0.$$

Where $C$ is the call price, $P$ the put price, $S_0$ current stock price, $r_f$ risk-free rate.

Why? The left side: buy a call, save enough cash at the risk- free rate to grow to $K$ at $T$. If $S_T > K$, exercise the call: you get the stock for $K$, paid with your saved cash. If $S_T < K$, let the call expire and keep your $K$ in cash. Final value: $\max(S_T, K)$.

The right side: buy a put and the stock. If $S_T > K$, the put expires worthless, you have the stock. If $S_T < K$, exercise the put to sell the stock for $K$. Final value: $\max(S_T, K)$ — same.

Two portfolios with identical payoffs must have the same cost (Law of One Price). Hence put-call parity.

Example. S_0 = \100$,$K = $100$,$T = 1$year,$r_f = 5%$. Suppose$C = $8$. Then put price:

$$P = C + K/(1+r_f)^T - S_0 = 8 + 100/1.05 - 100 = 8 + 95.24 - 100 = \$3.24.$$

If the market quotes the put at \5$, arbitrage exists: sell the put, buy the call, short the stock, lend$$95.24$to receive$$100$at$T. Net cash today: \5 - 8 + 100 - 95.24 = $1.76 of riskless profit. Real markets close this kind of mismatch in seconds via high-frequency arbitrage.

Intrinsic value and time value

The price of an option decomposes into:

• Intrinsic value: payoff if exercised today. $\max(S - K, 0)$ for a call, $\max(K - S, 0)$ for a put.
• Time value: the rest. Reflects the chance the option will be more valuable before expiration (more time = more uncertainty = more upside potential).

Time value is always non-negative. It’s largest at the money ($S \approx K$) and shrinks as expiration approaches (time decay).

Pricing by replication: the binomial model

The cleanest way to see why options have unique prices: build a replicating portfolio.

One-period setup. Today the stock is $S_0$. At time $T$ it will be either $S_u = uS_0$ (up state) or $S_d = dS_0$ (down state). Risk- free rate is $r_f$. A call with strike $K$ pays $C_u = \max(S_u - K, 0)$ in the up state and $C_d = \max(S_d - K, 0)$ in the down state.

Build a portfolio of $\Delta$ shares + $B$ dollars in bonds that exactly replicates the call:

• Up: $\Delta \cdot S_u + B(1+r_f) = C_u$.
• Down: $\Delta \cdot S_d + B(1+r_f) = C_d$.

Solve:

$$\Delta = \frac{C_u - C_d}{S_u - S_d}, \quad B = \frac{C_d - \Delta \cdot S_d}{1 + r_f}.$$

By the Law of One Price, the call must cost:

$$C_0 = \Delta \cdot S_0 + B.$$

Example. S_0 = \50$,$u = 1.2$,$d = 0.9$,$K = $50$,$r_f = 4%$.

• $S_u = 60$, $S_d = 45$, $C_u = 10$, $C_d = 0$.
• $\Delta = (10 - 0)/(60 - 45) = 0.667$.
• $B = (0 - 0.667 \times 45)/1.04 = -30/1.04 = -28.85$.
• C_0 = 0.667 \times 50 + (-28.85) = 33.33 - 28.85 = \4.48$.

(Negative $B$ means borrow.)

Why this matters for corporate finance

Two reasons options matter beyond derivatives trading:

1. Real options in capital budgeting (next lesson). Many projects have option-like features — defer, expand, abandon — that pure DCF undervalues.
2. Capital structure as options. Equity in a levered firm is a call option on firm value with strike equal to debt face value. Debt is a risk-free bond minus a put. Merton (1974) built credit-risk models on this insight, and modern banking regulation (Basel) still uses derivatives of it.

For undergraduate corporate finance, you mostly need to recognize option structures and know put-call parity. The Black-Scholes formula is one level deeper than this course goes.