---
title: "The Sell Side"
book: "Markets I: The Ecosystem and Exchange-Traded Markets"
subject: quant
language: en
chapter: 2
exercises: 8
source: https://one-course.com/books/quant/1/en/chapter/2-the-sell-side
---

# Chapter 2 — The Sell Side

At 11:40 a pension fund decides to sell two million shares of a stock that trades ten million a day. Worked patiently, the order would take two days and the price might fall all the while. The fund’s trader phones a bank instead. Forty seconds later she has a price for the whole block, a dollar and a half below the screen, and says “done”. The bank now owns a hundred million dollars of stock it never wanted. This chapter is about the firms that say yes to that call, how they decide the price, and why they are organised the way they are.

## 2.1 The markets division

**Definition 2.1 (Sell side and broker-dealer).**

The *sell side* is the set of firms that sell trading services to investors: execution, prices, financing, research and new issues. Its legal vehicle is the *broker-dealer*, a firm licensed both to execute clients’ orders as agent (broker) and to trade with them as principal (dealer).

The largest [broker-dealers](#def-m1-the-sell-side-sell-side) sit inside banks. A bank’s *markets division* is organised around the client, not around a strategy:

- **Sales** owns the client relationship and brings in the enquiry.
- **Trading** makes the price and manages the resulting risk.
- **Structuring** designs products that package several risks for a client — the subject of One Quant Book 5.
- **Research** publishes analysis that gives clients a reason to call.
- **Quants and technology** build the pricing, risk and execution systems the other four run on.

**Definition 2.2 (Sales-trader).**

A *sales-trader* takes clients’ orders and works them in the market on their behalf, as agent, advising on timing and tactics; a *trader*, by contrast, commits the bank’s capital.

![A markets division seen from the client. Everything in the middle column exists to bring business to the right-hand column, which carries the risk.](https://one-course.com/images/onecourse/chapters/quant-1/m1-the-sell-side/fig-d557feccc081.svg)

***Figure 2.1.** A markets division seen from the client. Everything in the middle column exists to bring business to the right-hand column, which carries the risk.*

**Definition 2.3 (Flow trading and franchise).**

*Flow trading* is market making to clients: the desk quotes on request, takes the other side, and manages the inventory that results. The desk’s *franchise* is the stream of client enquiry it sees; its value is both the spread earned on that flow and the information the flow carries about who wants to do what.

**Remark 2.4 (A bank desk is not a proprietary firm).**

An electronic [market maker](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-market-maker) ([Chapter 1](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#ch-m1-what-a-trading-firm-does)) competes on price and speed for anonymous orders. A bank desk competes for *relationships*: the client who is shown a good price in a difficult block also gives the bank its financing, its derivatives and its next bond issue. A trade that loses money in isolation can be good business; deciding when is the head of desk’s job (One Quant Book 16).

## 2.2 Pricing a block

**Definition 2.5 (Block trade and risk bid).**

A *block trade* is a transaction much larger than the market’s displayed size, negotiated away from the order book. In a *risk bid* the dealer buys the whole block as principal at a discount to the market price; the discount pays it for the cost and the risk of selling the shares afterwards.

The dealer’s problem has two parts. Selling a large quantity pushes the price down: an *impact* cost. And while it sells, the price moves for reasons of its own: a *risk*.

**Method 2.6 (The square-root rule of thumb for impact).**

The average cost of trading $Q$ shares of a stock with daily volatility $\sigma$ and daily volume $V$, as a fraction of their value, is close to

$$
I(Q) \;=\; \kappa\,\sigma\sqrt{Q/V},
$$

with $\kappa$ of order one and, within reasonable limits, little dependence on how long the execution takes. The empirical law and its theory belong to One Quant Book 10; here it is a tool, with $\kappa = 0.7$.

**Proposition 2.7 (Risk of a linear unwind).**

A dealer holds a block worth $N$ and sells it at a constant rate over $T$ days, while the price follows a driftless random walk with daily volatility $\sigma$. Ignoring impact, its profit relative to selling everything at the initial price has mean zero and standard deviation

$$
N\,\sigma\sqrt{T/3}.
$$

**Proof.** Let $W_t$ be the price return since purchase, a Brownian motion with variance $\sigma^2 t$. Selling at rate $1/T$, the dealer realises the average return $\bar W = \frac1T\int_0^T W_t\dd t$. Then $\E[\bar W]=0$ and

$$
\Var(\bar W) = \frac{1}{T^2}\int_0^T\!\!\int_0^T \sigma^2 \min(s,t)\dd s\dd t
 = \frac{\sigma^2}{T^2}\cdot\frac{T^3}{3} = \frac{\sigma^2 T}{3}.
\qedhere
$$

∎

The factor $1/3$ is worth remembering: a position that shrinks linearly to zero over $T$ days carries the risk of a constant position held for $T/3$ days.

**Proposition 2.8 (Break-even discount).**

With participation rate $\pi$ (the fraction of daily volume the dealer is willing to be), the unwind lasts $T = Q/(\pi V)$ days, and the discount at which the dealer loses money with probability $1-\Phi(z)$ only is

$$
d_z \;=\; \kappa\,\sigma\sqrt{Q/V} \;+\; z\,\sigma\sqrt{\frac{Q}{3\pi V}} .
$$

**Proof.** The dealer’s profit per unit of value is $d - I(Q) + \bar W$, normal with mean $d - I(Q)$ and standard deviation $\sigma\sqrt{T/3}$ by [Proposition 2.7](#prop-m1-the-sell-side-tover3). It is negative with probability $\Phi\bigl(-(d-I)/(\sigma\sqrt{T/3})\bigr)$; set this equal to $\Phi(-z)$. ∎

**Example 2.9 (The pension fund’s block).**

$Q = 2$ million shares at $50, $V = 10$ million, $\sigma = 2\%$ a day, $\pi = 10\%$. Then $T = 2$ days, $I = 0.7 \times 0.02 \times \sqrt{0.2} =
0.63\%$, and the risk term at 95% ($z = 1.645$) is $1.645 \times 0.02
\times \sqrt{2/3} = 2.69\%$. The break-even discount is $3.31\%$, or $1.66 a share: the dealer who bids $48.34 expects to make $2.69\%$ of $100 million, $2.7 million, and loses money one time in twenty. Both terms grow like $\sqrt{Q}$: *a block four times larger costs twice as much per share.*

![The 95% break-even discount of a risk bid, at 10% participation. Most of the discount pays for risk, not for impact. Data: computed by the chapter’s script from .](https://one-course.com/images/onecourse/chapters/quant-1/m1-the-sell-side/fig-17c6b4168e35.svg)

***Figure 2.2.** The 95% break-even discount of a risk bid, at 10% participation. Most of the discount pays for risk, not for impact. Data: computed by the chapter’s script from [Proposition 2.8](#prop-m1-the-sell-side-discount).*

**Remark 2.10 (What the model leaves out).**

Three things, all of which a desk prices by hand. *Information:* a client who sells because it knows something is worth a wider discount — [adverse selection](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-adverse-selection) again. *Hedging:* the dealer can sell index futures within seconds and keep only the stock-specific risk: a hedge that removes half of the variance shrinks $\sigma$ by 29%. *[Franchise](#def-m1-the-sell-side-flow):* the desk may bid through its break-even to win the client’s other business.

## 2.3 Financing, fees and new issues

Making prices is one of three businesses in the division.

**Definition 2.11 (Prime services).**

*Prime services* is the bank business that finances [hedge funds](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-hedge-fund): it lends them cash against their long positions, lends them securities to sell short, holds and settles their portfolios, and is paid a spread on every balance. [Chapter 6](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#ch-m1-financing) takes it apart.

**Definition 2.12 (Underwriting).**

In *underwriting* a bank buys newly issued shares or bonds from the issuer and resells them to investors, earning a percentage of the amount raised and bearing, for a few hours or days, the risk that the issue does not sell. With *advisory* work on mergers it forms *investment banking*, housed next to the markets division because the new issues feed the trading desks.

**As of September 2026 — One markets division in its own numbers.**

Goldman Sachs reported 2025 net revenues of $41.5 billion in Global Banking & Markets, out of $58.3 billion for the firm. Within it: fixed income, currencies and commodities $14.5 billion, of which $10.3 billion from intermediation (making prices) and $4.3 billion from financing; equities $16.5 billion, of which $9.3 billion intermediation and $7.2 billion financing; investment banking fees $9.3 billion (advisory $4.7 billion, equity [underwriting](#def-m1-the-sell-side-underwriting) $1.8 billion, debt [underwriting](#def-m1-the-sell-side-underwriting) $2.8 billion).

![Where one large markets division earned its revenue in 2025. Financing — lending against positions — is more than a third of the trading revenue. Data: the firm’s earnings release, as in .](https://one-course.com/images/onecourse/chapters/quant-1/m1-the-sell-side/fig-08e70991b27a.svg)

***Figure 2.3.** Where one large markets division earned its revenue in 2025. Financing — lending against positions — is more than a third of the trading revenue. Data: the firm’s earnings release, as in [Box 2.1](#dat-m1-the-sell-side-gs).*

## 2.4 What regulation did to the dealer

Before the 2008 crisis large dealers also traded for their own account on dedicated proprietary desks. Two sets of rules ended that and reshaped what remained.

**Definition 2.13 (Risk-weighted assets).**

A bank’s *risk-weighted assets* (RWA) are its exposures, each multiplied by a regulatory weight reflecting its risk; the bank must hold equity capital of at least a fixed percentage of their sum. For a trading desk, the market-risk part of RWA is computed from the riskiness of its positions under the Basel Committee’s market-risk standard.

**Method 2.14 (Judging a desk by its return on capital).**

1. Take the desk’s revenue less its direct costs.
2. Divide by the equity its positions consume: RWA times the bank’s target capital ratio.
3. Compare with the bank’s cost of equity, around ten percent.

A position that ties up capital for weeks must earn far more than one that turns over in minutes: this is why bank desks prefer flow and financing to large, slow risk.

**Example 2.15 (Two desks).**

Desk A earns $60 million on RWA of $2 billion; desk B earns $25 million on RWA of $400 million. At a 12% capital ratio A uses $240 million of equity and returns 25%; B uses $48 million and returns 52%. The bank grows B.

The second rule is a prohibition. Section 13 of the US Bank Holding Company Act, added by the Dodd–Frank Act of 2010 and known as the *Volcker rule*, forbids banking entities to engage as principal in short-term proprietary trading. Its exemptions define the modern dealer: [underwriting](#def-m1-the-sell-side-underwriting), *market making-related* activity, risk-mitigating hedging, trading on behalf of customers, and trading in government obligations remain allowed. A desk must therefore be able to show that its inventory is sized to the “reasonably expected near term demands of clients, customers, or counterparties” — that it is a shop, not a fund. Much of the proprietary talent that left the banks after 2010 founded or joined the firms of [Chapter 1](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#ch-m1-what-a-trading-firm-does).

## 2.5 Tutorial: pricing the risk bid

**Goal.** Price the pension fund’s block and check the $T/3$ rule by simulation. **End state:** the histogram below, with 5% of its mass left of zero.

1. **Code the two propositions.** `def unwind_days (b: Block) -> float : return b.shares / (b.participation * b.adv) def impact_cost (b: Block) -> float : return b.kappa * b.sigma * math.sqrt(b.shares / b.adv) def risk_std (b: Block) -> float : """Std of the unwind P&L as a fraction of value: sigma * sqrt(T / 3).""" return b.sigma * math.sqrt(unwind_days(b) / 3.0 ) def breakeven_discount (b: Block, z: float = Z95) -> float : """Discount at which the dealer loses money with probability 1 - Phi(z).""" return impact_cost(b) + z * risk_std(b)` **Listing 2.1.** Unwind time, impact, risk and the break-even discount. code/markets-1/02-the-sell-side/python/blockbid.py
2. **Simulate the unwind.** Two hundred equal slices over $T$ days, each sold at the then price; the dealer’s result is the discount, less impact, plus the average return of the slices. `def simulate_unwind (b: Block, discount: float , n_paths: int , seed: int , steps: int = 200 ): """P&L per unit of block value over n_paths, selling linearly over T days.""" rng = np.random.default_rng(seed) dt = unwind_days(b) / steps dw = rng.normal(0.0 , b.sigma * math.sqrt(dt), size=(n_paths, steps)) price = np.cumsum(dw, axis=1 ) # return of the stock since purchase avg_sale = price.mean(axis=1 ) # equal slices: average sale return return discount - impact_cost(b) + avg_sale` **Listing 2.2.** Monte Carlo of a linear unwind: the average of a random walk. code/markets-1/02-the-sell-side/python/blockbid.py
3. **Run and compare.** For the example’s block the simulated standard deviation is $1.64\%$ against a predicted $0.02\sqrt{2/3} = 1.63\%$ , and $4.9\%$ of 20 000 paths lose money at the 95% discount.
4. **Read the histogram.** The mean, $2.7\%$ , is the dealer’s expected profit; it is what the client pays for certainty.

**What to change next.** Raise `participation` to 0.25: the discount falls to $2.3\%$, and the model says faster is always better. Why does no desk sell at 50% of volume? (Exercise [2.8](#exo-m1-the-sell-side-8).) Then hedge: replace $\sigma$ by the stock’s residual volatility.

![Profit of the risk bid at the 95% break-even discount. The dashed line is zero: one path in twenty falls to its left. Data: the tutorial’s simulation, seed 2.](https://one-course.com/images/onecourse/chapters/quant-1/m1-the-sell-side/fig-02097bdab6ff.svg)

***Figure 2.4.** Profit of the risk bid at the 95% break-even discount. The dashed line is zero: one path in twenty falls to its left. Data: the tutorial’s simulation, seed 2.*

## 2.6 Build: the block pricer

**Purpose.** A pricing service the miniature firm’s bank-style desk calls when a client asks for a risk bid.

**Interface.** `quote(shares, price, adv, sigma, participation, confidence, hedge_ratio=0.0)` returning the unwind time, impact, risk, discount and bid price. `hedge_ratio` is the fraction of variance removed by an index hedge: the risk term uses $\sigma\sqrt{1-h}$.

**Rules.** Reject a participation above 30% and a block above five days’ volume (`ValueError`): outside that range the square-root rule is not to be trusted. Round the bid *down* to the cent.

**Acceptance tests.** `code/firm/blockbid/tests/`: reproduces [Example 2.9](#ex-m1-the-sell-side-block) to the cent; a fully hedged block is priced at impact only; limits raise.

**Stretch.** Post the desk’s realised profit on a simulated unwind to the ledger of [Chapter 1](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#ch-m1-what-a-trading-firm-does) under `position`.

Sources and further reading

- Goldman Sachs, *Full Year and Fourth Quarter 2025 Earnings Results* , Form 8-K, January 2026.
- Federal Deposit Insurance Corporation, “Volcker Rule” (overview of Section 13 of the Bank Holding Company Act and its implementing rules); *Federal Register* , 31 July 2020, final rule on covered funds.
- Basel Committee on Banking Supervision, *Minimum capital requirements for market risk* , January 2019.
- L. Harris, *Trading and Exchanges* , Oxford University Press, 2003, chapters 15 and 16 (block trading, dealers).
- J.-P. Bouchaud, J. Bonart, J. Donier and M. Gould, *Trades, Quotes and Prices* , Cambridge University Press, 2018, chapter 12 (the square-root law).

## 2.7 Exercises

**Exercise 2.1 ★.**

A dealer holds a $30 million position in a stock with daily volatility 1.5% and sells it linearly over three days. Give the standard deviation of its result in dollars.

**Solution of Exercise 2.1.**

$30 \times 10^6 \times 0.015 \times \sqrt{3/3} = \$450\,000$: three days of linear selling carry the risk of one day of the full position.

**Exercise 2.2 ★.**

A block of 500 000 shares in a stock trading 5 million a day is unwound at 10% participation. How long does the unwind take, and what is the impact cost in percent if $\sigma = 1.5\%$ and $\kappa = 0.7$?

**Solution of Exercise 2.2.**

$T = 500\,000/(0.1 \times 5\,000\,000) = 1$ day. $I = 0.7 \times 0.015 \times
\sqrt{0.1} = 0.33\%$.

**Exercise 2.3 ★.**

An issuer raises $800 million of new shares with an [underwriting](#def-m1-the-sell-side-underwriting) fee of 3.5%. What do the banks earn? The shares are placed at $40; on the first day they close at $46. How much did the issuer “leave on the table”, and who received it?

**Solution of Exercise 2.3.**

Fee: $0.035 \times 800 = \$28$ million. The issuer sold 20 million shares at $40 that the market valued at $46 that evening: $120 million left on the table, received by the investors who were allocated shares — the banks’ clients, which is why allocation is itself a [franchise](#def-m1-the-sell-side-flow) asset.

**Exercise 2.4 ★★.**

Compute the 95% break-even discount, in percent and in dollars a share, for a block of 3 million shares at $20 in a stock with $V = 6$ million and $\sigma = 2.5\%$, at 10% participation.

**Solution of Exercise 2.4.**

$T = 3/(0.1\times 6) = 5$ days. $I = 0.7 \times 0.025 \times \sqrt{0.5} =
1.24\%$; risk $= 0.025\sqrt{5/3} = 3.23\%$; $d = 1.24 + 1.645 \times 3.23 =
6.55\%$, that is $1.31 a share.

**Exercise 2.5 ★★.**

Show that at fixed participation the break-even discount is proportional to $\sqrt{Q}$. A dealer quotes 2% for a block; what should it quote, other things equal, for a block nine times larger?

**Solution of Exercise 2.5.**

$d_z = \sigma\sqrt{Q/V}\,\bigl(\kappa + z/\sqrt{3\pi}\bigr)$, and the bracket does not depend on $Q$. Nine times the size: three times the discount, 6%.

**Exercise 2.6 ★★.**

With the figures of [Box 2.1](#dat-m1-the-sell-side-gs): (a) what share of the division’s revenue came from financing, from intermediation and from investment banking fees? (b) Which of the five revenue sources of [Method 1.9](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#met-m1-what-a-trading-firm-does-sources) is each?

**Solution of Exercise 2.6.**

(a) Financing $4.251 + 7.195 = \$11.4$ billion, 28% of the division; intermediation $\$19.6$ billion, 47%; fees $\$9.3$ billion, 23% (the small remainder is other revenue). Financing is 37% of the FICC and equities total. (b) Financing is carry; intermediation is spread, with some position P&L; investment banking fees are fees.

**Exercise 2.7 ★★★.**

*Coding.* An index hedge removes 40% of the variance of the stock of [Example 2.9](#ex-m1-the-sell-side-block). (a) Compute the new break-even discount. (b) The hedge costs 2 basis points of the block’s value in futures commissions and spread. Is it worth it?

**Solution of Exercise 2.7.**

(a) The residual volatility is $0.02\sqrt{0.6} = 1.55\%$; risk $1.55\% \times \sqrt{2/3} = 1.26\%$; $d = 0.63 + 1.645 \times 1.26 =
2.71\%$. (b) The hedge lowers the break-even discount by 61 basis points for a cost of 2: it is worth it thirty times over, which is why every block is hedged within seconds of the trade.

**Exercise 2.8 ★★★.**

*Find the flaw.* A junior trader observes that in [Proposition 2.8](#prop-m1-the-sell-side-discount) the discount falls as participation rises, and proposes to unwind every block at 50% of volume. Give two reasons, one inside the model’s assumptions and one outside them, why the desk’s discount would not fall as the formula says.

**Solution of Exercise 2.8.**

Inside the model: the square-root rule was stated as insensitive to duration *within reasonable limits*; at half of the volume the dealer is the market, its selling is visible to everyone, and realised impact rises well above $\kappa\sigma\sqrt{Q/V}$, so the first term is no longer constant in $\pi$. Outside the model: volume is not a given. Other participants detect the seller and step back or sell ahead of it, so the volume against which 50% is measured shrinks, and the price path acquires a drift against the dealer that the driftless random walk excludes.

## 2.8 Problem: The Risk Bid

**Problem 2.1.**

Weekend problem — forty seconds to price a block

You run the cash equities risk book at a bank. At 11:40 a pension fund asks for a bid on 4 million shares of a stock trading at $25, with a daily volume of 8 million shares and a daily volatility of 1.8%. Use $\kappa = 0.7$.

**Part I — The mechanical price.**

1. What is the value of the block, and what fraction of daily volume?
2. At 10% participation, how many days does the unwind last?
3. Compute the impact cost in percent and in dollars.
4. Compute the standard deviation of the unwind result in percent and in dollars.
5. Compute the 95% break-even discount and the corresponding bid price, rounded down to the cent.

**Part II — Improving it.**

6. You are willing to be 20% of the volume. Recompute the unwind time and the discount.
7. An index hedge removes half the variance. Recompute the discount at 20% participation.
8. What is your expected profit in dollars if you win the block at that discount?
9. A competitor is known to bid at one standard deviation ( $z = 1$ ) instead of 1.645. What discount does it quote with the same hedge and participation, and how often does it lose money?

**Part III — The client.**

10. The alternative for the fund is an agency execution over the same $T$ days at a commission of 1 cent a share. Give its expected cost in percent (impact plus commission) and the standard deviation of its outcome.
11. By how much does the risk bid of question 7 exceed that expected cost? What does the fund buy with the difference?
12. The fund’s trader is measured against the price at the time of the decision. Explain why she may prefer the risk bid even at that premium.
13. Over the last year this client’s blocks were followed, on average, by a fall of 0.4% in the stock relative to the index over the next two days. What should you add to your discount, and what is this cost called?

**Part IV — The bank.**

14. The position adds $150 million of RWA for three days. At a 12% capital ratio and a 10% cost of equity, what does that capital cost in dollars (use 252 days)?
15. Compare with the expected profit of question 8. Is capital the binding cost of this trade?
16. The same client paid the bank $3 million last year in financing and derivatives. How low could you rationally bid?
17. You win the block and sell $100 million of index futures against it. In the first hour the index falls 1% and the stock 3%. Estimate the mark-to-market loss on the pair before any sale.
18. Under the Volcker rule, what would you need to show an examiner about this position?
19. State the *named result* : the discount, in cents a share, at which this desk breaks even with 95% confidence after hedging and the adverse-selection charge of question 13.
20. In one sentence, why is a risk bid an option the client owns?

**Solution of Problem 2.1.**

**1.** $100 million; 50% of daily volume. **2.** $T = 4/(0.1\times 8) = 5$ days. **3.** $0.7\times 0.018\times\sqrt{0.5} = 0.89\%$, $891 000. **4.** $0.018\sqrt{5/3} = 2.32\%$, $2.32 million. **5.** $d = 0.89 + 1.645\times 2.32 = 4.71\%$; bid $23.82. **6.** $T = 2.5$ days; $d = 0.89 + 1.645 \times 1.64 = 3.59\%$. **7.** Residual volatility $0.018/\sqrt2 = 1.27\%$; risk $1.16\%$; $d = 0.89 + 1.645\times 1.16 = 2.80\%$. **8.** The discount less the impact: $1.91\%$ of $100 million, $1.91 million. **9.** $0.89 + 1.16 = 2.05\%$; it loses money with probability $1-\Phi(1) = 16\%$. **10.** Expected cost $0.89 + 0.04 = 0.93\%$; standard deviation $0.018\sqrt{2.5/3} = 1.64\%$ (the fund does not hedge). **11.** $2.80 - 0.93 = 1.87\%$, $1.87 million: the fund buys the removal of a $1.64 million standard deviation and of all execution effort. **12.** Her benchmark is the decision price; against it an agency execution can end 3% or 4% adrift on a bad day, for which she answers personally, while the risk bid’s shortfall is known, agreed and final at 11:41. **13.** Add 0.4% (hedged blocks lose exactly the move relative to the index): this is the adverse-selection cost of the client’s flow. **14.** $150\times10^6 \times 0.12 \times 0.10 \times 3/252 = \$21\,429$. **15.** No: capital costs about 1% of the expected profit. Risk, not capital, prices a three-day position; capital binds for positions held for months. **16.** Bidding at the impact cost, a discount of 0.89%, gives up the whole expected profit of $1.9 million and still has zero expected loss; below that the desk pays for the relationship out of the $3 million, and should do so only if losing the block means losing the client. **17.** $-\$3$ million on the stock, $+\$1$ million on the futures: $-\$2$ million, already 1.7 standard deviations of the hedged unwind. **18.** That the position was taken to meet a client’s demand and that the desk’s inventory and risk limits are designed not to exceed the reasonably expected near-term demands of its clients: a documented client enquiry, limits calibrated on past flow, and an unwind consistent with them. **19.** $2.80 + 0.40 = 3.20\%$, that is **80 cents a share**. **20.** The client may sell at the bid or walk away, and it sells more readily when it believes the price is about to fall: the dealer has written a put whose exercise is correlated with bad news.

## 2.9 Interview questions

**Interview question 2.1 ★ bank, trader.**

What is the difference between a [sales-trader](#def-m1-the-sell-side-sales-trader) and a trader? Who is responsible for the P&L of a client order in each case?

**Solution of Interview question 2.1.**

The [sales-trader](#def-m1-the-sell-side-sales-trader) works the client’s order in the market as agent: the client owns every fill and the execution risk, and the [sales-trader](#def-m1-the-sell-side-sales-trader) is judged on execution quality against a benchmark. The trader buys or sells against the client as principal: once the price is agreed the P&L of the position belongs to the trader’s book.

*What the interviewer is looking for: principal versus agent stated in terms of who owns the risk after the phone call ends.*

**Interview question 2.2 ★★ trader, researcher, bank.**

You must sell a position evenly over nine days. Roughly how much risk are you carrying compared with holding it unchanged for the same nine days?

**Solution of Interview question 2.2.**

A linearly shrinking position has the variance of a constant one held a third as long: its standard deviation is $\sqrt{1/3} \approx 58\%$ of that of holding everything for nine days — the risk of holding it all for three.

*What the interviewer is looking for: the $T/3$ rule from memory or derived in a minute.*

**Interview question 2.3 ★★ trader, bank.**

A client asks for a two-way price in a block of a stock you know nothing about. What do you ask yourself in the ten seconds you have?

**Solution of Interview question 2.3.**

How big is it against daily volume and what is the volatility (the mechanical discount); can I hedge it, and with what; why is this client trading, and what happened after its previous blocks; is there news or an event (earnings, index change) within my unwind window; what is my current position and limit in the name and the sector; and what is the client worth to the bank.

*What the interviewer is looking for: [adverse selection](https://one-course.com/books/quant/1/en/chapter/1-what-a-trading-firm-does#def-m1-what-a-trading-firm-does-adverse-selection) and existing inventory mentioned, not only the formula.*

**Interview question 2.4 ★★ bank, researcher.**

Why did financing grow to more than a third of the trading revenue of large banks after 2010, while their proprietary risk shrank?

**Solution of Interview question 2.4.**

Proprietary trading was prohibited to banks and market-risk capital requirements rose, so slow directional risk earns too little per unit of capital. Financing is client-driven and collateralised, consumes little market-risk capital per dollar of revenue, recurs every day, and grew with the hedge-fund industry that the banks’ own former traders went to build.

*What the interviewer is looking for: return on capital as the organising idea, plus the regulatory cause.*

**Interview question 2.5 ★★★ researcher, trader.**

The cost of a block grows like the square root of its size. A client can give you one block of 4 million shares or four blocks of 1 million on four consecutive days. Which is cheaper for the client, and what assumption decides it?

**Solution of Interview question 2.5.**

One block of 4 million costs $2c$ per share if 1 million costs $c$: a total of $8c$ million. Four blocks of 1 million cost $4c$ million: half as much — *if* impact is fully forgotten from one day to the next and the market does not infer that three more blocks follow. If impact is permanent or the programme is detected, the later blocks are priced off a lower price and the advantage shrinks toward zero. The client also carries the price risk of the unsold shares for three more days.

*What the interviewer is looking for: the concavity argument, then the two caveats (impact decay, information leakage) without being asked.*

**Interview question 2.6 ★★★ bank, researcher.**

Derive the variance of the time-average of a Brownian motion over $[0,T]$.

**Solution of Interview question 2.6.**

$\Var\bigl(\tfrac1T\int_0^T W_t\dd t\bigr) = \tfrac{1}{T^2}\iint
\min(s,t)\dd s\dd t = \tfrac{2}{T^2}\int_0^T\!\int_0^t s\dd s\dd t =
\tfrac{2}{T^2}\cdot\tfrac{T^3}{6} = T/3$ (times $\sigma^2$).

*What the interviewer is looking for: the covariance $\min(s,t)$ and a clean double integral; bonus for the integration-by-parts route, $\int_0^T (T-t)\dd W_t$.*