No-Arbitrage Upper Bound for a European Call Option
Summary
The document asks why an arbitrage argument implies that a European call’s price cannot exceed the underlying share price. It compares a short call and long stock position with an alternative profit calculation that also includes investing the initial cash difference and accounting for the stock value at expiry. The question highlights that the derivation depends on defining the portfolio cash flows consistently across time.
The excerpt itself does not resolve the apparent discrepancy. It ends with a brief reply saying the questioner independently reached the same alternative calculation and had found a related discussion, but gives no derivation or conclusion. Thus, the material raises a useful option-pricing issue involving arbitrage bounds and profit accounting, yet supplies little evidence or practical guidance. It also does not address assumptions such as dividends, financing, or the precise timing and settlement of cash flows, which matter when formulating an arbitrage portfolio.
Key ideas
- A European call price is subject to a no-arbitrage upper bound relative to the underlying asset.
- Arbitrage profit calculations must account consistently for cash flows at initiation and expiry.
- The excerpt questions whether proceeds from selling an overpriced call should be invested until expiration.
- The provided reply does not complete the derivation or explain the assumptions behind the bound.
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Full text
# Derivation for call option upper bound
# Derivation for call option upper bound
In Euan Sinclair's book, Option Trading, he writes that $c <= S$, the price of a European call must be lower than the price of the underlying stock. To prove it, he applies the principle of no arbitrage:
> Imagine a European call is trading for more than the underlying. We then choose to sell the call and buy the underlying at a price of $S_0$. At expiration (time $T$) our profit will be $c - (S_0 - S_T)$. But the second term has to be less than $S_0$, and our assumption was that $c > S_0$, so this profit must be greater than zero. We have to make money. So for there to be no arbitrage we need to have $c <= S$.
When I put myself in the arbitrageur's shoes, I imagine myself selling the call, buying the underlying, invested the proceeds at $r$ until expiry, then selling the underlying, all of which gets me the following profit:
$$ PL = (c-S_0)\,e^{\,r\,T}+\text{min}\,(K,S_T) $$ I have looked in every book I can find in order to understand why Sinclair's $$ PL = c-(S_0-S_T) $$ is a better equation for arbitrage profits on an overpriced ($c>S_0$) European call than $$ PL = (c-S_0)\,e^{\,r\,T}+\min\,(K,S_T) $$ but most books do not even bother starting with the profit equation when deriving an upper bound for an overpriced European call, except Sinclair. I like his approach of systematically thinking in terms of arbitrage, and I can derive the other bounds, but this one has me flummoxed.
Why is my, more complicated profit equation wrong or inaccurate?
## Answer by bigrig78 (score 1)
https://quant.stackexchange.com/a/65495
I literally was going through this exact same problem going over it for too long. I came up with what you had independently so I think we are right.
Just found this:Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.