Replicating a Call Ladder to Identify an Arbitrage
Summary
The answer shows how to test a laddered option payoff for mispricing by expressing it as a portfolio of simpler calls. It first represents the payoff across strike-defined regions, then rewrites it using call payoffs that activate above each strike. For the example, the ladder is equivalent at expiry to a long lower-strike call and short calls at two higher strikes.
Under the no-arbitrage assumption, the ladder and its replicating portfolio should have the same price. The quoted prices violate that equality, so the strategy is to sell the more expensive ladder and buy the cheaper replicating calls, producing an initial credit while matching the terminal payoff. The argument depends on exact payoff replication and executable prices; transaction costs, bid-ask spreads, contract details, and funding can erase an apparent discrepancy. The example illustrates payoff algebra, not a general automated method for finding arbitrage.
Key ideas
- Piecewise option payoffs can often be rewritten as linear combinations of calls at their boundary strikes.
- Payoff equality at expiry implies equal prices under the no-arbitrage assumption.
- When the ladder is more expensive than its replicating calls, selling the ladder and buying the calls creates the stated arbitrage structure.
- Real implementation depends on executable prices, contract terms, and trading costs.
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Full text
# Finding arbitrage opportunity
# Finding arbitrage opportunity
Find an arbitrage opportunity in this market.
Can anyone explain how to mathematically solve this exercise with for example solving a system of linear equations?
## Answer by Daneel Olivaw (score 8, accepted)
https://quant.stackexchange.com/a/34888
Generally speaking, let us consider a problem where you have a series of simple payoffs $f_{K_i}(S_T)$ of strike $K_i$, $i \in I$, that depend on the value of $S_T$ at time $T$, as well as a more complex, laddered payoff $P_L(T)$ which pays a quantity $g_i(S_T)$ on regions of the form $\{K_i \leq S_T < K_{i+1}\}$ $-$ regions are delimited by the strikes of the simpler payoffs. Then the payoff of the ladder product can normally be written:
$$ P_L(T) = \sum_{i\in I} g_i(S_T)1_{\{K_i \leq S_T < K_{i+1}\}}$$
From the above representation, it is normally possible to rewrite the payoff with indicator functions that depend only on one strike:
$$ P_L(T) = \sum_{i\in I} h_i(S_T)1_{\{K_i \leq S_T\}}$$
Letting $a_i \in \mathbb{R}$ for all $i$, you will then normally observe that:
$$ h_i(S_T)1_{\{K_i \leq S_T\}} = a_if_{K_i}(S_T) $$
i.e. the payoff of the ladder product can be written as a linear combination of the simple payoffs.
In this case, note that:
$$ \begin{align} X_4(T) & =(S_T-60)\times1_{\{60 \leq S_T <80\}}+20\times 1_{\{80 \leq S_T <100\}}+(120-S_T)\times1_{\{100 \leq S_T\}} \\[6pt] &=(S_T-60)\times1_{\{60 \leq S_T\}}-(S_T-80)\times 1_{\{80 \leq S_T\}}-(S_T-100)\times1_{\{100 \leq S_T\}} \quad (1) \end{align}$$
Indeed:
$$ \begin{align} S_T \leq 0 \quad & \Rightarrow \quad X_4(T) = 0 \\[6pt] 60 \leq S_T < 80 \quad & \Rightarrow \quad X_4(T) = S_T-60 \\[6pt] 80 \leq S_T < 100 \quad & \Rightarrow \quad X_4(T) = 20 = (S_T-60)-(S_T-80) \\[6pt] 100 \leq S_T \quad & \Rightarrow \quad X_4(T) = 120-S_T = (S_T-60)-(S_T-80)-(S_T-100) \end{align}$$
$(1)$ can be rewritten as:
$$ \begin{align} (1) & = \max(S_T-60,0)-\max(S_T-80,0)-\max(S_T-100,0) \\[6pt] & = X_1(T) - X_2(T) - X_3(T) \end{align}$$
Thus:
$$ X_4(T) = X_1(T) - X_2(T) - X_3(T) $$
By no-arbitrage assumption, given the long call ladder payoff can be replicated with a portfolio constructed by buying a call $1$ and selling a call $2$ and a call $3$, both the long call ladder and the replicating portfolio should have the same price at time $t$. However this is not the case here:
$$ C_1(t)-C_2(t)-C_3(t) = 40-21.5-8.4 = 10.1 < 11 = C_4(t)$$
The arbitrage strategy consists on selling the call ladder for $11\$$ and buying the replicating portfolio for $10.1\$$, making a riskless profit of $0.9\$$ per contract.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.