Replicating a Bounded Linear Payoff with Call and Put Spreads
Summary
The document shows how to replicate a payoff that is zero below a lower strike, rises linearly between two strikes, and is capped above the higher strike. The call construction is a bull call spread: buy a call at the lower strike and sell a call at the upper strike. The difference between the calls produces the required rising segment and caps the payoff once the asset finishes above the upper strike.
It then applies put–call parity to express the same payoff using puts: buy the put at the lower strike, sell the put at the upper strike, and add a discounted cash amount tied to the difference between the strikes. This is a terminal payoff replication for European options. The explanation does not address transaction costs, early exercise, dividends, or practical trade execution; the parity expression depends on the stated discount factor and standard parity assumptions.
Key ideas
- Buying a call at the lower strike and selling a call at the upper strike creates the capped rising payoff.
- The call spread pays nothing below the lower strike and reaches its cap above the upper strike.
- Put–call parity expresses the same payoff as a put spread plus discounted cash.
- The construction describes European option payoffs at maturity, not execution costs or implementation details.
Tags
Full text
# Construct a portfolio of European call options with a certain payoff function
# Construct a portfolio of European call options with a certain payoff function
My question is similar to Replicate a Portfolio with Given Payoff but I am not quite sure how to apply this to my problem.
A portfolio of European call options on an asset $S_T$ has a payoff function given by $V_T$ where: $$V_T = 0,\ S_T < A$$ $$V_T = S_T - A , \ A \leq S_T \leq B$$ $$V_T = B - A, \ S_T > B$$
> (i) Construct a portfolio $H_1$ of European call options with this payoff function. (ii) Use Put-Call parity to construct a portfolio $H_2$ of European put options with this payoff function.
## Answer by Daneel Olivaw (score 2)
https://quant.stackexchange.com/a/45004
You can check my answer to this question for general details on how to solve this kind of problem.
Let $C_X(S_T)$ and $P_Y(S_T)$ be a call and a put option with strikes $X$ and $Y$ respectively, then: $$\begin{align} (\text{i}) \quad V_T &= (S_T-A)1_{\{A\leq S_T\leq B\}}+(B-A)1_{\{S_T>B\}} \\ &=(S_T-A)1_{\{S_T\geq A\}}+(B-S_T)1_{\{S_T\geq B\}} \\ &=(S_T-A)1_{\{S_T\geq A\}}-(S_T-B)1_{\{S_T\geq B\}} \\ &=\max(S_T-A,0)-\max(S_T-B,0) \\ &=C_A(S_T)-C_B(S_T) \\ &=H_1 \end{align}$$ Using Put-Call parity: $$\begin{align} (\text{ii}) \quad V_T &=C_A(S_T)-C_B(S_T) \\ &=\left(P_A(S_T)+S_T-D_TA\right)-\left(P_B(S_T)+S_T-D_TB\right) \\ &=(P_A(S_T)-P_B(S_T))+D_T(B-A) \\ &=H_2 \end{align}$$ where $D_T$ is the discount factor from maturity $T$ to the present time.
## Answer by AdB (score 1)
https://quant.stackexchange.com/a/45003
For question (i), you simply buy one EU call with strike A and sell one EU call with strike B - this is called a bull call spread.
Try using the put-call-parity to construct the corresponding bull put spread yourself.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.