Explanation

A butterfly spread using calls has the following structure:

• Buy 1 call @ X1 (low strike)
• Sell 2 calls @ X2 (middle strike)
• Buy 1 call @ X3 (high strike)

Risk Profile Analysis:

Maximum Profit:

• Occurs when the underlying price equals X2 at expiration
• Limited to: (X2 - X1) - Net premium paid

Maximum Loss:

• Occurs when the underlying price is below X1 or above X3
• Limited to the net premium paid for establishing the position

Why both upside and downside are limited:

1. Downside Risk (Price < X1):

   • All calls expire worthless
   • Loss = Net premium paid (limited)

2. Upside Risk (Price > X3):

   • The long call @ X1 and short calls @ X2 create offsetting positions
   • The long call @ X3 provides protection
   • The positions cancel out, resulting in limited loss equal to the net premium

Mathematical Proof:

• When S > X3:
  • Long call @ X1: S - X1
  • Short calls @ X2: -2(S - X2)
  • Long call @ X3: S - X3
  • Total payoff = (S - X1) - 2(S - X2) + (S - X3) = 0
  • Profit = Payoff - Net premium = -Net premium (limited loss)

Key Characteristics:

• Butterfly spreads are range-bound strategies
• They profit when the underlying stays within a specific range
• Both maximum profit and maximum loss are predetermined and limited
• This makes them suitable for low-volatility environments

Therefore, option B is correct: Both the upside and downside is limited.