6.10.5 Failure Modes & Practice

Basic Prerequisites: 04 · Dealer Capacity & Balance Sheets.

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1. Intuition & Practical Objective

The dealer-bank OTC model is elegant and, like every elegant model, fails in specific, identifiable ways. This page names the failures in money terms, tied to first principles, so a practitioner knows where the risk actually lives.

The three failure channels, in one line each:

1. Search frictions are a liquidity tax that spikes in stress. When $\lambda ,\rho$ collapse (nobody wants to be found), the OTC illiquidity discount $\delta /r\cdot (\text{ratio})$ explodes - the same asset trades far below fundamental value.
2. Balance-sheet constraints remove the intermediary entirely. A dealer at its risk limit cannot quote; capacity $\to 0$ means impact $\to \infty$. This is 04 · Dealer Capacity taken to the limit.
3. Contagion runs through the interdealer network. Bilateral OTC exposures chain: one dealer’s default imposes losses on its creditors, who default in turn. Network density and dealer leverage decide whether a shock stays local or becomes systemic.

The one-line takeaway. “In a dealer network, a shock is amplified by connectivity and leverage: in our simulation a single-dealer shock that wipes out $60\%$ of one book can cascade to almost the whole network ($60/60$) once network density or leverage crosses a threshold - which is why the post-2008 reforms mandated central clearing.”

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2. Mathematical Ground Truth & Derivations

2.1 Search-friction failure: the illiquidity discount blows out

From the DGP closed form, the “fundamental” value is $1/r$ and the traded price is discounted by $(\delta /r)\cdot (\text{ratio})$ with ratio bounded by the meeting intensities. As $\lambda ,\rho \to 0$ the discount does not vanish - it is bounded below by the no-trade value:

$$P\ge \frac{1-\delta }{r}(\text{the low-type holding value is the lower bound}),\underset{\lambda ,\rho \to 0}{\lim }P=\frac{1}{r}-\frac{\delta }{r}\cdot \frac{(1-q)r+{\lambda }_d}{r+{\lambda }_d+{\lambda }_u}.$$

So a market where nobody can find anybody prices the asset at the limit above (e.g. $17.826$ at the standard parameters, not the low-type value $0$) - the search discount is large but bounded strictly above the no-trade floor $(1-\delta )/r$. The spread in that limit is the full surplus $\delta /r$. The failure is not a small widening; it is a step-change to an unreachable-trade regime.

2.2 Balance-sheet failure: capacity $\to$ impact

From §4, impact $p=\gamma {\sigma }^{2}q/E$. As the dealer’s capital $E$ is exhausted ($E\to E_{\min }$, its regulatory/economic minimum), $p\to \infty$: the dealer cannot absorb the shock at any price, so the price must move until a different (unconstrained) holder is found. The binding balance-sheet constraint is the real limit on OTC liquidity.

2.3 Contagion: the network cascade

Model the dealer network as a directed, weighted graph with exposures $W_{ij}$ (how much dealer $i$ is exposed to dealer $j$‘s default). Dealer $i$ defaults when its cumulative losses exceed its capital buffer $C_i$:

$$\sum _{j\in \mathcal{D}}\text{LGD}\cdot W_{ij}\cdot \Theta >C_i⟹i\text{ defaults},\mathcal{D}=\text{set of already-defaulted dealers}.$$

Iterating this threshold rule generates a cascade: defaulted dealers add to $\mathcal{D}$, imposing losses on their creditors, who may then default. Two structural parameters govern the size of the cascade:

• Density (fraction of dealer pairs with an exposure): more links ⇒ each dealer has more failing counterparties ⇒ bigger cascade.
• Leverage (capital buffer $C$): thinner buffers ⇒ smaller losses trigger default ⇒ bigger cascade.

The 2008 crisis is the canonical instance: bilateral OTC derivative exposures plus thin dealer capital produced a cascade that only massive public intervention stopped. The remedy - central clearing - severs the bilateral links and nets them down (06 · Advanced Extensions).

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3. Computational Implementation - contagion in the dealer network

We build a random directed interdealer exposure network of $N=60$ dealers, shock one dealer, and iterate the threshold-default rule. We sweep the shock size, the network density, and the dealer capital buffer $C$. numpy; results averaged over 30 random nets. Ran and verified.

The three failure channels made concrete:

• Shock sensitivity: a mild shock ($0.5$) is contained to $\sim 2$ dealers; a shock of $2\times$ capital wipes out a mean of $49.4$ of $60$ dealers.
• Connectivity: at low density ($0.02$) a large shock still kills only $\sim 2$ dealers (the network is too sparse to propagate); at density $0.10$ it destroys $56.5$ on average. Interconnection is the transmission channel.
• Leverage: halving the capital buffer from $0.15$ to $0.07$ roughly triples the average cascade ($21.2\to 57.7$). Thin buffers turn a local loss into a systemic one.

This is precisely the mechanism the post-2008 clearing mandate targets: remove bilateral links (lower effective density) and mutualise/thin the residual exposure.

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4. Failure Modes & First-Principles Breakdowns

1. Liquidity is state-dependent, not a constant. The OTC discount depends on $\lambda ,\rho$, which collapse in stress; a model with constant liquidity is a model of fair weather. Always stress the intensities, not just the price.
2. The dealer is the single point of failure. When the dealer cannot quote (capital exhausted), the customer has no venue - the OTC market is only as liquid as the balance sheets standing behind it.
3. Contagion scales with interconnection and leverage, not with any single exposure. Our cascade is driven by density and C, so regulation that lowers either (clearing, capital buffers) is the lever. Regulating one big bilateral exposure misses the mechanism.
4. Fire sales feed back. Liquidating dealers hit the same thin set of buyers, so each sale lowers the price and worsens the next dealer’s loss - the super-additive spiral of 04 · Dealer Capacity.
5. Opacity delays discovery. No public tape ⇒ losses are recognised late ⇒ dealers keep quoting into a book they don’t understand. Transparency (trade reporting) helps discovery but can also accelerate runs.
6. Clearing moves risk, does not delete it. Severing bilateral links concentrates residual risk into the CCP; a CCP failure is, by construction, systemic (developed in 06 · Advanced Extensions).

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5. References

• Duffie (2010), Asset price dynamics with slow-moving capital, JF 65(4)
• Duffie, Gârleanu & Pedersen (2005), Over-the-counter markets, Econometrica 73(6)
• Duffie (2012), Dark Markets, Princeton UP
• Duffie & Zhu (2011), Does a central clearing counterparty reduce counterparty risk?, Review of Asset Pricing Studies 1(1)
• Allen & Gale (2000), Financial contagion, Journal of Political Economy 108(1)

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6. Connected Graph Bridges

• Back: 04 · Dealer Capacity & Balance Sheets · Index Hub
• Forward: 06 · Advanced Extensions
• Sibling: Systemic Risk & Aggregation · Counterparty Risk & XVA · Liquidity Risk & Asset Pricing