Approximating a Discontinuous Derivative Payoff with Calls and Puts

Summary

The document considers a derivative payoff that equals the underlying’s initial price above a threshold and the underlying’s maturity value at or below it. Because the payoff jumps at the threshold, the author asks how to approximate it using a constant, a linear position in the underlying, a call struck below the threshold, and a put struck above it.

The included response matches the proposed portfolio’s payoff to the target in two outer regions: below the put strike and above the call strike. Equating the constant and underlying terms in those regions gives four equations for the four coefficients. The response does not verify the resulting approximation in the interval between the option strikes, and it explicitly notes that the algebra has not been checked. Thus the construction offers boundary-matching conditions, not a confirmed replication across all underlying prices.

Key ideas

• The target payoff has a jump at its threshold, so a smooth linear approximation may not reproduce it exactly.
• The proposed approximation combines cash, the underlying, a lower-strike call, and a higher-strike put.
• The response derives coefficient equations by matching payoffs in the two outer regions.
• The intermediate region is not checked, and the suggested coefficients are not confirmed as a complete approximation.

Tags

# Discontinuous derivative payoff approximation


# Discontinuous derivative payoff approximation












Consider a derivative of digital type which pays this kind of payoff at time $T$: \begin{align*} g(S_T,k) &= \begin{cases} P_0,~S_T>k \\ S_T, ~S_T\leq k \end{cases} \end{align*}

with $S_T$ being the current price of the underlying at maturity time $T$, $P_0$ the price of the underlying at the issue time 0 and $k$ - kind of the strike price with barrier feature.

Apparently, function $g$ is discontinuous at $S_T=k$ and has a jump there. The idea is to approximate it with a set options, call $c(S_T,k_1)$ and put $p(S_T,k_2)$ that have strikes: $k_1 < k < k_2$. Then, to construct a linear piece-wise function that will look as following: $$ \hat g(S_T,k_1,k_2)=a_0+a_1 S_T+a_2 c(S_T,k_1) + a_3 p(S_T,k_2). $$

The question is how to get the coefficients. Which complementary equations may be used?

## Answer by Ali (score 3)

https://quant.stackexchange.com/a/45676

We should be able to replicate the payoff exactly in each of the two regions $S_{T}\leq k_{1}$ and $S_{T}\geq k_{2}$. From the first, $$a_{0}+a_{1}S_{T}+a_{3}(k_{2}-S_{T}) =S_{T}$$ so, matching coefficients, $a_{0}+a_{3}k_{2}=0$ and $a_{1}-a_{3}=1$. From the second, $$a_{0}+a_{1}S_{T}+a_{2}(S_{T}-k_{1})=P_{0}$$ so, matching coefficients, $a_{0}-a_{2}k_{1}=P_{0}$ and $a_{1}+a_{2}=0$.

Sorry I haven't time to check this works. Hope it helps.

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