Setup

We are given prices for European put options on the same underlying asset and maturity with the following strikes and prices:

\[ P(30) = 6, \quad P(20) = 4, \quad P(0) = 0 \]

Let \( P(K) \) be the price of the put option with strike \( K \). Then, the points \( (K, P(K)) \) lie on the line:

\[ P(K) = \frac{2}{3}K \]

This implies a linear relationship, contradicting the convexity of put prices as a function of strike price. This leads to an arbitrage opportunity.

Arbitrage Strategy

The arbitrage arises because the price of the put at strike 20 is too high compared to what convexity implies.

We can exploit this using the following zero-cost portfolio:

  • Long 2 puts with strike 30
  • Short 3 puts with strike 20

This results in no initial cost:

\[ 3 \cdot $4 - 2 \cdot $6 = $12 - $12 = $0 \]

Payoff at Maturity

Let \( S_T \) be the underlying asset price at maturity. The portfolio payoff is:

  • Long 2 puts at strike 30: \( 2 \cdot \max(30 - S_T, 0) \)
  • Short 3 puts at strike 20: \( -3 \cdot \max(20 - S_T, 0) \)

So the net payoff is:

\[ V(S_T) = 2 \cdot \max(30 - S_T, 0) - 3 \cdot \max(20 - S_T, 0) \]

Let’s analyze key intervals:

  • If \( S_T \geq 30 \): payoff = 0
  • If \( 20 \leq S_T < 30 \): payoff = \( 2(30 - S_T) - 3(20 - S_T) = 60 - 2S_T - 60 + 3S_T = S_T \)
  • If \( S_T < 20 \): payoff = \( 2(30 - S_T) - 3(20 - S_T) = 60 - 2S_T - 60 + 3S_T = S_T \)

Hence:

\[ V(S_T) = \begin{cases} 0 & \text{if } S_T \geq 30
S_T & \text{if } S_T < 30 \end{cases} \]

Conclusion

This portfolio costs nothing to set up and yields a non-negative payoff, strictly positive when \( S_T < 30 \). Thus, we have identified an arbitrage opportunity due to the violation of put price convexity.

Reference