Swaptions, the Black Model & Non-Rate Swaps

Branch: Derivatives (Finance-Plus Ring) · Node DV1.04 · ~18 focused hours · Mastery gate ≥ 85%

This Ring is additive institutional and CFA breadth, and it presents the toolkit and the plumbing alongside, not in place of, the program's process-over-P&L, concentration-over-diversification philosophy.

Twenty-five million dollars of floating-rate debt, and two quotes on the same screen. The first says the borrower can fix its rate at 4.9628% across the three quarters that are still open and pay nothing at all for the privilege. The second says it can cap the rate at 4.75%, keep every basis point of any fall, and pay $74,848.94 up front for that.

Both quotes come off the same curve. Neither is a forecast. The whole difference between them is that one is an obligation and the other is a choice, and the $74,848.94 is what a choice costs when the market is charging 22% volatility for it.

That distinction runs through every instrument here. A forward, a future and a swap are forward commitments: both sides are bound, nobody pays anything at inception, and the contract can be worth a large negative number to you later. An option, a cap, a swaption and a credit default swap are contingent claims: one side buys a right and pays for it, the other side sells the right and collects, and the buyer's worst case is the premium. Almost every argument a derivatives desk has is about which of those two shapes a particular risk deserves.

You have priced forwards, futures, vanilla swaps and equity options already. What is missing is the machinery for the instruments a treasurer and a portfolio manager actually meet. Nobody hedges an issuance with a European call on a stock. They buy a payer swaption. Nobody caps a floating-rate loan with a straddle. They buy a cap, which is a strip of options on forward rates, priced one at a time. And a great deal of what a multinational balance sheet contains is not a rate derivative at all: a currency swap that exchanges principal in two currencies, or an equity swap that rents index exposure without buying a share.

One substitution opens all of it. Replace the spot price with the forward as the underlying, and the option formula you already know collapses into something simpler. There is no dividend yield, no carry term, no convenience yield, because the forward already contains them. What comes out is the Black model, and it prices caps, floors, swaptions, options on futures and the swap-rate volatility surface with one shape.

By the end you will strip a cap into caplets and price each one off the forward curve, then prove that a cap minus a floor at the same strike is exactly a payer swap. You will price a payer swaption, carry it to expiry and make the exercise decision. You will price an option on a futures contract and check it against a binomial tree that permits early exercise. And you will build a fixed-for-fixed currency swap off two discount curves so that it is worth zero at inception, then mark it six months later with the rate move and the currency move separated. Every number below was produced by a Python script before it was written down, and the verification identities are stated so you can catch your own errors without asking anyone.

One warning, because this region rewards it. Cap and swaption premiums are the most volatility-sensitive numbers in the fixed-income world: the same cap on the same curve costs $48,519.81 at a 10% volatility and $128,904.05 at 45%, a swing larger than the entire premium at the level actually quoted. A price quoted without its volatility is not a price. Ask, every time.

Learning objectives

By the end you can:

1. Classify any derivative as a forward commitment or a contingent claim, state what each shape does to the payoff diagram and to the credit exposure, and place forwards, futures, swaps, options, caps, swaptions and credit default swaps in the right family.
2. Contrast exchange-traded and over-the-counter derivatives on five dimensions (standardisation, counterparty, clearing, margin and transparency) and say which dimension drives a given desk's choice.
3. Explain novation and the clearing house's default waterfall, distinguish initial from variation margin as institutions rather than formulas, and compute the illustrative scale of each on a stated swap.
4. State the purposes derivatives serve and the risks they carry, each with its mechanism and one public-record example, so that "derivatives are risky" becomes five separate, separately manageable claims.
5. Price with the Black model, explaining why replacing spot with the forward removes the carry and dividend terms, and apply it to an option on a futures contract with the American-exercise caveat identified.
6. Decompose a cap into caplets and a floor into floorlets, price each off the forward curve with its own expiry, and explain why a one-year quarterly cap contains three options and not four.
7. Prove and use rate-option parity: cap minus floor at a common strike equals the payer swap on the same schedule, with the at-the-money-forward strike and the zero-cost collar as its two special cases.
8. Price payer and receiver swaptions, apply payer-receiver parity to recover either from the other, compute a straddle's breakeven rate move, and carry a swaption to expiry and settle it in cash or in a swap.
9. Price and mark a fixed-for-fixed currency swap off two discount curves, and split the mark to market into carry, a rate component and a currency component that add back exactly.
10. Value an equity swap in its return-for-fixed, return-for-floating and return-for-return forms, computing each leg separately and reading the result as synthetic exposure rather than as a trade.

The understanding objective is what the gate rewards: you earn a pass by pricing the caplet by hand and by making the exercise decision yourself. Code and assistants make the strip, the surface and the mark-to-market fast, and the verification section states exactly which identity has to fire before any of that output is admissible.

Prerequisites & connections

Builds on. DV1.01 gave you the cost-of-carry argument under three compounding conventions, the forward rate agreement, the valuation of a swap as two bonds, and the par swap rate c = (1 - DF_final) / Σ DF_i. Everything here inherits those conventions and must not contradict them. DV1.02 gave you risk-neutral pricing, the binomial tree, Black-Scholes-Merton and the Greeks; the Black model below is that machinery with the forward substituted for spot. DV1.03 gave you option strategies, the collar, delta and gamma hedging, and one paragraph plus one practice item on caps, which this node finishes. FI1.03 gave you bootstrapping, so that a discount curve is an object you build rather than one you are handed. From the analyst core, M1.09 owns hedge accounting and M0.02 owns the clearing and settlement of cash equities.

Feeds forward. FI1.04 assumes an interest-rate volatility when it calibrates its tree, and the swaption market is where that number comes from; a callable bond's issuer option is a short receiver swaption the holder never chose to sell, and the option cost that module computes is the same quantity priced two ways. AA1.02 implements allocation shifts through overlays that are usually swaps, and the equity-swap valuation here is what its risk budget consumes. AL1.01 reads option and swap overlays inside a fund's return stream. MS1.04 owns the cross-currency basis as a funding phenomenon, which is the reason a currency swap priced off two clean domestic curves will not match a dealer's quote.

The material this node does not own is worth naming precisely, because four neighbouring files already teach it and rebuilding any of it here would create a contradiction rather than a lesson. DV1.01 owns forward, futures and FRA pricing and vanilla swap valuation, including the day-count and compounding simplifications carried here unchanged. DV1.02 owns Black-Scholes-Merton, put-call parity on a spot underlying, and the Greeks. DV1.03 owns option strategies, delta and gamma hedging, and transaction-level foreign-exchange hedging with forwards and risk reversals. DV1.05, Exchange-Traded Derivatives in Practice: Margins, Expiry, Tax and Portfolio Uses, owns exchange-traded contract mechanics, margin arithmetic in rupees, expiry and settlement practice, the tax overlay and the portfolio uses of futures. FI1.04 owns embedded options, option-adjusted spread and credit default swap pricing. M0.06 owns personal taxation of securities. Where those modules own a concept, this one cites the idea in plain language and moves on.

Core teaching content

4.3 The clearing house: novation, margin and the default waterfall

Central clearing is the single largest change in derivatives market structure since the 1990s, and it is usually taught as a formula for initial margin. The formula is the least interesting part.

Novation is the mechanism. When a trade is cleared, the original contract between buyer and seller is torn up and replaced by two contracts, each with the clearing house in the middle. The clearing house is now the buyer to every seller and the seller to every buyer, and its net position is flat by construction. Nothing about the market risk has changed. What has changed is that a chain of bilateral credit exposures, in which everyone had to assess everyone, has been replaced by a hub whose credit everyone assesses once.

That hub has to be made safe, and the tools are margin and a loss-absorbing stack.

Variation margin is the daily settlement of the position's change in value. Calling it a deposit or a buffer misses what actually happens. The party whose position lost value pays cash to the party whose position gained, every day, and the contract's value resets to zero. A $50 million five-year swap has an annuity of 4.126543, so a one basis point move in the swap rate changes its value by 50,000,000 × 4.126543 × 0.0001 = $20,632.71. On a ten basis point day, $206,327.13 moves between the two accounts, and no unpaid claim ever accumulates.

Initial margin is the buffer against what could happen between the last variation-margin payment and the moment the clearing house finishes closing out a defaulted member's position. It is sized off a model, and the parameters that matter are the assumed close-out period and the confidence level. To see the scale on the same swap: at a 6 basis point daily volatility of the swap rate, a five-day close-out and a 99% one-tailed confidence, initial margin is roughly 2.326348 × 6 × √5 × 20,632.71 = $643,972.33, or 1.2879% of notional. Real requirements come from the clearing house's own model rather than from that sketch, and the model is published; the sketch is here to show why the number is measured in a percent or two of notional rather than in a fraction of a percent.

When a clearing member fails, the losses are absorbed in a stated order, and the order is the thing to remember rather than any number in it. First the defaulter's own initial margin, then the defaulter's contribution to the guarantee fund, then a tranche of the clearing house's own capital, deliberately placed ahead of surviving members so that its incentives point the right way. Then the surviving members' guarantee-fund contributions, then any assessment rights the rulebook grants over those members, and finally the recovery tools of last resort. The name for that sequence is the default waterfall, and the reason it looks like a securitisation waterfall is that it is one: a strict priority ordering of who absorbs loss first.

Two consequences follow, and both are worth carrying. Clearing converts counterparty credit risk into liquidity risk, because a cleared position that moves against you demands cash today rather than a credit assessment later. And clearing concentrates risk rather than destroying it, which is why the clearing houses are themselves systemically important and supervised as such.

4.6 Who is on each side: issuers, investors, and the accounting seam

The same instrument does opposite work depending on which balance sheet it sits on, and getting that straight is most of what a footnote reader needs.

An issuer uses derivatives to shape the liability it has created. A company that issued fixed-rate debt and now wants floating enters a receive-fixed swap. A company planning to issue in six months and worried about a rate rise buys a payer swaption or enters a forward-starting swap. A company that issued callable debt owns the call rather than having sold it, because the right to redeem early belongs to the issuer, and an issuer that has decided it will not call can monetise that right by writing a receiver swaption of matching expiry and tenor. An exporter with dollar revenue and rupee costs uses a currency swap to align the liability with the cash that will service it.

An investor uses derivatives to shape the asset side. Adjusting equity exposure, adding or removing duration, taking a view on volatility, renting an index return through an equity swap, and buying or selling credit protection are all asset-side decisions. The instruments are identical; the intent, the accounting and the disclosure are not.

The accounting seam matters because the footnote is usually where an analyst meets these instruments. A derivative is carried at fair value, and the default is that the change in fair value hits profit or loss. Hedge accounting is an election that lets a company match the timing of the hedge's gain with the hedged item's loss, and it comes with documentation and effectiveness requirements that are stringent enough that many companies choose not to elect it. The accounting phase of the analyst core owns that machinery in full. What to carry from here is the reading habit: when a company's earnings swing on a line called fair value change on derivatives, find out whether the swing is an economic result or the absence of a hedge-accounting election, because those are very different facts.

4.15 The four checks that make any of this admissible

Every model in this node produces a number that looks reasonable at almost any value. A cap premium of $74,848.94 and one of $48,519.81 are equally plausible on a page, and the difference between them is nothing but an assumption about volatility that the curve does not constrain. The defence is not care. It is four identities that either fire or do not.

1. Does cap minus floor equal the payer swap on the same schedule, to machine precision? If it does not, the caplet strip has a date error, a discounting error or a mismatched schedule. This check is model-free in the volatility, so it isolates the plumbing from the assumption.
2. Does payer minus receiver equal the forward swap value, N·A·(S - K)? Same logic, same independence from sigma, and it also confirms that the annuity and the forward swap rate were built off the same curve.
3. Does the option converge to Black when early exercise is switched off? A tree that permits early exercise should reproduce the European price to within the discretisation error when the exercise test is removed. Worked example 4 shows a 2,000-step European lattice landing $0.000324 from Black on a $9.78 put, and that residual is what a converged tree looks like.
4. Is every assumption named? The volatility, the volatility convention (flat against a stripped strip), the day count, the compounding, the settlement style, and for a currency swap whether the cross-currency basis is in or out. An unnamed assumption is the mechanism by which a teaching price gets quoted as a dealer price.

The four take five minutes and they catch essentially every error this region produces. The reason the gate rewards the by-hand skill is that you cannot audit a machine's output with a machine you also cannot audit.

{
  "skill": "Stripping a cap into caplets and pricing a swaption grid without letting the model's confident output stand in for a check",
  "use": "Use a code assistant to build the Black-76 pricer, the caplet strip and the forward-swap-rate/annuity pair from a bootstrapped discount curve, then to generate the parity assertions you run yourself before reading any premium.",
  "tools": ["Chat assistants with code execution - Claude, ChatGPT, Gemini", "Python: NumPy and SciPy (norm.cdf, brentq for the implied-volatility and zero-cost-strike solves)", "A published curve to test against - the US Treasury par yield curve or SOFR OIS for USD, CCIL or FBIL for INR"],
  "prompt": "Given a set of zero rates, write Python that computes discount factors, simple forward rates for a stated accrual convention, and then: (a) a cap and a floor as strips of Black-76 caplets and floorlets, using the FIXING date inside d1/d2 and the PAYMENT date for discounting, excluding the already-fixed first period; (b) the forward-starting par swap rate and annuity for a stated expiry and tenor; (c) payer and receiver swaption prices. Then assert cap(K) - floor(K) equals the payer swap PV to within 1e-8, and payer - receiver equals N*A*(S-K) to within 1e-8. Finally solve for the floor strike that makes a collar zero-cost at a stated cap strike, and report when no such strike below the forward exists.",
  "verify": "Run both assertions and then break the code deliberately to confirm they can fail: discount the caplet at the fixing date instead of the payment date and check that the cap-floor parity assertion fires, and include the already-fixed first period and check that the premium jumps while parity still holds, which is the error parity CANNOT catch. Recompute one caplet by hand from d1 and d2. Confirm the volatility you used is the volatility of the FORWARD and name where it came from. Trace and Date-check every curve point to a named primary source, and state the volatility, the day count and the compounding alongside any premium you report (Guardrail 1-5).",
  "diy": "You must be able to price a single caplet with a calculator from F, K, T, sigma and the payment discount factor, and to say why a one-year quarterly cap holds three options rather than four. Parity is a check on the plumbing and cannot see a wrong volatility or a wrong period count, so the judgement about what the strip should contain stays yours.",
  "market": "US"
}

Common mistakes & how experts think differently

1. Getting the caplet schedule wrong, in either of the two available ways. The first is counting four caplets in a one-year quarterly cap: the first period's rate is already fixed when the cap is struck, so there is nothing contingent about it and no option is sold. Including it overstates the premium by roughly a quarter and, worse, cap-floor parity still passes, because the same phantom period appears on both legs. The second is discounting a caplet at its fixing date: the option expires on the fixing date but pays one period later, so the T inside d1 and d2 and the discount factor outside the bracket refer to different dates. The expert writes the fixing date and the payment date as two separate columns before writing any formula, so the two can never become the same variable, and states the number of live caplets out loud against the loan's fixing schedule rather than against the cap's tenor.

1. Quoting a swaption premium in basis points of notional and comparing it to a swap rate. A premium of 180.04 basis points up front and a rate of 5.25% a year are not comparable quantities. Dividing the premium by the underlying swap's annuity converts it into 43.63 basis points a year, which is. The expert converts every up-front premium into a running spread before comparing hedges, and treats a treasury memo that skips the conversion as unfinished.

1. Treating "at the money" as "at today's swap rate". A swaption is at the money when its strike equals the forward swap rate for its own expiry and tenor, not the current spot swap rate. On an upward-sloping curve those differ by a lot, and mislabelling the strike misprices the option and misreads the delta. The expert computes the forward swap rate first and quotes every strike as a spread to it.

1. Assuming the cap's flat volatility is a caplet's volatility. A quoted flat volatility is a single number chosen to reproduce a total premium, not an estimate of any individual forward rate's volatility. On the module's curve, caplet volatilities of 26%, 22% and 19% and a flat 21.786675% price the same cap and value the individual caplets differently by thousands of dollars. The expert never values a partial unwind, a caplet sale or a collar leg off a cap's flat volatility, and asks whether a quoted number is a flat or a stripped volatility before using it.

1. Pricing an American option on a futures contract with Black. Black is European by construction and contains no exercise test. Early exercise on a deep in-the-money futures put is a live decision, because exercising converts a discounted payoff into cash today. The expert prices the European value with Black for speed, then runs a tree with the exercise test whenever the option is in the money and the discounting is material, and reports the early-exercise premium as a separate number rather than folding it in.

1. Using the risk-neutral probability from a spot underlying in a futures tree. For a futures price the probability is (1 - d)/(u - d), with no growth term, because entering a futures position costs nothing. Carrying over (e^{rΔt} - d)/(u - d) from a stock tree biases every node. The expert derives the probability from the statement "this underlying has zero expected return under the risk-neutral measure" each time rather than reusing a remembered expression.

1. Marking a currency swap at the new spot without revaluing either leg, and then reporting only the net. The two legs are bonds in different currencies and both have moved. Multiplying an unchanged foreign leg by a new spot captures the currency move and misses the rate move entirely, which on the module's numbers would report plus ₹2.5387 crore instead of plus ₹26.26 lakh. The reporting half of the error is worse: a small net can sit on top of two large offsetting components, and a board that sees only the net has no idea how much rate risk and how much currency risk the position carries. The expert revalues each leg on its own curve first, applies the spot rate last, and reports carry, rates and currency as three numbers that add to the total, because those components are what the next quarter's mark will be made of.

1. Treating an equity swap's value as the index return. The equity leg is the index return, but the swap's value is that leg minus the funding leg, and the funding leg is a bond that has its own price. An index up 2.500000% can leave the receiver up only 1.296266% of notional, and a rise below the breakeven of 1.203734% leaves the receiver down. The expert computes both legs separately and quotes the breakeven index level alongside the mark, because that is the number that says whether the position is working.

1. Believing a premium quoted without a volatility. Every price in this node is a function of an assumption the curve does not pin down. The module's cap costs $48,519.81 at a 10% volatility and $128,904.05 at 45%, and both are correct given their input. The expert asks "at what volatility, flat or stripped, of what forward?" before recording any premium, and treats a quote that cannot answer as a quote from someone who did not build it.

1. Confusing initial margin with variation margin. Variation margin is settlement of a loss that has already happened; initial margin is a buffer against a loss that has not. Calling either "collateral" hides the difference and hides the liquidity demand. The expert models the two separately in a cash-flow forecast, because a hedge that is right and unfunded fails exactly as completely as a hedge that is wrong.

Worked examples

Every figure below was produced by a Python computation before it was written down. All entities are explicitly synthetic; all curves are illustrative but internally consistent, and the parity identities are stated so each answer is verified by an independent route.

Worked example 1: A one-year SOFR cap, caplet by caplet (US, $)

Ridgeline Logistics Inc. (synthetic) has drawn $25,000,000 on a one-year revolving facility that reprices quarterly against SOFR. It buys a cap struck at K = 4.75%, accrual factor tau = 0.25 per period, quoted at a flat volatility of 22%. The dollar zero curve, continuously compounded, is 4.60% at three months, 4.70% at six, 4.78% at nine and 4.85% at twelve. Price the cap, price the matching floor, and verify that cap minus floor is the payer swap at the same strike.

Solution. Step 1, discount factors and forward rates. D(t) = e^{-z·t} gives 0.988565872, 0.976773975, 0.964785000 and 0.952657339. The simple forward for each period is F = (D(T1)/D(T2) - 1)/0.25:

Step 2, count the options. The facility has four interest periods, but the first period's rate was fixed at drawdown. The cap therefore contains three caplets, not four.

Step 3, price each caplet with Black. Each uses its own time to fixing inside d1 and d2, and its own payment-date discount factor outside:

Cap premium = $74,848.94, which is 29.94 basis points of notional.

Step 4, the floor. The same three periods, same strike, same volatilities, put rather than call: floorlets of $10,559.24, $12,284.80 and $13,512.05, totalling $36,356.09, or 14.54 basis points. All three forwards sit above the strike, so every floorlet is pure time value, which is why the floorlet column reproduces the caplets' time-value column exactly.

Step 5, verify by parity. The annuity over the three payment dates is 0.25 × (0.976773975 + 0.964785000 + 0.952657339) = 0.723554079, and the floating leg's value factor is D(0.25) - D(1.00) = 0.035908533, so the par forward-starting swap rate is 4.962799%. The payer swap at 4.75% is worth 25,000,000 × Σ 0.25 × (F - 0.0475) × D(pay) = $38,492.86. Cap minus floor is 74,848.94 - 36,356.09 = 38,492.86, and the residual against the swap value is $2.9 × 10⁻¹¹.

Step 6, the payoff tie. If SOFR fixes at 6.00% for the final period, that caplet pays 25,000,000 × (6.00% - 4.75%) × 0.25 = $78,125.00, which is precisely the figure the DV1.03 practice item on caps computes for the same contract. The two modules price and settle the same instrument.

Read it. The cap costs 29.94 basis points of notional up front, which over the annuity of the three covered quarters is 41.38 basis points a year on the covered amount. The forward-starting swap at 4.962799% costs nothing and fixes the rate 21.28 basis points above the cap strike. What the borrower is buying for 41 basis points a year is the right to benefit from a fall, and whether that is worth it is a view about the distribution rather than about the level. What the number does not mean is that the market expects rates above 4.75%: it means the market charges 22% volatility for the uncertainty, and at 10% the same cap costs $48,519.81 while at 45% it costs $128,904.05.

Worked example 2: A payer swaption, priced and then exercised (US, $)

Cedar Point Manufacturing Inc. (synthetic) will issue $50,000,000 of five-year fixed-rate debt in one year and wants protection against a rate rise. It buys a 1y × 5y payer swaption struck at K = 5.25%, at a swap-rate volatility of 24%. The dollar zero curve, continuously compounded, is 4.60%, 4.70%, 4.78%, 4.85%, 4.90% and 4.94% at one through six years. Price it, then carry it to expiry under two rate outcomes.

Solution. Step 1, the forward swap rate and the annuity. The swap starts at year 1 and pays annually at years 2 through 6. Discount factors are 0.955041962, 0.910282762, 0.866407437, 0.823657904, 0.782704538 and 0.743489973.

A = D(2)+D(3)+D(4)+D(5)+D(6) = 4.126542614
S = [D(1) - D(6)] / A = (0.955041962 - 0.743489973) / 4.126542614 = 5.126616%

Step 2, Black on the forward swap rate. With S = 5.126616%, K = 5.25%, T = 1, sigma = 24%: d1 = 0.020907, d2 = -0.219093, N(d1) = 0.508340, N(d2) = 0.413289.

payer = 50,000,000 × 4.126542614 × [0.05126616 × 0.508340 - 0.0525 × 0.413289]
      = $900,201.39      (180.04 bp of notional)

Step 3, the receiver and the parity check. The receiver at the same strike is $1,154,776.27. Parity requires payer - receiver = N·A·(S - K) = 50,000,000 × 4.126542614 × (0.05126616 - 0.0525) = -$254,574.88, and 900,201.39 - 1,154,776.27 = -254,574.88. Residual: $1.5 × 10⁻⁹.

Step 4, expiry with rates up. The five-year par swap rate sets at 6.10%. On a flat 6.10% curve the five-year annual annuity is 4.200929113, which checks out because a 6.10% annual bond prices at exactly 100.000000000 on that curve. Exercise:

swap value = 50,000,000 × (6.10% - 5.25%) × 4.200929 = $1,785,394.87

The premium future-valued to expiry is 900,201.39 / 0.955041962 = $942,577.84, so the net gain is $842,817.03. Cedar Point's coupon on the new bond is 85 basis points above the 5.25% it bought protection at, and the swaption has handed back the present value of that difference.

Step 5, expiry with rates down. The rate sets at 4.60%. Exercising would produce a swap worth 50,000,000 × (4.60% - 5.25%) × 4.377770 = -$1,422,775.35, so the option is abandoned and the loss is the premium, $942,577.84 at expiry value. Cedar Point issues at 4.60% and is better off than the hedge implied.

Step 6, the breakeven. Because the payoff at expiry is N·(S_T - K)·A_T and the premium's future value is premium / D(1), and the forward annuity is A / D(1) = 4.320797, the annuity cancels and

breakeven S_T = K + premium / (N × A) = 5.2500% + 43.6298 bp = 5.686298%

That identity holds when the annuity at expiry equals today's forward annuity. On a flat curve at the breakeven the actual annuity is 4.248557 rather than 4.320797, so the payoff there is $926,818.66 against a premium of $942,577.84, a gap of 1.7% that is entirely the annuity approximation.

Read it. The premium of 180.04 basis points is the number the treasurer will be asked about, and it is the wrong number to argue with. Over the five-year annuity of the swap it is 43.63 basis points a year, and that is the figure to set against the alternative of locking at the forward for nothing. What the price does not say is anything about where rates are going. It says the market charges 24% volatility on a forward of 5.126616%, and at a lower volatility the same protection is cheaper without any change in the forecast.

Worked example 3: The swaption straddle as a pure volatility position (US, $)

Same curve and same expiry as Worked example 2, but now struck at the forward, K = S = 5.126616%, on $50,000,000, at 24% volatility. Price the straddle and find what has to happen for the buyer to make money.

Solution. Step 1, the two legs. At K = S the parity identity payer - receiver = N·A·(S - K) forces the two premiums to be equal, and they are: payer $1,010,339.00, receiver $1,010,339.00. Straddle premium $2,020,678.01, or 404.14 basis points of notional.

Step 2, how far the delta cancels. The payer's sensitivity to the forward swap rate is N·A·N(d1) and the receiver's is -N·A·N(-d1), so the straddle's is N·A·[2N(d1) - 1]. Striking at the forward sets d1 = 0.5 × sigma × sqrt(T) = 0.120000, which leaves 9.5517% of a forward swap's rate sensitivity, or $1,970.77 per basis point against a premium of $2,020,678.01. The position starts close to flat rather than exactly flat, and what dominates from there is how far the rate moves rather than which way it goes.

Step 3, the breakeven move. The payoff at expiry is N × |S_T - K| × A_T and the premium future-valued is premium / D(1). With A_T taken as today's forward annuity A / D(1), the discount factor cancels on both sides:

breakeven |S_T - K| = premium / (N × A) = 2,020,678.01 / (50,000,000 × 4.126543) = 97.9356 bp

The five-year swap rate must finish outside 4.147259% to 6.105972%.

Step 4, the rule of thumb. The standard approximation for an at-the-money straddle's breakeven is sqrt(2/pi) × sigma × sqrt(T) × S = 0.7979 × 0.24 × 1 × 5.126616% = 98.1707 bp. The exact answer is 97.9356, so the approximation is within 0.24%.

Step 5, what a volatility point is worth. Re-price the straddle at 22%, 24% and 26%: $1,852,997.10, $2,020,678.01 and $2,188,157.82. Between 23% and 25% the premium moves $83,791.21 per volatility point. On a $50 million notional that is 16.76 basis points of notional per point of volatility.

Read it. The straddle costs 404.14 basis points of notional, which over the annuity is 97.94 basis points a year, and that number is identical to the breakeven move in basis points. That is not a coincidence: the breakeven and the annuity-normalised premium are the same quantity, which is why market participants quote swaption straddles in "basis points of annualised premium" and read the breakeven straight off the quote. What the position is not is a view on rates. It is a view that realised volatility will exceed 24%, and it loses money in a market that drifts 90 basis points in either direction without hurrying.

Worked example 4: An option on a futures contract, Black against a tree (US, $)

A December crude oil futures contract trades at F = $78.40 per barrel, 1,000 barrels per contract. An $86.00-strike put expires in three months, implied volatility 32%, the risk-free rate 4.60% continuously compounded. Price it European with Black, then decide whether early exercise is worth anything.

Solution. Step 1, Black. D = e^{-0.046×0.25} = 0.988565872, d1 = -0.498271, d2 = -0.658271, N(d1) = 0.309147, N(d2) = 0.255182.

put  = 0.988565872 × [86.00 × (1 - 0.255182) - 78.40 × (1 - 0.309147)] = $9.778335 / bbl
call = 0.988565872 × [78.40 × 0.309147 - 86.00 × 0.255182]             = $2.265234 / bbl

Per contract, $9,778.34 and $2,265.23.

Step 2, the parity check. c - p = 2.265234 - 9.778335 = -7.513101, and D × (F - K) = 0.988565872 × (78.40 - 86.00) = -7.513101. Residual 8.9 × 10⁻¹⁶.

Step 3, the early-exercise question, stated as arithmetic. Exercising now pays 86.00 - 78.40 = 7.600000. The option's discounted intrinsic value is 0.988565872 × 7.60 = 7.513101. Exercise beats discounted intrinsic by 0.086899 per barrel, so the question is whether the remaining time value exceeds that.

Step 4, a three-step tree. With dt = 0.083333, u = e^{0.32√dt} = 1.096777, d = 0.911762, and the futures probability p = (1 - d)/(u - d) = 0.476922:

Rolling the American put back, comparing continuation against exercise at every node:

Step 5, convergence. Three steps is far too coarse to price with, and the table shows it:

Black's European put is $9.778335, and the 2,000-step European lattice lands $0.000324 away, which is what a converged tree looks like. The American put is $9.807122, so the early-exercise premium is $0.029111 per barrel, or $29.11 per contract, which is 0.298% of the European price.

Read it. Two lessons sit in that table and they point in opposite directions. Early exercise is worth almost nothing here in value terms, so pricing this put with Black and adding nothing costs three tenths of one percent. But early exercise is not rare. It fires at three of the five interior nodes in the three-step lattice, including node (2,1), where the futures price has come back to the 78.40 it started at. A risk system that assumes the position will survive to expiry is wrong about the position's life even though it is nearly right about its price. What would break the conclusion is a longer expiry or a higher rate, both of which raise the value of taking cash today, and the same table run at a one-year expiry would not be so forgiving.

Worked example 5: A USD/INR fixed-for-fixed currency swap, priced and marked (India, ₹ and $)

Kaveri Speciality Chemicals Ltd (synthetic) has issued $10,000,000 of three-year fixed-rate dollar debt and wants rupee liabilities instead. It enters a three-year fixed-for-fixed currency swap at spot ₹86.00/$: it delivers $10,000,000 and receives ₹86.00 crore at inception, receives USD fixed and pays INR fixed annually, and re-exchanges the principals at maturity at the same rate. Price both fixed rates so the swap is worth zero, then mark it six months later.

The zero curves at inception, continuously compounded:

Solution. Step 1, each leg's par fixed rate off its own curve. Dollar discount factors at 1, 2 and 3 years are 0.955041962, 0.910282762 and 0.866407437, summing to 2.731732161. Rupee factors are 0.936130864, 0.873715912 and 0.814240095, summing to 2.624086871.

c_USD = (1 - 0.866407437) / 2.731732161 = 4.890398%
c_INR = (1 - 0.814240095) / 2.624086871 = 7.079030%

The differential of 2.188633% is the interest-rate differential between the two curves and nothing else. No view about the rupee is present anywhere in it.

Step 2, the inception check. Each leg is now a par bond in its own currency: PV(USD leg) = $10,000,000.000000 and PV(INR leg) = ₹860,000,000.000000, both equal to face to machine precision. In rupees, V0 = 86.00 × 10,000,000 - 860,000,000 = ₹0.000000. A currency swap that does not price to zero at inception has a curve error or a rate error, and there is nowhere else for it to hide.

Step 3, six months later. Spot moves to ₹88.50/$, a 2.9070% weakening of the rupee. The dollar curve shifts up 40 basis points in parallel and the rupee curve down 25. The remaining cash flows sit at 0.5, 1.5 and 2.5 years from the new date.

Step 4, three revaluations rather than one.

Step 5, the decomposition.

The three add to the total with a residual of exactly zero. Cross-check the currency term independently: (88.50 - 86.00) × 10,154,723.27 = ₹25,386,808.17, the same figure.

Step 6, the first coupon exchange. At the first anniversary Kaveri receives $10,000,000 × 4.890398% = $489,039.76 and pays ₹860,000,000 × 7.079030% = ₹60,879,660.83. Converted at ₹88.50 the dollar receipt is worth ₹43,280,018.55, so the net cash out is ₹17,599,642.28 that year. The swap was never a source of cash. It was a change in the currency of a liability, and the rupee-denominated cost is exactly the 7.079030% Kaveri agreed to.

Read it. The mark is plus ₹26.26 lakh, or 0.3053% of the rupee notional, and it is the least informative number in the table. Underneath it, rates cost the position ₹1.3441 crore and the currency move made ₹2.5387 crore, so the net is the residue of two moves roughly ten times its own size. A treasurer who reports only the net has told the board nothing about what the next quarter will bring. What the numbers do not include is the cross-currency basis, which is a traded spread applied to one leg and would move the inception rates; pricing without it makes this a teaching price rather than a dealer quote. The onshore forward market and the offshore non-deliverable market can quote different forward points for the same tenor, and the gap moves with policy. Verify against the Reserve Bank of India's reference rate and the Clearing Corporation of India's traded data before using any of it live; checked at the time of writing.

Worked example 6: An equity swap valued after an index move (India, ₹)

Ashwattha Pension Trust (synthetic) wants Indian equity exposure without buying and custodying a basket. It enters a one-year quarterly equity swap on ₹50.00 crore notional: it receives the Nifty 50 total return and pays 7.20% fixed annually, so 1.8000% per quarter. At the last reset the index was 24,800.00. Value the swap 45 days later with the index at 25,420.00.

The rupee zero curve on the valuation date, continuously compounded, at the remaining times to each payment: 6.55% at 0.125 years, 6.62% at 0.375, 6.70% at 0.625 and 6.78% at 0.875.

Solution. Step 1, the equity leg. The index return since the reset is 25,420.00 / 24,800.00 - 1 = 2.500000%. The equity leg is the notional grown by that ratio:

equity leg = 50.00 crore × 1.025000000 = Rs 51.250000 crore

Step 2, the fixed leg, as a bond. The quarterly coupon is 1.8000% × 50.00 crore = ₹0.90 crore, and the final payment carries the notional back.

Step 3, the value. To the equity receiver, 51.250000 - 50.601867 = ₹0.648133 crore, or ₹64,81,331, which is 1.296266% of notional.

Step 4, decompose it, because the total misleads. The index contributed 50.00 × 2.500000% = ₹1.250000 crore. The fixed leg trades ₹0.601867 crore above par, because the 7.20% coupon sits above the current curve and because nearly a full quarter's coupon is still owed. 1.250000 - 0.601867 = 0.648133, with a residual of 1.5 × 10⁻⁸ crore.

Step 5, the breakeven index level. Setting the equity leg equal to the fixed leg, I* = 24,800.00 × 50.601867/50.00 = 25,098.53, a 1.203734% rise from the reset. Anywhere between 24,800 and 25,098.53 the index is up and the equity receiver is down.

Step 6, the two variants on the same trade. Under return-for-floating at an illustrative 6.68% three-month reference set at the reset, the floating leg mid-period is worth 50.00 × (1 + 6.68% × 0.25) × 0.991845926 = ₹50.420488 crore, so the equity receiver's value rises to ₹0.829512 crore: the floating leg has drifted less above par than the fixed one, because it will reset. Under return-for-return against a foreign index that moved from 3,100.00 to 3,162.00, a rise of 2.000000%, the swap is worth 50.00 × (2.500000% - 2.000000%) = ₹0.250000 crore, and no discount factor appears at all because both legs are returns on the same notional over the same window.

Read it. The index rose 2.5% and the position is up 1.296266% of notional. The missing 1.2% is the funding leg doing exactly what it is contracted to do, and the breakeven of 25,098.53 is the number to quote alongside the mark. What this swap does not give the trust is ownership: no voting rights, no securities lending revenue, and a counterparty exposure to the dealer that a share certificate would not carry. The futures route to the same exposure has a different cost structure and its own roll mechanics, worked through in the exchange-traded practice node that follows this one.

Worked example 7: A zero-cost interest-rate collar on MIBOR (India, ₹)

Sarvodaya Housing Finance Ltd (synthetic) has drawn ₹100.00 crore of one-year floating-rate funding priced off three-month MIBOR, resetting quarterly. It wants protection above 7.25% but will not pay a premium. Find the floor strike that makes the collar cost nothing, at a flat volatility of 18%.

The rupee zero curve, continuously compounded: 6.55% at three months, 6.65% at six, 6.75% at nine and 6.85% at twelve.

Solution. Step 1, the curve, the forwards and the reference rate. Discount factors 0.983758341, 0.967296705, 0.950635092 and 0.933793460 give quarterly forwards of 6.807275%, 7.010729% and 7.214286%. The annuity over the three payment dates is 0.712931314, and the par forward-starting swap rate is

S = (0.983758341 - 0.933793460) / 0.712931314 = 7.008372%

Step 2, the cap. Three caplets, the first period already fixed:

That is ₹18.31 lakh, or 18.31 basis points of notional, which over the annuity is 25.69 basis points a year.

Step 3, solve for the zero-cost floor strike. Find K_f such that the floor's premium equals ₹1,831,260.37. Solving numerically gives K_f = 6.812143%, with the floor priced at ₹1,831,260.37 and a residual of ₹2.5 × 10⁻⁷.

Step 4, verify by parity. At the cap strike of 7.25%, cap minus floor is -₹1,722,638.89, and the payer swap at 7.25% on the same schedule is worth -₹1,722,638.89. Residual: ₹5.6 × 10⁻⁹. The sign is negative because a payer swap struck above the forward is a liability to the fixed payer.

Step 5, read the band. The collar traps the reference rate in 6.812143% to 7.250000%, a band of 43.79 basis points around a forward of 7.008372%. Set against the three alternatives, on the present value of interest over the three covered quarters:

Read it. "Zero cost" bought a 43.79 basis point band. In the 5.50% scenario the collar costs ₹93.55 lakh more than staying unhedged, which is the premium paid in kind rather than in cash, and it is larger than the cap's up-front premium of ₹18.31 lakh. The cap is the cheaper hedge in every falling-rate outcome and the dearer one in every rising outcome above 7.25%. What the table does not price is that the collar leaves Sarvodaya with a sold option and therefore with margin or collateral obligations that a bought cap does not create, which for a housing finance company is a treasury constraint rather than a valuation input.

Verify. Every headline figure above, at full precision, so a reader who rebuilds this can find the exact line where their implementation diverges.

Practice set

Work each problem fully before reading its solution. A calculator is assumed, and answers within about ±1% score as correct unless the problem states otherwise. All entities are synthetic; all curves are illustrative but internally consistent. Guided problems come first and carry the scaffolding; the later ones do not.

P1 (guided). Sort the instruments. Classify each of the following as a forward commitment or a contingent claim, and for each state who pays at inception: a three-month USD/INR forward; a bought receiver swaption; a five-year interest-rate swap; a zero-cost collar on a floating-rate loan; a sold cap; a fixed-for-fixed currency swap; a bought credit default swap.

Solution. Forward commitments, nobody pays at inception because the terms are struck at fair value: the USD/INR forward, the interest-rate swap, the currency swap. Contingent claims, the buyer pays a premium: the bought receiver swaption (buyer pays), the sold cap (the seller here receives the premium), the bought credit default swap (the protection buyer pays, either up front or as a running spread, or both). The zero-cost collar is the interesting case: each leg is a contingent claim, one bought and one sold, and the strikes are chosen so the two premiums cancel, so no cash moves at inception even though the position contains two options. It is a forward commitment in aggregate only when the two strikes coincide, in which case it collapses to a payer swap.

P2 (guided). One caplet by hand. A caplet on a $20,000,000 notional covers a three-month period, tau = 0.25. The forward rate for the period is 5.12%, the strike is 5.00%, the fixing is in six months, the discount factor to the payment date is 0.9625, and the volatility is 20%. Price the caplet, then price the matching floorlet two ways.

Solution. d1 = [ln(0.0512/0.0500) + 0.5 × 0.20² × 0.50] / (0.20 × √0.50) = 0.238412, d2 = 0.238412 - 0.141421 = 0.096990. N(d1) = 0.594219, N(d2) = 0.538633.

caplet = 20,000,000 × 0.25 × 0.9625 × (0.0512 × 0.594219 - 0.0500 × 0.538633) = $16,807.03

Floorlet by parity: caplet - N·tau·D·(F - K) = 16,807.03 - 20,000,000 × 0.25 × 0.9625 × 0.0012 = $11,032.03. Direct: 20,000,000 × 0.25 × 0.9625 × [0.0500 × (1 - 0.538633) - 0.0512 × (1 - 0.594219)] = $11,032.03. The two routes agree, which is the check.

P3 (guided). Back out a floor from parity. A one-year cap on a floating-rate loan costs $184,500. The payer swap on the same schedule at the same strike is worth $61,800 to the fixed payer. What is the floor worth, and what does the sign of the swap value tell you about the strike relative to the forward?

Solution. floor = cap - swap = 184,500 - 61,800 = $122,700. The swap value is positive to the fixed payer, so the forward rates lie above the strike on average, which means the cap is in the money on a forward basis and the floor is out of it. That is consistent with the cap costing more than the floor. No volatility was needed anywhere in this problem, which is the point of the identity.

P4. Price a payer swaption. A 2y × 4y payer swaption on $100,000,000 is struck at 5.00%. The forward swap rate is 4.80%, the annuity of the underlying swap is 3.9210, and the volatility is 26%. Price the payer, then the receiver two ways.

Solution. d1 = [ln(0.0480/0.0500) + 0.5 × 0.26² × 2] / (0.26 × √2) = 0.072827, d2 = 0.072827 - 0.367696 = -0.294869. N(d1) = 0.529028, N(d2) = 0.384047.

payer = 100,000,000 × 3.9210 × (0.0480 × 0.529028 - 0.0500 × 0.384047) = $2,427,487.67

That is 242.75 basis points of notional. Receiver by parity: payer - N·A·(S - K) = 2,427,487.67 - 100,000,000 × 3.9210 × (-0.0020) = $3,211,687.67. Direct Black: $3,211,687.67. The receiver is dearer because the strike sits above the forward.

P5. Recover the forward swap rate from two premiums. A payer swaption costs $1,248,000 and the matching receiver costs $1,642,000 on a $75,000,000 notional at a strike of 4.50%, with an underlying annuity of 4.4120. What is the forward swap rate?

Solution. Parity gives payer - receiver = N·A·(S - K), so

S = K + (payer - receiver)/(N × A) = 0.0450 + (-394,000)/(75,000,000 × 4.4120) = 4.380931%

The forward sits 11.91 basis points below the strike, which is why the receiver is the more expensive of the two. The volatility never enters, so this works even when the two premiums were quoted at different volatilities by mistake, in which case the recovered forward will disagree with the curve and you have found an error.

P6. Straddle breakeven. An at-the-money swaption straddle on $80,000,000 costs $2,960,000. The strike is 5.05% and the underlying swap's annuity is 4.1500. What must the swap rate do for the buyer to break even?

Solution. breakeven move = 2,960,000 / (80,000,000 × 4.1500) = 0.0089157 = 89.1566 bp. The swap rate must finish outside 4.158434% to 5.941566%. Note what did not appear: the option's expiry, the volatility, and any discount factor. The premium expressed in annuity units is the breakeven, which is why the market quotes straddles that way.

P7. Exercise and settle. A payer swaption on $40,000,000, struck at 5.00%, expires today into a seven-year annual swap. The seven-year par swap rate has set at 5.65% and the curve is flat there. Exercise or not, and what is the cash settlement?

Solution. The forward is above the strike, so exercise. On a flat 5.65% curve the seven-year annual annuity is Σ (1.0565)^{-i} = 5.652500.

settlement = 40,000,000 × (5.65% - 5.00%) × 5.652500 = $1,469,649.93

A receiver swaption at the same strike expires worthless: 40,000,000 × (5.00% - 5.65%) × 5.652500 is negative, and the holder abandons rather than exercising into a losing swap. Cash settlement and physical settlement produce the same value here; the difference is that the physically settled holder now owns a seven-year swap and the cash-settled one owns nothing.

P8. An option on an index futures contract. A four-month option on an equity index futures contract quoted at 4,820.00 is struck at 4,900.00. The volatility is 19%, the risk-free rate is 4.50% continuously compounded, and the contract multiplier is 50. Price the European call and put, verify parity, and give the per-contract premiums.

Solution. T = 1/3, D = e^{-0.045/3} = 0.985112. d1 = -0.095214, d2 = -0.204910. Call 172.361254 index points, put 251.170209. Parity: c - p = -78.808955 and D × (F - K) = 0.985112 × (-80.00) = -78.808955, residual 2.8 × 10⁻¹⁴. Per contract at 50×: call $8,618.06, put $12,558.51. If the option is American, both figures are floors rather than prices, and the put is the one to re-run on a tree because it is in the money.

P9. Set the two fixed rates on a currency swap. A four-year fixed-for-fixed USD/INR swap has a $25,000,000 dollar notional at a spot of ₹87.20/$. Dollar zero rates, continuously compounded, are 4.10%, 4.25%, 4.35% and 4.42% at one through four years; rupee rates are 7.05%, 7.18%, 7.28% and 7.35%. Find both fixed rates and the annual coupon on each leg.

Solution. Dollar discount factors: 0.959829130, 0.918512284, 0.877656493, 0.837947357, summing to 3.593945264. Rupee factors: 0.931927740, 0.866234172, 0.803803856, 0.745276491, summing to 3.347242259.

c_USD = (1 - 0.837947357)/3.593945264 = 4.509046%
c_INR = (1 - 0.745276491)/3.347242259 = 7.609951%

The differential is 3.100905%. Coupons: $25,000,000 × 4.509046% = $1,127,261.48 and ₹2,180,000,000 × 7.609951% = ₹165,896,940.12. Both legs price at par by construction, so the swap is worth zero at inception in either currency.

P10. Mark that swap a year later. One year in, the rupee has strengthened to ₹84.90/$. The dollar curve at the remaining one, two and three years is 4.48%, 4.55% and 4.62%; the rupee curve is 7.42%, 7.50% and 7.58%. Value the swap to the party receiving USD and paying INR, and separate the currency effect.

Solution. With the fixed rates from P9, PV(USD leg) = $24,852,856.13 and PV(INR leg) = ₹2,165,568,178.96. At the new spot,

V = 84.90 × 24,852,856.13 - 2,165,568,178.96 = -Rs 55,560,693.51   (-5.5561 crore)

At the unchanged spot of ₹87.20 the same legs give +₹1,600,875.59, so the currency effect alone is (84.90 - 87.20) × 24,852,856.13 = -₹57,161,569.10. A rupee that strengthened 2.64% turned a small positive mark into a loss of ₹5.56 crore, and anyone reporting only the net has hidden the fact that the entire result is currency.

P11. Value an equity swap that is under water. A ₹30.00 crore equity swap pays 6.90% fixed semi-annually and receives a single stock's total return. At the last reset the stock was ₹21,450.00; it is now ₹20,760.00. Two coupon dates remain, at 0.25 and 0.75 years, and the rupee zero rates are 6.48% and 6.65% at those tenors. Value it to the equity receiver.

Solution. The semi-annual coupon is 3.45% × 30.00 = ₹1.035000 crore. Fixed leg: 1.035000 × e^{-0.0648×0.25} + (1.035000 + 30.00) × e^{-0.0665×0.75} = ₹30.543464 crore. Equity leg: 30.00 × (20,760/21,450) = ₹29.034965 crore. Value to the receiver = -₹1.508499 crore, which is -5.028329% of notional. The stock fell 3.217%, and the funding leg added the rest.

P12. Return-for-return, with no discounting. A manager holds a ₹40.00 crore return-for-return swap paying an Indian mid-cap index return and receiving a large-cap index return over one quarter. The mid-cap index rose 6.20% and the large cap rose 3.80%. What is the swap worth at the reset, and why does no discount factor appear?

Solution. 40.00 × (3.80% - 6.20%) = -₹0.96 crore to the manager. No discount factor appears because both legs accrue on the same notional over the same window and settle on the same date, so the discounting is common to both and cancels in the difference. The manager was long large caps against mid caps and mid caps won, which is precisely the relative view the structure expresses and the reason nobody had to sell a share to express it.

P13. Find the zero-cost floor strike, and find where none exists. On the rupee curve of Worked example 7 (forwards 6.807275%, 7.010729%, 7.214286%; par forward-starting swap rate 7.008372%; volatility 18%; notional ₹100.00 crore), the cap premiums at four strikes are ₹2,552,459.35 at 7.00%, ₹1,831,260.37 at 7.25%, ₹1,291,472.98 at 7.50% and ₹618,343.63 at 8.00%. Find the zero-cost floor strike for each, and explain the one that fails.

Solution. Solving floor(K_f) = cap(K_c) for each:

The 7.00% cap fails because its strike sits below the forward of 7.008372%, so the cap is worth more than the floor at the same strike, and by the parity identity no floor struck below the forward can be worth as much. The general rule falls out of the identity rather than out of the arithmetic: a zero-cost collar exists only when the cap strike is above the forward, and the further above it sits, the wider the band the borrower has to accept.

P14. Show that Black and Black-Scholes-Merton are the same formula. An index trades at 4,000.00 with a continuous dividend yield of 1.30%. A nine-month 4,100.00-strike call has a volatility of 17%, and the risk-free rate is 4.50% continuously compounded. Price it with Black-Scholes-Merton, then build the forward and price it with Black-76.

Solution. F = S·e^{(r-q)T} = 4,000.00 × e^{(0.045-0.013)×0.75} = 4,097.161272. Black-Scholes-Merton with the dividend yield gives 231.157412847. Black-76 on the forward, discounted at e^{-rT}, gives 231.157412847. The residual is -4 × 10⁻¹³, which is floating-point noise. The dividend yield did not disappear: it went into the forward, which is what "the forward is the right underlying" means operationally.

P15. Buy back the option you sold with the bond. An investor holds $200,000,000 of a callable bond whose embedded option is, in swap terms, a 3y × 5y receiver swaption struck at 5.00%. The forward swap rate for that expiry and tenor is 5.12%, the underlying annuity is 4.0500, and the swaption volatility is 28%. Price the receiver swaption, state what it costs as a running spread, and say what it means if the tree-based option cost on the bond disagrees.

Solution. d1 = [ln(0.0512/0.0500) + 0.5 × 0.28² × 3] / (0.28 × √3) = 0.291390, d2 = -0.193584, N(-d1) = 0.385377, N(-d2) = 0.576749. The receiver is 200,000,000 × 4.0500 × [0.0500 × 0.576749 - 0.0512 × 0.385377] = $7,376,010.14, which is 368.80 basis points of notional. The matching payer is $8,348,010.14, and parity checks: 8,348,010.14 - 7,376,010.14 = $972,000.00 = 200,000,000 × 4.0500 × (0.0512 - 0.0500). Spread over the annuity, the receiver costs 91.06 basis points a year, and that is the figure comparable to an option cost quoted as Z-spread minus OAS on the bond itself. If the tree says the option cost is 40 basis points a year and the swaption market says 91, the tree's volatility assumption is almost certainly too low, because the swaption price is a traded number and the option cost is a model output.

Applied mini-project

Hedge an issuance two ways, and say which one you would sign.

A synthetic issuer of your own construction plans to raise five-year fixed-rate debt in six months and is exposed to a rate rise between now and then. Your job is to price both hedges, show what each does across a stated range of outcomes, and write the recommendation. Four artefacts, each with a numbered test it must pass before it counts toward the score. Use a spreadsheet or Python. Every tie-out must be shown rather than asserted. Label the issuer synthetic and trace any real curve to a named primary source with the date recorded: the US Treasury par yield curve or SOFR OIS levels for dollars, the Clearing Corporation of India or the Financial Benchmarks India valuation curve for rupees.

The first artefact is the curve and the forward. Build a discount curve of at least six annual maturities, then compute the forward-starting par swap rate for a swap beginning at the hedge horizon and running five years, together with its annuity. Test 1: the curve reprices its own inputs, so that each instrument you bootstrapped from returns its quoted price. Test 2: the forward swap rate you computed, when used as the fixed rate on a forward-starting swap, gives that swap a present value of zero to within 1e-9. Report both residuals. A failed Test 1 makes every number downstream meaningless.

The second artefact is the two hedges, priced. Price a payer swaption at a strike of your choosing, stating the volatility and where you got it, and price the forward-starting swap struck at the forward. Test 3: payer minus receiver at your strike equals N × A × (S - K) to within 1e-8. Report the swaption premium three ways: in currency, in basis points of notional, and as a running spread over the underlying swap's annuity. That third figure is the only one comparable to a swap rate, and a memo that omits it is incomplete.

The third artefact is the payoff grid. Tabulate the outcome of each hedge across at least five rate outcomes at the horizon, spanning at least 150 basis points below and above the forward, and include the unhedged case as a column. Test 4: the two hedges' payoffs cross at exactly one rate, and you have solved for it rather than read it off the chart. Test 5: the swaption column is flat below the strike at a loss equal to the premium future-valued to the horizon, and your table shows that value rather than the premium itself. State the annuity assumption you used at the horizon and quantify the error it introduces, as Worked example 2 does.

The fourth artefact is a sensitivity you did not want to run. Re-price both hedges at a volatility 25% relative below and 25% relative above your base assumption, and re-price the swaption at a strike 25 basis points either side. Test 6: report how much of the swaption's premium is attributable to the volatility assumption alone, and say whether that swing is larger or smaller than the difference between the two hedges at the crossover rate. Then answer, in one paragraph, what you would have to believe about the distribution of rates for the option to be the better purchase.

Deliverable. One workbook or notebook plus a two-page recommendation carrying the four artefacts' headline numbers, the six test residuals, a named list of every assumption (volatility and its convention, day count, compounding, settlement style, the annuity treatment at the horizon), and a closing paragraph naming the single scenario in which your recommended hedge is the wrong one. That last paragraph is the point of the exercise. A recommendation that cannot name its own failure case has not been thought through.

Rubric (21 points; pass at 17 with no zero on any row).

Two points for a row at the "2" standard and three at the "3" standard. Any artefact whose test fails scores zero for that row regardless of how well the memo reads, because a hedge recommendation built on a curve that does not reprice its own inputs is not a cheaper hedge or a dearer one. It is a number with no relationship to the market it claims to describe.

Reading & resources

Treat each entry as an anchor to verify against and to learn from, never as a source to lift a number from without tracing it yourself.

The pricing spine:

• Fischer Black, "The Pricing of Commodity Contracts", Journal of Financial Economics, 1976. The original paper, short enough to read in a sitting, and the substitution it makes is exactly the one used throughout here. [Free] [Intermediate]
• John C. Hull, Options, Futures, and Other Derivatives. The standard treatment of the Black model, caps and floors, swaptions, and options on futures, with the trees worked. [Paid] [Intermediate]
• Frank J. Fabozzi (ed.), The Handbook of Fixed Income Securities. The desk reference for the interest-rate option chapters and for the market conventions this module simplifies. [Paid] [Intermediate to Advanced]
• CFA Institute Level II and Level III Derivatives readings on swaps, forwards and options, and on risk management applications. The exam-shaped version of the taxonomy, the valuation identities and the hedging applications. [Paid] [Intermediate]

Volatility, the surface, and what a quote means:

• Riccardo Rebonato, Volatility and Correlation. The book to read on why a cap's flat volatility is not a caplet's volatility and on what stripping a caplet surface actually involves. [Paid] [Advanced]
• Patrick Hagan and collaborators on the SABR model. The industry standard for the smile in swaption and cap markets, and the reason a single volatility number is a summary rather than a parameter. [Free] [Advanced]

Clearing, margin and market structure:

• The Bank for International Settlements Quarterly Review and Committee on Payments and Market Infrastructures papers on central clearing and the default waterfall. The primary description of the sequence, from the people who supervise it. [Free] [Intermediate]
• A clearing house's own rulebook and default-management framework, from any major central counterparty. The waterfall in its actual language, which is shorter and clearer than most summaries of it. [Free] [Intermediate]
• The ISDA Master Agreement and its Credit Support Annex, together with ISDA's published market surveys. The bilateral half of the market, and the document under which a currency or equity swap is actually written. [Free to read, membership for some material] [Advanced]

Primary-source anchors for the data you will use:

• The US Treasury daily par yield curve and the Federal Reserve Bank of New York's SOFR publications. The dollar curve inputs for everything in the worked examples. [Free] [Beginner to Advanced]
• The Clearing Corporation of India and Financial Benchmarks India. The rupee benchmark rates, the government securities valuation curve and the traded forward data that any Indian pricing must start from. [Free] [Beginner to Advanced]
• The Reserve Bank of India's Master Directions on risk management and interbank dealings, and its circulars on rupee interest-rate derivatives. What Indian residents may transact, in which market, and with whom. Perishable and amended often; check the current version and its date. [Free] [Intermediate]
• The Securities and Exchange Board of India and the exchanges' own contract specification pages, for the listed instruments the over-the-counter market is priced against. [Free] [Beginner]

Do this rather than only read. Build Worked example 1's three-caplet strip in a spreadsheet. Then break it on purpose. Discount each caplet at its fixing date instead of its payment date and watch the cap-floor parity residual go from machine noise to a number in the hundreds. Then restore that and instead add a fourth caplet for the already-fixed first period, and watch the premium jump while parity keeps passing. Ten minutes of watching one check catch an error and miss another will teach you more about what a verification identity does than any paragraph about verification can.

Teach it back & journal

Feynman prompt. Find a colleague who prices vanilla swaps and has never touched an option, and teach them the cap in three moves. Start with the loan: four quarterly resets, the first already fixed, so three things left to be uncertain about. Make them say out loud why that means three options and not four, because if they cannot say it they will build a four-caplet strip within a week. Then price one caplet in front of them, naming the fixing date and the payment date as you use each, and let them see that the two dates enter the formula in different places. Then hand them the identity: cap minus floor is the payer swap, volatility cancels, so the two legs together must reproduce a number they already know how to compute off the curve alone. Have them check it. When the residual comes back at ten to the minus eleven, ask them what error that check would have caught and what error it would have sailed past, and do not accept an answer until they name the phantom first caplet.

Then the second half, which is the one that changes how they read a term sheet. Take the payer swaption from Worked example 2 and ask them whether 180 basis points is expensive. They will not be able to answer, which is the whole reason for asking, because 180 basis points up front against a rate quoted per year is not a comparison. Divide by the annuity, get 43.63 basis points a year, and set that against the forward-starting swap's zero. Now they can argue. Finish by asking them to name the single scenario in which the more expensive hedge was the right purchase, and make them state a rate, not a direction.

Journal prompt. For two weeks, every time you meet a derivative price in a broker's report, a term sheet, a treasury memo or a footnote, write down four things before you use it: which family it belongs to, what its underlying actually is, what volatility produced it, and what identity you could run to check it. Then answer in writing at the end of the fortnight. Count how many of those prices arrived with their volatility attached, how many quoted a premium in a unit that could be compared to the alternative, and how many named the settlement style. Where the assumption was missing, what did you do, and what should you have done?

Then write a paragraph on this. The instruments here are the ones where the program's process-over-P&L philosophy is most visible, because almost nothing in the output is observable. A cap premium, a swaption premium, a currency swap's mark and an equity swap's value are all manufactured, and their quality is exactly the quality of the checks you ran. Of the four checks that make any of it admissible, which one are you most likely to skip when a deadline is close? Write down what specifically goes wrong when you skip it, who finds out and when, and what you will put in place this month to make skipping harder than doing.