option strategies - Replicate a Portfolio with Given Payoff

Looking for a convincing general strategy [not trial and error] to solve these kind of questions:

Any help will be super helpful!

Thanks a bunch!

Replicate a portfolio on an underlying asset $S$ with payoff at time $T$ equal to:

$$ \begin{align} V(T) & = 2S(T) + 30 & & \text{if } 0 \leq S(T) < 10 \\[6pt] V(T) & = -3S(T) + 80 & & \text{if } 10 \leq S(T) < 30 \\[6pt] V(T) &= S(T) − 40 & & \text{if } 30 \leq S(T) \end{align}$$

Answer

Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function $V \left( S_T \right)$ over $n$ intervals with $n + 1$ node points $S_i$ for $i = 0, 1, \ldots, n$. Without loss of generality, we assume that $S_0 = 0$ and write $V_i$ as short-hand for $V \left( S_i \right)$. We assume that the slope of the payoff function for $S > S_n$ is $x_{n + 1}$.

Take the following steps in order to replicate this payoff:

1. Buy zero-coupon bonds with a notional value of $V_0$.


2. For each $i \in 1, \ldots n$, buy $x_i = \left( V_i - V_{i - 1} \right) / \left( S_i - S_{i - 1} \right)$ European call options with a strike of $S_{i - 1}$ and sell the same amount withe a strike of $S_i$.


3. Buy $x_{n + 1}$ European call options with a strike of $S_n$.

All contracts mature at time $T$.

---

Applying this to your example, we have $n = 2$ and obtain the following portfolio:

1. Buy zero-coupon bonds with a notional value of 30 USD.


2. Buy 2 call options with a strike of 0 USD and sell 2 call options with a strike of 10 USD.


3. Sell 3 call options with a strike of 10 USD and buy 3 call options with a strike of 30 USD.


4. Buy one call option with a strike of 30 USD.

Our net positions are thus:

1. Long a zero-coupon bond with with a notional value of 30 USD.


2. Long 2 zero-strike call options.


3. Short 5 call options with a strike of 10 USD.


4. Long 4 call options with a strike of 30 USD.

---

Note that this decomposition is not unique as you can always apply put/call parity to any of the positions.