Bull Call Spread

Overview

A Bull Call Spread (also known as a Long Call Vertical Spread) is a two-legged trade designed to express a bullish view at a lower cost than buying a single call.

By selling a higher strike call against your long call, you use the premium received (the "credit" from the short leg) to subsidize the cost of your long leg. In exchange for this discount, you agree to cap your maximum profit.

[!IMPORTANT] This is a Debit Spread. You pay money to enter. Your risk is strictly limited to the net amount you paid.

The Setup

A Bull Call Spread consists of:

1. Long Call: Buy a call at Strike A ($K_L$).
2. Short Call: Sell a call at Strike B ($K_H$), where Strike B is higher than Strike A.

Mechanics: The Power of Offsetting

Why do professionals use spreads instead of naked calls?

• Reduced Cost: The short call lowers your total entry price.
• Lower Break-even: Because you paid less, the stock doesn't have to rally as far for you to start profiting.
• Lower Sensitivity: The Greeks of the two calls partially cancel each other out. This makes the trade less sensitive to sudden market panic (Vega) or time decay (Theta).

Payoff and Break-even

Max Profit

Profit is capped at the difference between the strikes, minus the net debit paid:

$$\text{Max Profit} = (K_H - K_L) - \text{Net Debit}$$

Break-even Point

The stock only needs to rise above your lower strike plus the net cost:

$$\text{Break-even} = K_L + \text{Net Debit}$$

The "Vertical" Greeks

Spreads are "Vertical" because both options have the same expiration but different strikes. This creates a unique Greek profile:

1. Delta: Moderate Directional Exposure

The spread has positive delta, but it is much lower than a single call. As the stock rallies past the higher strike ($K_H$), the Delta of the spread drops toward zero.

2. Vega: Stability in Chaos

Since you are both long and short a call, an increase in market volatility affects both sides. Your long call gains value, but your short call becomes more expensive to buy back. They mostly cancel out, giving you a very low net Vega.

3. Theta: Slower Decay

The short call you sold is decaying every day, which helps your position. This offsets the decay of your long call, making "Theta bleed" much less painful.

Strategic Selection

• Aggressive (OTM/OTM): Buy a high strike, sell an even higher strike. Very cheap, very high leverage, but low probability.
• Moderate (ATM/OTM): Buy at-the-money, sell out-of-the-money. The balanced choice for most traders.
• Conservative (ITM/ATM): Buy deep in-the-money, sell at-the-money. Highest probability of profit, but most expensive.

Checklist for Entry

• Is my target stock price above the Short Strike by expiration?
• Is the Max Risk/Reward ratio acceptable (usually looking for 1:2 or better)?
• am I okay with the fact that I won't profit more if the stock "moons"?

Mathematical Breakdown of SpreadsRead more

The value of the spread ($V$) at any time is the difference between the two calls:

$$V = C(S, K_L, T, \sigma) - C(S, K_H, T, \sigma)$$

The Delta of the spread is the difference between the individual deltas:

$$\Delta_{\text{spread}} = N(d_{1,L}) - N(d_{1,H})$$

This explains why the delta is highest when the stock is between the two strikes. If the stock is far above both, both $N(d_1)$ values approach 1.0, and the net delta becomes $1 - 1 = 0$.