No-Arbitrage Bounds & Strategies

Medium3–4 hrsArbitrageBoundsSpreadsConvexity

Overview

Option bounds require only no-arbitrage — no model needed. These bounds come from simple portfolio dominance arguments: if portfolio A always pays at least as much as portfolio B at expiry, then A must cost at least as much today. Violations mean free money. In interviews, the question is almost always: 'Are these prices consistent?' — and your answer is to check each bound systematically.

How to Recognize

→'Given these option prices, can you make money?'
→Asked for bounds on option prices without a model
→Calendar spreads, butterfly spreads, or vertical spreads
→Static arbitrage vs. dynamic hedging

Step-by-Step Approach

1.Intrinsic value bound: C ≥ max(S−K·e^(−rT), 0); P ≥ max(K·e^(−rT)−S, 0)
2.Monotonicity: C(K₁) ≥ C(K₂) for K₁ < K₂ (lower strike = more valuable call)
3.Convexity: option price is convex in strike — butterfly spread must cost ≥ 0
4.If any bound is violated: construct the explicit arbitrage portfolio

Key Formulas

C \geq \max(S - Ke^{-rT}, 0)

and

C \leq S

(call bounds)

P \geq \max(Ke^{-rT} - S, 0)

and

P \leq Ke^{-rT}

(put bounds)

03Butterfly spread ≥ 0:

C(K_1) - 2C(K_2) + C(K_3) \geq 0

for

K_1 < K_2 < K_3

04Calendar spread:

C(T_2) \geq C(T_1)

for

T_2 > T_1

(longer dated ≥ shorter dated)

Worked Examples

0Quick Example

Call with K=

90 costs

5 on stock at

100. Call with K=

100 also costs

5. Arbitrage: buy K=

100, sell K=

90, pocket spread. At expiry,

90 call is always at least as valuable.

Problem

Calls: K= $90→C=$ 15, K= $100→C=$ 7, K= $110→C=$ 4. Is this arbitrage-free?

Solution

Butterfly spread: buy K=90 call, sell 2×K=100 calls, buy K=110 call.

Cost = 15 − 2·7 + 4 = 15 − 14 + 4 = $5 > 0. No arbitrage here.

If cost were negative (say: K=90→ $15, K=100→$ 12, K=110→ $4): cost = 15−24+4 = −$ 5.

Negative butterfly means you get $5 today. At expiry, butterfly always pays ≥ 0.

Example payoff: S= $100 at expiry: (100−90)−2·(100−100)+0 =$ 10 ≥ 0. Free $5 today + nonneg at expiry = arbitrage.

Answer

Butterfly cost = C(90) − 2C(100) + C(110) = $5 ≥ 0: consistent. If butterfly were negative, you'd pocket the spread AND have non-negative payoff — pure arbitrage.

Common Mistakes

!
Forgetting to discount the strike. Bounds use K·e^(−rT), not K.
!
Confusing American and European bounds. American call on non-dividend stock: C_A ≥ S − K (undiscounted). European: C_E ≥ S − K·e^(−rT).
!
Confusing the butterfly spread direction. The standard butterfly buys the wings and sells the body — NOT the reverse.

Practice Problems

Click "Show Answer" to reveal

A call option trades at $12 on a stock that trades at$ 10. Strike is $5. Is this arbitrage-free?

Hint: Check: does C ≤ S hold?

A 6-month call costs $8. A 3-month call (same strike, same stock) costs$ 9. Is this a violation?

Hint: Calendar spread: longer-dated call ≥ shorter-dated call.

Back to Study Plan

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