AFM 322: Derivative Securities
Estimated study time: 33 minutes
Table of contents
Sources and References
Primary textbook — Hull, J. C. Options, Futures, and Other Derivatives, 11th Edition. Pearson, 2022.
Supplementary — McDonald, R. L. Derivatives Markets, 3rd Edition. Pearson, 2013. Berk, J., DeMarzo, P., and Stangeland, D. Corporate Finance, Sixth Canadian Edition (selected chapters on corporate risk management).
Online resources — CME Group educational resources; CBOE Options Institute; Bank for International Settlements (BIS) derivatives statistics; MIT OCW 18.S096 Topics in Mathematics with Applications in Finance.
Chapter 1: Introduction to Derivatives
What Is a Derivative?
Derivatives serve three primary functions:
- Hedging: Reducing exposure to an existing risk (e.g., an airline hedging jet fuel costs using commodity futures).
- Speculation: Expressing a directional view on price movements with leverage.
- Arbitrage: Exploiting price discrepancies between related instruments to earn risk-free profits.
The derivative market is enormous. As of 2023, the notional value of outstanding over-the-counter (OTC) derivatives exceeds $700 trillion globally.
Exchange-Traded vs. OTC Derivatives
| Feature | Exchange-Traded | OTC |
|---|---|---|
| Standardization | Highly standardized | Fully customizable |
| Counterparty risk | Cleared by central counterparty | Bilateral (or CCP-cleared post-2010) |
| Transparency | Public price discovery | Limited transparency |
| Liquidity | Generally higher | Varies widely |
| Examples | Futures, listed options | Forwards, swaps, exotic options |
Post-2008, the Dodd-Frank Act (US) and EMIR (Europe) mandated central clearing and reporting for most standardized OTC derivatives.
Chapter 2: Futures Markets
Futures Contracts
A futures contract is a standardized legal agreement to buy or sell a specified quantity of an asset at a specified price on a future date. Key features:
- Standardization: Contract size, delivery dates, and delivery specifications are fixed by the exchange.
- Margin system: Both buyer and seller post an initial margin (a performance bond). Daily marking-to-market credits or debits variation margin to each party’s account based on price movements.
- Daily settlement (mark-to-market): Gains and losses are realized daily, not at contract expiry.
- Delivery or cash settlement: Most futures are closed before delivery; index futures are cash-settled.
Cost of carry model (futures pricing): For a non-dividend-paying asset:
\[ F_0 = S_0 e^{rT} \]For a dividend-paying asset with continuous dividend yield \(q\):
\[ F_0 = S_0 e^{(r-q)T} \]For a commodity with storage costs \(u\) and convenience yield \(y\):
\[ F_0 = S_0 e^{(r+u-y)T} \]If the futures traded above this level, arbitrageurs would buy gold spot, short futures, and earn a risk-free profit. If it traded below, they would short gold and buy futures.
Chapter 3: Hedging with Futures
Cross-Hedging and the Optimal Hedge Ratio
When the asset being hedged is not identical to the futures contract’s underlying (e.g., a jet fuel hedge using crude oil futures), the hedge is imperfect — a cross-hedge. The optimal hedge ratio minimizes the variance of the hedged position:
\[ h^* = \rho \times \frac{\sigma_S}{\sigma_F} \]where \(\rho\) is the correlation between spot price changes \(\Delta S\) and futures price changes \(\Delta F\), and \(\sigma_S\), \(\sigma_F\) are the respective standard deviations.
The optimal number of contracts is:
\[ N^* = h^* \times \frac{\text{Portfolio Value}}{F \times \text{Contract Size}} \]Effectiveness of a Hedge
Hedge effectiveness, measured as the proportion of variance reduced:
\[ \text{Effectiveness} = \rho^2 \]A perfect hedge (\(\rho = 1\) for the same asset) eliminates all basis risk; imperfect cross-hedges retain residual variance proportional to \(1 - \rho^2\).
Rolling a Hedge
When the hedging horizon exceeds available futures maturities, rolling the hedge involves closing near-month futures as they approach expiry and opening contracts with later maturities. Rolling introduces additional basis risk from the shift in the futures curve shape.
Chapter 4: Option Markets
Option Definitions and Payoffs
Put option: The right (but not obligation) to sell an underlying asset at the strike price \(K\) on or before expiry \(T\).
European option: Can only be exercised at expiry.
American option: Can be exercised at any time up to and including expiry.
Payoff diagrams at expiry:
- Long call: \( \max(S_T - K, 0) \)
- Long put: \( \max(K - S_T, 0) \)
- Short call: \( -\max(S_T - K, 0) \)
- Short put: \( -\max(K - S_T, 0) \)
The profit of an option position subtracts the premium paid (or adds the premium received) from the payoff.
Factors Affecting Option Prices
| Factor | Call Price | Put Price |
|---|---|---|
| Underlying price \(S\) increases | Increases | Decreases |
| Strike price \(K\) increases | Decreases | Increases |
| Time to expiry \(T\) increases | Increases | Increases (usually) |
| Volatility \(\sigma\) increases | Increases | Increases |
| Risk-free rate \(r\) increases | Increases | Decreases |
| Dividend yield \(q\) increases | Decreases | Increases |
Put-Call Parity
For European options on a non-dividend-paying stock:
\[ C + K e^{-rT} = P + S_0 \]This is a no-arbitrage relationship: a portfolio of long call + long bond (with face value \(K\)) must equal a portfolio of long put + long stock. Violations create riskless arbitrage opportunities.
For dividend-paying stocks with present value of dividends \(D\):
\[ C + K e^{-rT} = P + S_0 - D \]Bounds on Option Prices
For European calls on non-dividend-paying stock:
\[ \max(S_0 - Ke^{-rT}, 0) \leq C \leq S_0 \]The lower bound reflects that a call is worth at least its intrinsic value in present-value terms. For American calls on non-dividend-paying stock, it is never optimal to exercise early (the American call equals the European call).
For European puts:
\[ \max(Ke^{-rT} - S_0, 0) \leq P \leq Ke^{-rT} \]Chapter 5: Hedging with Options
Protective Put
A protective put combines a long position in the underlying with a long put at strike \(K\). This creates a floor on the portfolio value: if the stock falls below \(K\), the put compensates. The cost is the put premium, which functions like an insurance premium.
\[ \text{Hedged Portfolio Value} = \max(S_T, K) - P_0 e^{rT} \]Covered Call
A covered call involves holding the underlying and selling a call. This generates immediate premium income but caps the upside at the strike price. Suitable for investors who expect modest appreciation and want to reduce cost basis.
Collar
A collar protects against downside by purchasing a put at strike \(K_1\) (below current price) and partially finances it by selling a call at \(K_2\) (above current price):
- When \(S_T < K_1\): Put provides protection; loss is capped.
- When \(K_1 \leq S_T \leq K_2\): Hold underlying unchanged.
- When \(S_T > K_2\): Call is exercised; upside is capped at \(K_2\).
A zero-cost collar sets the strikes such that call premium = put premium.
Chapter 6: Option Strategies
Spread Strategies
Bull spread (using calls): Buy call at \(K_1\), sell call at \(K_2 > K_1\). Profits if the underlying rises moderately. The sold call reduces the cost of the position but caps the maximum profit.
Bear spread (using puts): Buy put at \(K_2\), sell put at \(K_1 < K_2\). Profits if the underlying falls.
Butterfly spread: Buy call at \(K_1\), sell two calls at \(K_2\), buy call at \(K_3\) where \(K_1 < K_2 < K_3\). Profits if the underlying stays near \(K_2\); maximum loss is the net premium paid.
Volatility Strategies
Long straddle: Buy call and put at the same strike and expiry. Profits if the underlying makes a large move in either direction. The breakeven points are \(K + P + C\) and \(K - P - C\).
Long strangle: Buy out-of-the-money call and out-of-the-money put. Cheaper than a straddle but requires a larger price move to be profitable.
These strategies are used when expecting high volatility around a specific event (earnings, regulatory decision) without a directional view.
Chapter 7: The Greeks and Delta Hedging
Option Greeks
The Greeks measure the sensitivity of an option’s price to changes in underlying variables:
Gamma (\(\Gamma\)): Rate of change of delta with respect to the underlying price. Gamma is highest for at-the-money options near expiry.
\[ \Gamma = \frac{N'(d_1)}{S_0 \sigma \sqrt{T}} \]Theta (\(\Theta\)): Rate of change of option price with respect to time (time decay). Long options lose value as expiry approaches (all else equal).
Vega (\(\mathcal{V}\)): Rate of change of option price with respect to volatility. Long options gain value when volatility increases.
Rho (\(\rho\)): Rate of change of option price with respect to the risk-free interest rate.
Delta Hedging
A delta-neutral portfolio has a combined delta of zero — the position does not change in value for small movements in the underlying. A market maker who has sold options maintains a delta-neutral hedge by holding \(\Delta\) shares of the underlying per option sold.
Because delta changes as the stock price moves (measured by gamma), a delta hedge requires continuous rebalancing. The cost of maintaining the hedge reflects the option’s time value.
\[ \text{Hedge P&L over small interval} = \frac{1}{2} \Gamma (\Delta S)^2 - \Theta \Delta t \]This shows the interplay between gamma (profiting from large moves) and theta (cost of time decay). A long option position has positive gamma and negative theta — the holder benefits from volatility but pays time decay.
Chapter 8: Option Valuation Models
Binomial Tree Model
The binomial model discretizes the stock price path. At each step, the stock price moves up by factor \(u\) or down by factor \(d\):
\[ S_{up} = S \times u, \quad S_{down} = S \times d \]The risk-neutral probability of an up move:
\[ p = \frac{e^{r\Delta t} - d}{u - d} \]The option value at each node is the expected payoff under risk-neutral probabilities, discounted at the risk-free rate. For a one-step binomial:
\[ C = e^{-r\Delta t} \left[ p \times C_u + (1-p) \times C_d \right] \]American options are priced by comparing the early exercise value to the continuation value at each node and taking the maximum.
The Black-Scholes-Merton Model
The Black-Scholes-Merton (BSM) model (1973) provides closed-form option pricing under the following assumptions: continuous trading, lognormally distributed stock returns, constant volatility, no dividends, no transaction costs, and a constant risk-free rate.
where:
\[ d_1 = \frac{\ln(S_0/K) + (r + \sigma^2/2)T}{\sigma \sqrt{T}} \]\[ d_2 = d_1 - \sigma \sqrt{T} \]and \(N(\cdot)\) is the cumulative standard normal distribution function.
Implied volatility is the volatility level that equates the BSM model price to the observed market price. It is the market’s consensus expectation of future volatility embedded in the option price. The volatility smile or volatility skew refers to the empirical pattern where implied volatility varies with strike price, violating the BSM assumption of constant volatility.
Chapter 9: Interest Rate Futures and Duration
Interest Rate Derivatives
Interest rate risk is the risk that changes in interest rates adversely affect the value of bond portfolios, loan books, and financial statements. Interest rate derivatives — futures, swaps, options — are essential tools for managing this risk.
Treasury bond futures allow hedging of long-term interest rate risk. The cheapest-to-deliver (CTD) bond determines the futures price through the conversion factor system.
Duration and DV01
The Macaulay duration of a bond is the weighted average time to receive cash flows:
\[ D = \frac{\sum_{t=1}^{T} t \times C_t / (1+y)^t}{P} \]Modified duration \(D^*\) relates price changes to yield changes:
\[ \frac{\Delta P}{P} \approx -D^* \Delta y \]The DV01 (dollar value of 1 basis point) measures the dollar change in bond value for a 1 basis point (0.01%) change in yield:
\[ DV01 = D^* \times P \times 0.0001 \]To hedge a bond portfolio using futures:
\[ N^* = -\frac{DV01_{\text{portfolio}}}{DV01_{\text{futures}}} \]Chapter 10: Plain Vanilla Swaps and Forward Rate Agreements
Forward Rate Agreements (FRAs)
A Forward Rate Agreement (FRA) is an OTC agreement that fixes an interest rate for a future period. The buyer of an FRA benefits if the realized rate exceeds the contracted rate; the seller benefits otherwise. Settlement is in cash at the start of the rate period, discounted to present value.
Settlement cash flow (from buyer’s perspective):
\[ \text{Settlement} = \frac{(R_{actual} - R_{FRA}) \times L \times \alpha}{1 + R_{actual} \times \alpha} \]where \(L\) is the notional principal, \(\alpha\) is the accrual fraction, and the denominator discounts the payment back to the settlement date.
Interest Rate Swaps
A plain vanilla interest rate swap is an agreement between two parties to exchange fixed and floating interest payments on a notional principal:
- Fixed-rate payer: Pays fixed rate; receives LIBOR/SOFR floating rate.
- Floating-rate payer: Receives fixed rate; pays LIBOR/SOFR.
The notional principal is not exchanged — only net interest payments are settled.
Swap valuation: A fixed-for-floating swap can be valued as a portfolio of FRAs, or equivalently, as a fixed-rate bond minus a floating-rate bond (from the fixed-rate payer’s perspective):
\[ \text{Value to fixed-rate payer} = V_{float} - V_{fixed} \]At inception, the swap rate (fixed rate) is set so the swap’s initial value is zero:
\[ R_{swap} = \frac{1 - P(0, T_n)}{\sum_{i=1}^{n} \alpha_i P(0, T_i)} \]where \(P(0, T_i)\) is the discount factor for period \(i\).
Uses of Swaps
- Converting floating to fixed: A borrower with a floating-rate loan uses a pay-fixed swap to convert to effectively fixed-rate borrowing.
- Asset-liability management: Banks and insurers match the duration of assets and liabilities.
- Comparative advantage arbitrage: Two parties with different credit quality in fixed and floating markets can both achieve lower borrowing costs by swapping.
Chapter 11: Swaptions
A swaption is an option to enter into an interest rate swap at a future date. A payer swaption gives the holder the right to enter as fixed-rate payer; a receiver swaption gives the right to enter as fixed-rate receiver.
Swaptions are valued using adaptations of Black’s model for interest rates. The annuity factor (present value of $1 per period over the swap’s life) plays the role of the forward contract’s current price:
\[ C_{swaption} = A \left[ F \cdot N(d_1) - K \cdot N(d_2) \right] \]where \(F\) is the forward swap rate, \(K\) is the strike rate, and \(A\) is the swap annuity.
Chapter 12: Interest Rate Caps and Floors
An interest rate cap is a series of caplets — European call options on a floating rate — that compensate the holder when the floating rate exceeds the cap rate. A cap protects a floating-rate borrower against rising rates.
An interest rate floor is a series of floorlets — put options on a floating rate — compensating the holder when the rate falls below the floor rate. A floor protects a floating-rate investor (lender) against falling rates.
Each caplet or floorlet is independently priced using Black’s model:
\[ \text{Caplet}_i = L \cdot \alpha_i \cdot e^{-r_i T_i} \left[ F_i N(d_1^i) - R_K N(d_2^i) \right] \]Cap-floor parity (analogous to put-call parity):
\[ \text{Cap} - \text{Floor} = \text{Swap (pay fixed)} \]An interest rate collar buys a cap and sells a floor to limit borrowing rate exposure to a band.
Chapter 13: Credit Default Swaps
What Is a CDS?
A Credit Default Swap (CDS) is an OTC derivative that transfers credit risk from one party to another. The protection buyer pays periodic premiums (the CDS spread, in basis points per year) to the protection seller. If a credit event (typically default, restructuring, or failure to pay) occurs on the reference entity, the protection seller compensates the buyer for the loss.
\[ \text{CDS Premium (per year)} = S \times N \]where \(S\) is the CDS spread (e.g., 150 bps = 1.5%) and \(N\) is the notional.
Physical settlement: Buyer delivers the defaulted bonds to the seller and receives par value. Cash settlement: Seller pays \(N \times (1 - \text{Recovery Rate})\).
CDS Valuation
The CDS spread is set so the expected present value of premiums equals the expected present value of protection payments, under the risk-neutral measure:
\[ S = \frac{\text{PV(Protection Leg)}}{\text{PV(Premium Leg)}} \]The protection leg involves integrating the product of the default probability density and loss given default over the life of the contract.
The 2007-2009 Crisis and CDS
The JPMorgan “London Whale” episode (2012) illustrated how concentrated CDS positions in an index can distort credit markets. The trader built enormous positions in CDX credit index derivatives, ultimately costing JPMorgan approximately $6 billion. Key lessons: CDS creates concentrated counterparty risk; hedging strategies that are large relative to market size can backfire; risk model assumptions (particularly about correlation) must be stress-tested.
Chapter 14: Value at Risk and Expected Shortfall
Value at Risk (VaR)
A 1-day 99% VaR of $1 million means there is a 1% chance the portfolio will lose more than $1 million in a single day.
Three main approaches:
Historical simulation: Rank actual historical daily P&L; VaR is the observation at the \((1-\alpha)\) percentile. Simple and non-parametric but backward-looking.
Parametric (variance-covariance) method: Assume normally distributed returns. For a single asset:
For a portfolio: \(VaR = z_{1-\alpha} \sqrt{w^T \Sigma w}\) where \(\Sigma\) is the covariance matrix.
- Monte Carlo simulation: Generate thousands of simulated paths using distributional assumptions and compute the loss distribution empirically.
Limitations of VaR
VaR is widely criticized because it tells nothing about the magnitude of losses beyond the threshold. Two portfolios can have identical VaR but very different tail risk profiles.
Expected Shortfall (ES)
ES is more informative than VaR because it averages over the entire tail. Regulators (Basel III and FRTB — Fundamental Review of the Trading Book) have shifted from VaR to ES for regulatory capital calculations, using a 97.5% ES as the primary risk metric.
Backtesting checks whether realized losses exceed VaR with the expected frequency. If a 99% VaR is exceeded more than 1% of the time, the model is poorly calibrated.
Summary
AFM 322 develops a comprehensive toolkit for understanding, valuing, and applying financial derivatives. The course begins with the mechanics of futures and forward markets, extends to the rich theory of option pricing through binomial models and Black-Scholes-Merton, and then explores the universe of interest rate derivatives (swaps, swaptions, caps, floors) and credit derivatives (CDS). Throughout, the application to corporate risk management — hedging jet fuel costs, interest rate exposure, and credit risk — ties the theory to practice. The Greeks provide the language for managing derivative portfolios dynamically, and VaR/ES frameworks formalize how institutions measure and report aggregate market risk. Together these tools enable financial professionals to both construct hedges and critically evaluate the risk management practices of the institutions they advise or audit.
