Testing a Call Spread for Arbitrage After Financing

Summary

The example compares two calls on the same underlying and expiration: the lower-strike call costs less than the higher-strike call. Buying the lower-strike call and selling the higher-strike call produces a positive initial cash inflow, while the spread’s payoff at expiration is nonnegative across all underlying prices. The accepted explanation points out that the initial premium must be included in the terminal value, with the cash received earning interest until expiration. Combining that financed amount with the spread payoff makes the net outcome positive in every state, under the stated prices and assumptions.

The example illustrates how to check an options arbitrage by tracking cash flows from entry through expiration, rather than considering only the terminal option payoff. It is a compact theoretical case: the document does not discuss transaction costs, margin, exercise details, or whether the quoted prices can actually be traded. Those practical conditions matter when assessing a real market opportunity.

Key ideas

• A call spread’s expiration payoff is nonnegative when the lower-strike call is long and the higher-strike call is short.
• The initial net premium must be carried forward to expiration when evaluating arbitrage.
• Under the stated prices, the premium received earns interest and adds to the spread payoff.
• Real-world costs and trading constraints are not addressed in the example.

Tags

# Arbitrage opportunity between two call options with strike price \$40, \$30 and cost \$4, \$3 respectively?


# Arbitrage opportunity between two call options with strike price \$40, \$30 and cost \$4, \$3 respectively?












> Question: Given two call options $c_1$ and $c_2$ with strike price $30$ and $40$ respectively. If $c_1$ costs \$3 and $c_2$ costs \$4, is there an arbitrage opportunity?

My attempt:

Short $c_2$ and long $c_1.$ Then we make a profit of $\$4 -\$3 = \$1.$ At expiration, we have $$(S(T) - 30)^+ - (S(T) - 40)^+ = \begin{cases} 0 & \text{ if } S(T)\leq 30, \\ S(T) - 30 & \text{ if } 30\leq S(T)\leq 40, \\ 10 & \text{ if } S(T)\geq 40. \end{cases}$$ Since there is a positive probability that the payoff is nonnegative, so we have an arbitrage opportunity.

Is my attempt above correct?

## Answer by emcor (score 3, accepted)

https://quant.stackexchange.com/a/50162

No you need to subtract the cost of entering your position as well as the financing costs thereof. In this case you actually receive net \$1 option premiums which yields additional interest at maturity: $$+1\cdot e^{rT}$$ Hence the net value is always positive and represents an arbitrage.

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