Derivative

A financial security whose value is derived from an underlying asset.

• The derivative itself has no independent value
• Its payoff depends entirely on how the underlying asset performs

Common Underlyings

• Equities: individual stocks, equity indices, volatility
• Fixed Income: bonds, interest rates
• Commodities: oil, gold, agricultural products
• Currencies: FX rates
• Indices: stock, bond, or commodity indices

Analogy:
A derivative is like a shadow — it moves only because the object (underlying asset) moves.

Derivatives are used to:

• Hedge risk (reduce uncertainty)
• Speculate (take directional views)
• Arbitrage (exploit mispricing)
• Transfer risk to parties willing to bear it

Derivatives do not eliminate risk — they reallocate it.

• Derivative returns depend on:
  • Price level of underlying
  • Price changes
  • Volatility
  • Time
• Small changes in the underlying can lead to large % changes in derivative value

Analogy:
Derivatives are financial leverage in pure form — like using a crowbar instead of your hands.

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Exchange-Traded Derivatives

Key Characteristics

• Standardized contracts
• Trade on a centralized exchange
• Clearinghouse guarantees performance
• High liquidity and price transparency
• Low counterparty risk

Examples

• Futures
• Exchange-traded options

Analogy:
Exchange-traded derivatives are like buying a concert ticket from Ticketmaster — standardized, guaranteed, and regulated.

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Over-the-Counter (OTC) Derivatives

Key Characteristics

• Customized contracts
• No central trading location
• Trades negotiated directly between parties
• Dealers or market makers connect buyers and sellers
• No automatic clearinghouse guarantee

Risks

• Less transparency
• Less liquidity
• Higher counterparty risk

Examples

• Swaps
• Forward contracts
• Customized options

Analogy:
OTC derivatives are like private contracts — flexible, but you must trust the counterparty.

Counterparty Risk

Risk that the other party fails to honor contractual obligations.

• Much higher in OTC markets
• Minimal in exchange-traded markets due to clearinghouses

Why CCPs Were Created

• To reduce systemic risk
• To lower counterparty risk in OTC markets

How CCPs Work

• CCP becomes the buyer to every seller
• CCP becomes the seller to every buyer
• Requires:
  • Initial margin
  • Variation margin
• Enforces daily settlement

Analogy:
A CCP is like an escrow service — it steps in between both sides to guarantee the deal.

Hedgers

• Use derivatives to reduce existing risk
• Example: airline hedging fuel prices

Speculators

• Take views on price direction or volatility
• Accept risk for potential return

Arbitrageurs

• Exploit pricing inconsistencies
• Help enforce law of one price

Speculators and arbitrageurs provide liquidity, benefiting hedgers.

Risk Type	Description
Market Risk	Underlying price moves unfavorably
Counterparty Risk	Other party defaults
Liquidity Risk	Inability to exit position
Operational Risk	Failures in systems/processes
Legal Risk	Contract enforceability issues

Feature	Exchange-Traded	OTC
Contract Terms	Standardized	Customized
Liquidity	High	Lower
Transparency	High	Low
Counterparty Risk	Low	High
Flexibility	Low	High

Core Concepts:

Concept	Meaning
Derivative	Value depends on an underlying asset
Underlying	Asset determining derivative payoff
Exchange-Traded	Standardized, cleared, transparent
OTC	Customized, bilateral, higher risk
CCP	Central guarantor of trades

• Derivatives derive value from an underlying asset, not standalone worth
• Exchange-traded derivatives minimize counterparty risk via clearinghouses
• OTC derivatives offer flexibility but introduce counterparty and liquidity risk
• Central counterparties were introduced to reduce systemic risk
• Derivatives transfer risk, they do not eliminate it
• Market participants include hedgers, speculators, and arbitrageurs

Firm Commitment

A contract in which both counterparties are obligated to complete the transaction at maturity.

Key idea:
There is no choice at expiration — both sides must perform.

Analogy:
A firm commitment is like agreeing today to buy a house in 3 months — backing out is not an option.

Definition

An agreement to buy or sell an asset in the future at a price agreed upon today (the forward price).

Positions

• Buyer: Long position
• Seller: Short position

Settlement Date Outcomes

• If spot price > forward price:
  • Long gains
  • Short loses
• If spot price < forward price:
  • Short gains
  • Long loses

Settlement Methods

• Physical (actual) delivery
• Cash settlement (difference between spot and forward price)

Market Features

• Customized contracts
• Traded OTC
• No clearinghouse
• Higher counterparty risk

Analogy:
A forward is like a custom-tailored suit — fits perfectly, but you can’t easily resell it.

Definition

Similar to forwards, but with key differences:

Key Features

• Standardized contracts
• Exchange-traded
• Regulated
• Clearinghouse guarantees performance
• Daily settlement (mark-to-market)

Daily Settlement (Mark-to-Market)

• Gains/losses are realized daily
• Reduces accumulation of losses
• Limits counterparty risk

Margin System

Initial Margin

• Good-faith deposit to enter a futures position

Maintenance Margin

• Minimum account balance required
• If balance < maintenance margin → margin call
• Investor must deposit funds to return to initial margin

Analogy:
Margin is a security deposit — if it gets too low, you must top it up.

How Futures Avoid Defaults

Price Limits

• Restrict how much price can move in a day

Circuit Breakers

• Temporarily halt trading if extreme price moves occur

CFA Insight:
These mechanisms reduce panic and systemic risk.

Definition

A series of forward contracts where two parties exchange cash flows over time.

Key Characteristics

• No initial exchange of principal
• Customized OTC contracts
• Not traded on exchanges

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Plain Vanilla Interest Rate Swap

• One party pays fixed interest
• Other party pays floating interest
• Payments netted at each settlement date

Analogy:
A swap is like agreeing to trade lunch meals every day — you swap fixed for variable.

Definition

Contracts whose payoff depends on whether a future event occurs.

• Buyer has the right, not obligation
• Seller has the obligation

Examples

• Options
• Credit derivatives

Analogy:
Contingent claims are like insurance — you get paid only if something happens.

Option Basics

• Buyer (holder) pays a premium
• Buyer gets a right, not obligation
• Seller (writer) receives premium and takes obligation

Call Options

Call Option

Gives the holder the right to buy an asset at a strike price before expiration.

Party	Position
Buyer	Long call
Seller	Short call

Payoff Rules

• Exercised if stock price > strike price
• Not exercised if stock price ≤ strike price
• Break-even price:

$$\text{Breakeven}=\text{Strike Price}+\text{Premium}$$

Risk/Return

• Long call: unlimited upside, max loss = premium
• Short call: max gain = premium, unlimited loss

Analogy:
A call option is a down payment on upside potential.

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Put Options

Put Option

Gives the holder the right to sell an asset at a strike price.

Party	Position
Buyer	Long put
Seller	Short put

Payoff Rules

• Exercised if stock price < strike price
• Not exercised if stock price ≥ strike price

Risk/Return

• Long put: max gain when stock → 0; max loss = premium
• Short put: max gain = premium; max loss when stock → 0

Analogy:
A put option is price insurance — protection against falling prices.

Definition

Instruments that compensate bondholders if a credit event occurs.

Credit Events Include

• Borrower default
• Credit rating downgrade

Analogy:
Credit derivatives are insurance policies on bonds.

Forward vs Futures

Feature	Forward	Futures
Trading Venue	OTC	Exchange
Standardization	Custom	Standard
Clearinghouse	No	Yes
Settlement	At maturity	Daily
Counterparty Risk	High	Low

Options Payoff Summary

Position	Max Gain	Max Loss
Long Call	Unlimited	Premium
Short Call	Premium	Unlimited
Long Put	Strike − Premium	Premium
Short Put	Premium	Strike

• Firm commitments create obligations for both parties
• Forwards are customized but carry high counterparty risk
• Futures reduce default risk via daily settlement & margining
• Swaps are series of forward contracts with netted payments
• Contingent claims give rights to buyers and obligations to sellers
• Options have asymmetric payoffs
• Long options have limited downside; short options have potentially large losses
• Credit derivatives protect against credit events

Derivatives improve how financial markets function, price risk, and allocate capital.

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Greater Market Efficiency

• Allow faster incorporation of information into prices
• Facilitate arbitrage → enforce law of one price

Analogy:
Derivatives act like express lanes in markets — information travels faster.

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Hedging Risk

• Transfer unwanted risk to parties willing to accept it
• Stabilize cash flows and portfolio values

Example:

• Airline hedges fuel costs with oil futures
• Bank hedges interest rate exposure with swaps

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Information Gathering & Price Discovery

• Derivative prices embed market expectations
• Used to infer:
  • Expected future prices
  • Expected volatility (implied volatility)

Options markets often reveal volatility expectations before spot markets.

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Risk Allocation & Transfer

• Risk flows to investors most willing and able to bear it
• Improves overall economic efficiency

Analogy:
Derivatives are like insurance marketplaces — risk goes to specialists.

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Operational Advantages

Lower Trading Costs

• Fewer transactions vs cash markets

Higher Liquidity

• Easier to enter/exit positions

Lower Cash Requirements

• Margin vs full notional value

Ability to Short

• Easier than shorting physical assets

Analogy:
Derivatives are financial power tools — less effort, more impact.

Derivatives magnify both gains and losses.

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Speculation & Leverage Risk

• Small underlying moves → large derivative value changes
• Can amplify losses rapidly

Exam warning:
Leverage is the most commonly tested derivative risk.

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Lack of Transparency

• Especially in OTC markets
• Hard to observe prices, exposures, or risks
• Can obscure true financial positions

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Basis Risk

Occurs when the derivative does not perfectly offset the underlying exposure.

Causes:

• Differences in:
  • Underlying asset
  • Maturity
  • Location
  • Quality

Analogy:
Basis risk is like buying the wrong size umbrella — partial protection only.

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Liquidity Risk

• Inability to close positions without significant price impact
• More severe during market stress

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Timing (Cash Flow) Risk

• Derivative cash flows occur at different times than underlying exposures
• Creates mismatch risk

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Counterparty Risk

• Other party fails to perform
• More prominent in OTC derivatives

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Systemic Risk

• Failure of one major participant can affect entire system
• Interconnected derivative positions amplify shocks

CFA Insight:
This risk motivated the creation of central counterparties (CCPs).

Issuers use derivatives mainly for risk management, not speculation.

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Cash Flow Hedge

Purpose:
Offset variability in future cash flows.

Examples:

• Floating-rate debt → interest rate swap
• Forecasted commodity purchases → commodity futures

Analogy:
Cash flow hedges smooth monthly paychecks.

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Fair Value Hedge

Purpose:
Offset changes in the market value of assets or liabilities.

Examples:

• Fixed-rate bond value hedged with interest rate swap
• Inventory value hedged with commodity futures

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Net Investment Hedge

Purpose:
Hedge foreign exchange risk from foreign subsidiaries.

Instruments:

• Currency swaps
• Currency forwards

Analogy:
Net investment hedges protect overseas “branches” from currency storms.

Replicating Cash Market Strategies

• Use derivatives to mimic:
  • Equity exposure
  • Bond exposure
  • Commodity exposure

Advantage:
Lower cost and faster execution.

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Hedging Portfolio Risk

• Protect against:
  • Market downturns
  • Interest rate changes
  • Currency movements

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Modifying or Adding Exposures

• Increase/decrease:
  • Duration
  • Equity beta
  • Volatility exposure

Analogy:
Derivatives are portfolio “dials” — turn risk up or down instantly.

Benefits vs Risks

Benefits	Risks
Efficient pricing	Leverage risk
Risk transfer	Counterparty risk
Liquidity	Liquidity risk
Low cost	Basis risk
Short selling	Systemic risk

Issuer Hedge Types

Hedge Type	What It Protects	Typical Instruments
Cash Flow Hedge	Future cash flows	Swaps, futures
Fair Value Hedge	Market value	Swaps, futures
Net Investment Hedge	FX exposure	Currency swaps/forwards

• Derivatives enhance market efficiency and price discovery
• They allow precise risk transfer and hedging
• Leverage magnifies gains and losses
• Basis risk means hedges are rarely perfect
• OTC derivatives introduce transparency and counterparty risk
• Issuers hedge cash flows, values, and FX exposure
• Investors use derivatives to hedge, replicate, and modify exposures

What Is Arbitrage?

Arbitrage is earning a riskless profit from a mispricing by:

• Buying an asset at a lower price
• Selling the same (or economically identical) asset at a higher price

True arbitrage has no risk, no net investment, and non-negative profit.

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Law of One Price

If two assets or portfolios have identical cash flows, they must have the same price.

• If prices differ → arbitrage opportunity exists
• Markets tend to eliminate arbitrage quickly

Analogy:
Two identical phones shouldn’t sell for different prices — if they do, resellers step in.

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Arbitrage via Mispricing (Discounting Error)

If:

• Discounted future value ≠ present value

Then:

• Buy the cheaper version
• Sell the overpriced version
• Lock in profit

• Combine a derivative + underlying asset
• Structure eliminates uncertainty
• Produces a certain payoff at maturity

Result:
Return on this hedged portfolio must equal the risk-free rate.

• If return > risk-free rate → arbitrage exists

Analogy:
A hedged portfolio is like locking in a fixed paycheck regardless of market weather.

• Derivatives are priced so arbitrage is impossible
• Price = value that makes expected return = risk-free rate
• This is called risk-neutral pricing

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Forward Pricing (No-Arbitrage Condition)

$$S_0=\frac{F_0(T)}{(1+r_f{)}^T}$$

• Spot price today equals the discounted forward price
• If violated → arbitrage opportunity

Replication Principle

A derivative can be replicated using:

• The underlying asset
• Borrowing or lending at the risk-free rate

If two strategies produce the same payoff at time T, they must have the same price today.

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Replication Logic

$$\text{Asset}+\text{Derivative}=\text{Risk-free payoff at }T$$

• Discounting this payoff at the risk-free rate gives today’s value
• Equivalent to holding a risk-free asset

Analogy:
Replication is like rebuilding a Lego model two different ways — if the final model is identical, the cost must be the same.

What Is Cost of Carry?

The net cost (or benefit) of holding an asset until maturity.$$\text{Cost of Carry}=\text{Benefits}-\text{Costs}$$

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Benefits of Holding an Asset

• Dividends (equities)
• Interest payments
• Convenience yield (physical commodities)

Costs of Holding an Asset

• Storage
• Insurance
• Financing costs (implicit via interest rates)

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Spot Price with Cost of Carry

$$S_0=\frac{F_0(T)}{(1+r_f{)}^T}+PV(\text{Benefits})-PV(\text{Costs})$$

• Forward price adjusted for:
  • Holding benefits
  • Holding costs

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Interpretation

• High benefits → higher spot price
• High costs → lower spot price
• Forward price reflects net cost of carry

Analogy:
Owning a house has benefits (rent) and costs (maintenance) — net effect determines value.

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Arbitrage & Cost of Carry in FX Markets

If:

• You can borrow at a lower domestic risk-free rate
• Lend at a higher foreign risk-free rate

Then:

• FX forward prices adjust to prevent arbitrage

Higher future exchange rates offset interest rate differentials

• Arbitrage forces prices into no-arbitrage equilibrium
• Derivative pricing is based on replication, not expectations
• Cost of carry explains why forward prices differ from spot prices
• If risk-free returns differ → prices adjust to eliminate profit opportunities

Arbitrage Concepts

Concept	Meaning
Arbitrage	Riskless profit from mispricing
Law of One Price	Same cash flows → same price
Hedged Portfolio	Risk eliminated
Risk-Neutral Pricing	Return = risk-free rate

Cost of Carry Components

Component	Examples
Benefits	Dividends, interest, convenience yield
Costs	Storage, insurance
Net Carry	Benefits − Costs

• Arbitrage exists when identical cash flows have different prices
• Law of one price underpins all derivative pricing
• Hedged portfolios must earn the risk-free rate
• Risk-neutral pricing ensures no-arbitrage
• Replication ties derivative prices to underlying assets
• Cost of carry explains forward vs spot price differences
• FX forward prices adjust to eliminate interest rate arbitrage

Price of a Forward

• The contracted forward price agreed at initiation
• Never changes over the life of the contract

Analogy:
The forward price is like the sticker price written on a receipt — it doesn’t change.

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Value of a Forward

• The current economic worth of the contract
• Changes over time as spot prices and interest rates move
• Price = fixed number in the contract
• Value = what the contract is worth today

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Value of a Forward at Time t

$$V_t=S_t-P{V}_t\text{\hspace{0.17em}⁣}[F_0(T)]$$

• Difference between:
  • Current spot price
  • Present value of the contracted forward price

Present Value of Forward Price

$$P{V}_t\text{\hspace{0.17em}⁣}[F_0(T)]=\frac{F_0(T)}{(1+r_f{)}^{(T-t)}}$$

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At Initiation

$$V_0=0$$

• No arbitrage → neither party pays anything upfront

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At Expiration

$$V_T=S_T-F_0(T)$$

• No discounting needed at maturity

Analogy:
At expiration, the forward is settled like paying today’s price minus the price you locked in earlier.

If the underlying provides benefits (e.g., dividends) or costs (e.g., storage):$$V_t=S_t+PV(\text{Benefits})-PV(\text{Costs})-P{V}_t\text{\hspace{0.17em}⁣}[F_0(T)]$$

Interpretation:

• Benefits increase forward value
• Costs reduce forward value

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Spot Rates

• Interest rates for immediate borrowing/lending to a specific maturity

Forward Rates

• Implied future interest rates
• Derived from current spot rates

Forward rates are implied by today’s yield curve, not forecasts.

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Discount Factors

Definition

A discount factor is the present value of one unit of currency received in the future.
$$D{F}_t=\frac{1}{(1+s_t{)}^t}$$

• Where $s_t$st​ = spot rate to maturity $t$t

Uses

• Discount individual cash flows
• Build present values
• Derive forward rates
• Value derivatives with varying maturities

Analogy:
Discount factors are time-travel converters — they translate future money into today’s dollars.

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Forward Rates from Spot Rates (Intuition)

• If spot rates are known:
  • You can bootstrap discount factors
  • Then derive forward rates
• Ensures no-arbitrage consistency across maturities

What Is an FRA?

A derivative whose underlying is a future interest rate.

• Most commonly based on LIBOR
• Used to lock in a borrowing or lending rate in the future
• Parties exchange:
  • Fixed rate (agreed today)
  • Floating rate (market rate at settlement)

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Purpose of an FRA

• Hedge interest rate risk
• Protect against rate increases (borrowers) or rate decreases (lenders)

Analogy:
An FRA is like setting a thermostat today for a room you’ll use later — you lock in comfort regardless of weather.

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FRA Settlement & Net Payment

• Only net cash flow is exchanged
• No exchange of principal

Net Payment Formula

$$\text{Net Payment}=({\text{MRR}}_{b-a}-\text{Fixed Rate})\times \text{Notional Principal}\times \text{Period}$$

Where:

• ${\text{MRR}}_{b-a}$ = market reference rate for the period
• Period = year fraction

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Who Pays Whom?

• If market rate > fixed rate → fixed-rate receiver gets paid
• If market rate < fixed rate → fixed-rate payer gets paid

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Forward Contract: Price vs Value

Concept	Meaning
Forward Price	Fixed contract price
Forward Value	Current economic worth
Value at Initiation	0
Value at Expiration	ST – F0(T)

FRA Overview

Feature	Description
Underlying	Future interest rate (LIBOR)
Purpose	Hedge interest rate risk
Cash Exchange	Net settlement only
Risk Hedged	Interest rate movements

• Forward price is constant, forward value fluctuates
• At initiation, no-arbitrage → forward value = 0
• Forward value equals spot minus discounted forward price
• Costs and benefits must be included in valuation
• Discount factors convert future cash flows into present value
• Forward rates are implied by spot rates
• FRAs lock in future borrowing/lending rates and settle in cash

Although futures and forwards have similar payoffs at maturity, their pricing can differ before maturity due to daily settlement and interest rate effects.

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Forward Contracts (Recap)

• Customized / tailored
• Traded OTC
• No clearinghouse
• No daily settlement
• Gains/losses realized only at maturity
• Higher counterparty risk

Analogy:
A forward is like agreeing to settle the bill at the end of the month — no cash moves until the end.

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Futures Contracts (Recap)

• Standardized
• Exchange-traded
• Regulated
• Guaranteed by a clearinghouse
• Daily settlement (mark-to-market)
• Gains/losses credited or debited every day

Analogy:
A futures contract is like paying or getting paid daily as prices change.

Daily Settlement (Mark-to-Market)

• Futures positions are revalued each trading day
• Daily gains are:
  • Added to margin account
  • Available for withdrawal
• Daily losses:
  • Subtracted from margin account
  • May trigger margin calls

Because cash flows occur daily, the timing of gains and losses matters.

Key Relationship

Futures prices are positively correlated with interest rates

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Why This Happens

• When interest rates rise:
  • Futures prices tend to rise
  • Long futures positions receive daily gains
  • Gains can be reinvested at higher rates
• This makes futures more attractive than forwards in rising-rate environments

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Opposite Case (Intuition)

• When interest rates fall:
  • Futures prices tend to fall
  • Long futures holders receive losses earlier
  • Losses are financed at lower rates
• In this case, forwards may be relatively more attractive

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Analogy:
Futures are like getting paid early — if interest rates are high, early cash is more valuable.

Because of daily settlement:

• Futures prices may be:
  • Higher than forward prices when interest rates and futures prices are positively correlated
  • Lower than forward prices when they are negatively correlated

This distinction matters only when interest rates are stochastic (changing).

• At initiation:
  • Futures value = 0
• After initiation:
  • Value realized daily, not accumulated
• At maturity:
  • Futures and forwards converge to the same payoff

Futures vs Forwards

Feature	Forwards	Futures
Trading Venue	OTC	Exchange
Standardization	Customized	Standardized
Clearinghouse	No	Yes
Daily Settlement	No	Yes
Counterparty Risk	High	Low
Cash Flow Timing	At maturity	Daily

Interest Rates & Pricing

Scenario	More Attractive
Rising interest rates	Futures
Falling interest rates	Forwards
Stable rates	Roughly equal

• Futures and forwards differ due to daily settlement
• Futures prices are positively correlated with interest rates
• Rising rates favor futures because gains are reinvested sooner
• Timing of cash flows can cause futures prices ≠ forward prices
• At maturity, futures and forwards converge
• Clearinghouses reduce counterparty risk in futures markets

A swap is a derivative in which two parties agree to exchange a series of cash flows on periodic settlement dates over a specified time period.

• Payments are based on:
  • A notional principal (never exchanged)
  • A fixed rate, floating rate, or other reference

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Plain Vanilla Interest Rate Swap (Most Testable)

• One party pays fixed
• One party pays floating (market reference rate, e.g., LIBOR)
• Only the net payment is exchanged at each settlement date

Analogy:
A swap is like two roommates splitting rent where one pays a fixed amount every month and the other pays whatever the variable utilities bill turns out to be — only the difference changes hands.

• No principal exchanged
• Net settlement only
• No cash exchanged at initiation
• Value = 0 at initiation
• Exposed to counterparty risk
• OTC instruments (not exchange-traded)

Swaps and forwards share several important features:

• Lock in a fixed rate
• Net payment at settlement
• Symmetric payoff profiles
• No upfront payment
• Counterparty risk
• Zero value at initiatio

A swap is economically equivalent to a bundle of forward contracts.

Core Idea

A swap can be decomposed into a series of FRAs, one for each settlement period.

• Each FRA:
  • Covers a specific future period
  • Has its own forward rate
• The swap uses one single fixed rate applied across all periods

Analogy:
A swap is like buying a season pass instead of individual tickets — same total exposure, simplified pricing.

Swap Price = Fixed Swap Rate

At initiation, the swap rate is set so that:$$PV(\text{Fixed Leg})=PV(\text{Floating Leg})$$

This ensures:$${\text{Swap Value}}_0=0$$

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Two Ways to Price a Swap

Method 1: Using an n-Year Swap Rate

• Apply a single fixed rate over the entire swap life
• Discount fixed payments using spot rates

Method 2: Using a Series of FRAs

• Determine forward rates for each period
• Discount each expected cash flow using appropriate spot rates
• Sum present values

Once market rates change, the swap acquires positive or negative value.

Settlement Value Formula

$$\text{Settlement Value}=(\text{MRR}-\text{Fixed Swap Rate})\times \text{Notional Principal}\times \text{Period}$$

Where:

• MRR = Market Reference Rate for the period

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Who Pays Whom?

• If MRR > fixed rate → fixed-rate receiver gains
• If MRR < fixed rate → fixed-rate payer gains

Analogy:
If you locked in a cheap fixed rate and market rates rise, you’re “winning” every period.

• Fixed-rate side:
  • Benefits when market rates move away from fixed rate
• Floating-rate side:
  • Always resets to market → PV ≈ par at reset dates
• Value changes driven primarily by interest rate movements

Swap Basics

Feature	Description
Instrument Type	OTC derivative
Principal	Not exchanged
Settlement	Net cash flow
Value at Initiation	0
Risk	Counterparty risk

Swaps vs Forwards

Feature	Swaps	Forwards
Structure	Series of payments	Single payment
Rate Locked	Fixed	Fixed
Value at Initiation	0	0
Counterparty Risk	Yes	Yes

• A swap exchanges streams of cash flows, not principal
• Plain vanilla swaps exchange fixed for floating
• Swaps have zero value at initiation
• Pricing sets PV fixed = PV floating
• Swaps can be replicated with a series of FRAs
• Swap value changes as market rates change
• Only net payments are exchanged

Call Option

• Gives the holder the right (not obligation) to buy an underlying asset at the strike price (X)

Exercise Rule:

• Exercised when:$$S>X$$

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Put Option

• Gives the holder the right to sell the underlying asset at the strike price (X)

Exercise Rule:

• Exercised when:$$S<X$$

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Moneyness of Options (Highly Testable)

Call Option

Status	Condition
In the Money (ITM)	$S>X$S>X
At the Money (ATM)	$S=X$S=X
Out of the Money (OTM)	$S<X$S<X

Put Option

Status	Condition
In the Money (ITM)	$S<X$S<X
At the Money (ATM)	$S=X$S=X
Out of the Money (OTM)	$S>X$S>X

Analogy:
Think of moneyness like a coupon — if you can buy cheaper or sell higher right now, it’s valuable.

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Intrinsic Value (IV)

Definition:
Profit obtained if the option is exercised immediately

Formulas

• Call:$$IV=\max (0,S-X)$$
• Put:$$IV=\max (0,X-S)$$

Key Rule:

• IV = 0 when option is ATM or OTM

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Time Value & Option Premium

Option Value Decomposition

$$\text{Option Premium}=\text{Intrinsic Value}+\text{Time Value}$$

• Time Value reflects uncertainty before expiration
• At expiration:$$\text{Time Value}=0$$

Analogy:
Time value is like paying extra for flexibility — more time means more chances to win.

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Present Value of Strike Price

If exercise occurs at maturity:$$PV(X)=\frac{X}{(1+r{)}^T}$$

This is used when calculating lower bounds.

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European vs American Options

Feature	American	European
Exercise Timing	Anytime	Only at maturity
Flexibility	Higher	Lower
Value	≥ European	≤ American

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Upper and Lower Bounds of Option Prices

Call Option Bounds

• Upper Bound:$$C\le S$$
• Lower Bound:$$C\ge \max (0,S-\frac{X}{(1+r{)}^T})$$

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Put Option Bounds

• Upper Bound:$$P\le X$$
• Lower Bound:$$P\ge \max (0,\frac{X}{(1+r{)}^T}-S)$$

Options can never be worth more than their maximum possible payoff.

Price of Underlying Asset (S)

• ↑ Stock Price:
  • Call → more valuable
  • Put → less valuable

Analogy:
Calls love rallies, puts love crashes.

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Strike Price (X)

• Higher X:
  • Call → less valuable
  • Put → more valuable

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Risk-Free Rate (r)

• ↑ Interest Rates:
  • Call → more valuable
  • Put → less valuable

Reason:
Paying later (PV of X falls) benefits calls.

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Volatility (σ)

• ↑ Volatility:
  • Call ↑
  • Put ↑

Why?
More volatility = higher chance of expiring deep ITM

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Stock Benefits and Costs of Carry

• Call holders:
  • ❌ Do not receive dividends
  • ❌ Do not pay storage or financing costs
• Put holders:
  • Benefit from dividends reducing stock price

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Inverse Relationships (Exam Traps)

Call Value Decreases When:

• Strike price ↑
• Stock benefits ↑

Put Value Decreases When:

• Stock price ↑
• Risk-free rate ↑
• Cost of carry ↑

Option Payoff Summary

Option	Profits When
Call	( S > X )
Put	( S < X )

Factors Affecting Option Prices

Factor	Call	Put
Stock Price ↑	↑	↓
Strike Price ↑	↓	↑
Volatility ↑	↑	↑
Risk-Free Rate ↑	↑	↓

• Option value = intrinsic + time
• Time value goes to zero at expiration
• Calls benefit from higher prices and rates
• Puts benefit from lower prices
• Volatility increases all option values
• Option prices are bounded — arbitrage prevents violations
• American options ≥ European options

Put-Call Parity defines a no-arbitrage relationship between European calls, European puts, the underlying asset, and a risk-free bond.

If parity does not hold → arbitrage opportunity exists

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Standard Put-Call Parity Formula

$$C+\frac{X}{(1+r{)}^T}=P+S$$

Where:

• $C$ = Call price
• $P$ = Put price
• $S$ = Spot price of underlying
• $X$ = Strike price
• $r$ = Risk-free rate
• $T$ = Time to maturity

Call and put must have the same strike price and maturity.

Left Side (Fiduciary Call)

$$C+\frac{X}{(1+r{)}^T}$$

• Buy a call
• Invest PV of strike price in a risk-free bond
• Guarantees ability to buy the asset at maturity

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Right Side (Protective Put)

$$S+P$$

• Buy the stock
• Buy a put for downside protection
• Guarantees a minimum payoff equal to the strike price

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Key Insight

Both portfolios produce identical payoffs at maturity, so they must cost the same today.

Analogy:
Two different roads leading to the same destination must cost the same taxi fare — otherwise arbitrageurs step in.

Definition

$$\text{Protective Put}=S+P$$

• Long stock
• Long put

Payoff Behavior

• Upside from stock appreciation
• Downside limited to strike price

Analogy:
Owning a house with insurance — upside if prices rise, floor if disaster strikes.

Definition

$$\text{Fiduciary Call}=C+\frac{X}{(1+r{)}^T}$$

• Long call
• Risk-free bond funding the strike price

Payoff Behavior

• Same payoff as protective put
• Less capital tied up initially

Fiduciary call and protective put are economically equivalent.

Put-call parity allows replication of positions using other instruments.

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Synthetic Call

Rearrange parity:$$C=S+P-\frac{X}{(1+r{)}^T}$$

Interpretation:

• Long stock
• Long put
• Short risk-free bond

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Synthetic Put

$$P=C+\frac{X}{(1+r{)}^T}-S$$

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Synthetic Stock

$$S=C+\frac{X}{(1+r{)}^T}-P$$

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Synthetic Risk-Free Bond

$$\frac{X}{(1+r{)}^T}=S+P-C$$

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Negative sign = short position

Put-call parity can also be expressed using forwards:$$C-P=\frac{F_0}{(1+r{)}^T}$$

Where:

• $F_0$= Forward price of the underlying

Meaning:
The difference between call and put prices reflects the present value of the forward contract.

Parity concepts extend beyond options:

• Equity = Call option on firm assets
• Debt = Risk-free bond − Put option

This framework underlies structural credit risk models

• Using parity for American options ❌ (parity strictly applies to European options)
• Mismatched strikes or maturities ❌
• Forgetting to discount the strike price ❌
• Misinterpreting negative signs (short positions) ❌

Put-Call Parity Components

Portfolio	Composition
Protective Put	( S + P )
Fiduciary Call	( C + PV(X) )
Synthetic Call	( S + P – PV(X) )

Synthetic Position Reference

Want Exposure To	Use
Call	Stock + Put − Bond
Put	Call + Bond − Stock
Stock	Call + Bond − Put
Bond	Stock + Put − Call

• Put-call parity enforces no-arbitrage pricing
• Protective put = fiduciary call
• Synthetic positions replicate payoffs exactly
• Strike prices and maturities must match
• Negative signs indicate short positions
• Parity works strictly for European options
• Violations imply arbitrage opportunities

The one-period binomial model values a derivative by assuming the underlying asset price can move to one of two possible values at expiration:

• Up state
• Down state

This framework enforces no-arbitrage pricing and introduces risk-neutral valuation, which is foundational for later option-pricing models.

Analogy:
Think of tomorrow’s weather forecast: either it rains or it doesn’t. The model prices today based on both outcomes.

At time 0:$$S_0$$

At time 1:$$S_u=S_0\times u\text{(up move)}$$$$S_d=S_0\times d\text{(down move)}$$

Where:

• $u$u = up-factor
• $d$d = down-factor

Step 1: Value the Option at Expiration

For a call option:$$C_u=\max (0,S_u-X)$$ $$C_d=\max (0,S_d-X)$$

This gives the option’s payoff in each state.

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Step 2: Compute the Hedge Ratio (Δ)

$$\text{Hedge Ratio (HR)}=\frac{C_u-C_d}{S_u-S_d}$$

Interpretation:

• Number of shares needed per option to eliminate risk
• Also known as delta (Δ)

Analogy:
It’s the amount of stock needed to “balance” the option so gains in one cancel losses in the other.

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Step 3: Construct the Risk-Free Portfolio

• Long HR shares of stock
• Short 1 call option

Portfolio value today:$$V_0=HR\cdot S_0-C_0$$

Portfolio value tomorrow (both states):$$V_u=HR\cdot S_u-C_u$$$$V_d=HR\cdot S_d-C_d$$

Because risk is eliminated:$$V_u=V_d$$

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Step 4: Discount at the Risk-Free Rate

Since the portfolio is risk-free, it must earn the risk-free rate:$$V_0=\frac{V_1}{1+R_f}$$

Solve for the option price $C_0$.

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Step 5: No-Arbitrage Condition

The discounted future value must equal today’s value

If this condition fails → arbitrage opportunity exists

If probabilities of up and down moves are not given, use risk-neutral probabilities.

Risk-Neutral Probability of Up Move

$$q=\frac{(1+R_f)-d}{u-d}$$

Down Move Probability

$$1-q$$

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Risk-Neutral Pricing Formula

$$C_0=\frac{q{C}_u+(1-q)C_d}{1+R_f}$$

These are not real probabilities — they are mathematical tools to enforce no-arbitrage.

Analogy:
Risk-neutral probabilities are like adjusting the odds so the “casino” (market) makes no free money.

Key Intuition

• Investors are assumed to be risk-neutral
• Expected returns = risk-free rate
• Risk preferences are absorbed into prices, not probabilities
• This logic scales to multi-period trees and Black-Scholes

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Common CFA Exam Traps

• Forgetting to compute payoffs at expiration first
• Mixing real probabilities with risk-neutral probabilities
• Not discounting at the risk-free rate
• Misinterpreting hedge ratio sign
• Forgetting that arbitrage-free portfolios must earn $R_f$Rf​

Binomial Valuation Steps

Step	Action
1	Compute option payoffs
2	Calculate hedge ratio
3	Build risk-free portfolio
4	Discount at ( R_f )
5	Enforce no-arbitrage

Key Variables

Symbol	Meaning
( u )	Up-factor
( d )	Down-factor
( q )	Risk-neutral probability
( HR )	Hedge ratio (delta)
( Rf)	Risk-free rate

• Binomial models assume two future price states
• Option value comes from replication and no-arbitrage
• Hedge ratio eliminates risk
• Risk-free portfolios earn the risk-free rate
• Risk-neutral probabilities simplify valuation
• This is the foundation for all modern option pricing