Long Stock

Overview

Owning a stock (Long Stock) is the absolute baseline of the financial world. It is the reference point against which all option strategies are measured. When you own 100 shares of a stock, your exposure is linear: if the stock goes up $1, you make$1 per share. If it goes down $1, you lose$1 per share.

[!NOTE] All option strategies can be viewed as "modifiers" to this basic linear payoff. Options add curvature (convexity), time decay (theta), and volatility sensitivity (vega) to this flat baseline.

Mechanics of the Trade

1. The Entry: You pay the full market price ($S_0$) for each share.
2. The Hold: You own a piece of the company. You may receive dividends, and you have no expiration date forcing you to sell.
3. The Exit: You sell the shares back to the market at the current price ($S_T$).

Direct vs. Derivative

Because you own the actual asset, you aren't fighting a "ticking clock" (Theta) or a "volatility surface" (Vega). This simplicity is the primary advantage of stock over options.

Payoff and Break-even

The payoff of long stock is the definition of simplicity:

$$\text{Profit} = S_T - S_0$$

Break-even Point

Your break-even is exactly the price you paid ($S_0$). There are no premiums or "extrinsic values" that you need to overcome.

Bounds

• Max Loss: $S_0$ (if the company goes bankrupt and the stock goes to $0$).
• Max Profit: Theoretically unlimited (as long as the company grows).

Why Use Options Instead?

If stock is so simple, why bother with options? It comes down to Capital Efficiency and Risk Shaping.

The Greeks of Stock

From a mathematical perspective, a long stock position has very simple "Greeks":

1. Delta ($\Delta$): Constant 1.0 Buying 100 shares is equivalent to 100 "Deltas". It never changes, regardless of price or time.
2. Gamma ($\Gamma$): 0 There is no "curvature" in the payoff. A $1 move always results in a$1 change, whether the stock is at $10 or$1,000.
3. Theta ($\Theta$): 0 Stock doesn't expire. You can hold it for 50 years without losing value to time decay.
4. Vega ($\nu$): 0 Market panic (IV spikes) doesn't directly change the value of your shares (though it might indirectly affect the price people are willing to pay).

Strategic Use Cases

• Buy and Hold: Long-term investing for growth and dividends.
• Collateral: Holding shares so you can sell "Covered Calls" against them (generating income).
• Hedging: Holding shares and buying "Protective Puts" to create a synthetic floor.

Checklist for Entry

• Am I comfortable with the full capital requirements?
• Do I have an exit plan if the stock hits zero?
• Would I be better off using a synthetic long (Call + Short Put) to save capital?

The Replication PrincipleRead more

In derivative pricing, we often say that "Stock is a Call with a zero strike price and infinite time."

If you look at the Black-Scholes formula for a Call:

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

If you set the Strike ($K$) to $0$ and the Time ($T$) to infinity, the value simplifies perfectly to $S_0$. This is why the visualizer's P&L line for stock is a perfectly straight, 45-degree angle.