Paul Wilmott on Quantitive Finance 2nd Edition by Paul Wilmott 0470018704 9780470018705
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ISBN 10: 0470018704
ISBN 13: 9780470018705
Author: Paul Wilmott
Volume 1: Mathematical and Financial Foundations; Basic Theory of Derivatives; Risk and Return.
The reader is introduced to the fundamental mathematical tools and financial concepts needed to understand quantitative finance, portfolio management and derivatives. Parallels are drawn between the respectable world of investing and the not-so-respectable world of gambling.
Volume 2: Exotic Contracts and Path Dependency; Fixed Income Modeling and Derivatives; Credit Risk
In this volume the reader sees further applications of stochastic mathematics to new financial problems and different markets.
Volume 3: Advanced Topics; Numerical Methods and Programs.
In this volume the reader enters territory rarely seen in textbooks, the cutting-edge research. Numerical methods are also introduced so that the models can now all be accurately and quickly solved.
Throughout the volumes, the author has included numerous Bloomberg screen dumps to illustrate in real terms the points he raises, together with essential Visual Basic code, spreadsheet explanations of the models, the reproduction of term sheets and option classification tables. In addition to the practical orientation of the book the author himself also appears throughout the book—in cartoon form, readers will be relieved to hear—to personally highlight and explain the key sections and issues discussed.
Paul Wilmott on Quantitive Finance 2nd Table of contents:
PART ONE MATHEMATICAL AND FINANCIAL FOUNDATIONS; BASIC THEORY OF DERIVATIVES; RISK AND RETURN
1 Products and Markets
1.1 Introduction
1.2 The time value of money
1.3 Equities
1.3.1 Dividends
1.3.2 Stock splits
1.4 Commodities
1.5 Currencies
1.6 Indices
1.7 Fixed-income securities
1.8 Inflation-proof bonds
1.9 Forwards and futures
1.9.1 A first example of no arbitrage
1.10 Summary
2 Derivatives
2.1 Introduction
2.2 Options
2.3 Definition of common terms
2.4 Payoff diagrams
2.4.1 Other representations of value
2.5 Writing options
2.6 Margin
2.7 Market conventions
2.8 The value of the option before expiry
2.9 Factors affecting derivatives prices
2.10 Speculation and gearing
2.11 Early exercise
2.12 Put-call parity
2.13 Binaries or digitals
2.14 Bull and bear spreads
2.15 Straddles and strangles
2.16 Risk reversal
2.17 Butterflies and condors
2.18 Calendar spreads
2.19 LEAPS and FLEX
2.20 Warrants
2.21 Convertible bonds
2.22 Over the counter options
2.23 Summary
3 The Random Behavior of Assets
3.1 Introduction
3.2 The popular forms of ‘analysis’
3.3 Why we need a model for randomness: Jensen’s Inequality
3.4 Similarities between equities, currencies, commodities and indices
3.5 Examining returns
3.6 Timescales
3.6.1 The drift
3.6.2 The volatility
3.7 Estimating volatility
3.8 The random walk on a spreadsheet
3.9 The Wiener process
3.10 The widely accepted model for equities, currencies, commodities and indices
3.11 Summary
4 Elementary Stochastic Calculus
4.1 Introduction
4.2 A motivating example
4.3 The Markov property
4.4 The martingale property
4.5 Quadratic variation
4.6 Brownian motion
4.7 Stochastic integration
4.8 Stochastic differential equations
4.9 The mean square limit
4.10 Functions of stochastic variables and Itˆo’s lemma
4.11 Interpretation of Itˆo’s lemma
4.12 Itˆo and Taylor
4.12.1 The intuition
4.12.2 Simple generalization
4.13 Itˆo in higher dimensions
4.14 Some pertinent examples
4.14.1 Brownian motion with drift
4.14.2 The lognormal random walk
4.14.3 A mean-reverting random walk
4.14.4 And another mean-reverting random walk
4.15 Summary
5 The Black–Scholes Model
5.1 Introduction
5.2 A very special portfolio
5.3 Elimination of risk: delta hedging
5.4 No arbitrage
5.5 The Black–Scholes equation
5.6 The Black–Scholes assumptions
5.7 Final conditions
5.8 Options on dividend-paying equities
5.9 Currency options
5.10 Commodity options
5.11 Options on futures
5.12 Some other ways of deriving the Black–Scholes equation
5.12.1 The martingale approach
5.12.2 The binomial model
5.12.3 CAPM/utility
5.13 Summary
6 Partial Differential Equations
6.1 Introduction
6.2 Putting the Black–Scholes equation into historical perspective
6.3 The meaning of the terms in the Black–Scholes equation
6.4 Boundary and initial/final conditions
6.5 Some solution methods
6.5.1 Transformation to constant coefficient diffusion equation
6.5.2 Green’s functions
6.5.3 Series solution
6.6 Similarity reductions
6.7 Other analytical techniques
6.8 Numerical solution
6.9 Summary
7 The Black–Scholes Formulae and the ‘Greeks’
7.1 Introduction
7.2 Derivation of the formulae for calls, puts and simple digitals
7.2.1 Formula for a call
7.2.2 Formula for a put
7.2.3 Formula for a binary call
7.2.4 Formula for a binary put
7.3 Delta
7.4 Gamma
7.5 Theta
7.6 Speed
7.7 Vega
7.8 Rho
7.9 Implied volatility
7.10 A classification of hedging types
7.10.1 Why hedge?
7.10.2 The two main classifications
7.10.3 Delta hedging
7.10.4 Gamma hedging
7.10.5 Vega hedging
7.10.6 Static hedging
7.10.7 Margin hedging
7.10.8 Crash (platinum) hedging
7.11 Summary
8 Simple Generalizations of the Black–Scholes World
8.1 Introduction
8.2 Dividends, foreign interest and cost of carry
8.3 Dividend structures
8.4 Dividend payments and no arbitrage
8.5 The behavior of an option value across a dividend date
8.6 Commodities
8.6.1 Futures prices and arbitrage
8.6.2 Storage costs
8.6.3 Convenience yield
8.6.4 Cost of carry
8.6.5 Effect on options
8.7 Stock borrowing and repo
8.8 Time-dependent parameters
8.9 Formulae for power options
8.10 The log contract
8.11 Summary
9 Early Exercise and American Options
9.1 Introduction
9.2 The perpetual American put
9.3 Perpetual American call with dividends
9.4 Mathematical formulation for general payoff
9.5 Local solution for call with constant dividend yield
9.6 Other dividend structures
9.7 One-touch options
9.8 Other features in American-style contracts
9.8.1 Bermudan options
9.8.2 Make your mind up
9.9 Other issues
9.9.1 Non-linearity
9.9.2 Free-boundary problems
9.9.3 Numerical solution
9.10 Summary
10 Probability Density Functions and First-exit Times
10.1 Introduction
10.2 The transition probability density function
10.3 A trinomial model for the random walk
10.4 The forward equation
10.5 The steady-state distribution
10.6 The backward equation
10.7 First-exit times
10.8 Cumulative distribution functions for first-exit times
10.9 Expected first-exit times
10.10 Another example of optimal stopping
10.11 Expectations and Black–Scholes
10.12 A common misconception
10.13 Summary
11 Multi-asset Options
11.1 Introduction
11.2 Multi-dimensional lognormal random walks
11.3 Measuring correlations
11.4 Options on many underlyings
11.5 The pricing formula for European non-path-dependent options on dividend-paying assets
11.6 Exchanging one asset for another: A similarity solution
11.7 Quantos
11.8 Two examples
11.9 Other features
11.10 Realities of pricing basket options
11.10.1 Easy problems
11.10.2 Medium problems
11.10.3 Hard problems
11.11 Realities of hedging basket options
11.12 Correlation versus cointegration
11.13 Summary
12 How to Delta Hedge
12.1 Introduction
12.2 What if implied and actual volatilities are different?
12.3 Implied versus actual; delta hedging but using which volatility?
12.4 Case 1: Hedge with actual volatility, σ
12.5 Case 2: Hedge with implied volatility, σ
12.5.1 The expected profit after hedging using implied volatility
12.5.2 The variance of profit after hedging using implied volatility
12.5.3 Hedging with different volatilities
12.6 Portfolios when hedging with implied volatility
12.6.1 Expectation
12.6.2 Variance
12.6.3 Portfolio optimization possibilities
12.7 Hedging when implied volatility is stochastic
12.7.1 Case 1: Hedge with actual volatility, σ
12.7.2 Case 2: Hedge with implied volatility, σ?
12.8 How does implied volatility behave?
12.8.1 Sticky strike
12.8.2 Sticky delta
12.8.3 Time-periodic behavior
12.9 Summary
13 Fixed-income Products and Analysis: Yield, Duration and Convexity
13.1 Introduction
13.2 Simple fixed-income contracts and features
13.2.1 The zero-coupon bond
13.2.2 The coupon-bearing bond
13.2.3 The money market account
13.2.4 Floating rate bonds
13.2.5 Forward rate agreements
13.2.6 Repos
13.2.7 STRIPS
13.2.8 Amortization
13.2.9 Call provision
13.3 International bond markets
13.3.1 United States of America
13.3.2 United Kingdom
13.3.3 Japan
13.4 Accrued interest
13.5 Day-count conventions
13.6 Continuously and discretely compounded interest
13.7 Measures of yield
13.7.1 Current yield
13.7.2 The yield to maturity (YTM) or internal rate of return (IRR)
13.8 The yield curve
13.9 Price/yield relationship
13.10 Duration
13.11 Convexity
13.12 An example
13.13 Hedging
13.14 Time-dependent interest rate
13.15 Discretely paid coupons
13.16 Forward rates and bootstrapping
13.16.1 Discrete data
13.16.2 On a spreadsheet
13.17 Interpolation
13.18 Summary
14 Swaps
14.1 Introduction
14.2 The vanilla interest rate swap
14.3 Comparative advantage
14.4 The swap curve
14.5 Relationship between swaps and bonds
14.6 Bootstrapping
14.7 Other features of swaps contracts
14.8 Other types of swap
14.8.1 Basis rate swap
14.8.2 Equity swaps
14.8.3 Currency swaps
14.9 Summary
15 The Binomial Model
15.1 Introduction
15.2 Equities can go down as well as up
15.3 The option value
15.4 Which part of our ‘model’ didn’t we need?
15.5 Why should this ‘theoretical price’ be the ‘market price’?
15.5.1 The role of expectations
15.6 How did I know to sell 12 of the stock for hedging?
15.6.1 The general formula for
15.7 How does this change if interest rates are non-zero?
15.8 Is the stock itself correctly priced?
15.9 Complete markets
15.10 The real and risk-neutral worlds
15.10.1 Non-zero interest rates
15.11 And now using symbols
15.11.1 Average asset change
15.11.2 Standard deviation of asset price change
15.12 An equation for the value of an option
15.12.1 Hedging
15.12.2 No arbitrage
15.13 Where did the probability p go?
15.14 Counterintuitive?
15.15 The binomial tree
15.16 The asset price distribution
15.17 Valuing back down the tree
15.18 Programming the binomial method
15.19 The greeks
15.20 Early exercise
15.21 The continuous-time limit
15.22 No arbitrage in the binomial, Black–Scholes and ‘other’ worlds
15.23 Summary
16 How Accurate is the Normal Approximation?
16.1 Introduction
16.2 Why we like the Normal distribution: the Central Limit Theorem
16.3 Normal versus lognormal
16.4 Does my tail look fat in this?
16.4.1 Probability of a 20% SPX fall: empirical
16.4.2 Probability of a 20% SPX fall: theoretical
16.5 Use a different distribution, perhaps
16.6 Serial Autocorrelation
16.7 Summary
17 Investment Lessons from Blackjack and Gambling
17.1 Introduction
17.2 The rules of blackjack
17.3 Beating the dealer
17.3.1 Summary of winning at blackjack
17.4 The distribution of profit in blackjack
17.5 The Kelly criterion
17.6 Can you win at roulette?
17.7 Horse race betting and no arbitrage
17.7.1 Setting the odds in a sporting game
17.7.2 The mathematics
17.8 Arbitrage
17.8.1 How best to profit from the opportunity?
17.9 How to bet
17.10 Summary
18 Portfolio Management
18.1 Introduction
18.2 Diversification
18.2.1 Uncorrelated assets
18.3 Modern Portfolio Theory
18.3.1 Including a risk-free investment
18.4 Where do I want to be on the efficient frontier?
18.5 Markowitz in practice
18.6 Capital Asset Pricing Model
18.6.1 The single-index model
18.6.2 Choosing the optimal portfolio
18.7 The multi-index model
18.8 Cointegration
18.9 Performance measurement
18.10 Summary
19 Value at Risk
19.1 Introduction
19.2 Definition of value at risk
19.3 VaR for a single asset
19.4 VaR for a portfolio
19.5 VaR for derivatives
19.5.1 The delta approximation
19.5.2 Which volatility do I use?
19.5.3 The delta-gamma approximation
19.5.4 Use of valuation models
19.5.5 Fixed-income portfolios
19.6 Simulations
19.6.1 Monte Carlo
19.6.2 Bootstrapping
19.7 Use of VaR as a performance measure
19.8 Introductory extreme value theory
19.8.1 Some EVT results
19.9 Coherence
19.10 Summary
20 Forecasting the Markets?
20.1 Introduction
20.2 Technical analysis
20.2.1 Plotting
20.2.2 Support and resistance
20.2.3 Trendlines
20.2.4 Moving averages
20.2.5 Relative strength
20.2.6 Oscillators
20.2.7 Bollinger bands
20.2.8 Miscellaneous patterns
20.2.9 Japanese candlesticks
20.2.10 Point and figure charts
20.3 Wave theory
20.3.1 Elliott waves and Fibonacci numbers
20.3.2 Gann charts
20.4 Other analytics
20.5 Market microstructure modeling
20.5.1 Effect of demand on price
20.5.2 Combining market microstructure and option theory
20.5.3 Imitation
20.6 Crisis prediction
20.7 Summary
21 A Trading Game
21.1 Introduction
21.2 Aims
21.3 Object of the game
21.4 Rules of the game
21.5 Notes
21.6 How to fill in your trading sheet
21.6.1 During a trading round
21.6.2 At the end of the game
PART TWO EXOTIC CONTRACTS AND PATH DEPENDENCY
22 An Introduction to Exotic and Path-dependent Derivatives
22.1 Introduction
22.2 Option classification
22.3 Time dependence
22.4 Cashflows
22.5 Path dependence
22.5.1 Strong path dependence
22.5.2 Weak path dependence
22.6 Dimensionality
22.7 The order of an option
22.8 Embedded decisions
22.9 Classification tables
22.10 Examples of exotic options
22.10.1 Compounds and choosers
22.10.2 Range notes
22.10.3 Barrier options
22.10.4 Asian options
22.10.5 Lookback options
22.11 Summary of math/coding consequences
22.12 Summary
23 Barrier Options
23.1 Introduction
23.2 Different types of barrier options
23.3 Pricing methodologies
23.3.1 Monte Carlo simulation
23.3.2 Partial differential equations
23.4 Pricing barriers in the partial differential equation framework
23.4.1 ‘Out’ barriers
23.4.2 ‘In’ barriers
23.4.3 Some formulae when volatility is constant
23.4.4 Some more examples
23.5 Other features in barrier-style options
23.5.1 Early exercise
23.5.2 The intermittent barrier
23.5.3 Repeated hitting of the barrier
23.5.4 Resetting of barrier
23.5.5 Outside barrier options
23.5.6 Soft barriers
23.5.7 Parisian options
23.5.8 The emergency exit
23.6 First-exit time
23.7 Market practice: What volatility should I use?
23.8 Hedging barrier options
23.9 Slippage costs
23.10 Summary
24 Strongly Path-Dependent Derivatives
24.1 Introduction
24.2 Path-dependent quantities represented by an integral
24.2.1 Examples
24.3 Continuous sampling: The pricing equation
24.3.1 Example
24.4 Path-dependent quantities represented by an updating rule
24.4.1 Examples
24.5 Discrete sampling: The pricing equation
24.5.1 Examples
24.5.2 The algorithm for discrete sampling
24.6 Higher dimensions
24.7 Pricing via expectations
24.8 Early exercise
24.9 Summary
25 Asian Options
25.1 Introduction
25.2 Payoff types
25.3 Types of averaging
25.3.1 Arithmetic or geometric
25.3.2 Discrete or continuous
25.4 Solution methods
25.4.1 Monte Carlo simulation
25.5 Extending the Black–Scholes equation
25.5.1 Continuously sampled averages
25.5.2 Discretely sampled averages
25.5.3 Exponentially weighted and other averages
25.5.4 The Asian tail
25.6 Early exercise
25.7 Asian options in higher dimensions
25.8 Similarity reductions
25.8.1 Put-call parity for the European average strike
25.9 Closed-form solutions and approximations
25.9.1 Kemna and Vorst (1990)
25.9.2 Turnbull and Wakeman (1991)
25.9.3 Curran (1992)
25.9.4 Thompson (2000)
25.10 Term-structure effects
25.10.1 Some results
25.11 Some formulae
25.12 Summary
26 Lookback Options
26.1 Introduction
26.2 Types of payoff
26.3 Continuous measurement of the maximum
26.4 Discrete measurement of the maximum
26.5 Similarity reduction
26.6 Some formulae
26.7 Summary
27 Derivatives and Stochastic Control
27.1 Introduction
27.2 Perfect trader and passport options
27.2.1 Similarity solution
27.3 Limiting the number of trades
27.4 Limiting the time between trades
27.5 Non-optimal trading and the bene.ts to the writer
27.6 Summary
28 Miscellaneous Exotics
28.1 Introduction
28.2 Forward-start options
28.3 Shout options
28.4 Capped lookbacks and Asians
28.5 Combining path-dependent quantities: The lookback-Asian etc.
28.5.1 The maximum of the asset and the average of the asset
28.5.2 The average of the asset and the maximum of the average
28.5.3 The maximum of the asset and the average of the maximum
28.6 The volatility option
28.6.1 The continuous-time limit
28.6.2 Hedging variance swaps with vanilla options
28.7 Correlation swap
28.7.1 Dispersion trading
28.8 Ladders
28.9 Parisian options
28.9.1 Examples
28.10 Yet more exotics
28.11 Summary
29 Equity and FX Term Sheets
29.1 Introduction
29.2 Contingent premium put
29.3 Basket options
29.3.1 Simple basket option
29.3.2 Basket option with averaging over time
29.4 Knockout options
29.4.1 Double knockout
29.4.2 Instalment knockout
29.5 Range notes
29.5.1 A really simple range note
29.6 Lookbacks
29.7 Cliquet option
29.7.1 Path dependency, constant volatility
29.8 Passport options
29.9 Decomposition of exotics into vanillas
PART THREE FIXED-INCOME MODELING AND DERIVATIVES
30 One-factor Interest Rate Modeling
30.1 Introduction
30.2 Stochastic interest rates
30.3 The bond pricing equation for the general model
30.4 What is the market price of risk?
30.5 Interpreting the market price of risk, and risk neutrality
30.6 Tractable models and solutions of the bond pricing equation
30.7 Solution for constant parameters
30.8 Named models
30.8.1 Vasicek
30.8.2 Cox, Ingersoll & Ross
30.8.3 Ho & Lee
30.8.4 Hull & White
30.9 Equity and FX forwards and futures when rates are stochastic
30.9.1 Forward contracts
30.9.2 Futures contracts
30.9.3 The convexity adjustment
30.10 Summary
31 Yield Curve Fitting
31.1 Introduction
31.2 Ho & Lee
31.3 The extended Vasicek model of Hull & White
31.4 Yield-curve fitting: For and against
31.4.1 For
31.4.2 Against
31.5 Other models
31.6 Summary
32 Interest Rate Derivatives
32.1 Introduction
32.2 Callable bonds
32.3 Bond options
32.3.1 Market practice
32.4 Caps and floors
32.4.1 Cap/floor parity
32.4.2 The relationship between a caplet and a bond option
32.4.3 Market practice
32.4.4 Collars
32.4.5 Step-up swaps, caps and .oors
32.5 Range notes
32.6 Swaptions, captions and floortions
32.6.1 Market practice
32.7 Spread options
32.8 Index amortizing rate swaps
32.8.1 Other features in the index amortizing rate swap
32.9 Contracts with embedded decisions
32.10 When the interest rate is not the spot rate
32.10.1 The relationship between the spot interest rate and other rates
32.11 Some examples
32.12 More interest rate derivatives
32.13 Summary
33 Convertible Bonds
33.1 Introduction
33.2 Convertible bond basics
33.2.1 What are CBs for?
33.2.2 The issuers of CBs
33.2.3 Why issue a convertible?
33.2.4 Why buy a convertible?
33.2.5 Some statistics
33.3 Market practice
33.4 Converts as options
33.5 Pricing CBs with known interest rate
33.5.1 Call and put features
33.6 Two-factor modeling: Convertible bonds with stochastic interest rate
33.7 A special model
33.8 Path dependence in convertible bonds
33.9 Dilution
33.10 Credit risk issues
33.11 Summary
34 Mortgage-backed Securities
34.1 Introduction
34.2 Individual mortgages
34.2.1 Monthly payments in the fixed rate mortgage
34.2.2 Prepayment
34.3 Mortgage-backed securities
34.3.1 The issuers
34.4 Modeling prepayment
34.4.1 The statistics of repayment
34.4.2 The PSA model
34.4.3 More realistic models
34.5 Valuing MBSs
34.6 Summary
35 Multi-factor Interest Rate Modeling
35.1 Introduction
35.2 Theoretical framework for two factors
35.2.1 Special case: Modeling a long-term rate
35.2.2 Special case: Modeling the spread between the long and the short rate
35.3 Popular models
35.4 The market price of risk as a random factor
35.5 The phase plane in the absence of randomness
35.6 The yield curve swap
35.7 General multi-factor theory
35.7.1 Tractable affine models
35.8 Summary
36 Empirical Behavior of the Spot Interest Rate
36.1 Introduction
36.2 Popular one-factor spot-rate models
36.3 Implied modeling: One factor
36.4 The volatility structure
36.5 The drift structure
36.6 The slope of the yield curve and the market price of risk
36.7 What the slope of the yield curve tells us
36.8 Properties of the forward rate curve ‘on average’
36.9 Implied modeling: Two factor
36.10 Summary
37 The Heath, Jarrow & Morton and Brace, Gatarek & Musiela Models
37.1 Introduction
37.2 The forward rate equation
37.3 The spot rate process
37.3.1 The non-Markov nature of HJM
37.4 The market price of risk
37.5 Real and risk neutral
37.5.1 The relationship between the risk-neutral forward rate drift and volatility
37.6 Pricing derivatives
37.7 Simulations
37.8 Trees
37.9 The Musiela parameterization
37.10 Multi-factor HJM
37.11 Spreadsheet implementation
37.12 A simple one-factor example: Ho & Lee
37.13 Principal Component Analysis
37.13.1 The power method
37.14 Options on equities etc.
37.15 Non-in.nitesimal short rate
37.16 The Brace, Gatarek and Musiela model
37.17 Simulations
37.18 PVing the cashflows
37.19 Summary
38 Fixed-income Term Sheets
38.1 Introduction
38.2 Chooser range note
38.2.1 Optimal choice of ranges?
38.2.2 Pricing
38.2.3 Differences between optimal for the writer and the buyer
38.3 Index amortizing rate swap
38.3.1 Similarity solution
38.3.2 The code
PART FOUR CREDIT RISK
39 Value of the Firm and the Risk of Default
39.1 Introduction
39.2 The Merton model: Equity as an option on a company’s assets
39.2.1 Default before maturity
39.2.2 Probability of default
39.2.3 Stochastic interest rates
39.3 Modeling with measurable parameters and variables
39.4 Calculating the value of the firm
39.5 Summary
40 Credit Risk
40.1 Introduction
40.2 Risky bonds
40.3 Modeling the risk of default
40.4 The Poisson process and the instantaneous risk of default
40.4.1 A note on hedging
40.5 Time-dependent intensity and the term structure of default
40.6 Stochastic risk of default
40.7 Positive recovery
40.8 Special cases and yield curve .tting
40.9 A case study: The Argentine Par bond
40.10 Hedging the default
40.11 Is there any information content in the market price?
40.11.1 Implied hazard rate and duration
40.12 Credit rating
40.13 A model for change of credit rating
40.13.1 The forward equation
40.13.2 The backward equation
40.14 The pricing equation
40.14.1 Constant interest rates
40.14.2 Stochastic interest rates
40.15 Credit risk in CBs
40.15.1 Bankruptcy when stock reaches a critical level
40.15.2 Incorporating the instantaneous risk of default
40.16 Modeling liquidity risk
40.17 Summary
41 Credit Derivatives
41.1 Introduction
41.2 What are Credit Derivatives?
41.2.1 Uses of credit derivatives: Banks
41.2.2 Uses of credit derivatives: Investors
41.2.3 Uses of credit derivatives: Corporates
41.3 Popular credit derivatives
41.3.1 Asset swap
41.3.2 Total return swaps or total rate of return swaps
41.4 Derivatives triggered by default
41.4.1 Basic definitions
41.4.2 What defines a ‘credit event’?
41.4.3 Credit default swap
41.4.4 Limited recourse note
41.4.5 First to default
41.4.6 nth to default
41.5 Derivatives of the yield spread
41.5.1 Default calls and puts
41.5.2 Exchange options
41.5.3 Credit spread options
41.6 Payment on change of rating
41.7 Using default swaps in CB arbitrage
41.7.1 Exploiting your credit risk view
41.8 Term sheets
41.8.1 Put on credit spread on XYZ bond
41.8.2 Binary payoff bond
41.8.3 Digital spread option, one-year note linked to Venezuelan bond
41.8.4 Basket credit-linked note
41.8.5 Asset swap put option on one year ABC 6.65% bonds 10/2006 asset swap
41.9 Pricing credit derivatives
41.10 An exchange option
41.11 Default only when payment is due
41.11.1 The market’s estimate of default risk
41.11.2 Hedging
41.12 Payoff on change of rating
41.13 Multi-factor derivatives
41.14 Copulas: Pricing credit derivatives with many underlyings
41.14.1 The copula function
41.14.2 The mathematical definition
41.14.3 Examples of copulas
41.15 Collateralized Debt Obligations
41.16 Summary
42 RiskMetrics and CreditMetrics
42.1 Introduction
42.2 The RiskMetrics datasets
42.3 Calculating the parameters the RiskMetrics way
42.3.1 Estimating volatility
42.3.2 Correlation
42.4 The CreditMetrics dataset
42.4.1 Yield curves
42.4.2 Spreads
42.4.3 Transition matrices
42.4.4 Correlations
42.5 The CreditMetrics methodology
42.6 A portfolio of risky bonds
42.7 CreditMetrics model outputs
42.8 Summary
43 CrashMetrics
43.1 Introduction
43.2 Why do banks go broke?
43.3 Market crashes
43.4 CrashMetrics
43.5 CrashMetrics for one stock
43.6 Portfolio optimization and the Platinum Hedge
43.6.1 Other ‘cost’ functions
43.7 The multi-asset/single-index model
43.7.1 Assuming Taylor series for the moment
43.8 Portfolio optimization and the Platinum Hedge in the multi-asset model
43.8.1 The marginal effect of an asset
43.9 The multi-index model
43.10 Incorporating time value
43.11 Margin calls and margin hedging
43.11.1 What is margin?
43.11.2 Modeling margin
43.11.3 The single-index model
43.12 Counterparty risk
43.13 Simple extensions to CrashMetrics
43.14 The CrashMetrics Index (CMI)
43.15 Summary
44 Derivatives **** Ups
44.1 Introduction
44.2 Orange County
44.3 Proctor and Gamble
44.4 Metallgesellschaft
44.4.1 Basis risk
44.5 Gibson Greetings
44.6 Barings
44.7 Long-Term Capital Management
44.8 Summary
PART FIVE ADVANCED TOPICS
45 Financial Modeling
45.1 Introduction
45.2 Warning: Modeling as it is currently practiced
45.2.1 Models, a personal view
45.2.2 The find-and-replace school of mathematical modeling
45.3 Summary
46 Defects in the Black–Scholes Model
46.1 Introduction
46.2 Discrete hedging
46.3 Transaction costs
46.4 Overview of volatility modeling
46.5 Deterministic volatility surfaces
46.6 Stochastic volatility
46.7 Uncertain parameters
46.8 Empirical analysis of volatility
46.9 Stochastic volatility and mean-variance analysis
46.10 Asymptotic analysis of volatility
46.11 Jump diffusion
46.12 Crash modeling
46.13 Speculating with options
46.14 Optimal static hedging
46.15 The feedback effect of hedging in illiquid markets
46.16 Utility theory
46.17 More about American options and related matters
46.18 Advanced dividend modeling
46.19 Serial autocorrelation in returns
46.20 Summary
47 Discrete Hedging
47.1 Introduction
47.2 Motivating example: The trinomial model
47.3 A model for a discretely hedged position
47.4 A higher-order analysis
47.4.1 Choosing the best
47.4.2 The hedging error
47.4.3 Observations about the hedging error
47.4.4 Pricing the option
47.4.5 The adjusted and option value
47.5 The real distribution of returns and the hedging error
47.6 Total hedging error for the real distribution of returns
47.7 Which models allow perfect delta hedging
47.8 Summary
48 Transaction Costs
48.1 Introduction
48.2 The effect of costs
48.3 The model of Leland (1985)
48.4 The model of Hoggard, Whalley & Wilmott (1992)
48.5 Non-single-signed gamma
48.6 The marginal effect of transaction costs
48.7 Other cost structures
48.8 Hedging to a bandwidth: The model of Whalley & Wilmott (1993) and Henrotte (1993)
48.9 Utility-based models
48.9.1 The model of Hodges & Neuberger (1989)
48.9.2 The model of Davis, Panas & Zariphopoulou (1993)
48.9.3 The asymptotic analysis of Whalley & Wilmott (1993)
48.9.4 Arbitrary cost structure
48.10 Interpretation of the models
48.10.1 Nonlinearity
48.10.2 Negative option prices
48.10.3 Existence of solutions
48.11 Non-normal returns
48.12 Empirical testing
48.12.1 Black–Scholes and Leland hedging
48.12.2 Market movement or delta-tolerance strategy
48.12.3 The utility strategy
48.12.4 Using the real data
48.12.5 And the winner is. . .
48.13 Transaction costs and discrete hedging put together
48.14 Summary
49 Overview of Volatility Modeling
49.1 Introduction
49.2 The different types of volatility
49.2.1 Actual volatility
49.2.2 Historical or realized volatility
49.2.3 Implied volatility
49.2.4 Forward volatility
49.3 Volatility estimation by statistical means
49.3.1 The simplest volatility estimate: Constant volatility/moving window
49.3.2 Incorporating mean reversion
49.3.3 Exponentially weighted moving average
49.3.4 A simple GARCH model
49.3.5 Expected future volatility
49.3.6 Beyond close-close estimators: Range-based estimation of volatility
49.4 Maximum likelihood estimation
49.4.1 A simple motivating example: Taxi numbers
49.4.2 Three hats
49.4.3 The maths behind this: Find the standard deviation
49.4.4 Quants’ salaries
49.5 Skews and smiles
49.5.1 Sensitivity of the straddle to skews and smiles
49.5.2 Sensitivity of the risk reversal to skews and smiles
49.6 Different approaches to modeling volatility
49.6.1 To calibrate or not?
49.6.2 Deterministic volatility surfaces
49.6.3 Stochastic volatility
49.6.4 Uncertain parameters
49.6.5 Empirical analysis of volatility
49.6.6 Static hedging
49.6.7 Stochastic volatility and mean-variance analysis
49.6.8 Asymptotic analysis of volatility
49.6.9 Volatility case study: The cliquet option
49.7 The choices of volatility models
49.8 Summary
50 Deterministic Volatility Surfaces
50.1 Introduction
50.2 Implied volatility
50.3 Time-dependent volatility
50.4 Volatility smiles and skews
50.5 Volatility surfaces
50.6 Backing out the local volatility surface from European call option prices
50.7 A simple volatility surface parameterization
50.8 An approximate solution
50.9 Volatility information contained in an at-the-money straddle
50.10 Volatility information contained in a risk-reversal
50.11 Time dependence
50.12 A market convention
50.13 How do I use the local volatility surface?
50.14 Summary
51 Stochastic Volatility
51.1 Introduction
51.2 Random volatility
51.3 A stochastic differential equation for volatility
51.4 The pricing equation
51.5 The market price of volatility risk
51.5.1 Aside: The market price of risk for traded assets
51.6 The value as an expectation
51.7 An example
51.8 Choosing the model
51.9 Named/popular models
51.9.1 The Heston model
51.9.2 The REGARCH model and its diffusion limit
51.10 A note on biases
51.11 Stochastic implied volatility: The model of Sch¨onbucher
51.12 Summary
52 Uncertain Parameters
52.1 Introduction
52.2 Best and worst cases
52.2.1 Uncertain volatility: The model of Avellaneda, Levy & Par´as (1995) and Lyons (1995)
52.2.2 Example: An up-and-out call
52.2.3 Uncertain interest rate
52.2.4 Uncertain dividends
52.3 Uncertain correlation
52.4 Nonlinearity
52.5 Summary
53 Empirical Analysis of Volatility
53.1 Introduction
53.2 Stochastic volatility and uncertain parameters revisited
53.3 Deriving an empirical stochastic volatility model
53.4 Estimating the volatility of volatility
53.5 Estimating the drift of volatility
53.6 Out-of-sample results
53.7 How to use the model
53.7.1 Option pricing with stochastic volatility
53.7.2 The time evolution of stochastic volatility
53.7.3 Stochastic volatility, certainty bands and confidence limits
53.8 Summary
54 Stochastic Volatility and Mean-variance Analysis
54.1 Introduction
54.2 The model for the asset and its volatility
54.3 Analysis of the mean
54.4 Analysis of the variance
54.5 Choosing to minimize the variance
54.6 The mean and variance equations
54.7 How to interpret and use the mean and variance
54.8 Static hedging and portfolio optimization
54.9 Example: Valuing and hedging an up-and-out call
54.10 Static hedging
54.11 Other definitions of ‘value’
54.12 Summary
55 Asymptotic Analysis of Volatility
55.1 Introduction
55.2 Fast mean reversion and high volatility of volatility
55.3 Conditions on the models
55.4 Examples of models
55.4.1 Scott’s model
55.4.2 The Heston/Ball–Roma model
55.5 Notation
55.6 Asymptotic analysis
55.7 Vanilla options: Asymptotics for values
55.8 Vanilla options: Implied volatilities
55.9 Summary
56 Volatility Case Study: The Cliquet Option
56.1 Introduction
56.2 The subtle nature of the cliquet option
56.3 Path dependency, constant volatility
56.4 Results
56.4.1 Constant volatility
56.4.2 Uncertain volatility
56.5 Code: Cliquet with uncertain volatility, in similarity variables
56.6 Summary
57 Jump Diffusion
57.1 Introduction
57.2 Evidence for jumps
57.3 Poisson processes
57.4 Hedging when there are jumps
57.5 Hedging the diffusion
57.6 Hedging the jumps
57.7 Hedging the jumps and risk neutrality
57.8 The downside of jump-diffusion models
57.9 Jump volatility
57.10 Jump volatility with deterministic decay
57.11 Summary
58 Crash Modeling
58.1 Introduction
58.2 Value at risk
58.3 A simple example: The hedged call
58.4 A mathematical model for a crash
58.4.1 Case I: Black–Scholes Hedging
58.4.2 Case II: Crash Hedging
58.5 An example
58.6 Optimal static hedging: VaR reduction
58.7 Continuous-time limit
58.8 A range for the crash
58.9 Multiple crashes
58.9.1 Limiting the total number of crashes
58.9.2 Limiting the frequency of crashes
58.10 Crashes in a multi-asset world
58.11 Fixed and floating exchange rates
58.12 Summary
59 Speculating with Options
59.1 Introduction
59.2 A simple model for the value of an option to a speculator
59.2.1 The present value of expected payoff
59.2.2 Standard deviation
59.3 More sophisticated models for the return on an asset
59.3.1 Diffusive drift
59.3.2 Jump drift
59.4 Early closing
59.5 To hedge or not to hedge?
59.6 Other issues
59.7 Summary
60 Static Hedging
60.1 Introduction
60.2 Static replicating portfolio
60.3 Matching a ‘target’ contract
60.4 Vega matching
60.5 Static hedging: Non-linear governing equation
60.6 Non-linear equations
60.7 Pricing with a non-linear equation
60.7.1 Example: Non-linear model, unhedged
60.7.2 Static hedging: A first attempt
60.7.3 Static hedging: The best hedge
60.8 Optimal static hedging: The theory
60.9 Calibration?
60.10 Hedging path-dependent options with vanilla options, non-linear models
60.10.1 American options
60.10.2 Barrier options
60.10.3 Pricing and optimally hedging a portfolio of barrier options
60.11 The mathematics of optimization
60.11.1 Downhill simplex method
60.11.2 Simulated annealing
60.12 Summary
61 The Feedback Effect of Hedging in Illiquid Markets
61.1 Introduction
61.2 The trading strategy for option replication
61.3 The excess demand function
61.4 Incorporating the trading strategy
61.5 The influence of replication
61.6 The forward equation
61.6.1 The boundaries
61.7 Numerical results
61.7.1 Time-independent trading strategy
61.7.2 Put replication trading
61.8 Attraction and repulsion
61.9 Summary
62 Utility Theory
62.1 Introduction
62.2 Ranking events
62.3 The utility function
62.4 Risk aversion
62.5 Special utility functions
62.6 Certainty equivalent wealth
62.7 Maximization of expected utility
62.7.1 Ordinal and cardinal utility
62.8 Summary
63 More About American Options and Related Matters
63.1 Introduction
63.2 What Derivatives Week published
63.3 Hold these thoughts
63.4 Change of notation
63.5 And finally, the paper . . .
63.6 Introduction
63.7 Preliminary: Pricing and Hedging
63.8 Utility-Maximizing Exercise Time
63.8.1 Constant absolute risk aversion
63.8.2 Hyperbolic absolute risk aversion
63.8.3 The expected return
63.9 Profit from Selling American Options
63.10 Concluding Remarks
63.11 Who wins and who loses?
63.12 FAQ
63.13 Another situation where the same idea applies: Passport options
63.13.1 Recap
63.13.2 Utility maximization in the passport option
63.14 Summary
64 Advanced Dividend Modeling
64.1 Introduction
64.2 Why do we need dividend models?
64.3 Effects of dividends on asset prices
64.3.1 Market frictions
64.3.2 Term structure of dividends
64.4 Stochastic dividends
64.5 Poisson jumps
64.6 Uncertainty in dividend amount and timing
64.7 Summary
65 Serial Autocorrelation in Returns
65.1 Introduction
65.2 Evidence
65.3 The Telegraph equation
65.4 Pricing and hedging derivatives
65.5 Summary
66 Asset Allocation in Continuous Time
66.1 Introduction
66.2 One risk-free and one risky asset
66.2.1 The wealth process
66.2.2 Maximizing expected utility
66.2.3 Stochastic control and the Bellman equation
66.2.4 Constant relative risk aversion
66.2.5 Constant absolute risk aversion
66.3 Many assets
66.4 Maximizing long-term growth
66.5 A brief look at transaction costs
66.6 Summary
67 Asset Allocation under Threat of a Crash
67.1 Introduction
67.2 Optimal Portfolios under the Threat of a Crash: The single stock case
67.3 Maximizing Growth Rate under the Threat of a Crash: n stocks
67.4 Maximizing Growth Rate under the Threat of a Crash: An arbitrary number of crashes and other re
67.4.1 Arbitrary upper bound for the number of crashes
67.4.2 Changing volatility after a crash
67.4.3 Further possible refinements
67.5 Summary
68 Interest-rate Modeling Without Probabilities
68.1 Introduction
68.2 What do I want from an interest rate model?
68.3 A non-probabilistic model for the behavior of the short-term interest rate
68.4 Worst-case scenarios and a non-linear equation
68.4.1 Let’s see that again in slow motion
68.5 Examples of hedging: Spreads for prices
68.5.1 Hedging with one instrument
68.5.2 Hedging with multiple instruments
68.6 Generating the ‘Yield Envelope’
68.7 Swaps
68.8 Caps and floors
68.9 Applications of the model
68.9.1 Identifying arbitrage opportunities
68.9.2 Establishing prices for the market maker
68.9.3 Static hedging to reduce interest rate risk
68.9.4 Risk management: A measure of absolute loss
68.9.5 A remark on the validity of the model
68.10 Summary
69 Pricing and Optimal Hedging of Derivatives, the Non-probabilistic Model Cont’d
69.1 Introduction
69.2 A real portfolio
69.3 Bond options
69.3.1 Pricing the European option on a zero-coupon bond
69.3.2 Pricing and hedging American options
69.4 Contracts with embedded decisions
69.5 The index amortizing rate swap
69.6 Convertible bonds
69.7 Summary
70 Extensions to the Non-probabilistic Interest-rate Model
70.1 Introduction
70.2 Fitting forward rates
70.3 Economic cycles
70.4 Interest rate bands
70.4.1 Estimating from past data
70.5 Crash modeling
70.5.1 A maximum number of crashes
70.5.2 A maximum frequency of crashes
70.5.3 Estimating from past data
70.6 Liquidity
70.7 Summary
71 Modeling Inflation
71.1 Introduction
71.2 Inflation-linked products
71.2.1 Bonds
71.2.2 Inflation caps and floors
71.2.3 Swaps
71.2.4 Barriers
71.3 Pricing, first thoughts
71.4 What the data tell us
71.5 Pricing, second thoughts
71.6 Analyzing the data
71.7 Can we model inflation independently of interest rates?
71.8 Calibration and market price of risk
71.9 Non-linear pricing methods
71.10 Summary
72 Energy Derivatives
72.1 Introduction
72.2 The energy market
72.3 What’s so special about the energy markets?
72.4 Why can’t we apply Black–Scholes theory to energy derivatives?
72.5 The convenience yield
72.6 The Pilopovi´c two-factor model
72.6.1 Fitting
72.7 Energy derivatives
72.7.1 One-day options
72.7.2 Asian options
72.7.3 Caps and floors
72.7.4 Cheapest to deliver
72.7.5 Basis spreads
72.7.6 Swing options
72.7.7 Spread options
72.8 Summary
73 Real Options
73.1 Introduction
73.2 Financial options and Real options
73.3 An introductory example: Abandonment of a machine
73.4 Optimal investment: Simple example #2
73.5 Temporary suspension of the project, costless
73.6 Temporary suspension of the project, costly
73.7 Sequential and incremental investment
73.8 Ashanti: Gold mine case study
73.9 Summary
74 Life Settlements and Viaticals
74.1 Introduction
74.2 Life expectancy
74.2.1 Sex
74.2.2 Health
74.3 Actuarial tables
74.4 Death seen as default
74.5 Pricing a single policy
74.5.1 Internal rate of return
74.6 Pricing portfolios
74.6.1 Extension risk
74.7 Summary
75 Bonus Time
75.1 Introduction
75.2 One bonus period
75.2.1 Bonus depending on the Sharpe ratio
75.2.2 Numerical results
75.3 The skill factor
75.4 Putting skill into the equation
75.4.1 Example
75.5 Summary
PART SIX NUMERICAL METHODS AND PROGRAMS
76 Overview of Numerical Methods
76.1 Introduction
76.2 Finite-difference methods
76.2.1 Efficiency
76.2.2 Program of study
76.3 Monte Carlo methods
76.3.1 Efficiency
76.3.2 Program of study
76.4 Numerical integration
76.4.1 Efficiency
76.4.2 Program of study
76.5 Summary
77 Finite-difference Methods for One-factor Models
77.1 Introduction
77.2 Overview
77.3 Grids
77.4 Differentiation using the grid
77.5 Approximating θ
77.6 Approximating
77.6.1 One-sided differences
77.7 Approximating
77.8 Example
77.9 Final conditions and payoffs
77.10 Boundary conditions
77.10.1 Other boundary conditions
77.11 The explicit finite-difference method
77.11.1 The Black–Scholes equation
77.12 Convergence of the explicit method
77.13 The Code # 1: European option
77.14 The Code # 2: American exercise
77.15 The Code # 3: 2-D output
77.16 Bilinear interpolation
77.17 Upwind differencing
77.18 Summary
78 Further Finite-difference Methods for One-factor Models
78.1 Introduction
78.2 Implicit finite-difference methods
78.3 The Crank–Nicolson method
78.3.1 Boundary condition type I: V k+1 0 given
78.3.2 Boundary condition type II: Relationship between V k+1 0 and V k+ 1
78.3.3 Boundary condition type III: ∂2V/∂S2 = 0
78.3.4 The matrix equation
78.3.5 LU decomposition
78.3.6 Successive over-relaxation, SOR
78.3.7 Optimal choice of ω
78.4 Comparison of finite-difference methods
78.5 Other methods
78.6 Douglas schemes
78.7 Three time-level methods
78.8 Richardson extrapolation
78.9 Free boundary problems and American options
78.9.1 Early exercise and the explicit method
78.9.2 Early exercise and Crank–Nicolson
78.10 Jump conditions
78.10.1 A discrete cashflow
78.10.2 Discretely paid dividend
78.11 Path-dependent options
78.11.1 Discretely sampled quantities
78.11.2 Continuously sampled quantities
78.12 Summary
79 Finite-difference Methods for Two-factor Models
79.1 Introduction
79.2 Two-factor models
79.3 The explicit method
79.3.1 Stability of the explicit method
79.4 Calculation time
79.5 Alternating Direction Implicit
79.6 The Hopscotch method
79.7 Summary
80 Monte Carlo Simulation
80.1 Introduction
80.2 Relationship between derivative values and simulations: Equities, indices, currencies, commodit
80.3 Generating paths
80.4 Lognormal underlying, no path dependency
80.5 Advantages of Monte Carlo simulation
80.6 Using random numbers
80.7 Generating normal variables
80.7.1 Box–Muller
80.8 Real versus risk neutral, speculation versus hedging
80.9 Interest rate products
80.10 Calculating the greeks
80.11 Higher dimensions: Cholesky factorization
80.12 Calculation time
80.13 Speeding up convergence
80.13.1 Antithetic variables
80.13.2 Control variate technique
80.14 Pros and cons of Monte Carlo simulations
80.15 American options
80.16 Longstaff & Schwartz regression approach for American options
80.17 Basis Functions
80.18 Summary
81 Numerical Integration
81.1 Introduction
81.2 Regular grid
81.3 Basic Monte Carlo integration
81.4 Low-discrepancy sequences
81.5 Advanced techniques
81.6 Summary
82 Finite-difference Programs
82.1 Introduction
82.2 Kolmogorov equation
82.3 Explicit one-factor model for a convertible bond
82.4 American call, implicit
82.5 Explicit Parisian option
82.6 Passport options
82.7 Chooser passport option
82.8 Explicit stochastic volatility
82.9 Uncertain volatility
82.10 Crash modeling
82.11 Explicit Epstein–Wilmott solution
82.12 Risky-bond calculator
83 Monte Carlo Programs
83.1 Introduction
83.2 Monte Carlo pricing of a basket
83.3 Quasi Monte Carlo pricing of a basket
83.4 Monte Carlo for American options
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