Michael Brenndoerfer | Data & AI, Private Equity, Technology

Michael Brenndoerfer

Option Basics - Calls, Puts, and Payoffs › Leverage and Non-Linear Payoffs

Michael BrenndoerferNovember 15, 20253 min read

Quantitative FinanceAll chapters

Part 3 of 15Forwards, Futures, and Options

Chapter 18Option Basics - Calls, Puts, and Payoffs › Leverage and Non-Linear Payoffs

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Sections in this chapter15

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Options provide substantial leverage because a small premium controls exposure to a much larger position. This leverage works both ways: profits can multiply quickly, but so can losses (for sellers). Understanding leverage is essential for position sizing and risk management.

In[18]:

Code

S0 = 100  # Initial stock price
K = 100  # Strike price
C = 5  # Call premium

# Investment: \$100 in stock OR \$100 in calls (20 contracts)
stock_investment = 100
option_investment = 100
num_calls = option_investment / C  # 20 call contracts

# Scenarios at expiration
scenarios = [90, 95, 100, 105, 110, 115, 120]
stock_returns = []
option_returns = []

for S_T in scenarios:
    # Stock investment value
    stock_value = stock_investment * (S_T / S0)
    stock_returns.append(
        (stock_value - stock_investment) / stock_investment * 100
    )

    # Option investment value
    call_payoff = max(S_T - K, 0)
    option_value = num_calls * call_payoff
    option_returns.append(
        (option_value - option_investment) / option_investment * 100
    )

Out[19]:

Visualization

Bar chart comparing stock and option returns across different price scenarios. — Return comparison for stock versus call option investment. The call option offers substantial leverage, generating magnified percentage returns in upside scenarios but resulting in a 100% loss if the stock fails to rise, contrasting with the linear returns of stock ownership.

In[20]:

Code

# Compare returns at different expiration prices
S0 = 100
K = 100
C = 5
investment = 100

num_calls = investment / C  # $100 buys 20 calls at $5 each

# If stock goes to $110
S_T = 110
stock_return = (S_T - S0) / S0
option_payoff = max(S_T - K, 0) * num_calls
option_return = (option_payoff - investment) / investment
leverage_factor = option_return / stock_return

Out[21]:

Console

If stock moves from $100 to $110:
  Stock return: 10.0%
  Option return: 100.0%

  Leverage factor: 10.0x

The leverage cuts both ways. If the stock stays flat or declines, you lose your entire investment while the stock investor preserves most of their capital. This asymmetry makes options powerful tools for speculation and hedging, but they require careful position sizing and risk management.