Long Call

Overview

The Long Call is the quintessential "bullish" bet. It is an asymmetric contract that gives you the right—but not the obligation—to buy 100 shares of an underlying stock at a specific Strike Price ($K$) before a specific Expiration Date.

In exchange for this right, you pay an up-front Premium ($C_0$). This premium represents your Maximum Risk. Unlike buying stock directly, you cannot lose more than you paid, but your potential profit is theoretically unlimited.

[!IMPORTANT] A long call is a bet on Direction, Magnitude, and Timing. If the stock moves up too slowly, time decay might eat all your profits.

Mechanics of the Trade

When you buy a call:

1. The Entry: You pay $100 \times \text{Premium}$ to open the position.
2. The Hold: As the stock price rises, the call value increases. This is "leverage"—a 1% move in stock might result in a 10% move in the call.
3. The Exit: You can sell the call back to the market at any time, or wait until expiration to exercise (buy the shares).

The "Moneyness" Spectrum

Options are categorized by where the stock price ($S$) is relative to the strike ($K$):

Payoff and Break-even

At expiration, the value of your call is simple:

$$\text{Value} = \max(S_T - K, 0)$$

Your Profit is that value minus the premium you paid:

$$\text{Profit} = \max(S_T - K, 0) - C_0$$

Break-even Point

To make a profit at expiration, the stock must rise above:

$$\text{Break-even} = \text{Strike} + \text{Premium}$$

If the stock is at $100, strike is$100, and premium is $3, you need the stock at **$103** just to break even if you hold to the end.

The Greeks: Your Professional Dashboard

To trade calls like a pro, you must understand the "Greeks"—the mathematical sensitivities of the option's price.

1. Delta ($\Delta$): The Price Sensitivity

Delta tells you how much the call price changes for every $1 move in the stock.

• An ATM call usually has a Delta of 0.50.
• As the stock rises, Delta increases toward 1.00 (acting like 100 shares of stock).
• As the stock falls, Delta drops toward 0.00.

2. Theta ($\Theta$): The Silent Killer

Theta is the "rent" you pay to hold the position. It measures how much value the option loses every day.

• Theta is always negative for long calls.
• Critical Warning: Theta decay is not linear. It accelerates significantly in the last 30 days before expiration.

3. Vega ($\nu$): The Volatility Engine

Vega measures sensitivity to Implied Volatility (IV).

• If IV increases, the call price increases (even if the stock price is flat!).
• This is because higher volatility means a higher statistical chance of a massive price move.

Strategic Variations

Not all long calls are created equal. Your choice of strike determines your risk profile:

1. ITM Calls (Deep): High delta (0.80+), behaves almost like stock. Expensive, but safer from time decay.
2. ATM Calls: The "purest" bet. High liquidity and high sensitivity to price changes.
3. OTM Calls: The "lottery ticket". Cheap, high leverage, but very low probability of profit at expiration.

Checklist for Entry

Before you hit "Buy" on a Call option, ask yourself:

• Is my target price above Strike + Premium?
• Do I expect the move to happen faster than Theta erodes the value?
• Is Implied Volatility (IV) relatively low? (Avoid buying "expensive" volatility).
• Am I comfortable losing 100% of this premium?

The Black-Scholes FormulaRead more

The fair price of a call option ($C$) is calculated using:

$$C = S_0 N(d_1) - K e^{-rT} N(d_2)$$

Where:

• $d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma\sqrt{T}}$
• $d_2 = d_1 - \sigma\sqrt{T}$

This formula proves that the call price must increase when Volatility ($\sigma$) or Stock Price ($S_0$) increases, and must decrease as Time ($T$) approaches zero.