# 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?

• Note that \begin{align*} g &= P_0 \pmb{1}_{S_T > K} + S_T \pmb{1}_{S_T\le K}\\ &= P_0 \pmb{1}_{S_T > K} + S_T \left(1-\pmb{1}_{S_T> K}\right)\\ &=(P_0-K)\pmb{1}_{S_T > K} + S_T - (S_T-K)^+. \end{align*} – Gordon May 17 '19 at 15:44
• Moreover, $$\pmb{1}_{S>K} \approx \frac{(S_T-(K-\varepsilon))^+-(S_T-K)^+}{\varepsilon},$$ for a sufficiently small $\varepsilon$ (e.g., $10^{-4}K)$. – Gordon May 18 '19 at 11:36

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 $$a_{1}+a_{2}=0$$.
• Solved by $a_{0}=(P_{0}-k_{1}) k_{2}/(k_{2}-k_{1})$, $a_{1}=(k_{2}-P_{0})/(k_{2}-k_{1})$, $a_{2}=(P_{0}-k_{2})/(k_{2}-k_{1})$ and $a_{3}=(k_{1}-P_{0})/(k_{2}-k_{1})$. It would probably come out a bit nicer if you swapped the put for a call at the same strike. – Ali May 18 '19 at 12:36