Suppose the current price of a stock $X$ is $40$ and we are considering two $3$-month puts on $X$, strikes $K$ and $K+1$. What would be the upper bound for the price difference of these two puts?
I have tried to compute the difference of payoffs for the cases when $S < K$, $K <S<K+1$, and $S> K +1$.
$p = max(K-S, 0) $
| Scenario | Payoff $P_{K}$ | Payoff $P_{K+1}$ |
|---|---|---|
| $S < K$ | $(K-S)$ | $(K-S)+1$ |
| $K <S<K+1$ | $0$ | $(K-S)+1$ |
| $S> K +1$ | $0$ | $0$ |
How should I proceed to find the upper bound for the price difference of these two puts.
For European style put option with a strike price of $k$, interest rate $r$, and time to maturity $T$. We have that it cannot be worth more than the discounted value of the strike price.
$$P \leq k\exp(-rT).$$
Then the upper bound on the on price differences between two puts with different strikes would be the present value of the difference in strike prices
$$(k + 1 - k)\exp(-rT) = \exp(-rT).$$
If the options are American style puts it is just $(k + 1 - k) = 1$, we do not have to discount the value of the difference, because the American puts can be exercised immediately, not just at expiration, if there is a arbitrage opportunity.