Interpreting Floor Payoffs and Profit Across Expiry Prices
Summary
The document examines a position consisting of an index purchase, a put with the same strike, and borrowing to finance part of the initial cost. The asset and put together create a floor: at expiration, the position’s value is bounded below by the strike and otherwise follows the index price. The central question is how to draw its payoff and profit diagrams and whether one can substitute a single assumed future index price to calculate profit.
The answer emphasizes that the expiry price is unknown when the position is opened, so profit must be expressed as a function of the realized expiry price. A graph across possible prices makes that relationship clear. The brief response does not provide the requested graph or fully work through financing and profit calculations, so it serves mainly as a conceptual correction to treating one future price as certain.
Key ideas
- A long asset combined with a put at the floor strike creates a payoff bounded below by that strike.
- Profit depends on the realized asset price at expiration and should be shown across possible outcomes.
- A single assumed expiry price cannot determine the position’s future profit in advance.
- A payoff or profit graph helps distinguish terminal value from the initial investment and financing.
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Full text
# Floor and Cap problem
# Floor and Cap problem
So I have a problem from Marcel Finan's "A Basic Course in the Theory of Interest and Derivative Markets." We are going over floors and caps, covered puts and covered calls.
Consider the following combined position:
- Buy an index for 500
- Buy a 500-strike put with expiration date 3 months with a 3-month risk free rate of 1% and premium of 41.95.
- Borrow 495.05 with 3-month interest rate of 1%.
Graph the payoff diagram and the profit diagram of this position.
So, since we are simultaneously buying an asset and a purchased put, this makes this a floor. According to Broverman, the formula for a floor's payoff is $$\text{payoff}=\begin{cases} K, & \text{if $S_T\le{K}$} \\ S_T, & \text{if $S_T\gt{K}$} \\ \end{cases} $$ So in this case we know that the cost to purchase both the asset and the put is 541.95 and we are borrowing 495.05 from the start. So is the expiry price $S_T=500(1.01)=505$? This would make the payoff $505$, since $505\gt{500}$. What about the profit? Again Broverman has the profit function as $$\text{profit}=\begin{cases} K-(S_0+P_0)e^{rT}, & \text{if $S_T\le{K}$} \\ S_T-(S_0+P_0)e^{rT}, & \text{if $S_T\gt{K}$} \\ \end{cases} $$ So since $S_T\gt{K}$ is the profit just $505-(500+41.95)(1.01)=-42.37$?
## Answer by Bob Jansen (score 1, accepted)
https://quant.stackexchange.com/a/8169
As the asker already found out:
No this is not simply the answer. $S_T$ is not known at the moment of investment so the future profit is a function of the stochast $S_T$.
A graph is a useful tool to gain insight into the amount of profit for each realized value of $S_T$.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.