Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Option Basics - Calls, Puts, and Payoffs › Combining Options: Basic Strategies

Michael BrenndoerferNovember 15, 20256 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 18Option Basics - Calls, Puts, and Payoffs › Combining Options: Basic Strategies

Reading section 9 of 15 · Combining Options: Basic Strategies · 6 minSection 9 of 15 · 6 min

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Option Basics - Calls, Puts, and Payoffs › Combining Options: Basic StrategiesOne section · Sections in this chapter9 / 15 ↓

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Individual options can be combined to create strategies with specific risk-reward profiles. These combinations isolate different sources of return, including volatility and time decay, while limiting gains or losses to chosen price ranges. Rather than simply betting that a stock will go up or down, option strategies allow you to profit from volatility, benefit from time decay, or define precise profit zones.

StraddleLink Copied

A straddle involves buying both a call and a put at the same strike price and expiration. It profits from large moves in either direction. In a straddle, you are essentially betting that the stock will move significantly but are uncertain about the direction. This makes straddles popular around earnings announcements, FDA decisions, or other binary events where large moves are expected.

In[13]:

Code

import numpy as np

K = 100
C = 5
P = 4
total_cost = C + P
S_range = np.linspace(70, 130, 200)

call_payoff = np.maximum(S_range - K, 0)
put_payoff = np.maximum(K - S_range, 0)
straddle_profit = call_payoff + put_payoff - total_cost

Out[14]:

Visualization

V-shaped profit diagram showing straddle gains from volatility in either direction. — Long straddle profit diagram. The V-shaped payoff illustrates how the strategy profits from volatility in either direction, requiring the stock price to move beyond the breakeven points to cover the combined cost of the call and put.

The straddle's V-shape shows its volatility bet. You pay a combined premium of $C + P$ and need the stock to move beyond the breakeven points ( $K \pm (C + P)$ ) to profit. Maximum loss occurs if the stock finishes exactly at the strike, where both options expire worthless. The V-shape illustrates that the straddle buyer doesn't care which direction the stock moves, only that it moves enough to cover the combined premiums paid.

Bull Call SpreadLink Copied

A bull call spread involves buying a call at a lower strike $K_1$ (paying premium $C_1$ ) and selling a call at a higher strike $K_2$ (receiving premium $C_2$ ). It's a directional bet with limited risk and limited reward. By selling the higher-strike call, you reduce the cost of the position but also cap the potential upside.

In[15]:

Code

import numpy as np

K1 = 95  # Lower strike (buy)
K2 = 105  # Higher strike (sell)
C1 = 8  # Premium paid for K1 call
C2 = 3  # Premium received for K2 call
net_cost = C1 - C2

S_range = np.linspace(80, 120, 200)

long_call = np.maximum(S_range - K1, 0) - C1
short_call = C2 - np.maximum(S_range - K2, 0)
spread_profit = long_call + short_call

Out[16]:

Visualization

Line chart showing bull call spread with profit capped between two strike prices. — Bull call spread profit diagram. The strategy defines a precise profit zone between the two strike prices, capping both maximum potential gain and maximum risk for a lower initial cost than a straight call.

The spread's characteristics are:

Maximum loss: Net premium paid ( $C_1 - C_2$ ), occurring if $S_T \leq K_1$
Maximum profit: $(K_2 - K_1) - \text{net cost}$ , occurring if $S_T \geq K_2$
Breakeven: $K_1 + \text{net cost}$

Spreads reduce cost (and risk) compared to outright calls but also cap potential gains. They're appropriate when you expect a moderate move rather than a dramatic one.

Out[17]:

Visualization

Long call profit profile. The buyer pays a premium for unlimited upside potential if the stock rises, with risk limited to the initial premium paid.

Short call profit profile. The seller collects a premium but faces unlimited potential losses if the stock price rises significantly above the strike.

Long put profit profile. The buyer pays a premium for protection against price declines, with profit increasing as the stock falls below the strike.

Short put profit profile. The seller collects a premium but bears substantial risk if the stock price falls, with losses increasing as the price approaches zero.