---
title: "Margin"
book: "Markets I: The Ecosystem and Exchange-Traded Markets"
subject: quant
language: en
chapter: 20
exercises: 8
source: https://one-course.com/books/quant/1/en/chapter/20-margin
---

# Chapter 20 — Margin

A fund goes home on Friday long four hundred index futures, fully margined. On Monday it holds the same four hundred contracts and owes the clearing house cash twice over: once for what the contracts lost, and again because the clearing house has decided, looking at the same prices, that each contract is now more dangerous to hold. Over March 2020 the [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) required by central counterparties worldwide rose by roughly $300 billion, and on a single day, 9 March, they called $140 billion of [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin). None of that money was a loss for the system as a whole; all of it had to be found in cash within hours by someone. [Chapter 5](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#ch-m1-clearing-and-settlement) said what margin is for. This chapter computes it, the way the clearing house does, and looks at how the computation behaves on the day it matters.

## 20.1 Variation margin

[Variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) ([Definition 5.7](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin)) is arithmetic: position times the change in the settlement price times the multiplier, paid in cash, every day, in the contract’s currency. Three properties make it harder than it looks. It is *asymmetric in liquidity*: a hedged portfolio whose other leg is not at the same clearing house pays cash on the losing leg and receives none on the winning one. It is *fast*: a call made in the morning is due the same morning, and in stressed markets clearing houses call again during the day. And it is *unforgiving*: a member that does not pay is in default.

**Definition 20.1 (Performance bond and maintenance margin).**

A *performance bond* is the futures industry’s name for [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin): collateral deposited to guarantee performance, not a down payment. For a client account the broker sets an *initial* level, required to open a position, and a lower *maintenance margin*: when the account’s equity falls below the maintenance requirement the client is called to restore it to the initial level. For clearing members the two coincide.

**Definition 20.2 (Intraday margin call).**

An *intraday margin call* is a call for variation or [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) made by a clearing house between its scheduled daily cycles, on the basis of current prices and positions, payable within the hour or so that its rules specify.

## 20.2 Initial margin by scenarios

**Definition 20.3 (Scanning range and risk array).**

In a scenario-based margin system the clearing house sets, for each product, a *price scan range* and a *volatility scan range*: the largest moves in price and in implied volatility it wishes to cover over its horizon, the *scanning range*. The *risk array* of a contract is its gain or loss under each of a fixed list of scenarios built from those ranges; the *scan risk* of a portfolio is its largest total loss over the list.

The system that carried futures margining for three decades, SPAN, uses sixteen scenarios. Two move volatility alone, up and down. Twelve combine a price move of one third, two thirds or the whole of the price scan range, up or down, with volatility up or down. The last two are *extreme* moves, a multiple of the scan range, of which only a fraction of the loss is counted: they exist for short options far out of the money, which lose nothing in the first fourteen. For a future, the price scan range is the [maintenance margin](#def-m1-margin-bond) itself.

**Example 20.4 (The risk array of one future).**

A regulator’s review of the system prints the array of the large S&P 500 future on 12 April 2001, when its [maintenance margin](#def-m1-margin-bond) was $17 250: zero in the two volatility scenarios; losses of $\mp\$5\,750$, $\mp\$11\,500$ and $\mp\$17\,250$ at one, two and three thirds; and in the extreme scenarios, a move of three times the range with 30% of the loss counted, $\pm\$15\,525$. A long future’s [scan risk](#def-m1-margin-scan) is the full move down, $17 250; the extreme scenario, at $15 525, does not bind. It binds for the option seller.

**Method 20.5 (A scenario margin).**

For each product:

1. **[Scan risk](#def-m1-margin-scan)** : sum position times risk array over all contracts, scenario by scenario; take the largest loss.
2. **Inter-month charge** : futures of different expiries offset fully in the scan; add a charge per spread for the risk that they do not move together.
3. **Short option minimum** : a floor per short option, compared with the sum of the first two.

Across products: subtract an *inter-commodity spread credit* for pairs whose prices are correlated and whose positions offset. Add the results.

![Sixteen scenarios for a portfolio long 2 futures at 6 000, short 10 puts struck at 5 600 and short 10 calls at 6 400 (three months, 18% volatility, price scan 345 points, volatility scan 4 points, extreme moves of three ranges counted at 30%). Odd scenarios raise volatility, even ones lower it. The margin is set by scenario 16, the extreme fall. Data: the chapter’s build.](https://one-course.com/images/onecourse/chapters/quant-1/m1-margin/fig-af8f4e03f830.svg)

***Figure 20.1.** Sixteen scenarios for a portfolio long 2 futures at 6 000, short 10 puts struck at 5 600 and short 10 calls at 6 400 (three months, 18% volatility, price scan 345 points, volatility scan 4 points, extreme moves of three ranges counted at 30%). Odd scenarios raise volatility, even ones lower it. The margin is set by scenario 16, the extreme fall. Data: the chapter’s build.*

![What the scan sees. The same portfolio revalued over a wide range of prices; the fourteen standard scenarios are the dots, all inside the price scan range (thin lines). The losses that matter for this portfolio lie outside it, which is what scenarios 15 and 16 are for.](https://one-course.com/images/onecourse/chapters/quant-1/m1-margin/fig-541467037fc8.svg)

***Figure 20.2.** What the scan sees. The same portfolio revalued over a wide range of prices; the fourteen standard scenarios are the dots, all inside the price scan range (thin lines). The losses that matter for this portfolio lie outside it, which is what scenarios 15 and 16 are for.*

**Definition 20.6 (Portfolio margining).**

*Portfolio margining* sets the requirement on the net risk of all positions in an account, recognising offsets between them, in contrast with a sum of requirements computed position by position.

![Offsets. Margined separately the three legs require $213 000; together, $108 000, because the put and the call cannot both lose. Data: the chapter’s build.](https://one-course.com/images/onecourse/chapters/quant-1/m1-margin/fig-ac87baf78069.svg)

***Figure 20.3.** Offsets. Margined separately the three legs require $213 000; together, $108 000, because the put and the call cannot both lose. Data: the chapter’s build.*

## 20.3 Initial margin by value-at-risk

Sixteen scenarios per product and a table of credits between products are transparent and replicable, which is why the method lasted. They are also crude: the offsets are set by committee, and a portfolio of many products is margined as a sum of pairs. The large clearing houses have moved, or are moving, to portfolio value-at-risk.

**Definition 20.7 (Margin period of risk).**

The *margin period of risk* is the time assumed between a member’s last payment of [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) and the moment its portfolio has been hedged or liquidated by the clearing house: the horizon of the initial-margin calculation. It is a day or two for liquid listed futures and longer for cleared swaps and illiquid products.

**As of September 2026 — From scenarios to simulation.**

The Chicago clearing house’s successor to SPAN is built on historical value-at-risk: thousands of scenarios from at least ten years of data, with volatility and correlation scaling, and separate reporting of market, liquidity and concentration components. The Frankfurt clearing house’s portfolio method uses filtered historical simulation together with a stress-period value-at-risk, computed by groups of products that would be liquidated together, each with its own [margin period of risk](#def-m1-margin-mpor).

**Method 20.8 (Filtered historical simulation).**

(i) Estimate each day’s volatility $\sigma_t$ from past returns, for instance by exponential weighting. (ii) Standardise the last $n$ returns: $z_s = r_s/\sigma_s$. (iii) Rescale to today: $\tilde r_s = z_s\,\sigma_t$. (iv) Revalue the portfolio under each $\tilde r_s$, take the loss quantile at the chosen confidence, and scale to the [margin period of risk](#def-m1-margin-mpor).

Filtering makes the margin react within days to a change of regime, where a plain historical window reacts only as extreme days accumulate. It is the right property for covering tomorrow’s loss, and the wrong one for the stability of the system.

## 20.4 Procyclicality

**Definition 20.9 (Procyclicality).**

A margin system is *procyclical* to the extent that its requirements rise in stressed markets and fall in calm ones, adding to the demand for liquidity exactly when liquidity is scarce.

![Margin on a fixed position while daily volatility jumps from 0.8% to 3.5% for 25 days and decays (99%, two-day horizon, 250-day window). Over the worst ten days the plain margin rises by 109%, the filtered margin by 203%, the floored margin by 94%. The floor is paid for in calm times: 4.5% of the position against 3.0%. Data: the tutorial’s simulation.](https://one-course.com/images/onecourse/chapters/quant-1/m1-margin/fig-0f82ebef8e7d.svg)

***Figure 20.4.** Margin on a fixed position while daily volatility jumps from 0.8% to 3.5% for 25 days and decays (99%, two-day horizon, 250-day window). Over the worst ten days the plain margin rises by 109%, the filtered margin by 203%, the floored margin by 94%. The floor is paid for in calm times: 4.5% of the position against 3.0%. Data: the tutorial’s simulation.*

The joint review of March 2020 by the banking, market-infrastructure and securities standard setters found the rise in cleared [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) large and uneven across asset classes and clearing houses, and listed among its areas for further work the transparency of cleared margin, the liquidity preparedness of market participants, and the responsiveness of initial-margin models to stress. For a trading firm the operational conclusion does not wait for policy: the liquidity needed to hold a position is its [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) in a bad week *plus* the increase of its [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) in that same week, and both must be modelled.

## 20.5 Tutorial: a scenario margin for futures and options

**Goal.** Build risk arrays for futures and options on futures, scan a portfolio, add the charges, and compare a filtered with an unfiltered value-at-risk margin through a volatility shock. **End state:** the four data figures of this chapter.

1. **Sixteen scenarios** from two ranges and two extreme-move parameters. `STANDARD = tuple ((m / 3.0 , v) for m in (0 , 1 , -1 , 2 , -2 , 3 , -3 ) for v in (1 , -1 )) # scenarios 1 to 14 def scenarios (p: Params) -> list [tuple [float , float , float ]]: """(price move, vol move, fraction of the loss counted) for the 16 scenarios.""" std = [(m * p.price_scan, v * p.vol_scan, 1.0 ) for m, v in STANDARD] ext = p.extreme_mult * p.price_scan return std + [(ext, 0.0 , p.extreme_cover), (-ext, 0.0 , p.extreme_cover)]` **Listing 20.1.** The scenario list: price move, volatility move, fraction of the loss counted. code/firm/margin/firm_margin.py
2. **A risk array** by full revaluation with the Black (1976) formula for options on futures. `def value (pos: Position, mkt: Market, d_price: float = 0.0 , d_vol: float = 0.0 ) -> float : key = (pos.root, pos.expiry) f = mkt.future[key] + d_price if pos.strike is None : return f return black76(f, pos.strike, max (mkt.vol[key] + d_vol, 1e-4 ), mkt.years[key], pos.right) def risk_array (pos: Position, mkt: Market, p: Params) -> list [float ]: """Loss (positive = loss) of ONE LONG unit under each scenario, in currency.""" base = value(pos, mkt) return [-(value(pos, mkt, dp, dv) - base) * cover * p.multiplier for dp, dv, cover in scenarios(p)]` **Listing 20.2.** Revaluation and the risk array of one long unit. code/firm/margin/firm_margin.py
3. **Scan, charge, floor.** `def product_margin (positions: list [Position], mkt: Market, p: Params) -> MarginResult: losses = [0.0 ] * 16 for pos in positions: arr = risk_array(pos, mkt, p) for i in range (16 ): losses[i] += pos.qty * arr[i] worst = max (range (16 ), key=lambda i: losses[i]) scan = max (losses[worst], 0.0 ) net_by_expiry: dict [str , float ] = {} for pos in positions: # spreads are counted on futures only, for simplicity if pos.strike is None : net_by_expiry[pos.expiry] = net_by_expiry.get(pos.expiry, 0 ) + pos.qty longs = sum (q for q in net_by_expiry.values() if q > 0 ) shorts = -sum (q for q in net_by_expiry.values() if q < 0 ) inter = min (longs, shorts) * p.intermonth_charge som = sum (-pos.qty for pos in positions if pos.strike is not None and pos.qty < 0 ) * p.short_option_min return MarginResult(scan, worst + 1 , inter, som, max (scan + inter, som))` **Listing 20.3.** The margin of one product’s positions. code/firm/margin/firm_margin.py
4. **[Procyclicality](#def-m1-margin-procyclical).** Rescale the window to today’s volatility. `def fhs_var (returns: np.ndarray, vol: np.ndarray, t: int , lookback: int , q: float , horizon: int ) -> float : """Filtered historical simulation: past returns rescaled to today's volatility.""" window = returns[t - lookback:t] / vol[t - lookback:t] * vol[t] return float (-np.quantile(window, 1.0 - q)) * np.sqrt(horizon)` **Listing 20.4.** Filtered historical simulation. code/markets-1/20-margin/python/margin_demo.py

**What to change next.** Replace the two extreme scenarios by a full grid out to five ranges and see which portfolios were under-margined. Then give the filtered method a floor on $\sigma_t$ instead of a stressed add-on and compare the two tools at equal [average cost](https://one-course.com/books/quant/1/en/chapter/7-p-l-and-the-accounting-of-a-position#def-m1-pnl-and-positions-realised).

## 20.6 Build: the margin engine

**Purpose.** The miniature firm’s clearing component ([Chapter 5](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#ch-m1-clearing-and-settlement)) needs a number every night and on demand: the [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) of each account, recomputable by the member.

**Interface.** `Params(price_scan, vol_scan, extreme_mult, extreme_cover, intermonth_charge, short_option_min, multiplier)`; `Position(root, expiry, qty, strike, right)`; `Market`; `scenarios(params)`; `risk_array(position, market, params)`; `product_margin(positions, market, params)` returning [scan risk](#def-m1-margin-scan), the binding scenario, the inter-month charge, the short option minimum and the total; `portfolio_margin(by_root, market, params, credits)`.

**Rules.** Contract data from the master of [Section 18.7](https://one-course.com/books/quant/1/en/chapter/18-futures-contracts-and-their-exchanges#bld-m1-futures-contracts-and-exchanges-master). Full revaluation, no delta approximation. Futures of different expiries net in the scan and pay the inter-month charge per spread. The inter-commodity credit is given only when the two products’ binding scenarios are price moves in opposite directions.

**Acceptance tests.** `code/firm/margin/tests/`: the future’s array of [Example 20.4](#ex-m1-margin-array), to the dollar; a [calendar spread](https://one-course.com/books/quant/1/en/chapter/19-matching-algorithms-and-implied-spreads#def-m1-matching-algorithms-and-implied-spreads-calendar); a short far out-of-the-money put caught by the extreme scenario, and one caught by the minimum; a covered option; credits granted and refused.

**Stretch.** A second engine, portfolio value-at-risk by filtered historical simulation, behind the same interface, and a report of the difference between the two on every account.

Sources and further reading

- US Commodity Futures Trading Commission, *Review of Standard Portfolio Analysis of Risk (“SPAN”) Margin System* , April 2001.
- Basel Committee on Banking Supervision, Committee on Payments and Market Infrastructures and IOSCO, *Review of margining practices* , final report and press release, 29 September 2022.
- B. H. Cohen and K. Tracol, “Market turbulence and soaring margins: lessons from two recent episodes”, *BIS Quarterly Review* , March 2023.
- CME Group, *SPAN 2 methodology* ; Eurex Clearing, *Eurex Clearing Prisma* .
- F. Black, “The pricing of commodity contracts”, *Journal of Financial Economics* 3 (1976).

## 20.7 Exercises

**Exercise 20.1 ★.**

An account is long 25 E-minis. The settlement price goes from 6 012.50 to 5 968.25. Give the [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin), with its direction.

**Solution of Exercise 20.1.**

$-44.25$ points $\times\ \$50 \times 25 = -\$55\,312.50$: the account pays.

**Exercise 20.2 ★.**

A broker requires $23 100 of initial and $21 000 of [maintenance margin](#def-m1-margin-bond) per contract. A client holds 5 contracts with $120 000 of equity and loses $3 200 per contract. Is there a call, and for how much?

**Solution of Exercise 20.2.**

Equity falls to $120\,000 - 16\,000 = \$104\,000$, below the maintenance requirement of $5 \times 21\,000 = \$105\,000$: a call, to restore the initial level of $5 \times 23\,100 = \$115\,500$. The client must deposit $11 500.

**Exercise 20.3 ★.**

Write the sixteen-entry risk array of one long future with a multiplier of $50, a price scan range of 345 points, and extreme moves of three ranges counted at 30%. Which entry is its margin?

**Solution of Exercise 20.3.**

One third of the range is 115 points, $5 750. Scenarios 1–2: 0, 0. Then (loss positive) 3–4: $-5\,750$; 5–6: $+5\,750$; 7–8: $-11\,500$; 9–10: $+11\,500$; 11–12: $-17\,250$; 13–14: $+17\,250$; 15: $-15\,525$; 16: $+15\,525$ ($3 \times 17\,250 \times 30\%$). The margin is the largest loss: $17 250, scenarios 13 and 14.

**Exercise 20.4 ★★.**

In [Figure 20.1](#fig-m1-margin-scenarios), why do the odd scenarios lose more than their even neighbours? Why is scenario 16 worse than scenario 15 although the strangle is symmetric? What would make scenario 13 the binding one?

**Solution of Exercise 20.4.**

The portfolio is short options, hence short volatility: every odd scenario adds four volatility points and raises the value of what it has sold. Scenario 16 is worse than 15 because the portfolio is also long 2 futures: a fall hurts the futures and the short puts together, while in a rise the futures’ gain offsets part of the short calls’ loss. Scenario 13 (full range down, volatility up) would bind if the options were struck closer to the money, so that most of their loss occurs within one scan range, or if the extreme scenarios counted a smaller fraction.

**Exercise 20.5 ★★.**

From [Figure 20.3](#fig-m1-margin-offsets) give the saving from margining the portfolio together, in dollars and in percent. A broker margins each leg separately “for prudence”. What does that prudence cost a client who pays 5% a year for funding?

**Solution of Exercise 20.5.**

$213\,250 - 107\,770 = \$105\,480$, 49%. At 5% a year: $5 274 a year for this small portfolio, and, more importantly, half the position size for the same capital.

**Exercise 20.6 ★★.**

A one-day 99% value-at-risk is 2.1% of a position. Give the margin for margin periods of risk of two and five days under square-root scaling. Name one reason why the true five-day figure could be higher and one why it could be lower.

**Solution of Exercise 20.6.**

$2.1\% \times \sqrt2 = 2.97\%$; $2.1\% \times \sqrt5 = 4.70\%$. Higher: volatility clusters and returns in a crisis are positively autocorrelated, and the liquidation itself moves prices. Lower: mean reversion in some products, and a portfolio that is partly hedged on day one does not carry its full risk for five days.

**Exercise 20.7 ★★★.**

*Coding.* With `peak_to_trough` and `procyclical.csv` reproduce the three ten-day increases quoted in [Figure 20.4](#fig-m1-margin-procyclical) and the two calm-period averages. Find the weight of the stressed floor for which the ten-day increase is 50%, and its cost in calm times.

**Solution of Exercise 20.7.**

Ten-day increases: 109%, 203% and 94%. Calm averages: 3.0% (filtered) and 4.5% (floored at a weight of 25%). A weight of 47% brings the ten-day increase to 50%, at a calm-period margin of 5.8% of the position, nearly double the unfloored figure: stability is bought with collateral that sits idle most of the time.

**Exercise 20.8 ★★★.**

*Find the flaw.* “Our book is market neutral: long $200 million of shares at the [prime broker](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-pb), short $200 million of index futures at the clearing house. A 10% fall in the market costs us nothing, so [margin calls](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-margincall) are not a risk for us.” And a 10% rise?

**Solution of Exercise 20.8.**

In a 10% *rise* the futures lose $20 million, payable in cash the next morning, while the $20 million gained on the shares is unrealised: the [prime broker](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-pb) will lend against it, but at a haircut, on its own timetable, and not at all if it is cutting leverage at the same moment. In the fall the cash arrives from the clearing house, but the [prime broker](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-pb) may call margin on the shares. In both directions [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) on the futures rises with volatility. Market neutral is a statement about P&L; margin is a statement about cash, leg by leg, venue by venue.

## 20.8 Problem: A Fixed Portfolio in a Month Like March 2020

**Problem 20.1.**

Weekend problem — same contracts, more cash

The numbers are invented, in the proportions of a violent month. A fund is long 400 index futures (multiplier $50). At the start of the month the future is at 3 300 and the clearing house’s [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) is 4.5% of notional. Four weeks later the future is at 2 500 and the margin is 9% of notional. The fund keeps $12 million of cash and holds the rest of its assets in corporate bonds.

**Part I — Two calls.**

1. Give the notional and the [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) at the start.
2. Give the [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) paid over the month.
3. Give the [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) at the end, and the change.
4. Give the total cash paid, in dollars and as a percentage of the initial notional.
5. Compare with the fund’s cash. What must it do?

**Part II — The worst day.** On one day the future falls 9.5% from 2 750.

6. Give that day’s [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) .
7. The clearing house makes an intraday call at noon, when the future is down 6%. Give the amount and the deadline’s consequence.
8. The margin rate is raised that evening from 6% to 7.5% of notional. Give the additional [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) at the new price.
9. Give the total cash called in twenty-four hours.

**Part III — Models.**

10. Using [Figure 20.4](#fig-m1-margin-procyclical) , which model would have called the most additional margin in the first ten days, and which the least?
11. Which model leaves the clearing house best protected on day 15?
12. Explain why the floored model is unpopular in calm years.
13. A fund can choose between two brokers: one passes on the clearing house’s margin, the other charges 1.5 times it at all times. Which is safer for the fund in a crisis?
14. The clearing house doubles margins. Show with one sentence of arithmetic why the system as a whole needs more collateral although nobody has become poorer.

**Part IV — Judgement.**

15. The fund’s futures were a hedge of a short position in shares held elsewhere. Does that help with the calls?
16. What should the fund have measured before the month began?
17. Why can the fund not simply post its corporate bonds?
18. If many funds sell bonds to meet calls, what happens next?
19. State the *named result* : the additional [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) called on the unchanged portfolio over the month.
20. In one sentence: what is the difference between being solvent and being able to pay a [margin call](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-margincall) ?

**Solution of Problem 20.1.**

**1.** $400 \times 3\,300 \times 50 = \$66$ million; margin $4.5\% =
\$2.97$ million. **2.** $400 \times 50 \times 800 = \$16$ million. **3.** $400 \times 2\,500 \times 50 \times 9\% = \$4.5$ million: $+\$1.53$ million, although the notional has fallen by a quarter. **4.** $17.53 million, 26.6% of the initial notional. **5.** It has $12 million. It must raise $5.5 million by selling bonds, in the month when they are hardest to sell, or cut the position at the lows. **6.** The future falls to 2 488.75: $400 \times 50 \times 261.25 =
\$5.225$ million. **7.** $400 \times 50 \times 165 = \$3.3$ million, due within the hour: cash that is in a bond fund settling in two days does not count. A missed intraday call is a default like any other. **8.** From $6\% \times 55$ million $= \$3.3$ million to $7.5\% \times
49.775$ million $= \$3.733$ million: $+\$433\,125$. **9.** $5\,225\,000 + 433\,125 = \$5\,658\,125$, of which $3.3 million at noon. **10.** The filtered model the most (margin tripled in ten days); the floored model the least in relative terms, the plain historical model the least in absolute terms because it barely moved. **11.** The filtered and the floored models, which by then are near 9% of the position; the plain historical window is still near 5% while daily volatility is 3.5%, so a two-day move at 99% is about 11%: it is under-margined by half. **12.** It asks for half as much again in calm years, when members see no risk, competitors advertise lower margins, and the cost is visible while the benefit is not. **13.** The second, if the fund sizes its positions to it: its calls in the crisis are proportionally the same, but the fund has been holding less leverage and more collateral all along. The first is cheaper every day but one. **14.** Every long and every short posts [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin); doubling the rate doubles the collateral locked at the clearing house on both sides of every contract, while [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) merely moves cash from losers to winners. **15.** Not with the cash. The shares’ gain is at a [prime broker](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-pb), which decides how much can be withdrawn; the futures’ loss is due in cash at the clearing house in the morning. The hedge removes the loss, not the call. **16.** The cash needed to hold the position through a stress: a month of adverse [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) at a stressed volatility, plus the increase of [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) under the clearing house’s model in the same scenario, against cash and assets that are actually eligible and available in hours. **17.** Clearing houses accept a list of collateral, mostly cash and government bonds, with haircuts; corporate bonds are often ineligible or heavily haircut, and [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) is cash only. **18.** Bond prices fall, funds holding them face redemptions and their own calls, volatility rises, and the models of [Figure 20.4](#fig-m1-margin-procyclical) raise margins again: the loop that the word [procyclicality](#def-m1-margin-procyclical) names. **19.** **$1.53 million.** **20.** Solvency is about the value of assets against liabilities; a [margin call](https://one-course.com/books/quant/1/en/chapter/6-financing-repo-securities-lending-and-prime-brokerage#def-m1-financing-margincall) is about cash in the right currency at the right bank by ten o’clock.

## 20.9 Interview questions

**Interview question 20.1 ★ trader, researcher, developer.**

What is the difference between initial and [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin)?

**Solution of Interview question 20.1.**

[Variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) settles the day’s gain or loss in cash, so that no debt accumulates between the member and the clearing house; it passes from losers to winners. [Initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) is collateral held against the loss that could occur between a default and the close-out of the defaulter’s positions; it is returned when the position is closed. The first is P&L, the second a deposit whose size the clearing house can change.

*What the interviewer is looking for: the purpose of each, and that [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) can change with no change in position.*

**Interview question 20.2 ★ trader, bank.**

Describe how a scenario-based margin system such as SPAN computes the requirement for a portfolio of futures and options.

**Solution of Interview question 20.2.**

For each product the clearing house sets a price and a volatility scan range. Every contract has a risk array: its loss under sixteen scenarios, price moves of thirds of the range with volatility up and down, plus two extreme moves counted in part. Multiply by positions, sum by scenario, take the worst: the [scan risk](#def-m1-margin-scan). Add a charge for spreads between expiries, apply a minimum per short option, subtract credits between correlated products.

*What the interviewer is looking for: the extreme scenarios and the reason for the short option minimum.*

**Interview question 20.3 ★★ researcher, bank.**

What is [procyclicality](#def-m1-margin-procyclical) of margin, and what tools reduce it? What do they cost?

**Solution of Interview question 20.3.**

Margins that rise in stress add to the demand for cash when cash is scarcest. Tools: a floor from a stressed period, a weight on stressed value-at-risk, a long lookback that always contains a crisis, a buffer built in calm times and released in stress, limits on the speed of increases. All of them raise the average margin, which members pay for every day, and the last one leaves the clearing house under-covered while it catches up.

*What the interviewer is looking for: at least two tools and the cost of each.*

**Interview question 20.4 ★★ trader, researcher.**

You are short far out-of-the-money options. Which part of the margin calculation is aimed at you, and why does it exist?

**Solution of Interview question 20.4.**

The extreme scenarios and the short option minimum. Within the normal scan range a far out-of-the-money option loses almost nothing, so a scan alone would let a seller accumulate unlimited size for almost no margin, a position that loses catastrophically in exactly the move the scan does not reach. The extreme scenarios revalue at two or three ranges and count a fraction of the loss; the minimum puts a floor under every short option.

*What the interviewer is looking for: why the standard scenarios are blind to the position.*

**Interview question 20.5 ★★ researcher, mle.**

How would you include margin in a backtest of a leveraged futures strategy?

**Solution of Interview question 20.5.**

Track cash, not only P&L: daily [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin), [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) under the clearing house’s method with its parameters as they were on each date (margin rates rise in crises, precisely when a leveraged strategy is losing), the broker’s multiplier, and interest on posted collateral. Impose the constraint that cash never goes negative; size positions to the stressed margin, not the current one. Report the return on the capital actually needed, including the buffer.

*What the interviewer is looking for: historical margin rates and the cash constraint.*

**Interview question 20.6 ★★★ trader, bank, researcher.**

Your firm is hedged across two clearing houses. Describe its liquidity risk and how you would size the cash buffer.

**Solution of Interview question 20.6.**

P&L nets across the two houses; cash does not. On any large move one house calls [variation margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) on the losing leg in the morning while the gain at the other is paid out on that house’s cycle, possibly later and in another currency. [Initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin) rises at both. Size the buffer from a joint stress: the largest multi-day move in the common risk factor, with each house’s model re-run at stressed volatility, plus the timing gap between paying one and receiving from the other, plus currency. Pre-arrange credit lines and collateral transformation; test the intraday call operationally.

*What the interviewer is looking for: the timing gap and the simultaneous rise in [initial margin](https://one-course.com/books/quant/1/en/chapter/5-clearing-and-settlement#def-m1-clearing-and-settlement-margin).*