A Practitioner’s Guide to Pricing, Hedging, and Where the Money Actually Comes From
Julian R. Sterling
These are the companion files of Quantitative Finance by Julian R. Sterling.The book on Amazon →
The four Excel workbooks that go with the book, corrected where they were wrong. Every figure the book
prints is reproduced in them by a live formula rather than a typed constant, and each workbook ends with a
Checks sheet setting the printed figure beside the computed one: 84 controls in all,
every one green. Around them: the same four models with the inputs emptied, thirteen working documents
to print, forty questions that mark themselves, and three cases the book names and never takes to a number.
Nothing is locked, protected or watermarked.
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Twelve workbooks, the printable documents and the read-me. Each group below can also be
downloaded on its own. Last revised 24 September 2026.
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These carry the book’s own figures, so that every number it prints can be traced to a live
formula, and each closes on its checks sheet.
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Chapters 2, 4, 5 and 6
Kestrel 100 — three models
The closed form built from d1 and d2 with every intermediate on its own row, so the price of
11.454068 can be read as a construction rather than taken as an output. Put
by parity, the zero-volatility floor and the break-even daily move sit beside it as identities
that must hold.
The lattice is the sheet worth opening first. It is computed from the binomial distribution
rather than by laying out a tree, which means a thousand-step price is a single formula and the
entire convergence table — 10 steps at 11.220175 through 1000 at
11.451696 — recomputes when you change the volatility, the strike or the
maturity. The oscillation as the step count crosses the strike is visible rather than described,
and the halving of the error with each doubling of steps is a computed column. The simulation
results are the runs reported in the book; what is live beside them is the standard error, which
is the quantity that decides whether the number can be trusted at all. Revised: the error-halving
column and the three identities beside the price, described here before they were in the file, are now
in it, and each simulated price is shown in standard errors from the closed form: all four runs land below
it, the million-path run by 1.71.
All five Greeks from their formulas — delta 0.612816, gamma 0.015953,
vega 0.382882 per point, theta −0.018048 per day, rho 0.498276 —
and beside each of them a finite-difference estimate computed from the pricing function itself,
with the step size as an input.
That second column is the point of the workbook. Shrink the step and the numerical estimate
converges on the analytic one; shrink it further and it starts to disagree again, for reasons
of floating-point cancellation rather than of finance. Seeing where the agreement breaks is the
fastest way to understand what a Greek reported by somebody else’s system is actually
worth — and the sheet also converts each sensitivity into what it means on ten thousand
contracts, which is the form in which it reaches a risk report.
Revised: the finite-difference delta and vega read −4,980.55 and −4,982.83,
not 0.612816 and 0.382882: a missing bracket subtracted the strike term of the lower price instead of
adding it. Both now agree with the analytic value to the sixth decimal, rho has its own finite difference,
and each step is an input. The break-even move against theta net of financing is exactly 1.2562; against
raw theta it is 1.5042.
The input against the model, on one sheet. Eight points of volatility are worth
3.061643, or 26.7 per cent of the price; the three models disagree by
0.029701, or 0.2593 per cent; and the ratio between them —
103 times — is computed rather than quoted, so it can be recomputed on
whatever instrument you type in. On a short-dated far out-of-the-money option it is a different
number, and the sheet will tell you what it is.
Vega is shown across strikes and maturities so that its non-constancy is obvious, and implied
volatility is solved by iteration from a price you enter. The last sheet is the afternoon of
tests from Chapter 17 — parity, monotonicity, the boundary cases, the zero-volatility
limit, the deep-in-the-money limit — each with a PASS or a FAIL, ready to be pointed at a
pricing library you did not write and do not trust.
Revised: the implied-volatility iteration returned −227 per cent and a price of
−70.98 for a quote of 12.50, from the same missing bracket; six passes of Newton’s method now
reach 26.7302 per cent and reproduce the quote. Vega across strikes and maturities is on its own sheet,
and peaks at a strike of 107.12 at one year, not at the money.
The delta hedge laid out period by period on a single path: the share position, the cash
account, the financing, the rebalancing trade and the running profit, so the mechanics are
visible before the statistics arrive. Change the drift — the direction of the stock
— and watch the profit at expiry barely move. Change the realised volatility by two
points and watch it move a great deal. That is Chapter 13 in two keystrokes.
Across many paths the trade pays 1.5981 when realised volatility is 20 per
cent, 0.0004 when it is 24, and −1.6022 when it is 28,
against an analytic approximation of ±1.5315 that the sheet shows beside them rather
than in place of them. And when you are exactly right, the dispersion that remains —
2.3031 at twelve rebalances, 0.5265 at 252 — is what a
delta-hedged book actually owns. The square-root rule predicts a fall of 4.58 and the
measurement gives 4.37; the gap is shown, with the reason, rather than
smoothed away. Revised: the rebalancing count worked only at 52, and the table now runs to any count
from 1 to 252; and the shortfall is 85 per cent in the step from monthly to weekly, not
all of it, since weekly to daily falls short too, 2.1865 against 2.2014.
The four workbooks with their 51 inputs emptied. The checks sheets count what is still missing and test what holds for any inputs.
All chapters
Your option, your runs, your hedge
The three models, the sensitivities, volatility and the price, and the hedging simulator, with the same formulas and every input emptied: the option on the Inputs sheet, and the simulation runs a spreadsheet cannot produce, the four priced runs and the means and dispersions over many hedged paths. The lattice step counts, the volatility ladder, the vega grid and the finite-difference steps stay, because they are the axes of their tables or a method, not data.
The book’s 51 inputs are listed in a text file inside the archive. Type them back in and the book’s figures come back with them: 11.454068 for the price, 0.029701 across the three models, 4.37 against the rule’s 4.58.
Thirteen pages to print and use. A4, with margins wide enough for US Letter.
Chapters 2 to 17
Thirteen working documents
The pricing record, six inputs and the one nobody can observe, with Kestrel 100 beside a blank column. The closed form built on one page, with its three identities. The lattice convergence log. The simulation run report, with the standard error beside every price. The model comparison memo, noise taken out. The Greeks report with a second path to each. The risk report line, the lot in currency. The volatility note. The library acceptance tests. The hedge log, period by period. What the trade paid, and why. The rebalancing policy. The desk report line, with the questions to ask of it.
A workbook that marks itself. Answer in the yellow cells; the marking fills in as you go.
All parts of the book
Forty questions on the book
Seven on the three models, seven on the sensitivities, seven on volatility and the price, six on the hedge and what it pays, seven on discrete hedging, and six on the three cases below. Numbers are marked to a tolerance, multiple choice to the letter, and the score sheet counts your errors part by part: that is your revision list.
Nothing is hidden. The marking sheet holds every answer, every tolerance, and the reason for each answer.
New material, not a restatement. Three situations the chapters name and never take to a number. Each case is one workbook: the note is the first sheet, the model is the rest, and a checks sheet closes it. Type your own figures in and every finding recomputes.
Chapters 4, 5 and 7 · Case one
The spread that is mostly noise
Chapter 7 measures the three models 0.029701 apart; chapter 5 says the standard error decides whether a simulated number can be trusted. Put together: the lattice sits 0.002372 from the closed form, and the whole spread is the million-path run, 1.71 of its own standard errors low. On the models’ own disagreement the input outweighs the model 1,291 times, not 103. Pooled, the book’s four runs sit 2.36 standard errors low (if independent: ADDED), which is a question to put to whoever ran them.
Chapters 13 and 14 · Case two
Right about volatility, and still losing
Chapter 13 prices the edge, chapter 14 the residue; they never meet. Sold at 24 and realised at 22 (ADDED), hedged weekly, the trade expects 0.765765 an option and loses in 25.3 per cent of years. On the book’s own four-point edge, hedged monthly, it loses in 24.4 per cent. Keeping losing years to one in ten (ADDED) on a two-point edge takes 196 rebalances a year.
Chapter 14 · Case three
The rebalancing frequency, chosen
Chapter 14 gives the two-thirds-power rule and leaves the answer to the desk. At 15.00 a rebalance on the lot and a charge of 0.20 a unit of residue (both ADDED), the optimum is 67.71 a year, 3,047 in all. Weekly costs 1.7 per cent more, daily 58.6 per cent more; half or twice the optimal count costs 10.9 and 13.8 per cent more. The price of residue matters more than the argument over daily against weekly.
an input cell; everything else on the sheet is a formula
Black text
a formula — do not overtype these
Checks sheet
the printed figure beside the computed one, with a PASS or a FAIL; in the blank set and the cases, each control states its own verdict and the file ends ALL OK
ADDED
a figure the book does not give, used in the cases and marked wherever it appears
Why the checks matter more than the models
A workbook that agrees with a book proves nothing on its own — the author wrote both. What
the Checks sheets do is different: they force the model to reproduce a number that was printed
before the model existed, from a formula rather than from the number itself.
On this book the checks caught two things a reader would never have seen. The lattice was rebuilt
from the binomial distribution because a laid-out tree cannot reach a thousand steps in a
spreadsheet, and the closed form had to reproduce all seven printed convergence prices to six
decimal places before it could be trusted. And gamma, vega and theta came out wrong at first
because a defined name for the normal density collided with the one for the cumulative
distribution — spreadsheet names are case-insensitive, and the check sheet was the only thing
that noticed.
What a spreadsheet cannot do is stated rather than hidden: a million simulated paths and twenty
thousand hedged paths do not fit in one, so those results are the runs reported in the book. What
is computed live beside them is the quantity that decides whether they can be trusted.
Go beyond the companion files
Quantitative Finance develops the complete framework behind these models, with the assumptions, trade-offs and worked examples tied together.
The workbooks open in Microsoft Excel, LibreOffice Calc, Google Sheets and Numbers; the PDF
prints on A4 or US Letter. They use no macros and no add-ins, so nothing needs to be enabled or trusted. If your spreadsheet asks
to update links on opening, decline — there are none.
What changed, and when
24 Sep 2026
The four workbooks corrected, 84 checks. The finite-difference delta and vega read −4,980.55 and −4,982.83, not 0.612816 and 0.382882, from a missing bracket; the implied volatility of a 12.50 quote read −227 per cent, not 26.7302; the hedging simulator worked only at 52 rebalances; the square-root rule’s shortfall is 85 per cent in the first step, not all of it. The error-halving column, the identities beside the price and vega across strikes and maturities, described on this page, are now in the files. Added: a blank set, thirteen working documents, forty questions that mark themselves, and three cases.
How do you use put-call parity? A worked exampleP = C minus S plus K e^-rT. A call at 11.4541 with spot and strike at 100, 4% for a year, implies a put of 7.5330; a put at 8.10 is 0.5670 rich.
How do you calculate option theta per day?Black-Scholes theta divided by 365: -6.5877 a year is -0.018048 a day on a one-year at-the-money call. With 10 days left the same option loses 0.0847 a day.
Closing the DealTwo defensible bridges 20.70 million apart, a peg worth 7.00 million, and the six choices that remove 6.60 of a 12.00 earn-out.
CMBS and CRE CLOsWhere the loss actually lands, from appraisal reduction to realised severity, and what the B-piece is really being paid for.
How to Read a Commercial LeaseThe three refinements chapter 19 names and never performs, and the renewal rate below which the mark-to-market is worth nothing.
How to Read a Credit AgreementWhere the default actually comes from, the cure that costs 5.5 times the other, and the capacity nobody adds up.
How to Read a Real Estate Loan AgreementThe cure ratio in closed form, the four-point window in which the cheap cure works, and the cure sized to the wrong threshold.
Office Real EstateA six per cent yield that returns 3.2 per cent once the re-letting cycle is paid for, and the headline-to-net-effective rent arithmetic.
Private Equity Real EstateBoth worked waterfalls to the dollar, the two capital stacks, and the arithmetic of the promote made changeable.
Private Markets PerformanceThirty-one of the thirty-three figures chapter 19 publishes reproduce exactly — and the two that do not are named rather than quietly adopted.
Raising a Real Estate FundThe chapter 17 funnel run on a calendar — when the first close actually lands, and why more travel does not help.
Real Estate FinanceFour people look at one building and reach four numbers; the lender is whole only above 105,109,489, twelve per cent below today’s value rather than forty.
Real Estate Financial ModelingProperty, development and fund models built line by line, and the modelling test worked end to end.
Real Estate Fund ManagementThe waterfall of 6.11, the build-to-core of 8.7 and the proceeds gap, reproduced as live formulas rather than asserted.
REIT Analysis and ValuationFFO of 532.0, AFFO of 381.0, net asset value and dividend safety — every figure a formula you can change.
Retail Real EstateThe occupancy cost of every unit in a centre, the sixteen per cent of the rent roll no tenant can sustain, and the right-size-convert-or-hold decision priced.
Sale and LeasebackA €179.5 million transaction end to end, with rent cover measured on the entity that actually signs the lease.
Self-Storage Real EstateThe cohort engine behind a 590-unit store, and the rate increase on existing customers priced against the move-outs it causes.
The Fund Finance ProfessionalChapter 8 builds the reported-to-eligible NAV bridge; chapter 9 computes every ratio without it. Two points at every state — and what a subscription line does to the IRR.
The Growth Equity InvestorWhat a pro rata cheque really costs, and the band where defending your ownership loses money.
The Private Credit InvestorThe two coverage ratios are not measured on the same thing: the erosion is 47.7 per cent, not the 28.7 the headline implies.
The Private Equity Fund Controller PlaybookThe book defines IRR, DPI, RVPI and TVPI, tells you to update them at the exit, and prints not one value. Computed: a 1.833× deal inside a fund at 0.892 TVPI.
The Venture Capital AssociateWhat defending a position costs, and how many companies a reserve pool actually defends.
Financial Risk ManagementA fund inside every limit that cannot meet a redemption — and the number that decides it is the one with no currency attached.
Business ValuationThree advisers land 26.8 per cent apart on one company, and the whole gap turns out to be 1.96 points of perpetual growth.
Asset ManagementFour people quote four returns for one mandate, all correct and 2.7017 points apart — forty-eight times the manager’s net skill.
Alternative InvestmentsA manager reports 13.29 per cent and the endowment earns 6.26 — both correct, and only a third of the advertised advantage arrives.
Credit AnalysisFour defensible EBITDAs on one borrower give leverage from 3.19x to 6.47x — and the add-back argument is fifty times the covenant headroom.
Venture CapitalOne company out of twenty-eight returns 56.7 per cent of the fund, and half the capital goes in after the decision — at half the return.
Machine Learning for FinanceFive people quote the accuracy of one credit model, all five are right, and the number that decides how much money it makes is none of them.
Commercial Real Estate InvestingThe equity earned 8.6647 per cent and the investor received exactly 8.0000 — the preferred return, and nothing above it.
Mergers and AcquisitionsThe board paper says the deal creates 13,436,667 of value. The arithmetic says it destroys 17,530,855. Nobody is lying.
DerivativesThe treasury report says the hedge cost 1,233,698. That is the interest differential, not a cost.
Treasury ManagementFive cash balances for one company, all correct and 145,600,000 apart — and the revolver that is two-thirds of the liquidity leaves at a revenue fall of 8.4127 per cent.
Financial Planning and AnalysisRevenue 3.0190 per cent above budget and operating profit 16.3209 per cent below it, in the same quarter, with every figure correctly stated.
Energy TradingA position report that is 91.7031 per cent hedged and correctly computed, on a book that is short 2,542,000 MWh — and a margin call of 198,400,000 the next morning.
Construction Cost ControlA contract sum of 26,301,102 became a final account of 29,153,363 on the building that was drawn — and 85.8 per cent of what was lost was knowable on the day it was signed.
Trade FinanceSix routes to payment on one 4,200,000 export order cost between 178,040 and 223,268, a spread worth 14.8 per cent of the margin, and a day of buyer credit costs 1,031.76.
Cost AccountingOne factory costed twice on the same 13,440,000 of overhead, and 4,053,091 moves between four product families.
Pricing StrategyA list price of 148.00, a pocket price of 112.51, and the nine deductions in between — with what one point of price is actually worth.
Contract ManagementA bid 5.50 a unit cheaper, signed on paper worth 9.81 a unit more — and 94.68 per cent of it was knowable on the day of signature.
Capital MarketsFour defensible readings of one bond, the cost of an issue under each convention, and the placement discount computed at a constant discount.
Financial RegulationThree ratios pass and the fourth is in breach by 1,956.0 million: the output floor, the liquidity buffer after encumbrance, and the loan split.
Bank ManagementFour defensible readings of one balance sheet, the deposit beta measured two ways, and the liquidity that lasts 3.9 days.
The Real Estate WorkoutFour ways out of one broken loan, discounted to today, and the 58.7 cents at which a payoff beats enforcement.
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