Put Spread Bounds and Arbitrage Reasoning

Summary

The document asks whether the difference between prices of two European-style puts with strikes separated by an amount s must be at least the discounted strike difference. It frames the question by analogy with a call-spread bound: if the call price difference exceeded the discounted strike gap, a portfolio of calls and cash could create an arbitrage.

The excerpt contains the question and the proposed call-side argument, but no answer or proof for the put inequality. It therefore does not establish that the stated lower bound holds, nor specify assumptions such as exercise style, dividends, or market frictions. Its useful content is the setup for examining option price monotonicity and spread bounds through payoff comparisons and arbitrage arguments; resolving the put case requires a separate analysis under explicit pricing assumptions.

Key ideas

• The question concerns a lower bound on the price difference between puts with adjacent strikes.
• The document compares this with an upper bound for call price differences derived through arbitrage reasoning.
• A call portfolio and invested cash are offered as the intuition for the call-side bound.
• No answer is provided for the put inequality, so the claimed bound remains unverified in the excerpt.

Tags

# Is it necessary for $P(K, t) - P(K + s, t) \geq se^{-rt}$ to hold?


# Is it necessary for $P(K, t) - P(K + s, t) \geq se^{-rt}$ to hold?












Let $P(K, t)$ be a put option with strike price $K$ and expiration time $t$. Let $s > 0$. Is it necessarily true that the inequality

$$P(K, t) - P(K + s, t) \geq se^{-rt}$$

holds? I know that for a call option $C(K, t)$, it can be shown that the inequality $C(K, t) - C(K + s, t) \leq se^{-rt}$ must hold, otherwise there is an arbitrage. In particular, this arbitrage can be obtained by selling a call option with strike price $K$ and exercise time $t$, buying a $C(K + s, t)$ call option, and putting the remaining $C(K, t) - C(K + s, t) \geq se^{-rt}$ in a bank.

So, I was wondering whether there is a similar justification for the inequality I mentioned above to hold.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.