Why Call and Put Spreads with the Same Strikes Have Different Prices
Summary
The document compares bull call and bull put spreads built from options with the same two strikes. Their expiration payoffs differ by a constant equal to the strike gap: the call spread has a nonnegative payoff, while the put spread has a nonpositive one. Because the payoff difference is positive in every market state, the spreads cannot have the same price without creating an arbitrage opportunity.
The pricing intuition follows from their cash flows. Establishing the call spread costs a premium and can produce a positive payoff at expiry; establishing the put spread brings in a premium and can produce a negative payoff. The document adds that put-call parity can be used to construct an equivalent version of either spread, in which case the prices match. The discussion is qualitative and does not provide a full payoff table or account for transaction costs, early exercise, or other contract details.
Key ideas
- A bull call spread has a nonnegative expiry payoff, while a bull put spread with the same strikes has a nonpositive payoff.
- The two spreads’ payoffs differ by the strike gap in every market state.
- Equal prices for these distinct payoffs would allow an arbitrage under the stated setup.
- Put-call parity can be used to construct an equivalent spread with matching price.
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Full text
# Are the price of vanilla bull/bear spread constructed by calls and puts same? # Are the price of vanilla bull/bear spread constructed by calls and puts same? We know that both bull and bear can be constructed by either two calls or two puts. Say if given two strikes, will price of bull call equal to price of bull put? ## Answer by Kevin (score 3, accepted) https://quant.stackexchange.com/a/48989 There is a difference between a put spread and a call spread. In the bull case, both strategies expect a mediocre increase in the stock price but the investment idea is different. Look for instance at the two payoffs below for $K_1=75$ and $K_2=125$. As you see, the call strategy yields a non-negative payoff whereas the put strategy pays at most zero. The difference between both payoffs is $K_2-K_1$. Since this difference is constant for all states of the world, the strategies cannot have the same price (no arbitrage principle): suppose both strategies had the same price and you sold the put strategy and purchased the call strategy. Then, you have zero initial cost but you'll receive $K_2-K_1>0$ at maturity. The call spread always gives you a positive payoff and hence costs you a positive amount of money when setting the strategy up. I.e. you need to pay a premium (the premium of the lower strike call you buy is higher then the premium of the higher strike call which you sell). It is different for a put payoff which gives you a negative payoff. Hence, you also receive a premium when setting up the strategy. So, the difference between both strategies is receiving money at the beginning and pocketing it (put) and paying money at the beginning and receiving a payoff at maturity (call). P.S. Using the put-call-parity, it is of course possible to build an equivalent put spread with call options and then the prices are identical.
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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.