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Arbitrage Bound for a European Call: Price Cannot Exceed Spot

Article Quant Q&A · Author: Stijn D'hondt
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Summary

The document examines the no-arbitrage bound that a European call’s value cannot exceed the current stock price. Its proposed contradiction argument sells the call and buys the stock, using the received premium to leave an initial surplus if the call is overpriced. At expiry, the stock covers any call exercise, and the short answer expresses the remaining payoff as the lesser of the strike and terminal stock price.

A second answer reaches the bound by noting that a call’s value decreases as its strike rises and that a zero-strike call is worth the spot price. The argument relies on the standard frictionless no-arbitrage setup and the ability to trade the stock and option as described. The document does not discuss complications such as transaction costs, constraints, or market imperfections; its result is a basic pricing inequality rather than a full option valuation method.

Key ideas

  • A European call price is bounded above by the current stock price under no-arbitrage assumptions.
  • Selling an overpriced call and buying the underlying stock creates an initial surplus.
  • At expiry, the owned stock can be delivered if the call is exercised.
  • The terminal value left after meeting the call obligation is nonnegative.
  • The document also motivates the bound through the decreasing value of a call as strike rises.

Tags

OptionsEquitiesArbitrageDerivatives pricing
Full text
# Proof European call price is always less than stock price. (proof verification)


# Proof European call price is always less than stock price. (proof verification)












> Let $C_K(t,T)$ be the value of a European call with strike $K$ and maturity $T$ on a stock with value $S_t$ at time $t$. Then for all $t\leq T$ we have $$C_K(t,T)\leq S_t.$$

$\textbf{Proof}$: We derive an arbitrage opportunity by contradiction.

Assume $$C_K(t,T)>S_t$$ for some $t\leq T$. Consider the following:

At this time $t$: write a call option with strike $K$ and maturity $T$ on the stock and buy said stock. Since we cash the premium $C_K(t,T)$ of the call we have $C_K(t,T)-S_t>0$ left.

At time $t=T$, we do the following:

- If $S_T<K$, the call is worthless and nothing happens.

- If $S_T \geq K$, the call is made and we sell the holder the stock (which we still own).

In both cases, no money is lost and we end up with money. This is an arbitrage opportunity.

Can someone check if this proof is decent and correct?

> $\textbf{Edit:}$ I did not think this was necessary, but since a comment was made I feel obliged to add the following. This is a proof by contradiction. Since the principle of arbitrage is an axiom to financial theory, all true statements (in the theory) are implied by this principle. In other words: it is enough to show that the negation ($C_K(t,T)>S_t$) is in contradiction with the principle of arbitrage.

## Answer by Gordon (score 5)

https://quant.stackexchange.com/a/50749

What you need to note is the following: \begin{align*} S_T - \max(S_T-K, \,0) &= S_T + \min(K-S_T, \,0)\\ &=\min(K, \, S_T) >0. \end{align*}

## Answer by Valometrics.com (score 2)

https://quant.stackexchange.com/a/50748

The call price is decreasing with respect to the strike so for every strike the value of the option is inferior than the value for strike equal to zero which is the spot price.

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.