Derivatives Summary

Based on 2026 CFA Level II SchweserNotes, Book 4, Reading 28, Pricing and Valuation of Forward Commitments, and Reading 29, Valuation of Contingent Claims. The relevant material occupies PDF viewer pages 135 to 200, followed by the topic quiz notice on page 201, in the Book 4 attachment on CFA Lv2.

This guide develops the economic reasoning, formulas, applications, and common mistakes across both readings. Examples labeled Book example use the PDF’s inputs; examples labeled Illustration use separate inputs to clarify the same methods. Calculations retain precision until the final answer, so some results differ slightly from the book’s rounded answers.

1. The conceptual map

Forward commitments versus contingent claims

A forward commitment obligates both counterparties to perform. A contingent claim, such as an option, gives its buyer a right without the corresponding obligation to exercise.

That distinction changes the payoff shape. A forward payoff is linear in the underlying price, whereas an option payoff contains a maximum function that creates asymmetry and convexity.

Price, value, payoff, and profit are different

Price means the contractual delivery price or fixed rate that makes a new forward or swap fair. Value means what an existing position is worth now, given its original locked terms.

A stock forward can have a delivery price of 100 and an initial contract value of zero. If the stock later rises, the original contract can become valuable even though its delivery price remains 100.

Payoff is the amount received under the contract at a specified event or maturity. Profit also accounts for premiums, financing, and other costs, so an option payoff is not automatically an investment profit.

For a long asset forward with delivery price $K$:

For a purchased call and put:

Three questions before every calculation

1. What position is being valued? Write down exactly what it receives and pays.
2. When do the cash flows occur? Separate valuation date, fixing date, settlement date, and underlying maturity.
3. What convention applies? Identify currency, notional, accrual basis, discount curve, and compounding method.

2. Notation and discounting discipline

The discount factor notation unifies formulas that otherwise look unrelated:

Use the convention specified in each question. Reading 28 often uses a 360 day simple interest basis for reference rate contracts and a 365 day basis for equity and bond examples; the Black model discussion in Reading 29 uses its stated actual/365 treatment.

These are study conventions, not universal rules for every real contract. In particular, do not apply a blanket day count rule across all FRAs, swaps, and options without reading the question.

Reading 28: Pricing and Valuation of Forward Commitments

3. Carry arbitrage: the source of forward prices

The replication argument

Suppose an asset pays no income before delivery. Buying it today and financing the purchase produces the same asset at maturity as taking delivery through a forward.

The terminal cost of these strategies must agree in an ideal frictionless market. Otherwise, buy the cheaper strategy and sell the more expensive one.

With an annual effective rate, this becomes $F_0(T)=S_0(1+r{)}^T$. With a continuous rate, it becomes $F_0(T)=S_0{e}^{rT}$.

If the spot holder receives known cash income before delivery, that income helps finance the asset. It must therefore reduce the forward price:

Equivalently, compound the spot price to delivery and subtract the future value of the interim income. Storage or other carrying costs, when relevant, work in the opposite direction.

How to construct the arbitrage

Any income on a shorted asset must generally be compensated to its lender. That is why reverse carry calculations cannot ignore dividends or coupons.

Book example, Module 28.1. A zero coupon bond costs 500, the annual effective financing rate is 6%, and delivery is in three months. The fair price is $500(1.06{)}^{0.25}$, approximately 507.34.

A quoted forward price of 510 supports cash and carry, with approximately 2.66 of terminal profit per bond. A quoted price of 502 supports the reverse strategy, with approximately 5.34 of terminal profit.

What the forward price does not say

A forward price is not necessarily the market’s best forecast of the future spot price. It is an exchange price constrained by financing, income, and replication.

Expected investment returns can include risk premiums that do not appear in the carry formula. Substituting an analyst’s expected stock return for the financing rate confuses expected performance with arbitrage pricing.

Valuing an existing forward

For a contract originally struck at $K$:

Equivalently:

The second expression compares an old commitment with a new commitment on the same underlying and same delivery date. The price difference is a future cash flow, so it must be discounted.

The short position has the opposite value. These relationships omit counterparty credit effects and other market frictions, consistent with the reading’s basic framework.

4. Equity forwards: dividends belong to the spot holder

Discrete dividends

If dividends $d_i$ occur at dates $t_i$ before delivery:

At a later valuation date, include only dividends that remain between that date and delivery. A dividend already paid is not subtracted again, and a dividend after delivery does not belong in the carry adjustment.

The economic intuition is ownership. The stockholder receives the dividend, while the forward buyer does not own the shares before delivery.

Illustration. A stock costs 100, pays a dividend of 2 after 0.25 years, and has a forward delivery date in 0.50 years. With a 5% annual effective financing rate:

After 0.20 years, suppose the stock is 108 and the rate remains 5%. The dividend is now 0.05 years away and delivery is 0.30 years away:

The price used in the last term remains the original contract price. Replacing it with today’s fair forward price would value a new contract at zero rather than value the existing position.

Continuous dividend yield

For an index with continuous dividend yield $q$:

The first term is the prepaid forward value, the amount paid today for delivery of the asset later without receiving its interim distributions. The second term is the present value of the contractual payment.

If the dividend yield exceeds the financing rate, the fair forward price can be below the spot index. That is a carry result, not automatically an expectation of falling equity prices.

Exam traps

1. Subtracting dividends after the forward delivery date.
2. Using the original dividend schedule without removing payments already made.
3. Subtracting a future dividend without discounting it to the correct date.
4. Ignoring the contract multiplier when converting an index value into money.
5. Treating uncertain future dividends as perfectly known outside the question’s assumptions.

5. Bond forwards and futures: cash price first, quoted price second

Coupon bond forwards

The same reasoning applies to coupons:

Use the financing curve for the carry calculation, not automatically the bond’s yield to maturity. The bond yield describes the price of its own cash flows; it is not necessarily the cost of funding a carry trade.

Clean price and accrued interest

A quoted clean price is not the amount paid to acquire the bond. Financing the acquisition therefore starts from the full price.

Coupons paid before delivery reduce the holding cost. Accrued interest at delivery is a separate adjustment needed to translate the full economic delivery value into the contract’s quoted convention.

Conversion factors and delivery options

Bond futures may let the short choose from several eligible bonds. A conversion factor adjusts the invoice amount for the bond selected for delivery.

Under the reading’s carry setup, for a specified deliverable bond:

The corresponding invoice relationship is:

Keep all quantities on the same face value basis. Market quotes per 100 of principal cannot be mixed directly with dollar coupon amounts per 1,000 of principal.

Book example, Module 28.3. The deliverable bond has full price 1,040, coupons of 35 at 0.5 and 1.0 years, delivery at 1.2 years, financing at 5%, accrued interest at delivery of 14, and conversion factor 1.13.

The conversion factor does not eliminate the short’s delivery choice. In a full market valuation, changes in the cheapest deliverable bond and the delivery option itself can matter; the displayed formula is the reading’s simplified carry calculation.

6. Forward rate agreements: one future accrual period

Direction and timeline

The long FRA receives floating and pays fixed. Economically, it protects a future borrower against an increase in the reference rate.

The short FRA receives fixed and pays floating. It protects a future lender or investor against a decline in the reference rate.

A 1 × 4 FRA fixes today the rate for a loan beginning in one month and ending in four months. The accrual period is three months, not four months.

An FRA normally exchanges the interest difference, not the notional loan principal. Its hedge does not by itself provide the borrower with funding.

Pricing from discount factors

For an accrual period from $T_1$ to $T_2$:

This is the rate that makes investing to $T_2$ equivalent to investing to $T_1$ and then reinvesting over the forward period.

With simple spot rates:

Do not subtract the spot rates. You are comparing accumulation factors over different horizons.

Book example, Module 28.4. The 30 day rate is 4%, the 120 day rate is 5%, and the FRA covers the intervening 90 days:

The book rounds this to 5.32%. Whether a later calculation uses the exact rate or the rounded contractual rate affects the settlement amount slightly.

Settlement at the start of the loan

Let $L_{T_1}$ be the observed reference rate. The long’s interest advantage at the end of the accrual period is $N\alpha (L_{T_1}-K)$.

Because the standard FRA settles at the beginning of that period, discount the advantage over the accrual period:

With notional 1,000,000, an explicitly fixed rate of 5.32%, a realized reference rate of 6%, and a 90/360 accrual:

The undiscounted interest advantage is 1,700. It is not the amount paid at FRA expiry.

Valuation before the fixing date

In the reading’s single curve framework:

This discounts the economic interest difference from the loan end date, not merely from FRA expiry. Equivalently, settle the forward rate difference at loan start and then discount that settlement to today.

Book inputs, recomputed. Ten days after the original 1 × 4 FRA begins, there are 20 days to fixing and 110 days to loan maturity. If the corresponding spot rates are 5.7% and 5.9%, the current forward rate is approximately 5.92568%.

For a locked rate of 5.32% and notional 1,000,000:

Sign check: an increase in the relevant forward rate benefits the long FRA. An increase in an unrelated point on the curve need not have the same effect.

7. Futures: settlement changes the economics

A futures clearinghouse interposes itself between counterparties and uses margin and daily settlement to manage performance risk. Margin is a performance deposit, not an option premium or the purchase price of the underlying.

When the quoted futures price rises, the long receives variation margin. Once that gain is settled, the contract’s accumulated value is reset, even though the trader’s cash wealth has changed.

Forward versus futures

Analytical extension. With deterministic interest rates, forwards and futures generally have the same theoretical price under matching terms. With stochastic rates, the ability to reinvest gains and finance losses can create a difference.

For an asset whose futures gains tend to occur when rates are high, daily settlement is attractive to the long, which can make its futures price exceed the comparable forward price. The opposite covariance can reverse that relationship.

For interest rate futures quoted as 100 less the reference rate, a higher quote means a lower implied rate. A borrower hedging rising rates may therefore need to sell the futures even though the analogous FRA hedge is long.

This quote convention is a useful extension to the reading’s emphasis on settlement mechanics. Do not equate “long the quoted futures price” with “long the interest rate.”

8. Interest rate swaps: a strip of rate exchanges

From one FRA to many settlement dates

A plain interest rate swap exchanges fixed interest for floating interest over multiple periods. It can be understood as a collection of forward rate exchanges, although a swap generally uses one common fixed rate rather than a different fair rate for each period.

The fixed payer receives floating and benefits when the relevant market swap rate rises. The fixed receiver receives fixed and benefits when that rate falls.

For a reset date valuation in the reading’s framework:

The bond analogy adds the same hypothetical principal payment to each leg so that it cancels. Plain interest rate swaps generally do not exchange that principal.

Deriving the par swap rate

Define the discounted accrual sum:

At a floating reset date, the floating bond is worth par in the basic single curve framework. Equating the two legs gives:

For equal accrual fractions $\alpha$:

The annualized fixed rate equals the periodic rate divided by $\alpha$. A quarterly rate of 1.25% is an annualized simple rate of 5%, not 1.25% per year.

The par swap rate is also a weighted average of the period forward rates:

It is therefore not generally the arithmetic average of spot rates. The weights reflect both accrual lengths and payment timing.

Pricing illustration

Suppose a swap has four quarterly payments and discount factors of 0.990, 0.978, 0.965, and 0.950. Its notional is 10 million.

Each fixed payment is approximately 128,766.42. The first discount factor implies an annualized floating rate of $(1/0.990-1)/0.25=4.04040\%$, giving a floating payment of 101,010.10 and a first net payment of 27,756.32 from the fixed payer.

A zero initial swap value does not imply that its first payment is zero. It means the present values of the full remaining streams agree at initiation.

Valuing the swap after a reset

Let $K_{\text{old}}$ be the locked rate and $K_{\text{new}}$ the current par rate for the remaining schedule:

Continuing the illustration, suppose the remaining discounted accrual sum is 0.485 and the new par rate is 6%. The original fixed payer’s value is approximately 41,193.15.

Reading 28 focuses on interest rate swap valuation immediately after settlement. Between resets, the next floating coupon has already been fixed, so blindly treating the floating leg as par is incorrect.

Important precision point

A curve that is flat in annual simple spot rates does not imply an exactly identical multi payment par swap rate. Derive the par rate from discount factors rather than assuming the two rates must match.

The book’s flat curve example uses a simplifying shortcut and rounding. The discount factor formula above is the internally consistent general rule for the reading’s framework.

9. Currency swaps: two curves and an exchange rate

A currency swap combines cash flows in two different currencies. Unlike a standard interest rate swap, it often exchanges principal initially and returns the original principal amounts at maturity.

The initial notionals are chosen to have equal value at the initial spot exchange rate. Later exchange rate movements do not automatically reset those original notionals in the basic structure studied here.

Pricing the legs

Each fixed rate is derived from its own currency’s discount curve:

If $X_0$ is domestic currency per unit of foreign currency:

Do not use a dollar curve to discount sterling payments. First value each payment stream in the currency in which it is denominated.

Valuation in one reporting currency

For a position receiving domestic currency and paying foreign currency:

Use today’s spot exchange rate to translate the foreign leg’s present value. The initial exchange rate determines the original notionals, not today’s conversion of their values.

Book example, Module 28.6, recomputed without intermediate rounding. A position receives dollar fixed payments and pays sterling fixed payments, with 60 days remaining. The remaining amounts are 5,055,000 dollars and 2,542,500 pounds, with 60 day simple rates of 5.4% and 6.6%, respectively.

At 0.52 pounds per dollar, divide the sterling value by 0.52:

The book reports approximately 173,682 using rounded discount factors. The difference is numerical precision, not a different economic method.

Interpretation and limitations

Currency swap value can change because either interest rate curve moves or because the exchange rate changes. A large principal exchange can make currency exposure dominate the interest coupon differences.

A hedge should receive the currency cash flows needed to service the underlying liability. Matching a liability’s currency without matching payment dates and fixed or floating exposure can leave material residual risk.

The textbook calculation omits cross currency basis, collateral effects, and credit adjustments. Those are practical extensions, not extra terms to invent in an exam question that supplies a simpler setup.

10. Equity swaps: equity performance against financing

Fixed payment versus equity return

A position receiving equity returns and paying fixed combines an equity exposure with a financing obligation. In the reading’s setup, the fixed rate at inception is priced using the same discounted payment logic as a plain interest rate swap.

Between settlement dates, the accrued performance of the equity leg matters. For the book’s simplified fixed notional structure:

The reference index level is the most recent contractual reset level, not necessarily the level at original inception. The hypothetical principal in the two leg values is a replication device, not evidence that the equity swap exchanges principal.

Equity return exchanged for another equity return

In the reading’s simplified arrangement:

If A has returned 1.3% and B has returned negative 0.8%, a party paying A and receiving B loses 2.1% of notional. On 1 million, the position is worth negative 21,000.

Receiving an equity return does not mean receiving only positive returns. A negative return on the received leg creates a payment obligation.

Contract details that change the calculation

Distinguish a price return index from a total return index. A total return contract includes dividends, while a price return contract may treat them differently or exclude them.

Reset rules, notional adjustments, payment dates, and dividend treatment can change valuation outside the book’s simplified examples. The formula should be matched to the contract, not applied to every equity swap indiscriminately.

Reading 29: Valuation of Contingent Claims

11. Binomial valuation: replication creates the probabilities

Why a risky payoff can be discounted at a risk free rate

An option’s payoff depends on the state of the underlying asset. A portfolio of the underlying and risk free borrowing or lending can replicate those state dependent payoffs in the binomial model.

The replication cost determines the option value without requiring an estimate of investors’ expected stock return. Risk neutral probabilities are a mathematical way to express that same replication price.

They do not assert that actual investors are indifferent to risk. They are pricing probabilities under which the expected return on a traded asset, including its income, is consistent with the financing rate.

One period model

Let the underlying start at $S_0$ and finish at either $u{S}_0$ or $d{S}_0$. Let $R$ be the gross risk free accumulation factor over the period.

For an underlying without dividends:

The absence of arbitrage requires:

If the risk free accumulation factor lies outside the range of possible stock returns, the stock and financing account already admit an arbitrage. An invalid probability is therefore a warning about the model inputs, not an invitation to force the probability into a valid range.

For terminal option payoffs $V_u$ and $V_d$:

Under a continuous dividend yield representation, the corresponding probability uses the net carry factor $e^{(r-q)\Delta t}$ in the numerator. Discount the option value at $e^{r\Delta t}$, not at the net carry rate.

Illustration: one period call and put

Suppose $S_0=100$, $u=1.20$, $d=0.80$, $K=100$, and $R=1.05$. The terminal stock prices are 120 and 80.

The call pays 20 in the up state and zero in the down state:

The put pays zero in the up state and 20 in the down state:

The hedge ratio explains the valuation

The call’s hedge ratio is:

Buy 0.50 shares and borrow 38.0952. At maturity, the shares are worth 60 or 40, while the debt is 40 in either state.

The net payoff is therefore 20 or zero, exactly the call payoff. The initial replication cost is $50-38.0952=11.9048$.

This is why the risk neutral calculation works. It reproduces the price of a portfolio whose cash flows can be constructed from traded instruments.

Arbitrage if the market call costs 14

1. Sell the overpriced call for 14.
2. Buy 0.50 shares for 50.
3. Borrow 38.0952, to be repaid as 40 at maturity.
4. Retain the initial surplus of 2.0952; the stock and borrowing replicate the call liability in either state.

Investing the surplus at 5% produces terminal profit of 2.20. If the option is underpriced instead, buy the option and sell its replicating portfolio.

In a model with multiple periods, the replicating hedge generally must be adjusted at later nodes. The initial hedge ratio is not necessarily a permanent hedge.

12. Put call parity: equal terminal payoffs require equal prices

Protective put and fiduciary call

A protective put combines a stock and a put. A fiduciary call combines a call and a bond that pays the strike price at option expiry.

Both portfolios deliver the greater of the stock price and the strike at expiry. For European options on the same underlying, with identical strikes and expiry dates:

For a continuous dividend yield:

For known discrete dividends, replace the dividend adjusted spot term with $S_0-P{V}_0(\text{dividends})$.

Synthetic positions

For an underlying without dividends:

In the one period illustration:

Parity is both a pricing shortcut and an independent consistency check. It can identify an arbitrage without estimating volatility when observed instrument prices violate an exact payoff identity.

Boundaries of the identity

The simple equality applies to matching European contracts and consistent dividend and financing assumptions. American exercise rights complicate the relationship because one position may be exercised earlier than the other.

Call minus put produces a forward payoff with delivery price equal to the option strike. The combination has zero initial value only when that strike equals the fair forward price.

13. Two period trees: work backward from known payoffs

The valuation sequence

1. Build the stock price tree using the specified up and down factors.
2. Calculate the option payoff at every terminal node.
3. Calculate the value at each preceding node using the local expected payoff and discount factor.
4. Continue backward to today.
5. For an American option, compare continuation and immediate exercise at every eligible node.

If the same factors apply each period, an up move followed by a down move reaches the same price as a down move followed by an up move. The tree recombines, but the middle terminal price still represents two distinct paths.

An independent check

With constant transition probabilities and discount rates, the European option can also be valued directly from the terminal distribution:

The factor of two on the middle state is essential. Omitting it discards one of the two paths that reach the same terminal price.

This shortcut is not a substitute for backward induction when early exercise, changing node rates, or other path features matter.

14. American options: compare exercise with continuation

At each eligible node:

For a call, immediate exercise value is $\max (S-K,0)$. For a put, it is $\max (K-S,0)$.

The comparison is between exercise value and continuation value. Comparing exercise value with intrinsic value would compare the same quantity with itself.

Why a call on a stock without dividends is not normally exercised early

Under the standard assumptions with nonnegative interest rates, early exercise sacrifices the remaining option protection and requires payment of the strike sooner. Without a dividend to capture, waiting preserves the upside while postponing payment.

Therefore, the American and European call have the same value in this setting. A valuable early exercise right requires an economic benefit that offsets the sacrificed optionality and financing advantage.

Why a put can be exercised early

Exercising a put receives the strike sooner. For a sufficiently deep put, the extra benefit from retaining downside exposure may be small because the stock price cannot fall below zero.

Receiving and investing the strike earlier can then outweigh continuation. Dividend payments generally make retaining a put more attractive because they tend to reduce the underlying stock price.

Dividend paying calls

Exercising shortly before a dividend can make sense because the shareholder receives the dividend while the call holder does not. The decision still depends on the dividend benefit relative to remaining optionality and the cost of paying the strike early.

An American option cannot be worth less than an otherwise identical European option. The additional right may be worthless, but it cannot reduce the holder’s opportunity set.

15. Interest rate option trees: the discount rate changes by node

Rate options are not bond options

An interest rate call benefits when the reference rate rises. A call on a bond benefits when the bond price rises, which usually occurs when yields fall.

The word “call” identifies the payoff direction relative to the specified underlying. Always determine whether that underlying is a rate, a bond price, a futures quote, or an exchange rate.

For a rate option whose payment occurs at the accrual end date:

Backward induction on the interest rate tree

The book’s calibrated rate trees use equal risk neutral transition probabilities. The local rate at each node discounts cash flows over the following period:

A 50% probability is a feature of the tree construction used here, not a universal law for all possible interest rate models. Unlike the simple stock example, the discount rate is generally not constant across the tree.

Options on bonds

For a bond option, first value the underlying bond at the relevant tree nodes. Then calculate the option’s exercise value from those bond prices and work backward again.

The bond and the option can have different maturities. Using the bond’s face value as the underlying option payoff before the bond matures generally misses the remaining coupons and interest rate exposure.

16. Black Scholes Merton: the continuous time version of replication

The underlying logic

The binomial model forms a hedge that is riskless over the next modeled period. Black Scholes Merton, abbreviated BSM, forms a dynamically adjusted hedge that is locally riskless over an infinitesimal interval under the model assumptions.

The option’s price follows from the cost of that dynamic replication. The model is not based on forecasting the stock’s expected investment return.

Main assumptions and why they matter

The model assumes a lognormal price, not a normally distributed price. Its continuously compounded returns are normally distributed.

General equity formula with continuous dividends

Let $T$ be time to expiry, $q$ the continuous dividend yield, and $r$ the continuous interest rate:

Setting $q=0$ produces the version for a stock without dividends. Use annual volatility with time measured in years, and make the interest rate convention consistent with the exponential discount factors.

The leveraged stock interpretation

For a stock without dividends, the call resembles a position in $\Phi (d_1)$ shares financed partly by borrowing $K{e}^{-rT}\Phi (d_2)$. The put resembles a bond position combined with a short stock position.

These are the holdings required at the current instant. Maintaining replication through time generally requires rebalancing; buying the initial quantities and forgetting them does not reproduce every future option payoff.

What the normal probabilities mean

Under BSM, $\Phi (d_2)$ is the risk neutral probability that the call finishes in the money. $\Phi (d_1)$ is the call delta when there are no dividends, not that same exercise probability.

The two differ because the stock related payoff is larger in higher stock price states. Valuing the stock component requires more than multiplying a fixed stock amount by the chance of exercise.

Illustration

Suppose $S_0=100$, $K=100$, $T=1$, $r=5\%$, $q=2\%$, and $\sigma =20\%$, with continuous rates. Then:

Their difference equals $100e^{-0.02}-100e^{-0.05}$, as required by parity. The call delta is approximately 0.58685 and the put delta is approximately negative 0.39335.

Comparative statics

Holding other inputs constant, a higher stock price raises a call’s value and lowers a put’s value. A higher strike does the reverse.

Higher volatility raises both vanilla call and put values because the holder benefits from favorable outcomes while being able to decline exercise in unfavorable outcomes. Higher dividend yield lowers calls and raises puts because interim income belongs to the underlying holder rather than the option holder.

A higher financing rate raises a standard equity call’s value and lowers a put’s value, holding spot and yield fixed. The passage of time usually reduces option value, but theta and maturity effects require the qualifications discussed below.

17. Currency options: foreign interest acts like dividend yield

Let $X_0$ be the exchange rate in domestic currency per unit of foreign currency. A call on foreign currency gives the holder the right to pay domestic currency to acquire foreign currency at strike $K$.

Foreign currency held on deposit can earn the foreign rate $r_f$. This is economically analogous to dividend income on a stock.

The option values are domestic currency amounts per unit of foreign currency notional. Multiply by the relevant foreign notional to obtain the full premium.

The quotation test

If the exchange rate is dollars per euro, the domestic rate is the dollar rate and the foreign rate is the euro rate. Reversing the quote changes both the economic meaning of a call and the assignment of the interest rates.

For the stated quote and holding spot fixed, a higher domestic rate raises the call value. A higher foreign rate lowers it because holding the foreign currency directly becomes more attractive.

Connection to covered interest parity

The currency option and currency forward therefore share the same carry relationship. Optionality is added on top of the forward economics rather than replacing them.

18. Black model: choose the correct forward underlying

European options on futures or forwards

For a premium paid today and a compatible European payoff, the Black formula is:

Here the underlying input is the forward or futures price, not spot. The drift term inside $d_1$ no longer includes a separate $r-q$, because carry is already embedded in the forward price.

The reading abstracts from the futures settlement complications when applying this formula. Contract premium conventions and stochastic rate effects can require more care in practice.

One template, several underlyings

A model formula becomes dangerous when the wrong underlying is inserted into it. A spot reference rate is not the forward rate for a future caplet, and a current spot starting swap rate is not necessarily the forward swap rate for a future starting swaption.

Caplets and floorlets

A caplet fixes its reference rate at $T_1$ for an accrual period ending at $T_2$. If payment is in arrears, its value is:

There are three clocks here: $T_1$ determines uncertainty until fixing, $\alpha$ measures the interest accrual, and $T_2$ determines payment discounting. At a later valuation date, use the remaining times from that date.

Illustration. Let notional be 10 million, accrual fraction 0.25, fixing occur in one year, forward rate be 5%, strike be 4%, volatility be 20%, and the payment date discount factor be 0.95. The caplet value is approximately 25,158.29.

Caplet minus floorlet is the corresponding linear rate exchange value:

In the illustration, this difference is 23,750. Matching an FRA settled at loan start requires respecting its earlier discounted settlement convention rather than equating two cash payments made on different dates.

Caps, floors, and swaps

A cap is a portfolio of caplets over successive accrual periods. It protects a floating rate borrower against high reference rates while preserving the benefit of lower rates, in exchange for a premium.

A floor is a portfolio of floorlets. It protects a floating rate investor against low reference income while preserving the benefit of higher rates.

For identical notionals, strikes, dates, and settlement conventions:

Cap and floor values are equal when the common strike is the par fixed rate for the same covered swap schedule. It is not necessary for every individual caplet to equal its matching floorlet at that common strike.

Limits of the lognormal model

The standard Black formula requires positive forward and strike inputs because it contains their logarithmic ratio. Negative rate settings may require a shifted lognormal or normal model.

This is a practical limitation of the model family. It does not justify inserting negative rates into an undefined logarithm or changing the given exam inputs silently.

19. Swaptions: an option on a whole swap schedule

Payer and receiver directions

A payer swaption grants the right to pay fixed and receive floating at the strike rate. It is exercised when the market swap rate for the specified future swap exceeds the strike.

A receiver swaption grants the right to receive fixed and pay floating. It is exercised when the market swap rate is below the strike.

The option expiry date is the date on which the right is exercised. The underlying swap then has its own start date, payment dates, and final maturity.

The forward swap rate and annuity

For a swap beginning at $T_0$ and ending at $T_n$, with its first payment after $T_0$:

This is the forward starting version of the par swap formula. Using $1-D(0,T_n)$ would instead price a swap starting today under the basic reset assumptions.

Black swaption valuation

For the standard European swaption setup, with volatility referring to the forward swap rate and option expiry $T_0$:

The annuity $A_0$ already discounts every underlying fixed payment to today. Do not multiply by an additional expiry discount factor unless the annuity was instead defined at the exercise date and the formula explicitly requires that conversion.

The book writes its equal period version as accrual period × present value annuity × notional. The $A_0$ notation combines the first two components and also handles unequal accrual fractions.

Illustration and parity check

Let notional be 10 million, $A_0=2.7$, forward swap rate 5%, strike 4.5%, option expiry one year, and volatility 20%. The payer swaption is worth approximately 183,452.96, and the receiver is worth 48,452.96.

Their difference is:

That is the value of the forward starting payer swap at the same strike. The two swaptions have equal value when the strike equals the forward swap rate.

Embedded options in bonds

The reading links a callable bond to an ordinary bond plus a short receiver swaption exposure. When rates fall, the issuer’s opportunity to refinance becomes valuable, limiting the investor’s upside.

The receiver direction is intuitive: receiving a previously fixed high rate becomes valuable when market rates fall. The investor in a callable bond has effectively sold the issuer an option associated with that favorable rate decline.

This is a matching cash flow interpretation. Exact replication depends on the bond’s call schedule, strike conventions, amortization, and the swaption structure, especially when exercise is possible on multiple dates.

20. The Greeks: map the risks, not just the formula

Main sensitivities

Short option positions reverse the signs of the corresponding long option sensitivities. The table’s rho directions should not be transferred blindly to options on rates or to a calculation that holds a forward price fixed instead of spot.

Delta

For BSM equity options with continuous yield:

For a stock without dividends, call delta lies between zero and one, while put delta lies between negative one and zero. As the stock rises, both call and put deltas increase; the put delta becomes less negative.

Near expiry, a call clearly in the money approaches delta one and a call clearly out of the money approaches zero, assuming no dividends. For puts, the corresponding limits are negative one and zero.

At the strike, the expiry payoff has a kink. Do not assume a stable smooth hedge ratio exactly at that boundary.

Gamma

Gamma captures how quickly the hedge ratio changes. Vanilla calls and puts with matching underlying, strike, expiry, and model inputs have the same gamma.

Gamma is generally greatest near the money, especially near expiry. A cheap option close to expiry can therefore still create substantial hedge instability.

A second order approximation is:

Illustration. With delta 0.60, gamma 0.04, and a stock increase of 2, the delta only estimate is 1.20. Including gamma gives 1.28.

This is a local approximation, not an exact repricing formula for arbitrarily large moves. Gamma itself can change as the underlying moves.

Vega

Vega measures sensitivity to implied volatility rather than to the realized price move itself. Both long calls and long puts generally benefit from an increase in volatility, holding the other inputs fixed.

Vega is typically largest near the money and tends to be greater for longer dated options than for comparable options very close to expiry. High gamma near expiry does not automatically imply high vega.

Check the reporting units. A vega quoted per one percentage point of volatility must not be multiplied by a decimal volatility change without the appropriate conversion.

Theta

Theta is measured with respect to calendar time passing, not with respect to adding time to maturity. This convention explains why ordinary time decay is described as negative theta.

Negative theta is not universal for every European option and every carry environment. Deep puts can have positive theta, and dividends, interest rates, and moneyness can alter the usual patterns.

The strong acceleration of time decay near expiry is especially associated with options near the money. An option far from the strike does not necessarily exhibit the same pattern.

Rho

In standard equity BSM, a higher rate reduces the present value of paying the strike, helping a call and hurting a put when spot and dividend yield are held fixed. The same change can also affect forward values, so specify what is held constant.

The reading treats rho as less prominent than some other Greeks in common examples. It should not be interpreted as unimportant for every long maturity option or every interest rate derivative.

21. Dynamic hedging and gamma risk

Delta neutrality is local

A delta neutral position has zero first order exposure to a small change in the underlying at the current instant. It can still have gamma, volatility, time, rate, liquidity, and model risk.

For stock shares $n_S$, signed option contract count $n_O$, contract multiplier $m$, and option delta ${\Delta }_O$:

Choose the option or share position so that this sum is zero. A short option count is negative.

Rebalancing when delta changes

Suppose the call delta rises from 0.60 to 0.65 after the stock rises. Keeping the same 60,000 shares and 1,000 short contracts gives:

Buying 5,000 shares restores delta neutrality while keeping the option position unchanged. Alternatively, adjust the option exposure with appropriate attention to whole contract sizes.

The short call hedge tends to require buying more stock after a rise and selling stock after a fall. This unfavorable rebalancing pattern is part of being short gamma, rather than a mistake in the hedge calculation.

Gamma remains after delta is removed

A share has delta one and gamma zero. Adding stock can change delta, but it cannot remove an option portfolio’s gamma.

If a delta neutral portfolio has gamma negative 4,000 in the appropriate share based units, an immediate underlying move of either positive 2 or negative 2 creates an approximate curvature loss of:

This isolates the local gamma effect and omits theta, vega, financing, transaction costs, and higher order terms. It shows why a position can be delta neutral and still lose from a large move in either direction.

Practical meaning of gamma risk

In the ideal continuous BSM world, a continuously rebalanced hedge can eliminate the modeled diffusion risk. Gamma does not disappear; its effects are handled through continuous rebalancing and the model’s financing relationships.

In practice, trading is discrete and prices can jump. A hedge that was neutral before a move may be materially exposed afterward, especially near the strike close to expiry.

To change gamma directly, use options or other instruments with curvature, not stock alone. A gamma hedge may create new vega or maturity exposures that also require management.

22. Implied volatility: a price expressed in model units

Definition

Implied volatility is the volatility input that makes the chosen model reproduce the observed option price:

It is found numerically because the BSM price cannot generally be inverted into a simple explicit volatility formula. Historical volatility instead estimates variability from past returns.

Implied volatility is neither automatically a forecast nor automatically the volatility that will be realized. It is a model dependent transformation of an observed market price.

Why traders use it

Implied volatility makes it easier to compare options with different strikes and maturities in a common quotation language. A higher dollar premium can simply reflect a longer maturity or a more valuable strike, rather than unusually expensive volatility.

A smile or skew means that different strikes imply different volatilities under the same basic model. That pattern is evidence that a single constant volatility does not describe the full set of market prices.

A volatility term structure compares maturities. It can reflect event timing, uncertainty at different horizons, and compensation for bearing volatility risk.

From an opinion to a position

The book’s example compares market implied volatility of 22% with a trader’s estimate of 25%. Under that view, buying options expresses the belief that the optionality is inexpensive relative to expected future variability.

An outright call purchase also carries positive delta. To isolate a volatility view more closely, a trader may hedge delta, while still retaining gamma, theta, vega, and trading cost exposures.

A high implied volatility is not proof that an option is overpriced. An apparently high level may be justified by forthcoming events, tail risk, market frictions, or an inaccurate model assumption.

Realized volatility and hedging profit

As an analytical extension, a dynamically hedged long option can benefit when realized variation is large relative to the volatility embedded in the purchase price. That conclusion is conditional on hedge implementation, trading costs, the timing of price moves, and changes in implied volatility.

Before expiry, an option can gain from an increase in implied volatility even without a large realized stock move. Vega and gamma describe related but distinct channels.

Integration and Exam Application

23. Choose the instrument from the exposure

A swap can lock the reference rate without locking a borrower’s credit spread. Likewise, a currency swap can match expected debt payments while leaving liquidity demands from collateral or margin.

Hedging converts or reduces selected risks. It does not make every source of uncertainty disappear.

24. A formula selection and checking system

Step 1: classify the question

Is it asking for the fair terms of a new contract, the value of an existing contract, a settlement payment, a hedge quantity, or a risk sensitivity? Write that classification before choosing a formula.

A forward price and a forward value can have similar units but answer different questions. A swap rate and a swap value have different units altogether.

Step 2: write a receipt and payment statement

Examples include “receive asset, pay locked price,” “receive floating, pay fixed,” and “receive dollar leg, pay sterling leg.” Derive the sign from that statement rather than relying on an isolated label such as long or payer.

Step 3: draw the relevant dates

For an FRA or caplet, mark fixing and loan maturity separately. For a swaption, mark option expiry separately from the underlying swap’s final payment.

Step 4: use the smallest reliable valuation structure

Step 5: run economic checks

1. Initial value: does a newly fair forward or swap have zero value?
2. Maturity: does the forward reduce to $S_T-K$, and does the option reduce to its exercise payoff?
3. Direction: does a higher relevant forward rate benefit a long FRA and a fixed payer swap?
4. Opposite sides: do the two sides of a forward or swap have equal and opposite values in the basic framework?
5. Dividend treatment: does more interim income reduce the asset forward price and the call value, all else equal?
6. Optionality: is an American option at least as valuable as its otherwise identical European counterpart?
7. Parity: do call and put values satisfy the appropriate payoff identity?
8. Units: is the answer a rate, a price per unit, a cash amount, an index point amount, or a contract count?
9. Timing: have you discounted from the actual payment date rather than the most visible date in the question?
10. Precision: have you avoided rounding every intermediate discount factor and probability?

25. Common misconceptions to eliminate

26. Active recall questions

27. Source coverage and study priorities

Coverage of Reading 28

Coverage of Reading 29

Recommended order of mastery

1. Cash flow replication and discounting. Derive forward and swap formulas from what is received and paid.
2. Timing intensive calculations. Practice dividend adjustments, bond accrued interest, FRA settlement, and swap reset valuation.
3. Binomial mechanics. Solve European and American trees, then reproduce the answer through replication or parity where possible.
4. Model selection. Identify spot, forward rate, futures price, or forward swap rate before applying BSM or Black.
5. Risk interpretation. Explain the signs of delta, gamma, vega, theta, and rho, and distinguish local hedging from comprehensive risk removal.

The most transferable skill is not memorizing every formula separately. It is recognizing that carry, swaps, parity, trees, and option models are different applications of the same principle: identical cash flows must have consistent values.