Sources and References

Primary textbook — Hull, J. C. (2022). Options, Futures, and Other Derivatives (11th ed.). Pearson. Supplementary text — McDonald, R. L. (2013). Derivatives Markets (3rd ed.). Pearson. Additional references — Shreve, S. E. (2004). Stochastic Calculus for Finance I & II. Springer. | Wilmott, P. (2006). Paul Wilmott on Quantitative Finance (2nd ed.). Wiley.

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Chapter 1: Introduction to Derivatives

1.1 What Is a Derivative?

A derivative is a financial instrument whose value is derived from the value of some underlying asset, rate, or index. The underlying can be a stock, stock index, bond, interest rate, commodity, currency, or even another derivative. Derivatives are not new — forward contracts on agricultural commodities date to ancient Mesopotamia — but the modern derivatives market is vast: the Bank for International Settlements estimates notional outstanding over-the-counter derivatives at more than USD 600 trillion.

Derivatives serve three primary economic purposes:

1. Hedging — Reducing exposure to an already existing risk. A wheat farmer sells wheat futures to lock in a price before harvest, eliminating price uncertainty.
2. Speculation — Taking a leveraged position on the direction of prices, earning large gains (or losses) relative to the capital committed.
3. Arbitrage — Exploiting price discrepancies between markets or instruments to earn risk-free profit with no net investment.

1.2 Types of Derivative Instruments

Instrument	Exchange-Traded?	Payoff Type	Key Feature
Forward	No (OTC)	Linear	Customisable; counterparty credit risk
Futures	Yes	Linear	Mark-to-market daily; margin accounts
Option	Both	Non-linear	Buyer pays premium; right, not obligation
Swap	No (OTC)	Linear (periodic)	Exchange of cash flows over time
Credit Default Swap	No (OTC)	Contingent	Protection against credit events

1.3 Exchange-Traded vs. Over-the-Counter Markets

Exchange-traded derivatives (ETDs) trade on organised exchanges such as the CME Group, Intercontinental Exchange (ICE), or Eurex. A central counterparty (CCP) interposes itself between buyer and seller, eliminating bilateral counterparty credit risk. ETDs are standardised in terms of contract size, expiry, and settlement procedure.

Over-the-counter (OTC) derivatives are bilateral contracts negotiated directly between counterparties (banks, corporations, hedge funds). They are flexible — any notional amount, any maturity, any payoff structure — but carry counterparty credit risk. The Dodd-Frank Act (2010) and EMIR (2012) mandated central clearing for standardised OTC derivatives and trade reporting to repositories.

1.4 The Role of Derivatives in Corporate Risk Management

Corporations face multiple dimensions of financial risk:

• Market risk — Exposure to movements in prices (equities, commodities, FX, interest rates).
• Credit risk — The risk that a counterparty will default on an obligation.
• Liquidity risk — Inability to meet cash obligations or exit positions without significant price impact.
• Operational risk — Failures in internal processes, people, or systems.

Derivatives are primarily tools for managing market risk. Firms with natural exposures — an airline facing fuel cost uncertainty, a multinational with foreign-currency revenues, a bank with floating-rate liabilities — can use futures, forwards, options, and swaps to transfer these risks to counterparties willing to bear them, typically financial intermediaries or speculators.

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Chapter 2: Forwards and Futures

2.1 Forward Contracts

A forward contract is an agreement to buy (long) or sell (short) an asset at a pre-specified price \( F_0 \) (the forward price) on a future delivery date \( T \). No money changes hands at inception.

The payoff diagrams are linear and symmetric. The long position profits from rising prices; the short position profits from falling prices. At initiation, the forward price is set so that the contract has zero value: neither party pays a premium at \( t = 0 \).

2.2 Payoff Diagrams

It is essential to distinguish between payoff (cash received at expiry ignoring initial outlay) and profit (payoff minus initial investment, including any premium or borrowing cost).

For a forward contract entered at \( F_0 = 100 \):

\( S_T \)	Long payoff	Short payoff
80	\(-20\)	\(+20\)
90	\(-10\)	\(+10\)
100	\(0\)	\(0\)
110	\(+10\)	\(-10\)
120	\(+20\)	\(-20\)

The long forward is equivalent to being long the asset and short a zero-coupon bond with face value \( F_0 \). This synthetic decomposition is the backbone of derivatives pricing.

2.3 The Cost-of-Carry Model and Forward Pricing

The no-arbitrage forward price for a non-dividend-paying stock is derived by considering two strategies:

• Strategy A: Enter into a long forward at price \( F_0 \). Cost today: $0. Value at \( T \): \( S_T - F_0 \).
• Strategy B: Buy the stock today for \( S_0 \), funded by borrowing. At \( T \), repay \( S_0 e^{rT} \). Terminal value: \( S_T - S_0 e^{rT} \).

For no-arbitrage, both strategies must have the same payoff. Setting \( F_0 = S_0 e^{rT} \) equates the two:

When the underlying pays a continuous dividend yield \( q \) (as for equity indices or currencies):

For a currency forward, \( q \) is replaced by the foreign risk-free rate \( r_f \) (covered interest parity):

\[ F_0 = S_0 e^{(r - r_f)T} \]

For commodity futures, we include a convenience yield \( y \) (benefit from holding physical inventory) and storage cost \( u \):

\[ F_0 = S_0 e^{(r + u - y)T} \]

When the convenience yield exceeds storage costs plus the risk-free rate, futures prices fall below spot — the market is in backwardation. When storage costs dominate, futures exceed spot — the market is in contango.

2.4 Cash-and-Carry Arbitrage

2.5 Futures vs. Forwards

Futures are like forwards but with daily mark-to-market settlement. This creates a timing difference in cash flows: futures gains and losses are received or paid each day, whereas forward gains are settled only at maturity. For non-stochastic interest rates, forward and futures prices are identical. When interest rates are stochastic, a subtle correlation adjustment arises — the convexity adjustment — which becomes important for interest rate futures.

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Chapter 3: Hedging with Futures

3.1 Basic Hedging Concepts

A long hedge involves buying futures to hedge against a price increase in a commodity you need to purchase. A short hedge involves selling futures to hedge against a price decrease in a commodity you hold or will sell.

Basis risk arises because:

1. The asset being hedged may not be identical to the futures underlying (cross-hedge).
2. The hedge horizon may not match any futures expiry exactly.
3. The hedger may need to close out before expiry.

3.2 The Optimal Hedge Ratio

The minimum-variance hedge ratio minimises the variance of the hedged portfolio. Let:

• \( \Delta S \) = change in spot price over the hedge horizon
• \( \Delta F \) = change in futures price over the same period
• \( \sigma_S \) = standard deviation of \( \Delta S \)
• \( \sigma_F \) = standard deviation of \( \Delta F \)
• \( \rho \) = correlation between \( \Delta S \) and \( \Delta F \)

Derivation: Let \( V = \Delta S - h \Delta F \) be the P&L per unit of exposure of the hedged position. Then:

\[ \text{Var}(V) = \sigma_S^2 + h^2 \sigma_F^2 - 2h\rho\sigma_S\sigma_F \]

Taking the derivative with respect to \( h \) and setting to zero:

\[ \frac{d\,\text{Var}(V)}{dh} = 2h\sigma_F^2 - 2\rho\sigma_S\sigma_F = 0 \implies h^* = \rho\frac{\sigma_S}{\sigma_F} \]

The second derivative \( 2\sigma_F^2 > 0 \) confirms this is a minimum. Note that \( h^* \) is identical to the OLS slope coefficient in a regression of \( \Delta S \) on \( \Delta F \) — it can be estimated empirically from historical data.

3.3 Equity Index Futures and Beta Hedging

To hedge an equity portfolio, we use stock index futures. The CAPM beta \( \beta \) of the portfolio measures its sensitivity to the market index. The number of futures contracts needed to reduce beta to a target \( \beta^* \) is:

\[ N^* = (\beta^* - \beta) \times \frac{P}{A} \]

where \( P \) is the dollar value of the portfolio and \( A = F_0 \times \text{multiplier} \) is the dollar value of one futures contract. To fully hedge (target beta = 0), \( \beta^* = 0 \) and \( N^* = -\beta \times P/A \) (short position).

3.4 Currency Futures and FX Hedging

A Canadian exporter expecting USD 5 million in six months can sell USD/CAD currency futures to lock in the exchange rate. The forward price for a currency future is:

\[ F_0 = S_0 e^{(r_{CAD} - r_{USD}) \times T} \]

where \( S_0 \) is the spot rate (CAD per USD). If Canadian rates are 4% and US rates are 5%, and the spot is 1.35 CAD/USD:

\[ F_0 = 1.35 \times e^{(0.04 - 0.05) \times 0.5} = 1.35 \times e^{-0.005} = 1.35 \times 0.9950 = 1.3433 \text{ CAD/USD} \]

The exporter sells futures at 1.3433, locking in approximately CAD 6.72 million against the USD 5 million receivable. The USD discount relative to spot reflects the interest rate differential — US rates exceed Canadian rates, so the USD is at a forward discount.

3.5 Rolling Hedges

When the hedger’s exposure extends beyond the delivery date of available futures contracts, a rolling hedge is used: close out the near-term futures as they approach expiry and roll into later-dated contracts.

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Chapter 4: Options Fundamentals

4.1 Call and Put Options

Payoffs at expiry (European options):

\[ \text{Call payoff} = \max(S_T - K, 0) \equiv (S_T - K)^+ \]\[ \text{Put payoff} = \max(K - S_T, 0) \equiv (K - S_T)^+ \]

Profit at expiry (accounting for premium paid at \( t=0 \), future-valued):

\[ \text{Call profit} = \max(S_T - K, 0) - C e^{rT} \]\[ \text{Put profit} = \max(K - S_T, 0) - P e^{rT} \]

4.2 Moneyness

Moneyness	Call	Put
In-the-money (ITM)	\( S_0 > K \)	\( S_0 < K \)
At-the-money (ATM)	\( S_0 \approx K \)	\( S_0 \approx K \)
Out-of-the-money (OTM)	\( S_0 < K \)	\( S_0 > K \)

Deep ITM options behave like the underlying asset (delta near ±1). Deep OTM options are essentially lottery tickets — small premium, large payoff only if a large move occurs.

4.3 Intrinsic Value and Time Value

4.4 Bounds on Option Prices

For European options on a non-dividend-paying stock:

Upper bounds:

\[ C \leq S_0, \quad P \leq Ke^{-rT} \]

A call can never be worth more than the stock (you can always buy the stock directly). A put can never be worth more than the present value of the strike (the most you gain is \( K \) if the stock goes to zero).

4.5 Put-Call Parity

Proof: Consider two portfolios:

• Portfolio A: Long call \( C \) + long zero-coupon bond paying \( K \) at \( T \) (cost: \( C + Ke^{-rT} \))
• Portfolio B: Long put \( P \) + long stock (cost: \( P + S_0 \))

At expiry:

• If \( S_T > K \): Portfolio A pays \( (S_T - K) + K = S_T \); Portfolio B pays \( 0 + S_T = S_T \).
• If \( S_T \leq K \): Portfolio A pays \( 0 + K = K \); Portfolio B pays \( (K - S_T) + S_T = K \).

The payoffs are identical in all states, so by no-arbitrage the costs must be equal today: \( C + Ke^{-rT} = P + S_0 \). With continuous dividends \( q \), the relation becomes \( C + Ke^{-rT} = P + S_0 e^{-qT} \).

4.6 American vs. European Options

American options grant early exercise rights. Key results:

• American call on non-dividend-paying stock: It is never optimal to exercise early. The American call equals the European call: \( C_{\text{Am}} = C_{\text{Eu}} \). Intuition: early exercise gives \( S - K \), but selling the call gives at least \( S - Ke^{-rT} > S - K \). Early exercise destroys time value.
• American put: Early exercise can be optimal when the option is deep in the money. When \( S \) falls sufficiently, the interest on \( K \) received early outweighs the lost time value. Thus \( P_{\text{Am}} \geq P_{\text{Eu}} \), strictly so for some parameters.
• American call on dividend-paying stock: May be optimal to exercise just before an ex-dividend date to capture the dividend. This motivates binomial tree pricing with discrete dividend adjustments.

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Chapter 5: Option Strategies

5.1 Basic Positions: Building Blocks

Every option strategy is assembled from four elementary building blocks: long call, short call (written call), long put, short put (written put) — combined potentially with long or short positions in the underlying stock or bond.

The sign convention: long positions have positive payoffs as the instrument moves in the money; short (written) positions collect the premium but accept unlimited (call) or bounded (put) downside.

5.2 Vertical Spreads

A bull call spread involves buying a call at \( K_1 \) and selling a call at \( K_2 > K_1 \) (same expiry). Net premium: \( C_1 - C_2 > 0 \) (net debit). Payoff at expiry:

\[ \text{Payoff} = \max(S_T - K_1, 0) - \max(S_T - K_2, 0) = \begin{cases} 0 & S_T \leq K_1 \\ S_T - K_1 & K_1 < S_T \leq K_2 \\ K_2 - K_1 & S_T > K_2 \end{cases} \]

Maximum profit: \( K_2 - K_1 - (C_1 - C_2) \). Maximum loss: net premium \( C_1 - C_2 \). Break-even: \( K_1 + (C_1 - C_2) \).

A bear put spread is the mirror image for bearish views: buy a put at \( K_2 \), sell a put at \( K_1 < K_2 \). Profits when the stock falls below \( K_2 \) but the profit is capped at \( K_2 - K_1 - (P_2 - P_1) \).

5.3 Butterfly and Condor Spreads

A butterfly spread using calls: buy one call at \( K_1 \), sell two calls at \( K_2 \), buy one call at \( K_3 \), where \( K_1 < K_2 < K_3 \) and \( K_2 = (K_1+K_3)/2 \).

\[ \text{Payoff} = \max(S_T - K_1, 0) - 2\max(S_T - K_2, 0) + \max(S_T - K_3, 0) \]

The payoff peaks at \( K_2 \) (value \( K_2 - K_1 \) net of premium) and falls to zero outside \( [K_1, K_3] \). A butterfly is a bet on low volatility — that the stock stays near \( K_2 \).

A condor is a wider version with four different strikes \( K_1 < K_2 < K_3 < K_4 \), giving a flat profit zone between \( K_2 \) and \( K_3 \). Both spreads are used by options market makers to express and monetise views on the volatility smile.

5.4 Volatility Strategies: Straddles and Strangles

A straddle is a long call plus a long put at the same strike and expiry. Profits if the stock moves significantly in either direction. The break-even points are \( K - (C + P) \) (downside) and \( K + (C + P) \) (upside).

A strangle buys an OTM call \( (K_2 > S_0) \) and an OTM put \( (K_1 < S_0) \). Cheaper than a straddle (both options OTM) but requires a larger move to profit.

5.5 Protective Put and Covered Call

A protective put is holding a stock and buying a put. This creates a floor on the portfolio value: if \( S_T < K \), the put compensates for losses. The strategy mimics portfolio insurance at the cost of the put premium. The payoff profile resembles a long call plus a cash position.

By put-call parity: \( S_0 + P = C + Ke^{-rT} \). A protective put (stock + put) has the same payoff as a long call plus a bond — the option provides the same insurance structure.

A covered call is holding a stock and selling a call. This generates premium income but caps upside at \( K \). Payoff:

\[ \text{Payoff} = S_T - \max(S_T - K, 0) = \min(S_T, K) \]

By put-call parity, a covered call has the same payoff as a short put plus a bond: generating income in a flat-to-mildly-bullish market.

5.6 Collars

A collar combines a protective put at \( K_1 \) with a covered call at \( K_2 > K_1 \). The call premium offsets the put premium. A zero-cost collar is structured so that the premiums are exactly equal, eliminating out-of-pocket cost while providing downside protection at the expense of capped upside. Collars are widely used by corporate executives to hedge concentrated stock positions without triggering a taxable disposal.

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Chapter 6: Binomial Option Pricing

6.1 The One-Period Replicating Portfolio

The binomial model prices options by constructing a portfolio of the underlying stock and a risk-free bond that replicates the option payoff exactly in each state of the world.

Suppose the current stock price is \( S_0 \). Over one period of length \( T \), the stock either goes up to \( S_0 u \) (with real-world probability \( p \)) or down to \( S_0 d \) (probability \( 1 - p \)), where \( u > e^{rT} > d \) (to preclude arbitrage).

Let \( C_u \) and \( C_d \) be the call payoffs in the up and down states. We seek a portfolio of \( \Delta \) shares and a bond investment \( B \) (positive = lending) such that:

\[ \Delta \cdot S_0 u + B e^{rT} = C_u \]\[ \Delta \cdot S_0 d + B e^{rT} = C_d \]

Solving for \( \Delta \) (subtracting the two equations):

\[ \Delta = \frac{C_u - C_d}{S_0(u - d)} \]

And substituting back:

\[ B = e^{-rT}(C_u - \Delta S_0 u) \]

The option price is:

\[ C_0 = \Delta S_0 + B \]

Substituting into the option price formula:

\[ C_0 = e^{-rT}[q C_u + (1-q)C_d] \]

The option price is the present value of the expected payoff under the risk-neutral measure — the real-world probabilities \( p \) and \( 1-p \) drop out completely.

6.2 Multi-Period Binomial: Backward Induction

For \( n \) periods, the tree is built forward from \( S_0 \) to all terminal nodes at \( T \). Option values are computed by backward induction from expiry to today.

At each node at time step \( i \), \( j \) up-moves:

\[ S_{i,j} = S_0 u^j d^{i-j} \]

Terminal payoffs for a call: \( C_{n,j} = \max(S_{n,j} - K, 0) \), \( j = 0, 1, \ldots, n \).

Rolling back one step at each interior node:

\[ C_{i,j} = e^{-r\Delta t}[q C_{i+1,j+1} + (1-q) C_{i+1,j}] \]

where \( \Delta t = T/n \). The Cox-Ross-Rubinstein (CRR) parameterisation sets:

\[ u = e^{\sigma\sqrt{\Delta t}}, \quad d = e^{-\sigma\sqrt{\Delta t}} = 1/u, \quad q = \frac{e^{r\Delta t} - d}{u - d} \]

This ensures the tree is recombining (an up then down returns to the same node as down then up) and that in the limit \( n \to \infty \), the binomial tree converges to the Black-Scholes-Merton formula.

6.3 American Options: Early Exercise in the Binomial Model

For American options, at each node we compare the continuation value (hold) with the early exercise value (exercise immediately). The American option value is the maximum of the two:

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Chapter 7: The Black-Scholes-Merton Model

7.1 Assumptions

The Black-Scholes-Merton (BSM) model, published by Fischer Black and Myron Scholes (1973) and extended by Robert Merton (1973), rests on these assumptions:

1. The stock price follows geometric Brownian motion (GBM): \( dS = \mu S\,dt + \sigma S\,dW \), where \( W \) is a standard Brownian motion under the real-world measure.
2. The risk-free rate \( r \) and volatility \( \sigma \) are constant and known.
3. No dividends (base model; extensions add dividends).
4. No transaction costs or taxes; continuous trading is possible.
5. No short-selling restrictions; the underlying is perfectly divisible.
6. The market is complete (every payoff can be replicated).

7.2 The BSM Pricing Formula

Interpretation:

• \( N(d_2) \) is the risk-neutral probability that the call expires in the money (i.e., \( S_T > K \) under \( \mathbb{Q} \)).
• \( N(d_1) = e^{qT} \partial C / \partial S_0 \) — adjusted delta.
• \( S_0 e^{-qT} N(d_1) \) is the present value of receiving the stock conditional on exercise.
• \( Ke^{-rT} N(d_2) \) is the present value of paying the strike conditional on exercise.

7.3 BSM with Discrete Dividends

For stocks paying known discrete dividends \( D_i \) at times \( t_i < T \), subtract the present value of all dividends from the current stock price to obtain the prepaid forward price:

\[ S_0^* = S_0 - \sum_i D_i e^{-r t_i} \]

Apply BSM using \( S_0^* \) in place of \( S_0 \). This is Hull’s standard approximation for pricing European options near ex-dividend dates.

7.4 Black’s Model for Futures Options

A key extension of BSM applies to options on futures. Using the forward price \( F_0 = S_0 e^{(r-q)T} \), Black’s model (1976) gives:

\[ C = e^{-rT}[F_0 N(d_1) - K N(d_2)] \]\[ d_1 = \frac{\ln(F_0/K) + \sigma^2 T/2}{\sigma\sqrt{T}}, \quad d_2 = d_1 - \sigma\sqrt{T} \]

This is the standard pricing formula for interest rate caps, floors, and swaptions (with the forward rate or forward swap rate replacing \( F_0 \)).

7.5 BSM PDE and Risk-Neutral Pricing

The formal derivation proceeds via:

1. Applying Itô’s Lemma to \( V(S, t) \), yielding \( dV = \left(\frac{\partial V}{\partial t} + \mu S \frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2}\right)dt + \sigma S \frac{\partial V}{\partial S} dW \).
2. Constructing the delta-neutral portfolio \( \Pi = V - \frac{\partial V}{\partial S} S \). The \( dW \) terms cancel, making \( d\Pi \) deterministic.
3. Setting \( d\Pi = r\Pi\,dt \) (no-arbitrage).
4. This yields the Black-Scholes PDE:

\[ \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} = rV \]

5. Solving with terminal condition \( V(S_T, T) = (S_T - K)^+ \) gives the BSM call formula.

Equivalently, by the Feynman-Kac theorem, the PDE solution is the expectation under the risk-neutral measure \( \mathbb{Q} \):

\[ V(S_0, 0) = e^{-rT} \mathbb{E}^{\mathbb{Q}}[(S_T - K)^+] \]

Under \( \mathbb{Q} \), the stock drifts at the risk-free rate \( r \) (not the physical drift \( \mu \)), and integrating the log-normal density over the ITM region gives the BSM formula. The physical probability \( p \) of an up-move is irrelevant to pricing — a profound insight that underpins all of modern derivatives theory.

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Chapter 8: The Greeks

8.1 Overview

The Greeks quantify how an option’s price changes with respect to market inputs. They are the primary tools of options risk management. A derivatives desk manages its book by monitoring aggregate Greeks across all positions and using offsetting instruments to keep exposures within prescribed limits.

Greek	Symbol	Sensitivity
Delta	\( \Delta \)	Stock price \( S \)
Gamma	\( \Gamma \)	Rate of change of \( \Delta \) w.r.t. \( S \)
Theta	\( \Theta \)	Time \( t \) (passage of time)
Vega	\( \mathcal{V} \)	Volatility \( \sigma \)
Rho	\( \rho \)	Risk-free rate \( r \)

8.2 Delta

Delta is also the hedge ratio: to delta-hedge a position of \( -1 \) option (short one call), hold \( +\Delta \) shares. The portfolio \( \Pi = -C + \Delta S \) is instantaneously riskless.

Key behaviour:

• Deep ITM call: \( \Delta \to 1 \) (behaves like stock).
• ATM call: \( \Delta \approx 0.5 \).
• Deep OTM call: \( \Delta \to 0 \).

Delta is not constant — it changes as \( S \) changes, necessitating continuous rebalancing in theory and frequent rebalancing in practice.

8.3 Gamma

Gamma is always positive for long option positions (calls or puts) and always negative for short option positions. Gamma is largest for ATM options and for options with short time to expiry. Large positive gamma means the delta-hedge gains from large stock moves in either direction (convexity benefit). Large negative gamma means the delta-hedge loses from large moves.

The P&L of a delta-hedged position over a small interval \( dt \) is approximately:

\[ \delta\Pi \approx \frac{1}{2}\Gamma(\delta S)^2 + \Theta\,dt \]

where the first term is the gamma P&L (always positive for long gamma) and the second term is time decay.

8.4 Theta

Theta is typically negative for long positions (options lose value over time, all else equal). ATM options near expiry have the largest (most negative) theta — time decay accelerates as expiry approaches. This is the price paid for positive gamma.

8.5 The Gamma-Theta Tradeoff

From the BSM PDE, for a delta-neutral portfolio (\( \Delta = 0 \)):

\[ \Theta + \frac{1}{2}\sigma^2 S^2 \Gamma \approx 0 \]

This reveals the fundamental tension in options trading:

• Long gamma, short theta: Benefits from large moves (gamma P&L) but bleeds time value when the market is quiet. This is the long options / long volatility position.
• Short gamma, long theta: Earns time premium daily but suffers losses if the market moves sharply. This is the short options / short volatility position.

The gamma-theta tradeoff is the essence of volatility trading: you are betting whether realised volatility (actual market moves) will exceed or fall short of implied volatility (the level priced into the options).

8.6 Vega

Vega is largest for ATM options and longer-dated options. A vega-neutral portfolio has no first-order sensitivity to volatility changes. Achieving vega-neutrality requires adding options to the hedge, not just stock.

8.7 Rho

Rho is positive for calls (higher rates raise the cost-of-carry and thus the forward price, benefiting calls) and negative for puts. Rho is most material for long-dated options; for short-dated options, interest rate sensitivity is small.

8.8 Delta Hedging P&L

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Chapter 9: Volatility

9.1 Historical Volatility

Historical volatility (realised volatility) is estimated from past log-returns. Given daily closing prices \( S_0, S_1, \ldots, S_n \):

\[ u_i = \ln\!\left(\frac{S_i}{S_{i-1}}\right), \quad \bar{u} = \frac{1}{n}\sum_{i=1}^n u_i \]\[ \hat{\sigma}_{\text{daily}} = \sqrt{\frac{1}{n-1}\sum_{i=1}^n (u_i - \bar{u})^2} \]

Annualised: \( \hat{\sigma}_{\text{annual}} = \hat{\sigma}_{\text{daily}} \times \sqrt{252} \) (using 252 trading days per year).

The choice of lookback window involves a tradeoff: longer windows are more stable but less responsive to recent changes; shorter windows are more reactive but noisier. Common choices: 20-day, 60-day, or 252-day (1-year) windows.

9.2 Implied Volatility

Implied volatility is the market’s consensus forecast of future realised volatility over the option’s life. It is the single most important quantity for options pricing and risk management. Traders “trade vol” — buying options when IV is low relative to their forecast of realised vol, and selling when IV is high.

Newton-Raphson iteration for IV:

\[ \sigma_{n+1} = \sigma_n - \frac{C_{\text{BSM}}(\sigma_n) - C_{\text{mkt}}}{\mathcal{V}(\sigma_n)} \]

A good starting guess is \( \sigma_0 = \sqrt{2|\ln(F/K)|/T} \) (Brenner-Subrahmanyam approximation). Convergence is typically achieved in 3–5 iterations.

9.3 The Volatility Smile and Skew

If BSM were exactly correct, implied volatility would be constant across strikes and maturities — a flat volatility surface. In practice, it is not:

Equity skew (smirk): For equity options, IV is higher for low strikes (OTM puts, downside protection) than for high strikes (OTM calls). This “negative skew” reflects:

• Investor demand for downside protection.
• Leverage effect: stock price declines are associated with increased volatility.
• Fat left tails: equity return distributions have more left-tail probability than log-normal.

FX smile: Currency option IV is higher for both deep ITM and deep OTM options relative to ATM, producing a U-shape. This reflects symmetric fat tails (large currency moves in either direction are more likely than log-normal).

Term structure of volatility: ATM IV varies with expiry. During crises, short-dated IV spikes (term structure inverts). In normal markets, the term structure is upward sloping (long-dated vol > short-dated vol).

The volatility surface — IV as a bivariate function of strike and maturity — is the central object calibrated by options market-makers. Common parameterisations include the SABR model and the Heston stochastic volatility model.

9.4 The VIX Index

The CBOE Volatility Index (VIX) measures the market’s 30-day risk-neutral variance expectation for the S&P 500, derived from a model-free formula using SPX options across all strikes:

\[ \sigma^2_{\text{VIX}} = \frac{2}{T}\sum_i \frac{\Delta K_i}{K_i^2} e^{rT} Q(K_i) - \frac{1}{T}\left(\frac{F}{K_0} - 1\right)^2 \]

where \( Q(K_i) \) is the midpoint bid-ask of put (for \( K_i \leq F \)) or call (for \( K_i > F \)) with strike \( K_i \), \( F \) is the 30-day forward S&P 500 price, and \( K_0 \) is the first strike at or below \( F \). The VIX is \( 100 \times \sqrt{\sigma^2_{\text{VIX}}} \).

A VIX of 20 implies \( 20\%/\sqrt{12} \approx 5.8\% \) monthly volatility, or \( 20\%/\sqrt{252} \approx 1.26\% \) daily volatility. The VIX averaged around 15–20 in calm markets (2012–2019) and spiked above 80 in March 2020 (COVID) and above 60 in October 2008 (financial crisis).

9.5 GARCH Models for Volatility Forecasting

The GARCH(1,1) model (Bollerslev, 1986) captures volatility clustering and mean-reversion:

\[ \sigma_t^2 = \omega + \alpha u_{t-1}^2 + \beta \sigma_{t-1}^2 \]

where \( u_{t-1} \) is yesterday’s return shock, \( \sigma_{t-1}^2 \) is yesterday’s variance estimate, and \( \alpha + \beta < 1 \) ensures stationarity. The long-run variance is:

\[ \bar{\sigma}^2 = \frac{\omega}{1 - \alpha - \beta} \]

GARCH parameters are estimated by maximum likelihood. Typical S&P 500 estimates: \( \alpha \approx 0.05 \), \( \beta \approx 0.92 \) — high persistence (\( \alpha + \beta = 0.97 \)) consistent with observed slow decay of volatility spikes. The half-life of a volatility shock is approximately \( \ln(0.5)/\ln(\alpha+\beta) \approx 23 \) days.

GARCH is widely used for risk management (VaR estimation) and as a volatility input for options pricing. Extensions include EGARCH (captures the leverage effect) and GJR-GARCH (asymmetric response to negative shocks).

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Chapter 10: Interest Rate Derivatives

10.1 Forward Rate Agreements (FRAs)

A forward rate agreement is a bilateral OTC contract to exchange fixed-for-floating interest payments on a specified notional over a future period \( [T_1, T_2] \). The fixed rate is the FRA rate \( R_K \); the floating rate is SOFR (US), CORRA (Canada), or €STR (Europe).

If the floating rate at reset date \( T_1 \) is \( R_M \), the payoff to the party receiving floating at settlement time \( T_1 \) is:

\[ \text{Payoff (receive float)} = \frac{(R_M - R_K)(T_2 - T_1) \times L}{1 + R_M(T_2 - T_1)} \]

The fair FRA rate is the forward rate implied by the yield curve between \( T_1 \) and \( T_2 \). In continuously compounded terms:

\[ R_K = \frac{r_2 T_2 - r_1 T_1}{T_2 - T_1} \]

10.2 Interest Rate Swaps: Fixed-for-Floating

An interest rate swap exchanges periodic cash flows: one party pays a fixed rate \( R_{\text{swap}} \) (the par swap rate) and receives a floating rate (SOFR), or vice versa. The notional \( L \) is not exchanged.

Net cash flow to the fixed-rate payer at each payment date \( t_i \):

\[ CF_i = L \times (R_{\text{float},i} - R_{\text{swap}}) \times \delta_i \]

where \( \delta_i = t_i - t_{i-1} \) is the accrual fraction.

Valuation via Bond Equivalence

A fixed-for-floating swap can be decomposed:

• Fixed-rate payer = Long a floating-rate bond + Short a fixed-rate bond.
• Fixed-rate receiver = Long a fixed-rate bond + Short a floating-rate bond.

A floating-rate bond always reprices to par at each reset date. Therefore, the value of the floating leg to the receiver is always \( L \) at inception (and at any reset date). The value to the fixed-rate payer at inception is:

\[ V_{\text{swap}} = L - B_{\text{fix}} \]

where \( B_{\text{fix}} \) is the value of a fixed-rate bond with coupon \( R_{\text{swap}} \) and face \( L \). Setting \( B_{\text{fix}} = L \) gives the par swap rate:

\[ R_{\text{swap}} = \frac{1 - e^{-r_n T_n}}{\sum_{i=1}^n \delta_i e^{-r_i t_i}} \]

Post-Inception Valuation

After inception, as rates change, the swap has non-zero value. The value to the fixed-rate payer is:

\[ V_{\text{swap}} = B_{\text{float}} - B_{\text{fix}} = L - \left[R_{\text{swap}} \sum_{i=k}^n \delta_i e^{-r_i^* t_i} + L e^{-r_n^* T_n}\right] \]

where \( r_i^* \) are the current zero rates and the sum is over remaining payment dates.

10.3 Day Count and Market Conventions

Market convention matters for swap pricing. Key conventions:

Market	Floating Leg	Fixed Leg	Day Count
USD (SOFR)	Actual/360	30/360 or Actual/360	—
GBP (SONIA)	Actual/365(Fixed)	Actual/365(Fixed)	—
EUR (€STR)	Actual/360	30/360	—

10.4 Interest Rate Caps and Floors

An interest rate cap is a portfolio of caplets — European call options on the floating rate. It pays out whenever SOFR exceeds the cap rate \( R_{\text{cap}} \). A floor is a portfolio of floorlets — European put options on the floating rate.

Each caplet on the reset period \( [t_{i-1}, t_i] \) pays at \( t_i \):

\[ \text{Caplet}_i = L \cdot \delta_i \cdot \max(R_{\text{float},i} - R_{\text{cap}}, 0) \]

Priced using Black’s model:

\[ \text{Caplet}_i = L \delta_i e^{-r_i t_i} [F_i N(d_1) - R_{\text{cap}} N(d_2)] \]

where \( F_i \) is the forward rate for \( [t_{i-1}, t_i] \), \( \sigma_i \) is the caplet volatility, and:

\[ d_1 = \frac{\ln(F_i/R_{\text{cap}}) + \sigma_i^2 t_{i-1}/2}{\sigma_i\sqrt{t_{i-1}}}, \quad d_2 = d_1 - \sigma_i\sqrt{t_{i-1}} \]

Cap-Floor Parity (analogous to put-call parity):

\[ \text{Cap} - \text{Floor} = \text{Swap (receive float, pay fixed at } R_{\text{cap}}\text{)} \]

A zero-cost interest rate collar combines a long cap at \( R_{\text{cap}} \) and a short floor at \( R_{\text{floor}} \) (or vice versa), with premiums designed to offset. A corporate fixed-rate borrower buys a cap (protection against rate rises) and sells a floor (giving up benefit from rate falls) to achieve a collar at zero cost.

10.5 Swaptions

A swaption is an option on an interest rate swap. A payer swaption gives the holder the right to enter a swap as fixed-rate payer at a pre-specified swap rate \( K \). A receiver swaption gives the right to receive fixed.

Payer swaption payoff at expiry \( T_0 \):

\[ \text{Payoff} = A \cdot \max(R_{\text{swap,fwd}} - K, 0) \]

where \( A = L\sum_{i=1}^n \delta_i e^{-r_i^* t_i} \) is the annuity factor (swap’s DV01 per unit notional) and \( R_{\text{swap,fwd}} \) is the current forward swap rate.

Pricing via Black’s model:

\[ \text{Payer swaption} = L \cdot A \cdot [R_{\text{fwd}} N(d_1) - K N(d_2)] \]\[ d_1 = \frac{\ln(R_{\text{fwd}}/K) + \sigma_{\text{sw}}^2 T_0/2}{\sigma_{\text{sw}}\sqrt{T_0}}, \quad d_2 = d_1 - \sigma_{\text{sw}}\sqrt{T_0} \]

Swaption Put-Call Parity:

\[ \text{Payer swaption} - \text{Receiver swaption} = L \cdot A \cdot (R_{\text{fwd}} - K) \cdot e^{-r T_0} \]

Swaptions are used to hedge optionality in callable bonds (the issuer has an embedded call option equivalent to a receiver swaption) and mortgage prepayment risk.

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Chapter 11: Credit Derivatives

11.1 Credit Default Swaps

A credit default swap (CDS) is a contract in which the protection buyer pays periodic premiums (the CDS spread) to the protection seller. If the reference entity experiences a credit event, the protection seller compensates the buyer.

Upon a credit event, physical settlement (buyer delivers defaulted bonds for par) has largely been replaced by cash settlement via an ISDA-run auction determining the recovery rate.

11.2 CDS Spread as a Measure of Credit Risk

Under the simplifying assumption of a constant hazard rate \( \lambda \) (instantaneous default probability), the risk-neutral default probability over period \( [0, T] \) is:

\[ P(\text{default by }T) = 1 - e^{-\lambda T} \]

The survival probability to time \( t \) is \( e^{-\lambda t} \). The fair CDS spread equates the PV of the premium leg to the PV of the protection leg. Approximately:

\[ s \approx (1 - R) \times \lambda \]

A 5-year CDS trading at 150 bps with recovery 40% implies:

\[ \lambda \approx \frac{0.0150}{1 - 0.40} = \frac{0.0150}{0.60} = 2.5\% \text{ per annum hazard rate} \]

The probability of default in 5 years: \( 1 - e^{-0.025 \times 5} = 1 - e^{-0.125} = 11.75\% \).

11.3 Upfront CDS Convention

Post-2009 CDS reform standardised coupons: 100 bps (investment grade) or 500 bps (high yield). The difference between the fair spread and the standard coupon is settled upfront. The upfront amount (as a fraction of notional) is approximately:

\[ \text{Upfront} \approx (s_{\text{fair}} - s_{\text{coupon}}) \times \text{duration} \]

where “duration” here is the spread DV01 — the risky annuity factor (PV of $1 coupon payments conditional on survival).

11.4 CDS Applications

• Hedging credit exposure: A bank holding corporate bonds buys CDS protection to neutralise credit risk while retaining the bonds (useful for relationship lending where the bank cannot sell the loan).
• Synthetic short: Buying CDS without holding the underlying bonds is a bet that the reference entity will default or its credit quality will deteriorate (widening spreads increase CDS mark-to-market value for the protection buyer).
• Basis trading: The CDS-bond basis is \( s - z \), where \( z \) is the Z-spread of the bond (yield over the benchmark). Arbitrageurs exploit deviations from theoretical parity.

11.5 Synthetic CDOs

A collateralised debt obligation (CDO) tranches the credit risk from a reference portfolio. A synthetic CDO uses a portfolio of CDS exposures:

• Equity tranche (0-3%): Absorbs first losses, receives highest spread, highest risk.
• Mezzanine tranche (3-6%): Second loss position.
• Senior tranche (6-9%): Near-investment-grade, receives small spread.
• Super-senior tranche (9-100%): Rated AAA, receives minimal spread.

The key insight is that tranching redistributes correlation risk: equity tranches suffer from high default correlation (contagion leads to simultaneous defaults), while senior tranches are most exposed to systematic risk (they survive idiosyncratic defaults but not systemic crises).

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Chapter 12: Exotic Options

12.1 Overview of Exotics

Exotic options have payoffs that are more complex than standard European or American puts and calls. They are primarily traded OTC and serve specific hedging or structured product needs.

12.2 Barrier Options

Key types:

• Down-and-out call: Standard call that ceases to exist if \( S \) falls below \( H < S_0 \). Cheaper than a vanilla call since it offers less protection.
• Down-and-in call: Activated only if \( S \) falls below \( H \). Has value only if a prior drop occurs.
• Up-and-out call: Extinguished if \( S \) rises above \( H > S_0 \). Used by structured products to reduce cost.
• Up-and-in call: Activated if \( S \) rises above \( H \).

In-out parity (model-free):

\[ \text{Knock-in option} + \text{Knock-out option} = \text{Vanilla option} \]

Barrier options have closed-form BSM-style formulas but pose hedging challenges: near the barrier, delta and gamma can be extremely large and change sign rapidly, requiring frequent rebalancing and creating “pin risk” (the hedger’s P&L depends sensitively on whether the barrier is hit).

12.3 Asian Options

Types:

• Average price call: \( \max(\bar{S} - K, 0) \) where \( \bar{S} \) is the arithmetic or geometric average of \( S \) observed at discrete monitoring dates.
• Average strike call: \( \max(S_T - \bar{S}, 0) \) — pays out if the terminal price exceeds the average (you “bought” at the average).

Asian options are cheaper than vanilla options because the average is less volatile than the terminal price (by a factor of approximately \( \sqrt{1/3} \) for a geometric average under GBM). They are widely used by corporations hedging average commodity prices. Geometric-average options admit closed-form BSM solutions; arithmetic-average options require Monte Carlo simulation or log-normal approximations (Levy, 1992; Turnbull-Wakeman).

12.4 Lookback Options

• Floating-strike lookback call: \( S_T - \min_{0 \leq t \leq T} S_t \). Buy at the lowest price observed — no regret option.
• Fixed-strike lookback call: \( \max(\max_{0 \leq t \leq T} S_t - K, 0) \). Exercise at the highest price — always optimally timed.

Lookback options are expensive (typically 2–3× the ATM vanilla price) and have closed-form solutions under GBM (Goldman, Sosin, Gatto, 1979). They are used in structured products marketed with “get the best of the market” language.

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Chapter 13: Interest Rate Futures and Duration

13.1 Treasury Bond Futures

Treasury bond futures (CME/CBOT) allow participants to lock in the price of a Treasury bond for future delivery. The short party can deliver any bond from a basket of eligible bonds, choosing the cheapest-to-deliver (CTD) bond.

Invoice price received by the short at delivery:

\[ \text{Invoice price} = F_{\text{futures}} \times CF_{\text{CTD}} + \text{Accrued interest} \]

The conversion factor \( CF \) is the price that a bond would have per dollar of face value if it were to yield 6% (the CBOT standard coupon). It normalises the heterogeneous basket to a common standard.

The CTD bond minimises the short’s cost:

\[ \text{CTD} = \arg\min_i \left[ \text{Quoted price}_i - F \times CF_i \right] \]

In a normal yield curve environment, the CTD tends to be the bond with the longest duration and lowest coupon (highest convexity). As rates change, the CTD can switch — creating delivery option value for the short.

13.2 Duration-Based Hedging

To hedge a bond portfolio against a parallel yield shift using Treasury futures:

\[ N^* = -\frac{P \cdot D_P^*}{V_F \cdot D_F^*} \]

where \( D_F^* = D_{\text{CTD}}^* / CF_{\text{CTD}} \) adjusts for the conversion factor, and \( V_F \) is the futures contract dollar value.

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Chapter 14: Value at Risk and Expected Shortfall

14.1 Value at Risk

For a portfolio with normally distributed daily P&L (mean \( \mu \approx 0 \), standard deviation \( \sigma \)):

\[ \text{VaR}_\alpha = z_\alpha \sigma \]

where \( z_{0.99} = 2.326 \), \( z_{0.95} = 1.645 \). For multi-day VaR under i.i.d. normality:

\[ \text{VaR}_{h\text{-day}} = \text{VaR}_{1\text{-day}} \times \sqrt{h} \]

Basel III requires banks to compute 10-day, 99% VaR for market risk capital. The Fundamental Review of the Trading Book (FRTB, 2019) shifts to 10-day, 97.5% Expected Shortfall.

14.2 Limitations of VaR

1. Not subadditive: \( \text{VaR}(A+B) \leq \text{VaR}(A) + \text{VaR}(B) \) can fail in general, violating the diversification principle for a coherent risk measure.
2. Tail-blind: VaR says nothing about the magnitude of losses beyond the threshold.
3. Distribution assumption: Under fat-tailed distributions (Student-t, empirical), normal-VaR severely underestimates tail risk.
4. Procyclicality: VaR-based limits can force simultaneous de-risking across institutions, amplifying market crises.

14.3 Expected Shortfall

For the example above (\( \sigma = 1.5\% \), 99% confidence): \( N'(2.326) = 0.0267 \), \( \text{ES}_{0.99} = 0.015 \times 0.0267/0.01 = 0.015 \times 2.67 = 4.01\% \). Dollar ES: \( 4.01\% \times \$100M = \$4.01M \), which is larger than the $3.5M VaR as expected.

ES captures information about the entire tail, making it a more robust risk metric. FRTB mandates ES at 97.5% confidence for the trading book.

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Chapter 15: Corporate Hedging Applications

15.1 FX Hedging for Multinationals

Multinationals face three dimensions of currency risk:

• Transaction exposure: Contractual cash flows in foreign currency (e.g., USD receivable from an export sale). Most directly hedgeable with forwards, futures, or options.
• Translation exposure: Impact on consolidated balance sheet of converting foreign subsidiary financials into the parent’s reporting currency (IFRS or US GAAP).
• Economic exposure: Long-run impact of exchange rates on competitive position. Requires operational hedging (production geography, pricing strategy) in addition to financial instruments.

15.2 The Lufthansa Hedging Decision

The 1985 Lufthansa case is a classic teaching case. Lufthansa ordered Boeing aircraft priced in USD. The CFO hedged 50% of the exposure with forward purchases of USD, leaving the other 50% unhedged. When the USD subsequently fell against the DEM, the unhedged portion benefited (USD purchased more cheaply at spot), while the forward hedge generated losses relative to spot. The CFO faced public criticism despite having demonstrably reduced risk.

The Lufthansa case illustrates the evaluation problem in corporate hedging:

1. Hedging outcomes are judged ex post against the alternative (unhedged outcome or spot), not ex ante against the counterfactual risk.
2. A hedge that protects against adverse moves will, by construction, underperform when the market moves favourably.
3. The correct benchmark is the reduction in variance of cash flows, not whether the hedge generated a gain or loss in hindsight.

15.3 Commodity Hedging for Producers

An oil producer with planned output of 1 million barrels faces price risk. With \( S_0 = \$80 \)/bbl and 6-month futures at \( F_0 = \$82 \)/bbl:

1. Short futures: Sell 1,000 NYMEX WTI futures (1,000 bbl/contract). Revenue locked in approximately $82/bbl (subject to basis risk if the producer’s location differs from the Cushing delivery point).

2. Buy puts: Purchase $75 put options for a premium of $3/bbl. This establishes a $75 price floor while retaining upside above $75. Effective floor net of premium: $72/bbl.

3. Collar: Buy $75 puts, sell $90 calls. Premiums roughly offset. Revenue bounded between $75 and $90/bbl.

The choice depends on the firm’s views, balance sheet constraints, and the cost of financial distress. Airlines and utilities frequently hedge 60–80% of near-term fuel costs.

15.4 The Decision to Hedge

In the Modigliani-Miller world with frictionless markets, hedging does not add value — shareholders can diversify commodity and FX risk in their own portfolios. In practice, hedging can create value by:

• Reducing expected costs of financial distress.
• Smoothing taxable income, reducing expected taxes when the tax schedule is convex.
• Avoiding underinvestment: stable operating cash flows reduce reliance on costly external financing for capital expenditure.
• Reducing agency problems: predictable cash flows reduce information asymmetries between managers and outside investors.

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Chapter 16: BSM Extensions — Stochastic Volatility and Jumps

16.1 Jump-Diffusion Models

BSM assumes continuous price paths. In practice, asset prices jump discontinuously on news. Merton (1976) extended BSM by adding a Poisson jump process:

\[ dS = (\mu - \lambda \bar{k})S\,dt + \sigma S\,dW + J\,S\,dN_t \]

where \( N_t \) is a Poisson process with intensity \( \lambda \) (jumps per year), \( J \) is the random jump size (often log-normal), and \( \bar{k} = \mathbb{E}[J] - 1 \). The Merton jump-diffusion model generates fatter tails and a volatility smile, better matching observed OTM option prices. The formula is a mixture of BSM prices with different effective volatilities.

16.2 The Heston Stochastic Volatility Model

The Heston (1993) model allows variance to follow a mean-reverting CIR process:

\[ dS = \mu S\,dt + \sqrt{v}\,S\,dW_1 \]\[ dv = \kappa(\theta - v)\,dt + \xi\sqrt{v}\,dW_2, \quad d\langle W_1, W_2\rangle = \rho\,dt \]

Parameters: \( \kappa \) (mean-reversion speed), \( \theta \) (long-run variance), \( \xi \) (vol of vol), \( \rho \) (correlation). The correlation \( \rho < 0 \) for equities (stocks fall when vol rises) generates the left skew observed in equity options. The Heston model has a semi-analytic pricing formula via characteristic functions / Fourier inversion and is widely used for calibrating the vol surface.

16.3 Local Volatility

Dupire (1994) showed that given any arbitrage-free implied volatility surface \( \sigma_{\text{imp}}(K, T) \), there exists a unique local volatility function \( \sigma_{\text{loc}}(S, t) \) consistent with all observed option prices:

\[ \sigma_{\text{loc}}^2(K, T) = \frac{\partial C/\partial T + (r-q)K\,\partial C/\partial K + qC}{\frac{1}{2}K^2\,\partial^2 C/\partial K^2} \]

Local vol models perfectly calibrate to the market smile and are tractable via PDEs, making them popular for pricing path-dependent exotics. However, their forward vol dynamics differ from empirical observations — future smiles implied by local vol models are flatter than current smiles — limiting their use for forward-starting options and cliquets.

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Summary: Key Formulas Reference

Forwards and Futures

\[ F_0 = S_0 e^{(r-q)T} \quad \text{(general cost-of-carry)} \]

Put-Call Parity

\[ C + Ke^{-rT} = P + S_0 e^{-qT} \]

Black-Scholes-Merton

\[ C = S_0 e^{-qT} N(d_1) - Ke^{-rT} N(d_2), \qquad P = Ke^{-rT}N(-d_2) - S_0 e^{-qT}N(-d_1) \]\[ d_1 = \frac{\ln(S_0/K) + (r-q+\sigma^2/2)T}{\sigma\sqrt{T}}, \qquad d_2 = d_1 - \sigma\sqrt{T} \]

Greeks (European, BSM)

Greek	Call	Put
Delta \( \Delta \)	\( e^{-qT}N(d_1) \)	\( -e^{-qT}N(-d_1) \)
Gamma \( \Gamma \)	\( \frac{N'(d_1)e^{-qT}}{S_0\sigma\sqrt{T}} \)	Same as call
Vega \( \mathcal{V} \)	\( S_0\sqrt{T}N'(d_1)e^{-qT} \)	Same as call
Rho \( \rho \)	\( KTe^{-rT}N(d_2) \)	\( -KTe^{-rT}N(-d_2) \)

Hedging

Application	Formula
Min-variance hedge ratio	\( h^* = \rho\,\sigma_S/\sigma_F \)
Optimal contracts	\( N^* = h^* \times Q_A/Q_F \)
Beta hedge	\( N^* = (\beta^* - \beta) \times P/A \)
Duration hedge	\( N^* = -P D_P^* / (V_F D_F^*) \)

Swap Rate

\[ R_{\text{swap}} = \frac{1 - e^{-r_n T_n}}{\displaystyle\sum_{i=1}^n \delta_i e^{-r_i t_i}} \]

CDS Spread (Approximation)

\[ s \approx (1-R)\,\lambda, \qquad \lambda \approx \frac{s}{1-R} \]

VaR and ES (Normal)

\[ \text{VaR}_\alpha = z_\alpha\,\sigma, \qquad \text{ES}_\alpha = \sigma\,\frac{N'(z_\alpha)}{1-\alpha} \]

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