CFA Level 1 — Derivatives

Intuition-First Study Guide · All 10 Readings

Every concept explained from first principles · Formulas with worked examples · Exam traps highlighted

Reading 66

Derivative Instrument and Derivative Market Features

What is a derivative, where do they trade, and who's on the other side of your trade?

🗺 The Big Picture

A derivative is simply a contract. Its value derives from some other asset (the "underlying"). Understanding derivatives starts with one insight: a derivative is a zero-sum game — every dollar gained by the long position is a dollar lost by the short position. The underlying asset doesn't change; the derivative just transfers risk between two parties.

What is a Derivative?

A derivative is a financial instrument whose value depends on the value of an underlying asset, rate, index, or other variable. The derivative itself is not the underlying — it is a contract about the underlying.

An insurance policy on your car is a derivative. Its value depends on your car (the underlying). If the car is totalled (underlying event occurs), the policy pays off. If nothing happens, the policy expires worthless. You didn't buy the car again — you bought a contract about the car.

Underlying Assets

Category	Examples	Key Derivatives Traded
Equities	Individual stocks, stock indices (S&P 500)	Equity futures, index options, equity swaps
Fixed Income	Government bonds, T-bills	Bond futures, interest rate swaps, FRAs
Currencies	EUR/USD, GBP/JPY	Currency forwards, FX futures, currency swaps
Commodities	Oil, gold, wheat, natural gas	Commodity futures, commodity swaps
Credit	Corporate bond spreads, loan pools	Credit default swaps (CDS), CDOs
Other Derivatives	Options on futures ("options on futures")	Swaptions, options on futures

Exchange-Traded vs. OTC Derivatives Critical

Exchange-Traded (ETD)

Standardised contracts (fixed size, expiry, terms)
Trade on organised exchanges (CME, Eurex)
Central counterparty clearing (CCP) eliminates bilateral default risk
Margin requirements: initial + variation margin
Daily mark-to-market (marking to market)
High liquidity, transparent prices
Cannot customise to exact needs

Over-the-Counter (OTC)

Customised — any terms the two parties agree on
Traded directly between dealers and end-users
Bilateral counterparty credit risk (unless cleared)
Less transparent — prices are negotiated
Larger notional amounts typical
Post-2008 reforms: more OTC now centrally cleared (ISDA/Dodd-Frank)
ISDA Master Agreement governs most OTC derivatives

Think of exchange-traded vs OTC like buying a stock vs buying a custom bond. A stock is standardised — 100 shares of Apple at a quoted price, guaranteed by the exchange. A custom bond is negotiated — you and the issuer agree on the exact coupon, maturity, and covenants. More flexibility, but more counterparty risk.

The Role of the Derivatives Dealer

Most OTC derivatives are initiated through a dealer (typically a large bank). The dealer acts as the market-maker: they take the opposite side of the client's trade, then hedge their own exposure elsewhere. The dealer earns a bid-ask spread, not by speculating on direction.

🎯 Likely Exam Question

Which of the following best describes a key difference between exchange-traded and OTC derivatives? A) Exchange-traded derivatives have no counterparty risk. B) OTC derivatives cannot be used for hedging. C) Exchange-traded derivatives always have longer maturities.

Answer: A. Exchange-traded derivatives use a central counterparty (CCP), which effectively eliminates bilateral counterparty default risk — the CCP guarantees both sides. OTC derivatives still carry counterparty (credit) risk unless they are centrally cleared. B and C are incorrect.

🎯 Likely Exam Question

An investor enters a derivative contract. Over the life of the contract, the investor gains €5,000. The total gain or loss across both sides of this contract is:

Answer: Zero. Derivatives are zero-sum instruments — every gain by one party is exactly offset by a loss to the other party. The contract redistributes wealth between parties; it does not create wealth. This is a fundamental and frequently tested concept.

Reading 67

Forward Commitment and Contingent Claim Features and Instruments

The derivative family tree: two branches — obligations and rights.

🗺 The Big Picture

Every derivative falls into one of two families. Forward commitments (forwards, futures, swaps) obligate both parties to perform — neither can walk away. Contingent claims (options) give the holder a right but not an obligation — the holder only exercises if it's advantageous. This distinction drives everything about how they are valued and used.

Forward Commitments

1. Forward Contracts

A privately negotiated (OTC) agreement to buy or sell an asset at a fixed price (forward price) on a specific future date. No money changes hands at initiation — the contract has zero cost to enter.

Long forward: agrees to BUY at the forward price — profits if spot price rises above forward price at expiry
Short forward: agrees to SELL at the forward price — profits if spot price falls below forward price at expiry
Settled by delivery of the underlying (physical settlement) or cash payment of the difference (cash settlement)

You agree today to buy 100 barrels of oil in 3 months at $80/barrel (the forward price). In 3 months, if oil is at $90, you profit $10/barrel (you buy at $80 what's worth $90). If oil is at $70, you lose $10/barrel (you're forced to pay $80 for something worth $70). You cannot walk away.

2. Futures Contracts

An exchange-traded, standardised forward contract. Key feature: daily marking to market (variation margin). Both parties post initial margin and gains/losses are settled daily in cash — you can't accumulate large losses without the exchange noticing.

Feature	Forward	Futures
Trading venue	OTC (bilateral)	Exchange-traded
Standardisation	Fully customisable	Standardised terms
Settlement	Single settlement at expiry	Daily mark-to-market
Counterparty risk	Yes — bilateral credit risk	Minimal — CCP guaranteed
Liquidity	Illiquid (hard to exit)	Highly liquid
Margin requirement	Typically none at initiation	Initial + variation margin
Price transparency	Private	Public market price

3. Swaps

A series of forward contracts packaged together. The two parties exchange a series of cash flows over a specified period based on different references (e.g., fixed vs floating interest rates, or two different currencies).

Interest Rate Swap (IRS): most common — exchange fixed payments for floating (SOFR-based) payments on a notional principal. Notional is NEVER exchanged — only the net difference in cash flows is paid.
Currency Swap: exchange cash flows in different currencies, typically WITH exchange of principal at start and end.
Equity Swap: exchange return on equity index for a fixed or floating rate.
Credit Default Swap (CDS): protection buyer pays periodic premiums; protection seller pays the notional if the reference entity defaults.

A plain-vanilla interest rate swap is like splitting a fixed-rate mortgage into two parts and exchanging one part with someone else. One party wants to convert their fixed-rate exposure to floating; the other wants the opposite. They swap — and both get what they want without either refinancing their actual debt.

Contingent Claims

4. Options

A contract giving the holder the right, but not the obligation, to buy (call) or sell (put) an underlying at a specified price (exercise/strike price) on or before a specified date. The buyer pays a premium upfront for this right.

Type	Right Granted	Buyer Profits When...	Seller (Writer) Profits When...
Call Option	Right to BUY underlying at strike price	Underlying price rises above strike + premium	Underlying price stays below strike (option expires worthless — they keep premium)
Put Option	Right to SELL underlying at strike price	Underlying price falls below strike − premium	Underlying price stays above strike (option expires worthless — they keep premium)

The option buyer has LIMITED downside (maximum loss = premium paid) and theoretically unlimited upside. The option writer (seller) has LIMITED upside (maximum gain = premium received) and potentially unlimited downside. This asymmetry is why options require a premium payment at initiation — unlike forwards/futures.

European vs. American Options

European Options

Can ONLY be exercised at expiration
Easier to value analytically (Black-Scholes)
Most index options and many OTC options

American Options

Can be exercised at ANY time before expiration
More valuable than European (additional flexibility)
Most exchange-traded stock options

5. Credit Derivatives Exam Focus

A Credit Default Swap (CDS) is the most important credit derivative. The protection buyer makes periodic premium payments (the "CDS spread") to the protection seller. If the reference entity defaults, the protection seller pays par value and receives the defaulted bonds.

A CDS is like credit insurance. You own a corporate bond but are worried the company might default. You buy CDS protection, paying a quarterly premium. If the company defaults, the CDS seller compensates you — just like an insurance payout. You've transferred the credit risk, but you still own the bond.

🎯 Likely Exam Question

An investor buys a call option with a strike price of $50, paying a premium of $3. The stock price at expiry is $56. What is the investor's profit/loss?

Answer: Profit = $56 − $50 − $3 = $3 per share. The intrinsic value at expiry is $6 ($56 − $50), but you subtract the $3 premium paid upfront. Break-even is at $53 ($50 + $3). The investor exercises the call and profits $3 per share.

🎯 Likely Exam Question

Which of the following is a forward commitment (not a contingent claim)? A) Put option on a stock. B) Interest rate cap. C) Fixed-for-floating interest rate swap.

Answer: C. A swap is a series of forward contracts — both parties are obligated to make payments. Options (calls, puts) and interest rate caps/floors are contingent claims — the holder has the right but not the obligation to exercise. Caps and floors are packages of options on interest rates (caplets/floorlets).

Reading 68

Derivative Benefits, Risks and Issuer and Investor Uses

Why derivatives exist — and why they can also cause catastrophe.

Why Derivatives Exist: The Economic Benefits

Benefit	Explanation	Example
Risk Allocation / Transfer	Those who don't want risk can transfer it to those who do (or can bear it better)	Airline buys oil futures to lock in fuel costs; speculator takes the other side, bearing price risk in exchange for potential profit
Price Discovery	Futures markets aggregate information from many participants, producing forward-looking prices	Crude oil futures prices reflect market expectations of future supply/demand, visible to everyone
Operational Efficiency	Gaining exposure is faster and cheaper than transacting in the underlying	Buying S&P 500 futures is cheaper and faster than buying all 500 stocks
Leverage	Control large notional exposure with a small capital outlay (margin)	$5,000 margin controls $100,000 of futures contracts — 20:1 leverage
Short Selling Made Easy	Express bearish views without borrowing the underlying	Short equity futures to benefit from a market decline without stock-borrowing costs
Completing Markets	Enable payoff profiles that can't be replicated in spot markets	Volatility trades (e.g., buying straddles) are impossible without options

Risks of Derivatives

1. Leverage Risk (Amplification)

Because a small margin deposit controls a large notional position, gains and losses are magnified relative to the capital invested. A 5% move in the underlying can wipe out 100% of your margin if the leverage is 20:1.

Derivatives don't create risk — they redistribute it. But leveraged derivatives can cause the holder to lose more than they intended to risk, or even more than their initial investment (in the case of short options).

2. Counterparty Credit Risk

In OTC derivatives, if your counterparty defaults before the contract expires, you lose the mark-to-market gain you were owed. Exchange-traded derivatives eliminate this via CCP, but OTC exposure remains (partially mitigated by netting agreements and collateral under ISDA CSA).

3. Liquidity Risk

OTC derivatives can be very difficult to exit. Unlike exchange-traded contracts you can close with an offsetting trade, an OTC contract must be novated (assigned to a new counterparty) or terminated early, often at unfavourable prices.

4. Interconnectedness / Systemic Risk

Derivatives create webs of interdependence. If one major dealer defaults (like Lehman Brothers in 2008), the losses cascade through all counterparties. This is why post-2008 regulation pushed OTC derivatives toward central clearing.

Uses by Issuers (Corporations)

Use Case	Derivative Used	Why
Hedge FX exposure on foreign revenues	FX forwards or options	Lock in exchange rate to remove earnings uncertainty
Fix interest costs on floating-rate debt	Pay-fixed interest rate swap	Convert variable-rate borrowing to predictable fixed payments
Lock in commodity input prices	Commodity futures or forwards	Stabilise margins by fixing cost of raw materials (e.g., fuel, metals)
Manage pension fund duration	Interest rate futures or swaps	Match liability duration without selling/buying actual bonds

Uses by Investors

Use Case	Derivative Used	Why
Hedge equity portfolio against market decline	Buy index put options or sell index futures	Downside protection without liquidating the portfolio
Enhance yield on existing holdings	Write (sell) covered call options	Collect premium; caps upside but generates income
Gain market exposure quickly	Buy equity futures or total return swap	Faster and cheaper than buying a basket of stocks
Express a view on volatility	Buy a straddle (call + put, same strike)	Profit if underlying moves sharply in either direction
Access otherwise hard-to-reach markets	Total return swaps, commodity swaps	Get commodity or EM equity exposure without holding actual assets

🎯 Likely Exam Question

A US company expects to receive €10 million from a European client in 90 days. The company is concerned about EUR weakening against USD. What derivative strategy would best hedge this exposure?

Answer: Sell EUR forward (short EUR/USD forward). The company will receive €10M and needs to convert to USD. If EUR weakens, they receive fewer USD. By selling EUR forward, they lock in the current forward exchange rate — eliminating the FX risk. Alternatively, buying a EUR put / USD call option would provide a floor on the USD received while keeping upside.

Reading 69

Arbitrage, Replication and the Cost of Carry in Pricing Derivatives

Why derivative prices are what they are — the iron logic of no free lunches.

🗺 The Big Picture

Derivative pricing rests on one principle: no arbitrage. If a derivative is mispriced, traders will buy the cheap asset and sell the expensive one simultaneously, earning a riskless profit. This pressure immediately corrects the mispricing. The "correct" forward price is therefore exactly the price at which no arbitrage is possible — and this is derived from the cost of carry model.

The Law of One Price

If two portfolios produce identical future cash flows in all states of the world, they must have the same price today. If they don't, an arbitrage exists.

A €100 note in London and a €100 note in Paris are worth exactly the same (ignoring conversion costs). If they weren't, you'd buy cheap notes in Paris and sell them in London for a riskless profit. That trade instantly eliminates the price difference.

Replication

You can replicate the payoff of a forward contract with a portfolio of the underlying asset and a risk-free borrowing/lending position. Since the replicating portfolio has the same payoff as the forward, they must have the same price. This is how we derive the forward price.

Replication Portfolio for a Forward Contract

Consider a forward contract to buy stock S at price F in time T:

Payoff at T: S_T − F (long forward)
Replication: Buy the stock today at S₀, borrow S₀ at the risk-free rate r. At T, you own the stock (worth S_T) and owe S₀(1+r)^T. Net payoff: S_T − S₀(1+r)^T
For no arbitrage: F = S₀(1+r)^T

\[ F_0 = S_0 \times (1 + r_f)^T \]

Forward price for a non-dividend-paying asset (discrete compounding)

\[ F_0 = S_0 \times e^{r_f T} \]

Forward price (continuous compounding)

Cost of Carry — The General Framework Critical

The forward price equals the spot price plus the net cost of carry: all costs of holding the underlying asset minus all benefits received from holding it.

\[ F_0 = S_0 \times (1 + r_f)^T + \text{Costs of Carry} - \text{Benefits of Carry} \]

Component	Effect on F₀	Examples
Risk-free rate (r)	Increases F₀	The financing cost of buying the underlying today instead of at T
Storage costs (c)	Increases F₀	Warehouse fees for commodities (oil, gold, wheat)
Dividends (q) or coupon income	Decreases F₀	Dividends on stock; coupon payments on bonds
Convenience yield (y)	Decreases F₀	Benefit of having physical commodity on hand (e.g., oil in reserve for unexpected demand)

\[ F_0 = (S_0 + c - i) \times (1 + r_f)^T \]

Where c = PV of storage costs, i = PV of income/dividends received

Worked Example: Forward Price with Dividends

A stock trades at €50. The risk-free rate is 5% p.a. The stock pays a €2 dividend in 3 months. What is the 6-month forward price?

PV of dividend = €2 / (1.05)^(0.25) = €1.976
Adjusted spot = €50 − €1.976 = €48.024
F₀ = €48.024 × (1.05)^0.5 = €48.024 × 1.0247 = €49.21

Why do dividends reduce the forward price? Because if you own the stock, you receive dividends during the holding period. The forward contract buyer misses out on those dividends — so the forward price is reduced to compensate. The forward seller effectively "gives up" the dividends to the spot holder.

Arbitrage Mechanics: Cash-and-Carry and Reverse Cash-and-Carry

Cash-and-Carry (F₀ too HIGH)

Actual F₀ > Fair F₀
Action: Buy spot, borrow funds, sell (go short) the overpriced forward
At expiry: deliver the asset, repay the loan
Lock in a riskless profit = Actual F₀ − Fair F₀

Reverse Cash-and-Carry (F₀ too LOW)

Actual F₀ < Fair F₀
Action: Short sell the asset, invest proceeds, buy (go long) the underpriced forward
At expiry: receive delivery, return borrowed asset
Lock in a riskless profit = Fair F₀ − Actual F₀

🎯 Likely Exam Question

A non-dividend paying stock trades at $100. The risk-free rate is 4%. The 1-year forward price is quoted in the market at $108. Describe the arbitrage and calculate the riskless profit per share.

Fair forward price = $100 × 1.04 = $104. The market quote of $108 is too high. Cash-and-carry arbitrage: (1) Borrow $100 at 4%, buy the stock. (2) Simultaneously sell the forward at $108. At expiry: deliver the stock at $108, repay loan of $104. Riskless profit = $108 − $104 = $4 per share.

Convenience yield is only relevant for physical commodities — it captures the benefit of having the actual physical good on hand (e.g., a refinery that needs oil immediately cannot use a futures contract). It is NOT applicable to financial assets like stocks or bonds.

Reading 70

Pricing and Valuation of Forward Contracts

Pricing sets the forward price at inception; valuation tracks its worth as markets move.

🗺 Critical Distinction

Price vs. Value — this is the most important distinction in derivatives: The forward price (F₀) is fixed at initiation and never changes. The value of the forward contract changes continuously as the underlying spot price moves. At initiation, value = 0 by design. After initiation, value can be positive or negative.

Forward Price at Initiation

The forward price F₀ is set so that the initial value of the contract is zero — no money changes hands at initiation.

\[ F_0(T) = S_0 \times (1 + r_f)^T \quad \text{(no income or costs)} \]

\[ F_0(T) = (S_0 - PV_0(\text{Benefits}) + PV_0(\text{Costs})) \times (1 + r_f)^T \]

Value of a Forward Contract During Its Life Critical

After initiation, as the spot price changes and time passes, the forward contract acquires value. Let t be the current time (0 < t < T), S_t be the current spot price.

\[ V_t(\text{Long}) = S_t - PV_t(\text{Future Benefits}) - \frac{F_0}{(1 + r_f)^{T-t}} \]

\[ V_t(\text{Short}) = -V_t(\text{Long}) \]

Simplified (no benefits or costs): the value of the long forward at time t is:

\[ V_t = S_t - \frac{F_0}{(1 + r_f)^{T-t}} \]

Think of it this way: if you're long a forward at F₀, you've locked in buying at F₀. The contract is now worth the difference between what the asset is worth NOW (S_t) and what you've locked in paying, discounted back from expiry. If S_t has risen above your locked-in F₀, the contract is worth positive money to you. If S_t has fallen, the contract is a liability.

Worked Example: Valuing a Forward Mid-Life

6 months ago, you entered a 1-year forward to buy a non-dividend stock at F₀ = $104 (spot was $100, r = 4%). Now (at t = 6 months), the stock is at $112. What is the value of your long forward position?

T − t = 0.5 years remaining
PV of F₀ = $104 / (1.04)^0.5 = $104 / 1.0198 = $101.98
V_t = S_t − PV(F₀) = $112 − $101.98 = $10.02

At expiration (t = T): V_T = S_T − F₀. The forward settles at the difference between the spot price at expiry and the locked-in forward price. If S_T > F₀, long profits; if S_T < F₀, long loses.

Forward Rate Agreements (FRAs) Exam Focus

An FRA is a forward contract on an interest rate. The buyer of an FRA locks in a borrowing rate; the seller locks in a lending rate. Settlement is at the beginning of the reference period (not the end).

An FRA(1×4) is a 3-month rate, starting in 1 month. Notation: FRA(m×n) means the rate starts in m months and ends in n months, so the underlying period is (n − m) months.

\[ \text{FRA Payment (at settlement)} = \frac{(\text{Floating Rate} - \text{FRA Rate}) \times \text{Days}/360}{1 + \text{Floating Rate} \times \text{Days}/360} \times \text{Notional} \]

The FRA settles at the beginning of the interest period (discounted), not at the end. This is because interest itself is paid at the end, so the settlement is the present value of what the rate difference would produce at the end of the period. Always discount the payment back.

Forward Currency Pricing (Covered Interest Rate Parity)

The forward FX rate is determined by the interest rate differential between two countries. Higher interest rate currency trades at a forward discount.

\[ F_0 = S_0 \times \frac{(1 + r_d)^T}{(1 + r_f)^T} \]

Where r_d = domestic interest rate, r_f = foreign interest rate, quoted as domestic/foreign

🎯 Likely Exam Question

The EUR/USD spot rate is 1.10 (1 EUR = 1.10 USD). USD 1-year interest rate = 5%, EUR 1-year rate = 2%. What is the 1-year EUR/USD forward rate?

F₀ = 1.10 × (1.05/1.02) = 1.10 × 1.02941 = 1.1324. The USD has a higher interest rate, so it trades at a forward discount (EUR buys more USD in the forward than the spot). This reflects covered interest rate parity — if it didn't hold, you could borrow EUR cheaply, convert to USD, invest at 5%, then convert back via forward and earn a riskless profit.

🎯 Likely Exam Question

A forward contract on a non-dividend-paying stock is entered at F₀ = $50. Three months later, with 3 months remaining to expiry, the stock is at $54. The annual risk-free rate is 4%. What is the value of the long forward position?

V = S_t − F₀/(1+r)^(T−t) = $54 − $50/(1.04)^0.25 = $54 − $50/1.00995 = $54 − $49.51 = $4.49. The long forward has gained $4.49 in value because the stock has risen. If this were marked to market, the short party would owe $4.49 to the long party.

Reading 71

Pricing and Valuation of Futures Contracts

Like forwards, but with daily cash flows and some important differences.

Futures Price vs. Forward Price

In theory, if interest rates are deterministic (non-stochastic), futures prices equal forward prices for the same underlying, maturity, and terms. In practice, they can differ slightly due to the daily marking-to-market of futures.

When Futures Price > Forward Price

When the underlying price is positively correlated with interest rates
Gains on long futures position happen when rates are high (good reinvestment) → futures more valuable
Example: T-bond futures — bond prices inversely correlate with rates, so actually futures < forwards for bonds

When Futures Price < Forward Price

When the underlying price is negatively correlated with interest rates
Example: Long-term bond futures — when rates fall, bond prices rise; margin gains happen when rates are low

For the CFA exam, treat futures prices as equal to forward prices unless told otherwise. The difference is a second-order effect typically ignored at Level 1.

Marking to Market and Margin

Initial Margin

A good-faith deposit required to enter a futures position. It is a fraction (typically 5–15%) of the contract's notional value. This is NOT a down payment — the full notional is still at risk.

Variation Margin (Daily Settlement)

Each day, gains and losses are credited or debited to your margin account. If your account falls below the maintenance margin, you receive a margin call and must deposit funds to restore the initial margin level.

Imagine a daily settling scorecard. At the end of each day, the "score" is tallied. If you've lost, cash leaves your account. If you've gained, cash enters. You can never owe more than today's price move times the notional — the CCP resets each day.

Worked Example: Mark-to-Market

Day	Futures Price	Daily P&L (Long)	Margin Balance
0 (enter)	$1,000	—	$5,000 (initial)
1	$1,010	+$10 × 100 = +$1,000	$6,000
2	$985	−$25 × 100 = −$2,500	$3,500
2 (margin call)	—	Must deposit $1,500	$5,000 (restored)
3	$990	+$5 × 100 = +$500	$5,500

Contango and Backwardation Exam Focus

Contango (Normal Market)

Futures price > Spot price
The normal case for financial assets (cost of carry is positive)
Later expiry contracts priced higher
Upward-sloping forward curve

Backwardation (Inverted Market)

Futures price < Spot price
Common for commodities with high convenience yield
Occurs when immediate demand is very strong
Downward-sloping forward curve

Backwardation in oil futures: oil producers need to sell forward to lock in prices, and convenience yield is high (everyone wants oil NOW). The pressure of producers selling forward drives futures below spot. This is also called "normal backwardation" in some contexts — don't confuse the terms.

Basis and Basis Risk

\[ \text{Basis} = \text{Spot Price} - \text{Futures Price} \]

Basis is not constant — it changes as time passes and as spot and futures prices move differently. Basis risk is the risk that the basis changes unexpectedly, causing a hedge to be imperfect.

Basis converges to zero at expiry: as the futures contract approaches its delivery date, the futures price converges to the spot price (otherwise cash-and-carry arbitrage would exist). At expiry, Basis = 0.

🎯 Likely Exam Question

An investor is long 10 crude oil futures contracts (100 barrels each). The initial margin is $5,000 per contract. At the end of Day 1, the futures price falls from $80/barrel to $78/barrel. What happens to the investor's margin account?

Loss = ($78 − $80) × 100 × 10 = −$2 × 1,000 = −$2,000. The margin account is debited $2,000. If this falls below the maintenance margin, a margin call is issued. Note: the total initial margin was $5,000 × 10 = $50,000. A $2,000 single-day loss on a $80,000 notional position is a 2.5% move that causes a 4% hit to the margin balance.

Reading 72

Pricing and Valuation of Interest Rate and Other Swaps

A swap is just a bundle of forwards — price it the same way.

🗺 The Big Picture

An interest rate swap can be decomposed into a series of FRAs (forward rate agreements), one for each settlement date. Alternatively, it can be viewed as a fixed-rate bond vs. a floating-rate bond — the fixed-rate payer is short a fixed bond, long a floating bond. Both frameworks give the same answer for pricing and valuation.

Plain Vanilla Interest Rate Swap Structure

In a fixed-for-floating swap with notional N:

Fixed-rate payer pays the swap rate (FS) × N × period fraction, receives SOFR × N × period fraction
Floating-rate payer pays SOFR × N × period fraction, receives FS × N × period fraction
Only the net payment is exchanged (netting reduces credit exposure)
Notional principal is NEVER exchanged (unlike currency swaps)

A company has issued a $100M floating-rate bond at SOFR + 200bps. It now wants to lock in a fixed payment. It enters a "pay fixed, receive floating" swap. The swap's floating receipts cancel out the floating bond payments — net effect: the company pays a fixed all-in rate. The swap has converted floating borrowing to fixed, without touching the original bond.

Pricing the Swap: The Swap Rate Critical

The swap rate (fixed rate) is set so that the swap's initial value is zero. This means the present value of all fixed payments must equal the present value of all expected floating payments.

\[ \text{Swap Rate} = FS = \frac{1 - Z_N}{\sum_{i=1}^{N} Z_i} \]

Where Z_i = discount factor for period i (PV of $1 received at time i)

Or equivalently:

\[ \sum_{t=1}^{N} FS \cdot Z_t + Z_N = 1 \]

PV(fixed payments) + PV(final notional exchange) = par (=1 on a per-unit basis)

Worked Example: Pricing a 2-Year Annual Pay Swap

Given spot rates: S₁ = 4%, S₂ = 5%. Calculate the swap rate.

Z₁ = 1/(1.04)¹ = 0.9615
Z₂ = 1/(1.05)² = 0.9070
FS = (1 − Z₂) / (Z₁ + Z₂) = (1 − 0.9070) / (0.9615 + 0.9070) = 0.0930 / 1.8685 = 4.978% ≈ 4.98%

The swap rate is just the coupon rate on a par bond with the same maturity and yield structure. A par bond has value exactly equal to its face value — meaning the PV of coupons + PV of face value = 1. The swap rate is the coupon rate that makes this true, given the current yield curve.

Valuing a Swap During Its Life Exam Focus

After initiation, interest rates change, and the swap acquires value. From the fixed-rate payer's perspective, the swap is like being long a floating-rate bond (worth par, since it resets) and short a fixed-rate bond.

\[ V_{\text{fixed payer}} = V_{\text{floating}} - V_{\text{fixed}} \]

\[ V_{\text{floating}} = 1 \quad \text{(resets to par at each coupon date)} \]

\[ V_{\text{fixed}} = FS \cdot \sum Z_i + Z_N \]

If rates have risen since the swap was initiated, the fixed-rate payer benefits (they locked in the old, lower fixed rate, and the market now demands more). The swap is worth positive to the fixed-rate payer.

Currency Swaps

Unlike interest rate swaps, currency swaps involve the exchange of principal in two different currencies at the start and end of the swap. During the swap, interest payments in both currencies are exchanged.

Feature	Interest Rate Swap	Currency Swap
Principal exchange	No (never exchanged)	Yes — at inception AND maturity
Currency of cash flows	Single currency (net settlement)	Two different currencies
FX risk	None	Significant — payments are in different currencies
Credit exposure	Only interest differential	Principal + interest (larger notional at risk)

🎯 Likely Exam Question

Six months into a 2-year annual pay fixed-for-floating swap (you pay fixed 5%), market rates have risen and the new 1.5-year swap rate is 6%. Has the swap gained or lost value to you as the fixed payer? Why?

The swap has GAINED value to the fixed payer. You locked in paying 5% fixed when the market now requires 6%. You are paying below-market rates — the swap is "in the money" for you. Equivalently: you are short a fixed-rate bond (you receive fixed = 5%). When rates rise, the fixed bond falls in value. Short position on a falling bond = profit. The fixed payer gains when rates rise.

Reading 73

Pricing and Valuation of Options

Six forces drive every option price — master these and you can price anything.

🗺 The Big Picture

Unlike forward contracts (which have a zero initial value by design), options have a positive premium because they give one-sided rights. The option price has two components: intrinsic value (what it's worth if exercised right now) and time value (the extra premium for the possibility of future favourable moves). Time value always erodes to zero at expiry.

Option Value Components

\[ \text{Option Price} = \text{Intrinsic Value} + \text{Time Value} \]

Component	Call	Put
Intrinsic Value	max(0, S − X) → profit if exercised NOW	max(0, X − S) → profit if exercised NOW
Time Value	Option Price − Intrinsic Value	Option Price − Intrinsic Value
At Expiry	Time Value = 0; Price = max(0, S_T − X)	Time Value = 0; Price = max(0, X − S_T)

Moneyness Exam Focus

Status	Call (right to buy)	Put (right to sell)	Intrinsic Value
In-the-money (ITM)	S > X	S < X	Positive
At-the-money (ATM)	S = X	S = X	Zero
Out-of-the-money (OTM)	S < X	S > X	Zero (but time value still positive)

The Six Factors Affecting Option Prices Critical

Factor	↑ Factor Effect on Call	↑ Factor Effect on Put	Why
Underlying Price (S)	↑ Call Value	↓ Put Value	Higher S → call more ITM, put more OTM
Strike Price (X)	↓ Call Value	↑ Put Value	Higher X → call harder to reach, put easier to reach
Time to Expiry (T)	↑ Call Value (usually)	↑ Put Value (usually)	More time = more chance of favourable move (American always; European generally)
Volatility (σ)	↑ Call Value	↑ Put Value	Higher vol = bigger possible moves both up and down; option holder benefits from big moves, not harmed by bad moves (floor at zero)
Risk-Free Rate (r)	↑ Call Value	↓ Put Value	Higher r → PV of strike lower → call more valuable; higher r → PV of put payoff lower
Dividends/Distributions (D)	↓ Call Value	↑ Put Value	Dividend reduces stock price on ex-date → call less valuable, put more valuable

Volatility affects both calls AND puts in the SAME direction — more volatility = more valuable for BOTH. This is the most commonly tested and counterintuitive fact. The holder benefits from large moves but loses a maximum of the premium regardless of direction.

Boundary Conditions for European Options

Lower Bounds

\[ c \geq \max\!\left(0,\; S_0 - \frac{X}{(1+r_f)^T}\right) \]

\[ p \geq \max\!\left(0,\; \frac{X}{(1+r_f)^T} - S_0\right) \]

The call's lower bound is the current spot minus the PV of the strike. The put's lower bound is the PV of the strike minus the spot. Options can never have negative value (no one exercises an option at a loss).

Upper Bounds

\[ c \leq S_0 \quad \text{(a call can't be worth more than the stock itself)} \]

\[ p \leq \frac{X}{(1+r_f)^T} \quad \text{(a European put can't be worth more than PV of strike)} \]

Early Exercise of American Options

American Call on Non-Dividend Stock

Early exercise is NEVER optimal
Exercising destroys time value — better to sell the option
Therefore: American call = European call (same price)

American Put on Any Stock

Early exercise CAN be optimal for deep ITM puts
If S → 0, waiting adds little but you forego the time value of strike X
American put > European put always

Why is early exercise of an American call on a non-dividend stock never optimal? Suppose you own a call with strike $50 and the stock is at $100. You could exercise and receive $50. But the call is worth at least $50 (the lower bound), and actually more, because it has time value left. You're better off selling the call than exercising it — you collect time value by selling. Only dividends change this: if the stock is about to pay a large dividend that reduces the stock price, early exercise before the ex-date can be worth it.

🎯 Likely Exam Question

A European call on a non-dividend stock has S = €60, X = €55, T = 3 months, r = 4%, σ = 25%. The option's intrinsic value is €5. If the option trades at €7, what is the time value?

Time value = Option price − Intrinsic value = €7 − €5 = €2. The option is in-the-money (S = €60 > X = €55). The €2 time value reflects the probability of further favourable moves before expiry and the optionality that protects against a price reversal below the strike.

🎯 Likely Exam Question

If a stock's volatility increases significantly, what is the effect on the price of an at-the-money put option?

The put price increases. Higher volatility increases the probability of extreme moves in either direction. The put holder benefits from large downward moves but is protected (by paying only the premium) on upward moves. Higher volatility always benefits option holders regardless of whether the option is a call or a put.

Reading 74

Option Replication Using Put-Call Parity

One equation that connects calls, puts, stocks and bonds — and lets you create synthetic positions.

🗺 The Big Picture

Put-call parity is the cornerstone of option pricing theory. It establishes a precise no-arbitrage relationship between call prices, put prices, the underlying stock, and a risk-free bond. If this relationship is violated, you can earn a riskless profit by buying the underpriced combination and selling the overpriced one. More importantly, it means you can replicate any one instrument using the other three.

Put-Call Parity: The Formula Critical

\[ c + \frac{X}{(1+r_f)^T} = p + S_0 \]

For European options on non-dividend-paying stocks

Verbally: Call + PV(Strike) = Put + Stock. Both sides create the same payoff profile at expiry.

Proof: Why Both Sides Have the Same Payoff at Expiry

Scenario	Portfolio A: Call + PV(X) invested at r	Portfolio B: Put + Stock
S_T > X (call ITM)	Call pays S_T − X; bond pays X → total = S_T	Put expires worthless (0); stock worth S_T → total = S_T
S_T < X (put ITM)	Call expires worthless (0); bond pays X → total = X	Put pays X − S_T; stock worth S_T → total = X

Portfolio A and Portfolio B always produce the same payoff in all future states. By the law of one price, they must have the same value today. This proves put-call parity: c + PV(X) = p + S₀.

Synthetic Positions from Put-Call Parity Exam Focus

You can rearrange put-call parity to solve for any one instrument in terms of the others:

Synthetic Position	Components	Formula
Synthetic Long Call	Long put + Long stock + Borrow PV(X)	c = p + S₀ − PV(X)
Synthetic Long Put	Long call + Short stock + Lend PV(X)	p = c − S₀ + PV(X)
Synthetic Long Stock	Long call + Short put + Lend PV(X)	S₀ = c − p + PV(X)
Synthetic Risk-Free Bond	Long stock + Long put + Short call	PV(X) = p + S₀ − c
Synthetic Long Forward	Long call + Short put (same strike)	F₀ equivalent payoff = c − p

Synthetic positions are like building a chair from other furniture. You can't find the exact chair you want, so you combine two stools and a cushion to get the same thing. Put-call parity tells you exactly which pieces to combine and how many of each.

Put-Call Parity for Options on Forwards/Futures

For European options on a forward or futures contract (where the underlying is a forward price F₀):

\[ c + \frac{X}{(1+r_f)^T} = p + \frac{F_0}{(1+r_f)^T} \]

Or equivalently: c − p = PV(F₀ − X). The difference between a call and a put price equals the PV of the difference between the forward price and the strike price.

Arbitrage Using Put-Call Parity

Example: Detecting Mispricing

S = $100, X = $100, T = 1 year, r = 5%. Call trades at $10.00. What must the put trade at?

PV(X) = $100/1.05 = $95.24
Put = c − S₀ + PV(X) = $10.00 − $100 + $95.24 = $5.24
If put actually trades at $7.00 (too expensive):
Arbitrage: Sell the put, sell the stock short, buy the call, invest PV(X) = $95.24
Riskless profit = $7.00 − $5.24 = $1.76 per share (at initiation)

🎯 Likely Exam Question

A European call option (S=$80, X=$85, T=6 months, r=4%) trades at $3.50. Using put-call parity, what is the no-arbitrage price of the corresponding European put?

p = c − S₀ + PV(X) = $3.50 − $80 + $85/(1.04)^0.5 = $3.50 − $80 + $85/1.0198 = $3.50 − $80 + $83.35 = $6.85. The put should trade at $6.85. Since the call is out-of-the-money (S < X), the put is in-the-money (S < X for a put) and commands a higher price — consistent with the calculation.

Put-call parity holds exactly ONLY for European options. American options have early exercise rights, so the simple parity does not hold exactly. For American options, put-call parity gives an inequality (bounds), not an exact equality.

Reading 75

Valuing a Derivative Using a One-Period Binomial Model

The most elegant idea in derivatives: value options without knowing probabilities.

🗺 The Big Picture

The binomial model is the conceptual foundation of all options pricing. Its core insight is profound: you don't need to know what probability the stock will go up to price an option. Instead, you construct a replicating portfolio that exactly mimics the option's payoff. Since the portfolio and the option produce identical cash flows, they must have the same price — regardless of actual probabilities.

The One-Period Binomial Framework

The stock can move in only two ways over the next period:

Up: S moves to S⁺ = S × u (where u > 1)
Down: S moves to S⁻ = S × d (where 0 < d < 1)

\[ S^+ = S_0 \cdot u \qquad S^- = S_0 \cdot d \]

Risk-Neutral Probability Critical

We use a risk-neutral probability π (not the real-world probability) such that the expected return on the stock equals the risk-free rate. This is the key trick — by repricing probabilities to make all assets earn the risk-free rate, we can discount option payoffs at the risk-free rate.

\[ \pi = \frac{(1 + r_f) - d}{u - d} \qquad \text{(1 − π)} = \frac{u - (1+r_f)}{u - d} \]

π is NOT the real-world probability of the stock going up. It is a synthetic probability that makes the model internally consistent in a no-arbitrage world. The real-world probability is irrelevant to option pricing — what matters is the possible payoffs and the risk-free rate.

Option Price: The Binomial Formula Critical

\[ c = \frac{\pi \cdot c^+ + (1 - \pi) \cdot c^-}{1 + r_f} \]

\[ p = \frac{\pi \cdot p^+ + (1 - \pi) \cdot p^-}{1 + r_f} \]

Where c⁺, c⁻ are the option payoffs in the up and down states respectively

Worked Example: Pricing a Call Option

Stock: S = $100, u = 1.10, d = 0.90, r_f = 5%, X = $100 (ATM call).

Step 1: Calculate up and down stock prices.

S⁺ = $100 × 1.10 = $110
S⁻ = $100 × 0.90 = $90

Step 2: Calculate option payoffs at expiry.

c⁺ = max(0, $110 − $100) = $10
c⁻ = max(0, $90 − $100) = $0

Step 3: Calculate risk-neutral probability.

π = (1.05 − 0.90) / (1.10 − 0.90) = 0.15 / 0.20 = 0.75
1 − π = 0.25

Step 4: Calculate option price.

\[ c = \frac{0.75 \times \$10 + 0.25 \times \$0}{1.05} = \frac{\$7.50}{1.05} = \mathbf{\$7.14} \]

The Replicating Portfolio Approach (Alternative Method)

Construct a portfolio of Δ shares of stock and a bond B that exactly replicates the option payoffs:

\[ \Delta = \frac{c^+ - c^-}{S^+ - S^-} = \frac{\text{Change in option value}}{\text{Change in stock value}} \]

This Δ is the hedge ratio (option delta) — the number of shares needed to hedge one option.

Continuing the Example:

Δ = ($10 − $0) / ($110 − $90) = $10 / $20 = 0.50
Portfolio: Long 0.50 shares, short 1 call. This portfolio is riskless.
Up state: 0.50 × $110 − $10 = $55 − $10 = $45
Down state: 0.50 × $90 − $0 = $45 − $0 = $45 ✓ (same in both states)
PV of $45 = $45 / 1.05 = $42.86
Value of hedge portfolio today: 0.50 × $100 − c = $42.86 → c = $50 − $42.86 = $7.14 ✓

The hedge portfolio is riskless — it pays $45 regardless of whether the stock goes up or down. You've engineered a synthetic T-bill using a mix of the stock and the call option. Since the payoff is riskless, it must earn exactly the risk-free rate. This constraint determines the call price exactly — no probabilities needed.

No-Arbitrage Constraint on u and d

For no-arbitrage to hold, we need:

\[ d \lt (1 + r_f) \lt u \]

If the risk-free rate is outside the range [d, u], an arbitrage exists — you could earn more than the risk-free rate with zero risk by combining the stock and the riskless asset.

🎯 Likely Exam Question

A stock is priced at $50. In one period, it either rises to $60 (u = 1.20) or falls to $40 (d = 0.80). The risk-free rate is 6% per period. A put with strike $50 is written. Calculate the put's price using the binomial model.

Step 1: Put payoffs. p⁺ = max(0, $50 − $60) = $0; p⁻ = max(0, $50 − $40) = $10.
Step 2: Risk-neutral prob. π = (1.06 − 0.80)/(1.20 − 0.80) = 0.26/0.40 = 0.65; 1−π = 0.35.
Step 3: Put price. p = (0.65 × $0 + 0.35 × $10)/1.06 = $3.50/1.06 = $3.30.
Verify with delta: Δ = ($0 − $10)/($60 − $40) = −0.50 (short 0.50 shares to replicate the put).

🎯 Likely Exam Question

In the one-period binomial model, which of the following is used to determine the risk-neutral probability? A) The investor's risk aversion. B) The actual real-world probability of the up move. C) The up and down factors and the risk-free rate.

Answer: C. π = (1 + r_f − d)/(u − d). The risk-neutral probability is derived entirely from u, d, and r_f — not from any real-world probabilities or investor preferences. This is the crucial insight of risk-neutral pricing: the actual probability of an up move is irrelevant to option valuation.

Reference

Master Formula Sheet — All 10 Derivative Readings

Every formula you need, in one place.

Formula	Description	Reading
$F_0 = S_0 \cdot (1+r_f)^T$	Forward price — no income/costs	69/70
$F_0 = (S_0 - PV(\text{benefits}) + PV(\text{costs})) \cdot (1+r_f)^T$	Forward price — with carry costs/benefits	69/70
$F_0 = S_0 \cdot e^{r_f T}$	Forward price — continuous compounding	69/70
$V_t = S_t - F_0 / (1+r_f)^{T-t}$	Long forward value mid-life	70
$V_T = S_T - F_0$	Forward payoff at expiry (long)	70
$F_0 = S_0 \cdot (1+r_d)^T / (1+r_f)^T$	Forward FX rate (covered interest parity)	70
$\text{Basis} = S - F$	Futures basis (converges to 0 at expiry)	71
$FS = \frac{1-Z_N}{\sum Z_i}$	Swap rate (fixed rate at initiation)	72
$V_{\text{fixed payer}} = V_{\text{float}} - V_{\text{fixed bond}}$	Swap value during life	72
$c = \max(0, S - X)$	Call option intrinsic value	73
$p = \max(0, X - S)$	Put option intrinsic value	73
$c \geq \max(0, S_0 - X/(1+r)^T)$	Call lower bound (European)	73
$p \geq \max(0, X/(1+r)^T - S_0)$	Put lower bound (European)	73
$c + X/(1+r)^T = p + S_0$	Put-call parity (European, no dividends)	74
$\pi = \frac{(1+r_f) - d}{u - d}$	Risk-neutral probability (up)	75
$c = \frac{\pi \cdot c^+ + (1-\pi) \cdot c^-}{1+r_f}$	Call price — binomial model	75
$\Delta = \frac{c^+ - c^-}{S^+ - S^-}$	Hedge ratio / option delta (binomial)	75
$\text{FRA settlement} = \frac{(\text{FR}-\text{FRA rate}) \cdot \text{Days}/360}{1+\text{FR} \cdot \text{Days}/360} \cdot N$	FRA cash settlement payment	70

Common Exam Traps — Derivatives Critical

Trap	The Mistake	The Correct Thinking
Swap notional exchange	Assuming notional is exchanged in an interest rate swap	Notional is NEVER exchanged in IRS; it IS exchanged at start and end in currency swaps
Zero initial value	Saying forwards/futures cost something to enter	Forwards/futures have zero value at initiation (no money changes hands); options require a premium
Put-call parity scope	Applying c + PV(X) = p + S₀ to American options	Put-call parity holds exactly only for European options; gives bounds for American options
Volatility and puts	Saying higher volatility hurts puts	Higher volatility ALWAYS increases both call and put values
Risk-neutral probability	Using actual (real-world) probability in binomial model	The binomial model uses risk-neutral probabilities derived from u, d, r — not actual probabilities
Early call exercise	Saying American call = European call always	Only true for non-dividend-paying stocks. With dividends, early exercise can be optimal just before ex-date
Futures vs. forward pricing	Always assuming futures price ≠ forward price	Treat as equal for exam purposes unless specifically told there is a correlation effect
FRA settlement timing	Collecting FRA settlement at the end of the reference period	FRA settles at the BEGINNING of the reference period (discounted PV of the payment)
Basis direction	Confusing basis as always positive	Basis = Spot − Futures. In contango (futures > spot), basis is negative. Basis converges to 0.
Fixed-rate payer in IRS	Thinking the fixed payer gains when rates fall	Fixed payer gains when rates RISE (they locked in a low rate; market now demands more)

Formula	Description	Reading
\(F_0 = S_0 \cdot (1+r_f)^T\)	Forward price — no income/costs	69/70
\(F_0 = (S_0 - PV(\text{benefits}) + PV(\text{costs})) \cdot (1+r_f)^T\)	Forward price — with carry costs/benefits	69/70
\(F_0 = S_0 \cdot e^{r_f T}\)	Forward price — continuous compounding	69/70
\(V_t = S_t - F_0 / (1+r_f)^{T-t}\)	Long forward value mid-life	70
\(V_T = S_T - F_0\)	Forward payoff at expiry (long)	70
\(F_0 = S_0 \cdot (1+r_d)^T / (1+r_f)^T\)	Forward FX rate (covered interest parity)	70
\(\text{Basis} = S - F\)	Futures basis (converges to 0 at expiry)	71
\(FS = \frac{1-Z_N}{\sum Z_i}\)	Swap rate (fixed rate at initiation)	72
\(V_{\text{fixed payer}} = V_{\text{float}} - V_{\text{fixed bond}}\)	Swap value during life	72
\(c = \max(0, S - X)\)	Call option intrinsic value	73
\(p = \max(0, X - S)\)	Put option intrinsic value	73
\(c \geq \max(0, S_0 - X/(1+r)^T)\)	Call lower bound (European)	73
\(p \geq \max(0, X/(1+r)^T - S_0)\)	Put lower bound (European)	73
\(c + X/(1+r)^T = p + S_0\)	Put-call parity (European, no dividends)	74
\(\pi = \frac{(1+r_f) - d}{u - d}\)	Risk-neutral probability (up)	75
\(c = \frac{\pi \cdot c^+ + (1-\pi) \cdot c^-}{1+r_f}\)	Call price — binomial model	75
\(\Delta = \frac{c^+ - c^-}{S^+ - S^-}\)	Hedge ratio / option delta (binomial)	75
\(\text{FRA settlement} = \frac{(\text{FR}-\text{FRA rate}) \cdot \text{Days}/360}{1+\text{FR} \cdot \text{Days}/360} \cdot N\)	FRA cash settlement payment	70