Replicating a Discrete Stock State with Call or Put Spreads

Summary

The document examines how to isolate the state price for a particular expiration value when the underlying can finish only at integer prices. It corrects a proposed expression involving calls and puts and gives equivalent portfolios: a second difference of call prices at strikes around the target state, or the corresponding second difference of put prices. These portfolios create a payoff concentrated at the selected state, allowing its price to be inferred from traded option prices under frictionless-market assumptions.

The discussion is brief and provides no derivation of the payoff or numerical example. Its central lesson is that the proposed mixed call-put expression is incorrect, while the call-only and put-only spreads yield the same payoff. The result depends on the stated discrete set of possible prices and European options spanning the relevant strikes; it does not address transaction costs, discrete strike availability, or more general continuous outcome spaces.

Key ideas

• A second difference of call prices at neighboring strikes isolates a discrete terminal-price state.
• The equivalent second difference of put prices produces the same state-contingent payoff.
• The proposed expression combining calls at neighboring strikes with a put at the target strike is identified as incorrect.
• The argument assumes a frictionless market with European options across the required strikes and discrete integer outcomes.

Tags

# State price of a stock expresed by a portfolio of calls and puts


# State price of a stock expresed by a portfolio of calls and puts












> Suppose a competitive, frictionless market provides European call options on an asset with current price $S0$ for all strike prices $K$ at market price $C(K)$ and European put options for all strike prices $K$ at market price $P(K)$. The asset at the date of the option expiration can only take discrete values $S=1,2,\cdots$ . What is the state price for state s? The answer is: $C(s−1)+C(s+1)−2P(s)$

Question: I have been thinking about this question but I cannot see why that is the right answer. How would that expression replicate the pay-off? I do not think this would require the binomial tree approach.

Thanks in advance.

## Answer by MrLCh (score 2, accepted)

https://quant.stackexchange.com/a/77721

I think the correct solution should be:

$$C(s-1) + C(s+1) - 2C(s)$$ or (yielding the same payoff function):

$$P(s+1) + P(s-1) - 2P(s)$$

The answer you were given does not replicate the correct payoff.

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