Bull Call Spread Payoff Bounds

Article Quant Q&A · Author: M00000001

Summary

The document examines the payoff bounds of a bull call spread, formed by buying a call at a lower strike and selling a call at a higher strike. It states that the spread’s terminal payoff cannot exceed the difference between the strikes, which also bounds its initial value after discounting. The accepted explanation establishes a second payoff bound: the payoff is no greater than the terminal stock price multiplied by the strike difference divided by the higher strike.

The argument checks three regions for the terminal stock price: below the lower strike, between the strikes, and at or above the higher strike. In each region, it compares the spread payoff with the proposed stock-price-based bound. The document raises separate questions about whether geometric Brownian motion is equivalent to a lognormal distribution and whether Itô’s lemma requires that assumption, but it provides no answers to them. The payoff inequalities are shown for a standard spread with positive strikes; the discussion does not address broader option-pricing assumptions or valuation beyond these bounds.

Key ideas

A bull call spread buys a call at a lower strike and sells a call at a higher strike.
Its terminal payoff is capped by the difference between the two strikes.
The payoff is also bounded by terminal stock price times the strike difference divided by the higher strike.
The second bound follows by checking the payoff separately across three terminal-price regions.
The document raises, but does not resolve, questions about geometric Brownian motion and Itô’s lemma.

Tags

Options Derivatives pricing

Full text

# Boundaries for Call Spread


# Boundaries for Call Spread












I'm reading an interview book called A Practical Guide to Quantitative Finance Interview and I have some doubts regarding part of its solution and highlighted them in bold:

Question:

What are the price boundaries for a bull call spread?

Solution:

A bull call spread is a portfolio with two options: long a call $c_1$ with strike $K_1$ and short a call $c_2$ with strike $K_2$ and $(K_1<K_2)$. The cash flow a bull spread is summarized in the attached screenshot.

Since $(K_1<K_2)$, the initial cash flow is negative. Considering that the final payoff is bounded by $K_2-K_1$, the price of the spread, $c_1-c_2$, is bounded by $e^{-rT}(K_2-K_1)$.

But it also says (here is where my doubt is), the payoff is also bounded by $\frac{(K_2-K_1)S(T)}{K_2}$, but why? Could anyone share some advice on it? Really appreciate it!

By the way, another two questions (might be a stupid one) that is not related to the above question:

Question A:

As we know, one assumption for Black Scholes equation is underlying asset follows geometric Brownian motion, but can we say it is equivalent to "underlying follows lognormal distribution"? In other words: geometric Brownian motion is equivalent to lognormal distribution

Question B

And in order to use Ito lemma for those products that are derivatives on underlying asset, we need to make sure the underlying asset follows geometric Brownian motion?

## Answer by Canardini (score 3, accepted)

https://quant.stackexchange.com/a/50122

If $S_T<K_1$, the payoff is zero, and we have $\frac{(K_2-K_1)S(T)}{K_2} \geq0$

If $K_1 \leq S_T<K_2$, the payoff is $(S_T -K_1)$. We have $$K_1K_2 \geq S_TK_1$$

and $$S_TK_2+K_1K_2 \geq S_TK_2+S_TK_1$$

Thus, $$S_TK_2-S_TK_1 \geq S_TK_2-K_1K_2$$

Finally, $$S_T(K_2-K_1) \geq (S_T-K_1)K_2$$ $$\frac{S_T(K_2-K_1)}{K_2} \geq (S_T-K_1)$$

If $S_T \geq K_2$,we have that $\frac{S_T}{K_2} \geq1$, and because the payoff is $K_2 -K_1$, therefore $K_2 -K_1 \leq (K_2 -K_1)\frac{S_T}{K_2}$

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

Payoff Bounds for a Bull Call Spread

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