23Request for Quote and Dealer Markets in Theory

A client who asks five dealers for a price on a corporate bond gets a better price than one who asks two, up to a point: every dealer asked learns that someone is selling, and the ones who lose the request can trade on it. This chapter gives the theory behind the markets that have no central book: a search-and-bargaining model of dealer spreads, the request for quote as an auction with private values, common values and leakage, the number of dealers a client should ask, and the networks and relationships through which dealers trade.

23.1 Search and bargaining

Duffie, Gârleanu and Pedersen (2005) computed the prices at which investors trade with dealers in a dynamic model with search and bargaining, and found that bid–ask spreads are lower when investors can more easily find other investors or reach several dealers. This chapter solves a version of their dealer market (firm_otcsearch.dgp). An asset pays one a year; investors are of a high type or a low type, the low type paying a holding cost $\delta$; types switch at rates $\lambda_u$ (to high) and $\lambda_d$ (to low). Low-type owners sell to dealers and high-type non-owners buy from them, meeting a dealer at rate $\rho$ (a Poisson process, One Quant Book 4, chapter 6); dealers lay off every trade in an inter-dealer market and keep a share $z$ of each trade’s surplus. With a supply $s$ below the high types’ share $\lambda_u/(\lambda_u+\lambda_d)$, buyers are the long side and the inter-dealer price is their reservation value.

In steady state the low-type owners flow in from switching and out to dealers and to the high type: $\lambda_d\mu_{ho}=(\lambda_u+\rho)\mu_{lo}$, with $\mu_{ho}+\mu_{lo}=s$. The reservation values $\Delta V_\sigma$ (the value of owning over not owning, for type $\sigma$) solve two linear equations (the Bellman equations of a continuous-time Markov chain, One Quant Book 4, chapter 8):

and the ask is $\Delta V_h$, the bid $z\Delta V_l+(1-z)\Delta V_h$, the spread $z(\Delta V_h-\Delta V_l)$.

    phi = lam_u / (lam_u + lam_d)
    if s >= phi:
        raise ValueError("this version needs s below the high types' share")
    mu_lo = lam_d * s / (lam_u + lam_d + rho)   # in from switching, out to high and dealers
    mu_ho = s - mu_lo
    mu_hn = phi - mu_ho
    mu_ln = 1 - phi - mu_lo
    # r dVh = 1 + lam_d (dVl - dVh) - rho (dVh - A), A = M = dVh
    # r dVl = 1 - delta + lam_u (dVh - dVl) + rho (1 - z) (M - dVl)
    k = rho * (1 - z)
    a = np.array([[r + lam_d, -lam_d], [-lam_u - k, r + lam_u + k]])
    dvh, dvl = np.linalg.solve(a, [1.0, 1.0 - delta])
    ask = dvh
    bid = z * dvl + (1 - z) * dvh
    return {"mu": (mu_ho, mu_hn, mu_lo, mu_ln), "dV": (dvh, dvl), "bid": bid, "ask": ask,
            "spread": ask - bid, "spread_share": (ask - bid) / ((ask + bid) / 2)}

With $r=5\%$, $\lambda_u=2$ and $\lambda_d=0.2$ a year, $s=0.8$, $\delta=2$ and $z=0.8$, the spread falls from 210 basis points of the price when investors meet a dealer ten times a year to 36.7 at a hundred and 4.0 at a thousand (Figure 23.1): faster search leaves low types less time to bear their holding cost, which is the surplus dealers bargain over.

23.2 Dealer networks

Li and Schürhoff (2019) found that municipal bond dealers form such a network, with 10 to 30 hubs and over 2 000 peripheral broker-dealers; bonds flow from the periphery to the core and partly back; central dealers charge investors up to double the round-trip markups of peripheral ones but provide immediacy by matching buyers and sellers more directly. The chapter’s toy network (core_periphery) has 10 core dealers and 200 peripheral ones linked to two core dealers each; a bond passes along the shortest path from the seller’s dealer to the buyer’s, and each dealer on the path charges a markup, 4 basis points in the core and 2 in the periphery. A client of a core dealer is reached in 1.77 hops on average and pays 9.2 basis points; a client of a peripheral dealer, 2.58 hops and 10.4. The chain’s length is part of the price; the empirical finding that the core charges more comes from its markups, which this toy sets by hand.

23.3 The request for quote as an auction

A request for quote (One Quant Book 2, chapter 22) is a first-price sealed-bid auction among the dealers asked (One Quant Book 4, chapter 29). Each dealer’s value for the bond has a private part (its inventory, its axes: standard deviation $\sigma_p$) and it estimates the common value with an error ($\sigma_c$). The client sells to the best bid, and each losing dealer, who now knows that a seller is about, costs the client a leakage $\ell$ on average. A dealer who wins when its estimate of the common value is the highest has overestimated it: it shades its bid by $\sigma_c\E[\max\text{ of }n]$ (the winner’s curse, One Quant Book 2, chapter 30, and chapter 22’s risk bids). The client’s expected cost below the common value is

with $m$ the dealers’ markup: competition helps through the private values, the common part mostly cancels against the shading, and every extra dealer adds leakage (rfq_cost, on Book 2’s firm.rfq; a simulation of the auction agrees to 0.001 basis points).

def rfq_cost(n: int, markup: float, sigma_p: float, sigma_c: float, leak: float) -> float:
    sd = math.sqrt(sigma_p**2 + sigma_c**2)
    return expected_cost(n, markup, sd, leak) + sigma_c * expected_max_normal(n)


def best_n(markup: float, sigma_p: float, sigma_c: float, leak: float,
           n_max: int = 20) -> int:
    def cost(n):
        return rfq_cost(n, markup, sigma_p, sigma_c, leak)
    return min(range(1, n_max + 1), key=cost)

23.4 How many dealers to ask

With a markup of 10 basis points, $\sigma_p=5$ and $\sigma_c=2$, Figure 23.2 draws $C(n)$ for four leakage costs, and Table 23.1 gives the optimum.

The optimum falls fast: from 16 dealers when a loser costs a tenth of a basis point to 4 at half a basis point and a single dealer from 2 basis points. The common-value part matters too: at half a basis point of leakage, a client should ask 6 dealers if the dealers’ estimates had no common error, 4 with $\sigma_c=2$ and 3 with $\sigma_c=5$, because the common part adds noise that the winner shades away without adding competition. Riggs, Onur, Reiffen and Zhu (2020) found that customers on the two largest dealer-to-customer swap execution facilities for index CDS typically contact few dealers, and modelled why: the benefit of wider exposure is mitigated by the winner’s curse and by customer–dealer relationships.

23.5 Relationships and information chasing

In a competitive quote, a dealer’s spread to a client is its cost plus the adverse selection it expects, less the value of what it learns. With a cost of 2 basis points and adverse selection of 1.5 on an informed client’s trade, the informed client is quoted 3.5 when its information is worth nothing to the dealer, the same 2.0 as an uninformed client when it is worth 1.5, and 0.5 when it is worth 3 (chase): the informed client gets the better price. Relationships work the same way: Hendershott, Li, Livdan and Schürhoff (2020) found in insurers’ corporate bond trades that execution prices first improve with the number of dealers in a client’s network and worsen beyond about twenty, as repeat business with a few dealers trades off against competition among many; Di Maggio, Kermani and Song (2017) found that dealers charge lower spreads to the dealers with whom they have the strongest ties, more so in turmoil, and that systemically important dealers charge peripheral dealers and clients more.

23.6 Tutorial: dealers, searches and auctions

23.7 Build: OTC search and auctions

23.8 Exercises

23.9 Problem: How Many Dealers to Ask

23.10 Interview questions