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pages: 443 words: 51,804

Handbook of Modeling High-Frequency Data in Finance by Frederi G. Viens, Maria C. Mariani, Ionut Florescu

algorithmic trading, asset allocation, automated trading system, backtesting, Black-Scholes formula, Brownian motion, business process, buy and hold, continuous integration, corporate governance, discrete time, distributed generation, fixed income, Flash crash, housing crisis, implied volatility, incomplete markets, linear programming, mandelbrot fractal, market friction, market microstructure, martingale, Menlo Park, p-value, pattern recognition, performance metric, principal–agent problem, random walk, risk tolerance, risk/return, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process

J Finance 1997;52:975–1005. Ball C, Roma A. Stochastic volatility option pricing. J Financ Quant Anal 1994;29(4):589–607. Beran J. Statistics for long-memory processes. Chapman and Hall, USA; 1994. Black F, Scholes M. The valuation of options and corporate liability. J Polit Econ 1973;81:637–654. Breidt FJ, Crato N, De Lima P. The detection and estimation of long-memory in stochastic volatility. J Econometrics 1998;83:325–348. Casas I, Gao J. Econometric estimation in long-range dependent volatility models: theory and practice. J Econometrics 2008;147:72–83. Chronopoulou A, Viens F. Estimation and pricing under long-memory stochastic volatility. Ann Finance 2010. References 231 Comte F, Renault E. Long memory in continuous-time stochastic volatility models. Math Finance 1998;8(4):291–323. De Lima P, Crato N.

Probab Theor Relat Field 2001;120:346–368. Deo RS, Hurvich CM. On the log periodogram regression estimator of the memory parameter in long memory stochastic volatility models. Economet Theor 2001;17:686–710. Ding ZC, Granger WJ, Engle RF. A long memory property of stock market returns and a new model. J Empir Finance 1993;1:1. Florescu I, Viens FG. Stochastic volatility: option pricing using a multinomial recombining tree. Appl Math Finance 2008;15(2):151–181. Fouque JP, Papanicolaou G, Sircar KR. Derivatives in financial markets with stochastic volatility. Cambridge University Press; 2000a. Fouque JP, Papanicolaou G, Sircar KR. Mean-reverting stochastic volatility. Int J Theor Appl Finance 2000b;3(1):101–142. Fox R, Taqqu MS. Large-sample properties of parameter estimates for strongly dependent stationary Gaussian time series.

Heston SL. A closed-form solution for options with stochastic volatility with applications to bond and currency options. Rev Financ Stud 1993;6(2):327–343. 14. Avellaneda M, Zhu Y. Risk neutral stochastic volatility model. Int J Theor Appl Finance 1998;1(2):289–310. 15. Berestycki H, Busca J, Florent I. Computing the implied volatility in stochastic volatility models. Commu Pure and Appl Math 2004;57(10):1352 –1373. 16. Andersen L, Andreasen J. Jump-diffusion processes: volatility smile fitting and numerical methods for option pricing. Rev Deriv Res 2000;4:231–262. 17. Cont R, Tankov P. Financial modelling with jumps processes. CRC Financial mathematics series. Chapman & Hall, Boca Raton, Florida; 2003. References 381 18. Florescu I. Stochastic volatility stock price: approximation and valuation using a recombining tree. sharp estimation of the almost sure lyapunov exponent estimation for the anderson model in continuous space.

The Concepts and Practice of Mathematical Finance by Mark S. Joshi

Black-Scholes formula, Brownian motion, correlation coefficient, Credit Default Swap, delta neutral, discrete time, Emanuel Derman, fixed income, implied volatility, incomplete markets, interest rate derivative, interest rate swap, London Interbank Offered Rate, martingale, millennium bug, quantitative trading / quantitative finance, short selling, stochastic process, stochastic volatility, the market place, time value of money, transaction costs, value at risk, volatility smile, yield curve, zero-coupon bond

Stochastic-volatility smiles tend to be shallow relative to jump-diffusion smiles for short maturities and relatively steep for long maturities. 16.9 Further reading The transform approach developed here is based on that in Option Valuation Under Stochastic Volatility by A. Lewis, [100] where much more general stochasticvolatility models are studied and solved. If you want to implement transform-based solutions to stochastic volatility models this is the book to buy. 400 Stochastic volatility The transform approach to stochastic-volatility pricing was started by Heston, [72]. A quite general jump-diffusion and stochastic-volatility model, which probably pushes the transform technique as far as it will go in this direction, has been developed by Duffle, Pan & Singleton, [50]. In [101], the transform technique is extended to cover a large class of models. An alternate approach to stochastic volatility models using ideas from ergodic theory has been developed by Fouque, Papanicolaou & Sircar, [55].

15.10 Log-type models 15.11 Key points 15.12 Further reading 15.13 Exercises 14.6 14.7 14.8 14.9 14.10 14.11 14.12 14.13 14.14 14.15 15 xi 300 300 302 309 314 316 317 317 319 319 323 330 333 335 337 340 342 345 349 352 355 358 358 359 361 361 362 364 367 369 375 377 379 381 382 385 386 387 xii 16 17 18 Contents Stochastic volatility 16.1 Introduction 16.2 Risk-neutral pricing with stochastic-volatility models 16.3 Monte Carlo and stochastic volatility 16.4 Hedging issues 16.5 PDE pricing and transform methods 16.6 Stochastic volatility smiles 16.7 Pricing exotic options 16.8 Key points 16.9 Further reading 16.10 Exercises Variance Gamma models 17.1 The Variance Gamma process 17.2 Pricing options with Variance Gamma models 17.3 Pricing exotic options with Variance Gamma models 17.4 Deriving the properties 17.5 Key points 17.6 Further reading 17.7 Exercises Smile dynamics and the pricing of exotic options 18.1 Introduction 18.2 Smile dynamics in the market 18.3 Dynamics implied by models 18.4 Matching the smile to the model 18.5 Hedging 18.6 Matching the model to the product 18.7 Key points 18.8 Further reading Appendix A Financial and mathematical jargon Appendix B Computer projects Introduction B.

In particular, if the boundary condition is 1 the solution is Q( ,V,i)=e -Q, 2X tank( `Ziivt.r (16.27) In fact, since (16.26) does not involve any differentiations in , if we want the boundary condition to be an arbitrary function f (4), for example P, then we just multiply Q by f. The function Of has the correct boundary value by construction and since multiplication by f (1;) commutes with differentiation in the other variables, (16.26) is satisfied by Of. 398 Stochastic volatility We can now price any option for which we know the fundamental transform. We simply numerically invert the Fourier transform at the appropriate value of T and obtain a price. 16.6 Stochastic volatility smiles Since the possibility of stochastic volatility getting large increases the probability of large movements in the underlying stock, stochastic-volatility models lead to fatter tails for the distribution of the final stock price. This leads to implied-volatility smiles which pick up out-of-the-money; that is, smile-shaped smiles! If we allow correlation between the underlying and the volatility then a skew is introduced.

Analysis of Financial Time Series by Ruey S. Tsay

Asian financial crisis, asset allocation, Bayesian statistics, Black-Scholes formula, Brownian motion, business cycle, capital asset pricing model, compound rate of return, correlation coefficient, data acquisition, discrete time, frictionless, frictionless market, implied volatility, index arbitrage, Long Term Capital Management, market microstructure, martingale, p-value, pattern recognition, random walk, risk tolerance, short selling, statistical model, stochastic process, stochastic volatility, telemarketer, transaction costs, value at risk, volatility smile, Wiener process, yield curve

Table 10.4 gives the point forecasts of the return and its volatility for five forecast horizons starting with December 1999. Both the GARCH model in Eq. (10.26) and the stochastic volatility model in Eq. (10.27) are used in the forecasting. The volatility forecasts of the GARCH(1, 1) model increase gradually with the forecast horizon to the unconditional variance 3.349/(1 − 0.086 − 0.735) = 18.78. The volatility forecasts of the stochastic volatility model are higher than those of the GARCH model. This is understandable because the stochastic volatility model takes into consideration the paramTable 10.4. Volatility Forecasts for the Monthly Log Return of S&P 500 Index. The Data Span Is From January 1962 to December 1999 and the Forecast Origin Is December 1999. Forecasts of the Stochastic Volatility Model Are Obtained by a Gibbs Sampling with 2000 + 2000 Iterations. (a) Horizon GARCH SVM Log return 1 0.66 0.53 2 0.66 0.78 (b) Horizon GARCH SVM 3 0.66 0.92 4 0.66 0.88 5 0.66 0.84 4 18.34 19.65 5 18.42 20.13 Volatility 1 17.98 19.31 2 18.12 19.36 3 18.24 19.35 441 EXERCISES eter uncertainty in producing forecasts.

It is much faster in computation, but may produce spurious detections when multiple outliers are present. For the data in Example 10.2, the SCA program also identifies t = 323 and t = 201 as the two most significant additive outliers. The estimated outlier sizes are −0.39 and 0.36, respectively. 10.7 STOCHASTIC VOLATILITY MODELS An important financial application of MCMC methods is the estimation of stochastic volatility models; see Jacquier, Polson, and Rossi (1994) and the references therein. We start with a univariate stochastic volatility model. The mean and volatility equations of an asset return rt are rt = β0 + β1 x1t + · · · + β p x pt + at , at = h t t (10.20) ln h t = α0 + α1 ln h t−1 + vt (10.21) where {xit | i = 1, . . . , p} are explanatory variables available at time t − 1, β j s are parameters, {t } is a Gaussian white noise sequence with mean 0 and variance 1, {vt } is also a Gaussian white noise sequence with mean 0 and variance σv2 , and {t } and {vt } are independent.

The Ljung–Box statistics of the standardized residuals and their squared series fail to indicate any model inadequacy. Next, consider the stochastic volatility model r t = µ + at , at = h t t ln h t = α0 + α1 ln h t−1 + vt , (10.27) where vt s are iid N (0, σv2 ). To implement the Gibbs sampling, we use the prior distributions µ ∼ N (0, 9), α ∼ N [αo , diag(0.09, 0.04)], 5 × 0.2 ∼ χ52 , σv2 423 0 2 density 4 6 density 0.0 0.5 1.0 1.5 2.0 8 STOCHASTIC VOLATILITY MODELS 0.0 0.5 1.0 alpha0 1.5 2.0 0 2 4 sigvsq 6 0 1 density 2 3 4 5 density 0.0 0.5 1.0 1.5 2.0 6 -1.0 -0.5 0.0 0.5 1.0 alpha1 1.5 2.0 -10 -5 0 mu 5 10 Figure 10.4. Density functions of prior and posterior distributions of parameters in a stochastic volatility model for the monthly log returns of S&P 500 index. The dashed line denotes prior density and solid line denotes the posterior density, which is based on results of Gibbs sampling with 5000 iterations.

pages: 345 words: 86,394

Frequently Asked Questions in Quantitative Finance by Paul Wilmott

Albert Einstein, asset allocation, beat the dealer, Black-Scholes formula, Brownian motion, butterfly effect, buy and hold, capital asset pricing model, collateralized debt obligation, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discrete time, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, iterative process, lateral thinking, London Interbank Offered Rate, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, margin call, market bubble, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, quantitative trading / quantitative finance, random walk, regulatory arbitrage, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, transaction costs, urban planning, value at risk, volatility arbitrage, volatility smile, Wiener process, yield curve, zero-coupon bond

This is called the actuarial approach because it is how the insurance business works. You can’t hedge the lifespan of individual policyholders but you can figure out what will happen to hundreds of thousands of them on average using actuarial tables. The other way of pricing is to make options consistent with each other. This is commonly used when we have stochastic volatility models, for example, and is also often seen in fixed-income derivatives pricing. Let’s work with the stochastic volatility model to get inspiration. Suppose we have a lognormal random walk with stochastic volatility. This means we have two sources of randomness (stock and volatility) but only one quantity with which to hedge (stock). That’s like saying that there are more states of the world than underlying securities, hence incompleteness. Well, we know we can hedge the stock price risk with the stock, leaving us with only one source of risk that we can’t get rid of.

McGraw-Hill Heath, D, Jarrow, R & Morton, A 1992 Bond pricing and the term structure of interest rates: a new methodology. Econometrica 60 77-105 Heston, S 1993 A closed-form solution for options with stochastic volatility with application to bond and currency options. Review of Financial Studies 6 327-343 Ho, T & Lee, S 1986 Term structure movements and pricing interest rate contingent claims. Journal of Finance 42 1129-1142 Hull, JC & White, A 1987 The pricing of options on assets with stochastic volatilities. Journal of Finance 42 281-300 Hull, JC & White, A 1990 Pricing interest rate derivative securities. Review of Financial Studies 3 573-592 Lewis, A 2000 Option valuation under Stochastic Volatility. Finance Press Merton, RC 1973 Theory of rational option pricing. Bell Journal of Economics and Management Science 4 141-83 Merton, RC 1974 On the pricing of corporate debt: the risk structure of interest rates.

The interpretation of an option’s value as the present value of the expected payoff under a risk-neutral random walk also carries over. Unfortunately the Black-Scholes closed-form formulæ are no longer correct. This is a simple and popular model, but it does not capture the dynamics of implied volatility very well. Stochastic volatility: Since volatility is difficult to measure, and seems to be forever changing, it is natural to model it as stochastic. The most popular model of this type is due to Heston. Such models often have several parameters which can either be chosen to fit historical data or, more commonly, chosen so that theoretical prices calibrate to the market. Stochastic volatility models are better at capturing the dynamics of traded option prices better than deterministic models. However, different markets behave differently. Part of this is because of the way traders look at option prices.

pages: 313 words: 34,042

Tools for Computational Finance by Rüdiger Seydel

bioinformatics, Black-Scholes formula, Brownian motion, commoditize, continuous integration, discrete time, implied volatility, incomplete markets, interest rate swap, linear programming, London Interbank Offered Rate, mandelbrot fractal, martingale, random walk, stochastic process, stochastic volatility, transaction costs, value at risk, volatility smile, Wiener process, zero-coupon bond

Köln (2005). D.J. Higham: An algorithmic introduction to numerical solution of stochastic differential equations. SIAM Review 43 (2001) 525-546. D.J. Higham: An Introduction to Financial Option Valuation. Cambridge, Univ. Press, Cambridge (2004). N. Hilber, A.-M. Matache, C. Schwab: Sparse Wavelet Methods for Option Pricing under Stochastic Volatility. Report, ETH-Zürch (2004). N. Hofmann, E. Platen, M. Schweizer: Option pricing under incompleteness and stochastic volatility. Mathem. Finance 2 (1992) 153–187. P. Honoré, R. Poulsen: Option pricing with EXCEL. in [Nie02]. J.C. Hull: Options, Futures, and Other Derivatives. Fourth Edition. Prentice Hall International Editions, Upper Saddle River (2000). P.J. Hunt, J.E. Kennedy: Financial Derivatives in Theory and Practice. Wiley, Chichester (2000).

Simulation rt of the Cox-Ingersoll-Ross model (1.40) with β = 0.5 for R = 0.05, α = 1, σr = 0.1, y0 = 0.15, ∆t = 0.01 1.7 Stochastic Differential Equations 39 The SDE (1.40) is of a different kind as (1.33). Coupling the SDE for rt to that for St leads to a system of two SDEs. Even larger systems are obtained when further SDEs are coupled to define a stochasic process Rt or to calculate stochastic volatilities. A related example is given by Example 1.15 below. 1.7.5 Vector-Valued SDEs The Itô equation (1.31) is formulated as scalar equation; accordingly the SDE (1.33) represents a one-factor model. The general multifactor version can be (1) (n) written in the same notation. Then Xt = (Xt , . . . , Xt ) and a(Xt , t) are n-dimensional vectors. The Wiener processes of each component SDE need not be correlated.

Example 1.15 (mean-reverting volatility tandem) We consider a three-factor model with stock price St , instantaneous spot volatility σt and an averaged volatility ζt serving as mean-reverting parameter:  (1)   dS = σSdW   dσ = −(σ − ζ)dt + ασdW (2) dζ = β(σ − ζ)dt Here and sometimes later on, we suppress the subscript t, which may be done when the role of the variables as stochastic processes is clear from the context. The rate of return µ of S is zero; dW (1) and dW (2) may be 40 Chapter 1 Modeling Tools for Financial Options correlated. The stochastic volatility σ follows the mean volatility ζ and is simultaneously perturbed by a Wiener process. Both σ und ζ provide mutual mean reversion, and stick together. The two SDEs for σ and ζ may be seen as a tandem controlling the dynamics of the volatility. We recommend numerical tests. As motivation see Figure 3.1. Computational Matters Stochastic differential equations are simulated in the context of Monte Carlo methods.

pages: 354 words: 26,550

High-Frequency Trading: A Practical Guide to Algorithmic Strategies and Trading Systems by Irene Aldridge

algorithmic trading, asset allocation, asset-backed security, automated trading system, backtesting, Black Swan, Brownian motion, business cycle, business process, buy and hold, capital asset pricing model, centralized clearinghouse, collapse of Lehman Brothers, collateralized debt obligation, collective bargaining, computerized trading, diversification, equity premium, fault tolerance, financial intermediation, fixed income, high net worth, implied volatility, index arbitrage, information asymmetry, interest rate swap, inventory management, law of one price, Long Term Capital Management, Louis Bachelier, margin call, market friction, market microstructure, martingale, Myron Scholes, New Journalism, p-value, paper trading, performance metric, profit motive, purchasing power parity, quantitative trading / quantitative finance, random walk, Renaissance Technologies, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, short selling, Small Order Execution System, statistical arbitrage, statistical model, stochastic process, stochastic volatility, systematic trading, trade route, transaction costs, value at risk, yield curve, zero-sum game

The deterministic condition can be restrictive and fail to reflect the dynamic nature of volatility. A different class of volatility estimators, known as stochastic volatility estimators, have been developed to allow modeling of heteroscedasticity and volatility clustering without the functional form restrictions on volatility specification. The simplest stochastic volatility estimator can be specified as follows: vt = σt ξt = ς exp(αt /2) ξt (8.36) where αt = φαt−1 + ηt is the parameter modeling volatility persistence, |φ| < 1, ξt is an identically and independently distributed random variable with mean 0 and variance 1, and ζ is a positive constant. While stochastic volatility models reflect well the random nature underlying volatility processes, stochastic volatility is difficult to estimate. The parameters of equation (8.36) are often estimated using an econometric technique known as maximum likelihood or its close cousins.

The parameters of equation (8.36) are often estimated using an econometric technique known as maximum likelihood or its close cousins. Given the randomness of the stochastic volatility estimator, the estimation process is quite complex. Estimation of GARCH can seem trivial in comparison with the estimation of stochastic volatility. 108 HIGH-FREQUENCY TRADING NONLINEAR MODELS Overview As their name implies, nonlinear models allow modeling of complex nontrivial relationships in the data. Unlike linear models discussed in the first section of this chapter, nonlinear models forecast random variables that cannot be expressed as linear combinations of other, contemporaneous or lagged, random variables with well-defined distributions. Instead, nonlinear models can be expressed as some functions f (.) of other random variables. In mathematical terms, if a linear model can be expressed as shown in equation (8.37), reprinted here for convenience, then nonlinear models are best expressed as shown in equation (8.38) which follows: yt = α + ∞ βi xt−i + εt (8.37) i=0 yt = f (xt , xt−1 , xt−2 , · · ·) (8.38) where {yt } is the time series of random variables that are to be forecasted, {xt } is a factor significant in forecasting {yt }, and α and β are coefficients to be estimated.

Various extensions to the GARCH specification include additional explanatory right-hand side variables controlling for external events, an exponential “EGARCH” specification that addresses the asymmetric response of returns to positive and negative shocks (bad news is typically accompanied by a higher volatility than good news), and a “GARCH-M” model in which the return of a security depends on the security’s volatility, among numerous other GARCH extensions. In addition to the moving window and GARCH volatility estimators, popular volatility measurements include the intraperiod volatility estimator, known as the “realized volatility;” several measures based on the intraperiod range of prices; and a stochastic volatility model where volatility is thought to be a random variable drawn from a prespecified distribution. The realized volatility due to Andersen, Bollerslev, Diebold, and Labys (2001) is computed as the sum of squared intraperiod returns obtained by breaking a time period into n smaller time increments of equal duration: RVt = n 2 rt,i (8.30) i=1 The range-based volatility measures are based on combinations of open, high, low, and close prices for every period under consideration.

pages: 819 words: 181,185

Derivatives Markets by David Goldenberg

Black-Scholes formula, Brownian motion, capital asset pricing model, commodity trading advisor, compound rate of return, conceptual framework, correlation coefficient, Credit Default Swap, discounted cash flows, discrete time, diversification, diversified portfolio,, financial innovation, fudge factor, implied volatility, incomplete markets, interest rate derivative, interest rate swap, law of one price, locking in a profit, London Interbank Offered Rate, Louis Bachelier, margin call, market microstructure, martingale, Myron Scholes, Norbert Wiener, Paul Samuelson, price mechanism, random walk, reserve currency, risk/return, riskless arbitrage, Sharpe ratio, short selling, stochastic process, stochastic volatility, time value of money, transaction costs, volatility smile, Wiener process, yield curve, zero-coupon bond, zero-sum game

Black–Scholes per se no longer holds, but in many cases one can derive an explicit option pricing formula. 3. σ=σ(St,t), meaning the combination of 1. and 2. in that σ is both time-dependent and stock price level dependent. In the literature case 3. is known as the deterministic volatility Dupire (DV) model. The main virtue of the DV approach in modeling varying volatility is that it generates option pricing models that are complete in the sense we described. 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models Stochastic volatility (SVOL) models are beyond the scope of this text. However, a few comments may indicate the flavor of this approach. In the deterministic volatility model, randomness in volatility is purely a result of randomness in the underlying stock price process. This is not an independent source of randomness, since it is induced by the stock price. SVOL models introduce new source(s) of randomness into the volatility process.

11.7 Further Implications of European Put-Call Parity 11.7.1 Synthesizing Forward Contract from Puts and Calls 11.8 Financial Innovation using European Put-Call Parity 11.8.1 Generalized Forward Contracts 11.8.2 American Put-Call Parity (No Dividends) 11.9 Postscript on ROP CHAPTER 12 OPTION TRADING STRATEGIES, PART 2 12.1 Generating Synthetic Option Strategies from European Put-Call Parity 12.2 The Covered Call Hedging Strategy 12.2.1 Three Types Of Covered Call Writes 12.2.2 Economic Interpretation of the Covered Call Strategy 12.3 The Protective Put Hedging Strategy 12.3.1 Puts as Insurance 12.3.2 Economic Interpretation of the Protective Put Strategy CHAPTER 13 MODEL-BASED OPTION PRICING IN DISCRETE TIME, PART 1: THE BINOMIAL OPTION PRICING MODEL (BOPM, N=1) 13.1 The Objective of Model-Based Option Pricing (MBOP) 13.2 The Binomial Option Pricing Model, Basics 13.2.1 Modeling Time in a Discrete Time Framework 13.2.2 Modeling the Underlying Stock Price Uncertainty 13.3 The Binomial Option Pricing Model, Advanced 13.3.1 Path Structure of the Binomial Process, Total Number of Price Paths 13.3.2 Path Structure of the Binomial Process, Total Number of Price Paths Ending at a Specific Terminal Price 13.3.3 Summary of Stock Price Evolution for the N-Period Binomial Process 13.4 Option Valuation for the BOPM (N=1) 13.4.1 Step 1, Pricing the Option at Expiration 13.4.2 Step 2, Pricing the Option Currently (time t=0) 13.5 Modern Tools for Pricing Options 13.5.1 Tool 1, The Principle of No-Arbitrage 13.5.2 Tool 2, Complete Markets or Replicability, and a Rule of Thumb 13.5.3 Tool 3, Dynamic and Static Replication 13.5.4 Relationships between the Three Tools 13.6 Synthesizing a European Call Option 13.6.1 Step 1, Parameterization 13.6.2 Step 2, Defining the Hedge Ratio and the Dollar Bond Position 13.6.3 Step 3, Constructing the Replicating Portfolio 13.6.4 Step 4, Implications of Replication 13.7 Alternative Option Pricing Techniques 13.8 Appendix: Derivation of the BOPM (N=1) as a Risk-Neutral Valuation Relationship CHAPTER 14 OPTION PRICING IN DISCRETE TIME, PART 2: DYNAMIC HEDGING AND THE MULTI-PERIOD BINOMIAL OPTION PRICING MODEL, N>1 14.1 Modeling Time and Uncertainty in the BOPM, N>1 14.1.1 Stock Price Behavior, N=2 14.1.2 Option Price Behavior, N=2 14.2 Hedging a European Call Option, N=2 14.2.1 Step 1, Parameterization 14.2.2 Step 2, Defining the Hedge Ratio and the Dollar Bond Position 14.2.3 Step 3, Constructing the Replicating Portfolio 14.2.4 The Complete Hedging Program for the BOPM, N=2 14.3 Implementation of the BOPM for N=2 14.4 The BOPM, N>1 as a RNVR Formula 14.5 Multi-period BOPM, N>1: A Path Integral Approach 14.5.1 Thinking of the BOPM in Terms of Paths 14.5.2 Proof of the BOPM Model for general N CHAPTER 15 EQUIVALENT MARTINGALE MEASURES: A MODERN APPROACH TO OPTION PRICING 15.1 Primitive Arrow–Debreu Securities and Option Pricing 15.1.1 Exercise 1, Pricing B(0,1) 15.1.2 Exercise 2, Pricing ADu(ω) and ADd(ω) 15.2 Contingent Claim Pricing 15.2.1 Pricing a European Call Option 15.2.2 Pricing any Contingent Claim 15.3 Equivalent Martingale Measures (EMMs) 15.3.1 Introduction and Examples 15.3.2 Definition of a Discrete-Time Martingale 15.4 Martingales and Stock Prices 15.4.1 The Equivalent Martingale Representation of Stock Prices 15.5 The Equivalent Martingale Representation of Option Prices 15.5.1 Discounted Option Prices 15.5.2 Summary of the EMM Approach 15.6 The Efficient Market Hypothesis (EMH), A Guide To Modeling Prices 15.7 Appendix: Essential Martingale Properties CHAPTER 16 OPTION PRICING IN CONTINUOUS TIME 16.1 Arithmetic Brownian Motion (ABM) 16.2 Shifted Arithmetic Brownian Motion 16.3 Pricing European Options under Shifted Arithmetic Brownian Motion with No Drift (Bachelier) 16.3.1 Theory (FTAP1 and FTAP2) 16.3.2 Transition Density Functions 16.3.3 Deriving the Bachelier Option Pricing Formula 16.4 Defining and Pricing a Standard Numeraire 16.5 Geometric Brownian Motion (GBM) 16.5.1 GBM (Discrete Version) 16.5.2 Geometric Brownian Motion (GBM), Continuous Version 16.6 Itô’s Lemma 16.7 Black–Scholes Option Pricing 16.7.1 Reducing GBM to an ABM with Drift 16.7.2 Preliminaries on Generating Unknown Risk-Neutral Transition Density Functions from Known Ones 16.7.3 Black–Scholes Options Pricing from Bachelier 16.7.4 Volatility Estimation in the Black–Scholes Model 16.8 Non-Constant Volatility Models 16.8.1 Empirical Features of Volatility 16.8.2 Economic Reasons for why Volatility is not Constant, the Leverage Effect 16.8.3 Modeling Changing Volatility, the Deterministic Volatility Model 16.8.4 Modeling Changing Volatility, Stochastic Volatility Models 16.9 Why Black–Scholes is Still Important CHAPTER 17 RISK-NEUTRAL VALUATION, EMMS, THE BOPM, AND BLACK–SCHOLES 17.1 Introduction 17.1.1 Preliminaries on FTAP1 and FTAP2 and Navigating the Terminology 17.1.2 Pricing by Arbitrage and the FTAP2 17.1.3 Risk-Neutral Valuation without Consensus and with Consensus 17.1.4 Risk-Neutral Valuation without Consensus, Pricing Contingent Claims with Unhedgeable Risks 17.1.5 Black–Scholes’ Contribution 17.2 Formal Risk-Neutral Valuation without Replication 17.2.1 Constructing EMMs 17.2.2 Interpreting Formal Risk-Neutral Probabilities 17.3 MPRs and EMMs, Another Version of FTAP2 17.4 Complete Risk-Expected Return Analysis of the Riskless Hedge in the (BOPM, N=1) 17.4.1 Volatility of the Hedge Portfolio 17.4.2 Direct Calculation of σS 17.4.3 Direct Calculation of σC 17.4.4 Expected Return of the Hedge Portfolio 17.5 Analysis of the Relative Risks of the Hedge Portfolio’s Return 17.5.1 An Initial Look at Risk Neutrality in the Hedge Portfolio 17.5.2 Role of the Risk Premia for a Risk-Averse Investor in the Hedge Portfolio 17.6 Option Valuation 17.6.1 Some Manipulations 17.6.2 Option Valuation Done Directly by a Risk-Averse Investor 17.6.3 Option Valuation for the Risk-Neutral Investor Index FIGURES 1.1 Canada/US Foreign Exchange Rate 1.2 Intermediation by the Clearing House 1.3 Offsetting Trades 1.4 Gold Fixing Price in London Bullion Market (USD$) 2.1 Graphical Method to Get Hedged Position Profits 2.2 Payoff Per Share to a Long Forward Contract 2.3 Payoff Per Share to a Short Forward Contract 2.4 Profits per bu. for the Unhedged Position 3.1 Profits Per Share to a Naked Long Spot Position 3.2 Payoffs Per Share to a Naked Long Spot Position 3.3 Payoffs (=Profits) Per Share to a Naked Long Forward Position 3.4 Payoffs Per Share to a Naked Long Spot Position and to a Naked Long Forward Position 5.1 Order Flow Process (Pit Trading) 5.2 The Futures Clearing House 5.3 Offsetting Trades 5.4 Overall Profits for Example 2 6.1 Long vs.

This includes a discussion of the difference between hedging stock portfolios with forwards and hedging with futures; 11. an entry into understanding swaps, by viewing them as structured products, based on the forward concept; 12. the difference between commodity and interest rate swaps, and a detailed explanation of what it means to pay fixed and receive floating in an interest rate swap; 13. understanding Eurodollar futures strips, notation shifts, and the role of the quote mechanism; 14. discussion of swaps as a zero-sum game, and research challenges to the comparative advantage argument; 15. swaps pricing and alternative interpretations of the par swap rate; 16. a step-by-step approach to options starting in Chapter 9 with the usual emphasis on the quote mechanism, as well as incorporation of real asset options examples; 17. an American option pricing model in Chapter 9, and its extension to European options in Chapter 11; 18. the importance of identifying short, not just long, positions in an underlying asset and the hedging demand they create; 19. two chapters on option trading strategies; one basic, one advanced, including the three types of covered calls, the protective put strategy, and their interpretations; 20. a logical categorization of rational option pricing results in Chapter 11, and the inclusion of American puts and calls; 21. neither monotonicity nor convexity, which are usually assumed, are rational option results; 22. partial vs. full static replication of European options; 23. working backwards from payoffs to costs as a method for devising and interpreting derivatives strategies; 24. the introduction of generalized forward contracts paves the way for the connection between (generalized) forward contracts and options, and the discussion of American put-call parity; 25. the Binomial option pricing model, N=1, and why it works—which is not simply no-arbitrage; 26. three tools of modern mathematical finance: no-arbitrage, replicability and complete markets, and dynamic and static replication, and a rule of thumb on the number of hedging vehicles required to hedge a given number of independent sources of uncertainty; 27. static replication in the Binomial option pricing model, N=1, the hedge ratio can be 1.0 and a preliminary discussion in Chapter 13 on the meaning of risk-neutral valuation; 28. dynamic hedging as the new component of the BOPM, N>1, and a path approach to the multi-period Binomial option pricing model; 29. equivalent martingale measures (EMMs) in the representation of option and stock prices; 30. the efficient market hypothesis (EMH) as a guide to modeling prices; 31. arithmetic Brownian motion (ABM) and the Louis Bachelier model of option prices; 32. easy introduction to the tools of continuous time finance, including Itô’s lemma; 33. Black–Scholes derived from Bachelier, illustrating the important connection between these two models; 34. modeling non-constant volatility: the deterministic volatility model and stochastic volatility models; 35. why Black–Scholes is still important; 36. and a final synthesis chapter that includes a discussion of the different senses of risk-neutral valuation, their meaning and economic basis, and a complete discussion of the dynamics of the hedge portfolio in the BOPM, N=1. I would like to thank the giants of the derivatives field including: Louis Bachelier, Fischer Black, John Cox, Darrell Duffie, Jonathan Ingersoll, Kiyoshi Itô, Robert Merton, Paul Samuelson, Myron Scholes, Stephen Ross, Mark Rubinstein, and many others.

Mathematical Finance: Theory, Modeling, Implementation by Christian Fries

Black-Scholes formula, Brownian motion, continuous integration, discrete time, fixed income, implied volatility, interest rate derivative, martingale, quantitative trading / quantitative finance, random walk, short selling, Steve Jobs, stochastic process, stochastic volatility, volatility smile, Wiener process, zero-coupon bond

However, this insight is almost useless, since: • Most models are not able to reproduce arbitrarily given prices exactly. – – Extended models (e.g. models with stochastic volatility), which allow for a calibration to more than one option price per maturity, do this in an approximative way, i.e. the residual error of the fitting is minimized, but not necessarily 0. However, this may also be a desired effect, since such a fitting is more robust against errors in the input data. • Extended models, that fit to more than one option price per maturity, usually require a great effort to find the corresponding model parameters. These models are intended for the pricing of complex derivatives (which justifies the effort), but not primarily as interpolation method to price European options. Examples of such model extensions are stochastic volatility or jump-diffusion extensions of the LIBOR Market Model, [21]. • Some models even require a continuum of European option prices K 7→ V(T, K) as input.

We denote the simulation time parameter of the stochastic process by t. 256 This work is licensed under a Creative Commons License. Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 17.1. LIBOR MARKET MODEL Further generalization of the model consider non deterministic σi , i.e. stochastic volatility models. In this case the terminal LIBOR distribution no longer correspond to the ones of the Black model, which is, of course, intended. Equation (17.1) is to be seen as a starting point (the umbilic point) of a whole model family. The starting point has been chosen in the form of (17.1), because (historically) the lognormal (Black)model is well understood, especially by traders.3 C| Remark 193 (Interest Rate Structure): Equation (17.1) models the evolution of the LIBOR L(T i , T i+1 ).

The remaining decrees of freedoms are then derived from market prices. C| Choice of the Initial Conditions Reproduction of Bond Market Prices Let PMarket (T i ) ∈ [0, 1] denote a market observed (i.e. given) price of a T i -bond. If we set Li,0 := 6 PMarket (T i ) − PMarket (T i+1 ) , PMarket (T i+1 )(T i+1 − T i ) The parameters σi may well be stochastic processes. In this cases σi is called a stochastic volatility model. 265 This work is licensed under a Creative Commons License. Comments welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 CHAPTER 17. MARKET MODELS then the model reproduces the given market prices of the bonds PMarket . This is ensured by the model having the “right” drift and it is independent of the other parameters.

pages: 537 words: 144,318

The Invisible Hands: Top Hedge Fund Traders on Bubbles, Crashes, and Real Money by Steven Drobny

Albert Einstein, Asian financial crisis, asset allocation, asset-backed security, backtesting, banking crisis, Bernie Madoff, Black Swan, Bretton Woods, BRICs, British Empire, business cycle, business process, buy and hold, capital asset pricing model, capital controls, central bank independence, collateralized debt obligation, commoditize, Commodity Super-Cycle, commodity trading advisor, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, currency peg, debt deflation, diversification, diversified portfolio, equity premium, family office, fiat currency, fixed income, follow your passion, full employment, George Santayana, Hyman Minsky, implied volatility, index fund, inflation targeting, interest rate swap, inventory management, invisible hand, Kickstarter, London Interbank Offered Rate, Long Term Capital Management, market bubble, market fundamentalism, market microstructure, moral hazard, Myron Scholes, North Sea oil, open economy, peak oil, pension reform, Ponzi scheme, prediction markets, price discovery process, price stability, private sector deleveraging, profit motive, purchasing power parity, quantitative easing, random walk, reserve currency, risk tolerance, risk-adjusted returns, risk/return, savings glut, selection bias, Sharpe ratio, short selling, sovereign wealth fund, special drawing rights, statistical arbitrage, stochastic volatility, stocks for the long run, stocks for the long term, survivorship bias, The Great Moderation, Thomas Bayes, time value of money, too big to fail, transaction costs, unbiased observer, value at risk, Vanguard fund, yield curve, zero-sum game

Rather than finding excuses for poorly constructed models, risk managers would be much better served by an examination of stochastic volatility (see box). Granted, it is a complicated topic, but when juxtaposed against the possibility of losing billions again, most plans would be well-served by making an investment to study it. It turns out that liquid markets and instruments, such as the S&P 500, were not exceedingly “fat” or nonnormal in 2008; rather, they exhibited nonconstant volatility, which is not the same thing. A risk system capable of capturing short-term changes in risk would have gone a long way to reducing losses in 2008. Stochastic Volatility Stochastic volatility models are used to evaluate various derivative securities, whereby—as their name implies—they treat the volatility of the underlying securities as a random process. Stochastic volatility models attempt to capture the changing nature of volatility over the life of a derivative contract, something that the traditional Black-Scholes model and other constant volatility models fail to address.

See Risk premia payment Price/earnings (P/E) multiples, exchange rate valuation (relationship) Primary Dealer Credit Facility, placement Prime broker risk Princeton University (endowment) Private equity cash flow production tax shield/operational efficiency arguments Private sector debt, presence Private-to-public sector risk Probability, Bayesian interpretation Professor, The bubble predication capital loss, avoidance capital management cataclysms, analysis crowding factor process diversification efficient markets, disbelief fiat money, cessation global macro fund manager hedge fund space historical events, examination idea generation inflation/deflation debate interview investment process lessons LIBOR futures ownership liquidity conditions, change importance market entry money management, quality opportunities personal background, importance portfolio construction management positioning process real macro success, personality traits/characteristics (usage) returns, generation risk aversion rules risk management process setback stocks, purchase stop losses time horizon Titanic scenario threshold trades attractiveness, measurement process expression, options (usage) personal capital, usage quality unlevered portfolio Property/asset boom Prop shop trading, preference Prop trader, hedge fund manager (contrast) Protectionism danger hedge process Public college football coach salary, public pension manager salary (contrast) Public debt, problems Public pensions average wages to returns endowments impact Q ratio (Tobin) Qualitative screening, importance Quantitative easing (QE) impact usage Quantitative filtering Random walk, investment Real annual return Real assets Commodity Hedger perspective equity-like exposure Real estate, spread trade Real interest rates, increase (1931) Real macro involvement success, personality traits/characteristics (usage) Real money beta-plus domination denotation evolution flaws hedge funds, differentiation impacts, protection importance investors commodity exposure diversification, impact macro principles management, change weaknesses Real money accounts importance long-only investment focus losses (2008) Real money funds Commodity Hedger operation Equity Trader management flexibility frontier, efficiency illiquid asset avoidance importance leverage example usage management managerial reserve optimal portfolio construction failure portfolio management problems size Real money managers Commodity Investor scenario liquidity, importance long-term investor misguidance poor performance, usage (excuse) portfolio construction valuation approach, usage Real money portfolios downside volatility, mitigation leverage, amount management flaws Rear view mirror investment process Redemptions absence problems Reflexivity Rehypothecation Reichsmarks, foreign holders (1922-1923) Relative performance, inadequacy Reminiscences of a Stock Operator (Lefèvre) Renminbi (2005-2009) Repossession property levels Republic of Turkey examination investment rates+equities (1999-2000) Reserve currency, question Resource nationalism Returns forecast generation maximization momentum models targets, replacement Return-to-worst-drawdown, ratios (improvement) Reward-to-variability ratio Riksbank (Sweden) Risk amount, decision aversion rules capital, reduction collars function positive convexity framework, transition function global macro manager approach increase, leverage (usage) measurement techniques, importance parameters Pensioner management pricing reduction system, necessity Risk-adjusted return targets, usage Risk assets, decrease Risk-free arbitrage opportunities Risk management Commodity Hedger process example game importance learning lessons portfolio level process P&L, impact tactic techniques, importance Risk premia annualization earning level, decrease specification Risk/reward trades Risk-versus-return, Pensioner approach Risk-versus-reward characteristics opportunities Roll yield R-squared (correlation) Russia crisis Russia Index (RTSI$) (1995-2002) Russia problems Savings ratio, increase Scholes, Myron Sector risk, limits Securities, legal lists Self-reinforcing cycles (Soros) Sentiment prediction swings Seven Sisters Sharpe ratio increase return/risk Short-dated assets Short selling, ban Siegel’s Paradox example Single point volatility 60-40 equity-bond policy portfolio 60-40 model 60-40 portfolio standardization Smither, Andrew Socialism, Equity Trader concern Society, functioning public funds, impact real money funds, impact Softbank (2006) Soros, George self-reinforcing cycles success Sovereign wealth fund Equity Trader operation operation Soybeans (1970-2009) Special drawing rights (SDR) Spot price, forward price (contrast) Spot shortages/outages, impact Standard deviation (volatility) Standard & Poor’s 500 (S&P500) (2009) decrease Index (1986-1995) Index (2000-2009) Index (2008) shorting U.S. government bonds, performance (contrast) Standard & Poor’s (S&P) shorts, coverage Stanford University (endowment) State pension fund Equity Trader operation operation Stochastic volatility Stock index total returns (1974-2009) Stock market increase, Predator nervousness Stocks hedge funds, contrast holders, understanding pickers, equity index futures usage shorting/ownership, contrast Stops, setting Stress tests, conducting Subprime Index (2007-2009) Sunnies, bidding Super Major Survivorship bias Sweden AP pension funds government bond market Swensen, David equity-centric portfolio Swiss National Bank (SNB) independence Systemic banking crisis Tactical asset allocation function models, usage Tactical expertise Tail hedging, impact Tail risk Take-private LBO Taleb, Nassim Tax cut sunset provisions Taxes, hedge Ten-year U.S. government bonds (2008-2009) Theta, limits Thundering Herd (Merrill Lynch) Time horizons decrease defining determination shortening Titanic funnel, usage Titanic loss number Titanic scenario threshold Topix Index (1969-2000) Top-line inflation Total credit market, GDP percentage Total dependency ratio Trade ideas experience/awareness, impact generation process importance origination Traders ability Bond Trader hiring characteristics success, personality characteristics Trades attractiveness, measurement process hurdle money makers, percentage one-year time horizon selection, Commodity Super Cycle (impact) time horizon, defining Trading decisions, policy makers (impact) floor knowledge noise level ideas, origination Tragedy of the commons Transparency International, Corruption Perceptions Index Treasury Inflation-Protected Securities (TIPS) trade Triangulated conviction Troubled Asset Relief Program (TARP) Turkey economy inflation/equities (1990-2009) investment rates+equities (1999-2000) stock market index (ISE 100) Unconventional Success (Swensen) Underperformance, impact Undervaluation zones, examination United Kingdom (UK), two-year UK swap rates (2008) United States bonds pricing debt (1991-2008) debt (2000-2008) home prices (2000-2009) hyperinflation listed equities, asset investment long bonds, market pricing savings, increase stocks tax policy (1922-1936) trade deficit, narrowing yield curves (2004-2006) University endowments losses impact unlevered portfolio U.S.

pages: 257 words: 13,443

Statistical Arbitrage: Algorithmic Trading Insights and Techniques by Andrew Pole

algorithmic trading, Benoit Mandelbrot, constrained optimization, Dava Sobel, George Santayana, Long Term Capital Management, Louis Pasteur, mandelbrot fractal, market clearing, market fundamentalism, merger arbitrage, pattern recognition, price discrimination, profit maximization, quantitative trading / quantitative finance, risk tolerance, Sharpe ratio, statistical arbitrage, statistical model, stochastic volatility, systematic trading, transaction costs

Consider just the simple spread modeling that provides much of the background of the discussion in this book: The variance of the return stream determines the richness of potential bets (the basic viability of candidate raw material for a strategy), variability of mark to market gains and losses while a bet is extant (the risk profile of a strategy, stop loss rules), and return stretching by stochastic resonance (see Section 3.7). Generalized autoregressive conditional heteroscedastic (GARCH) and stochastic volatility models loom large in the modeling of volatilities. The derivatives literature is replete with variants of the basic GARCH model with acronyms ranging from AGARCH through EGARCH to GJR GARCH, IGARCH, SGARCH and TGARCH. GARCH models are linear regression models with a nonlinear structural specification for the error variance. The error variance, in other models assumed to be a constant or a known function of some aspect of the model or time (see the discussion of variance laws in Pole, et al.), is specified to be a linear function of the error term in the basic regression function.

Knowing that once a spread has ‘‘returned’’ to its mean it will henceforth exhibit essentially random variation about that mean suggests that the reversion exit rule can be modified from the basic ‘‘exit when the forecast is zero’’ to ‘‘exit a little on the other Structural Models 59 side of the zero forecast from which the trade was entered.’’ Here the ‘‘little’’ is calibrated by analysis of the range of variability of the spread in recent episodes of wandering about the mean before it took off (up or down). Volatility forecasting models, GARCH, stochastic volatility, or other models may be useful in this task. The phenomenon of ‘‘noise at rest,’’ the random wandering about the local mean just exemplified, is known as stochastic resonance. As you read the foregoing description, you may feel a sense of deja vu. The description of modeling the variation about the mean during periods of zero forecast activity is quite the same as the general description of the variation of the spread overall.

.), 3, 189 Shell, see Royal Dutch Shell (RD)–British Petroleum (BP) spread Sinusoid, 19–20, 170 Spatial model analogy, 200n1 Specialists, 3, 156–157 Speer, Leeds & Kellog, 189 Spitzer, Elliot, 176, 180 Spread margins, 16–18. See also specific companies Standard & Poor’s (S&P): S&P 500, 28 futures and exposure, 21 Standard deviations, 16–18 Stationarity, 49, 84–85 Stationary random process, reversion in, 114–136 amount of reversion, 118–135 frequency of moves, 117 movements from other than median, 135–136 Statistical arbitrage, 1–7, 9–10 Stochastic resonance, 20, 50, 58–59, 169, 204 Stochastic volatility, 50–51 Stock split, 13n1 Stop loss, 39 Structural change, return decline and, 179–180 Structural models, 37–66 accuracy issues, 59–61 classical time series models, 47–52 doubling and, 81–83 exponentially weighted moving average, 40–47 factor model, 53–58, 63–66 stochastic resonance, 58–59 Stuart, Alan, 63 Student t distribution, 75, 124–126, 201 Sunamerica, Inc. (SAI)–Federal Home Loan Mortgage Corp.

pages: 119 words: 10,356

Topics in Market Microstructure by Ilija I. Zovko

Brownian motion, computerized trading, continuous double auction, correlation coefficient, financial intermediation, Gini coefficient, information asymmetry, market design, market friction, market microstructure, Murray Gell-Mann, p-value, quantitative trading / quantitative finance, random walk, stochastic process, stochastic volatility, transaction costs

Provided agents prefer a lower expected execution time, their model predicts a positive relationship between volatility and limit order placement. Copeland and Galai (1983); Glosten and Milgrom (1985); Easley and O’Hara (1987); Glosten (1995); Foucault (1999); Easley et al. (2001) examine asymmetric information effects on order placement. Andersen (1996) modifies the Glosten and Milgrom (1985) model with the stochastic volatility and information flow perspective. Other models of trading in limit order markets include Cohen et al. (1981); Angel (1994); Harris (1998); Chakravarty and Holden (1995); Seppi (1997); Rock (1990); Parlour and Seppi (2003); Parlour (1998); Foucault et al. (2001); Domowitz and Wang (1994). Empirical research is equally rich. Roll (1984); Choi et al. (1998) estimate spreads from transaction prices.

This seems to be at odds with the interpretation of informed vs. uninformed trades, and we propose that it may be more likely it is the lack of liquidity associated with small orders that moves the price. 1 Firm order size is the number of Pounds a firm trades during a day or hour, depending on the aggregation scale. 98 Bibliography Y. Amihud and H. Mendelson. Dealership market: Market-making with inventory. Journal of Financial Economics, 8(1):31–53, 1980. T. G. Andersen. Return volatility and trading volume: An information flow interpretation of stochastic volatility. Journal of Finance, 51(1):169–204, 1996. J. J. Angel. Limit versus marker orders. In Working Paper Series. School of Business Administration, Georgetown University, 1994. L. Bachelier. Théorie de la spéculation, 1900. In H. Cooper, P., editor, The Random Character of Stock Prices. MIT Press, Cambridge, 1964. P. Bak, M. Paczuski, and M. Shubik. Price variations in a stock market with many agents.

pages: 320 words: 33,385

Market Risk Analysis, Quantitative Methods in Finance by Carol Alexander

asset allocation, backtesting, barriers to entry, Brownian motion, capital asset pricing model, constrained optimization, credit crunch, Credit Default Swap, discounted cash flows, discrete time, diversification, diversified portfolio,, fixed income, implied volatility, interest rate swap, market friction, market microstructure, p-value, performance metric, quantitative trading / quantitative finance, random walk, risk tolerance, risk-adjusted returns, risk/return, Sharpe ratio, statistical arbitrage, statistical model, stochastic process, stochastic volatility, Thomas Bayes, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-sum game

Otherwise we need to use a numerical method – usually simulation – to compute the VaR of a portfolio. The yield on a bond is the constant discount rate that, when applied to the future cash flows from the bond, gives its market price. Given the market price of a typical bond, we can only compute its yield using a numerical method. When we make realistic assumptions about the evolution of the underlying price, such as that the price process has a stochastic volatility, then the only way that we can find a theoretical price of an American option is using a numerical method such as finite differences or Monte Carlo simulations. Numerical methods are resolution techniques, i.e. techniques for resolving a problem. They need to be applied to virtually all problems in financial analysis once we progress beyond the most basic assumptions about returns processes.

But few other financial optimization problems have analytic solutions and in the vast majority of financial applications we need to use a numerical 8 Basis splines – commonly called B splines – are splines where each spline function is not a cubic but is a weighted sum of certain pre-defined basis functions. Discount rates are defined in Section III.1.2. Wikipedia has a good entry on Hermite polynomials. 9 Numerical Methods in Finance 201 method, often a gradient algorithm, to find the maximum or minimum value of the function in the feasible domain. Other financial applications of constrained optimization include the calibration of stochastic volatility option pricing models using a least squares algorithm and the estimation of the parameters of a statistical distribution. In the majority of cases no analytic solution exists and we must find a solution using a numerical method. I.5.4.1 Least Squares Problems Many statistical optimization problems involve changing the parameters of an objective function so that the function fits a given set of data as closely as possible.

Finally, we fill down the possible asset prices, starting with the current price S0 = 100, say, using (I.5.48). The computations are done in the case study spreadsheet. Figure I.5.18 shows four possible paths for the asset price, but each time you press F9 a new set of random numbers is generated. For pricing and hedging options it is common to simulate price paths of assets following alternative asset price diffusions, perhaps with mean reversion, stochastic volatility or jumps. Simulations are particularly useful when volatility is assumed to be stochastic.34 Simulation is a crude but sure method to obtain option prices and hedge ratios. It allows one to consider a huge variety of processes and price and hedge virtually any type of path-dependent claim. 33 Assuming, for simplicity, no dividend or carry cost on the asset; otherwise we would subtract this from the risk free rate to obtain the risk neutral drift.

pages: 1,088 words: 228,743

Expected Returns: An Investor's Guide to Harvesting Market Rewards by Antti Ilmanen

Andrei Shleifer, asset allocation, asset-backed security, availability heuristic, backtesting, balance sheet recession, bank run, banking crisis, barriers to entry, Bernie Madoff, Black Swan, Bretton Woods, business cycle, buy and hold, buy low sell high, capital asset pricing model, capital controls, Carmen Reinhart, central bank independence, collateralized debt obligation, commoditize, commodity trading advisor, corporate governance, credit crunch, Credit Default Swap, credit default swaps / collateralized debt obligations, debt deflation, deglobalization, delta neutral, demand response, discounted cash flows, disintermediation, diversification, diversified portfolio, dividend-yielding stocks, equity premium, Eugene Fama: efficient market hypothesis, fiat currency, financial deregulation, financial innovation, financial intermediation, fixed income, Flash crash, framing effect, frictionless, frictionless market, G4S, George Akerlof, global reserve currency, Google Earth, high net worth, hindsight bias, Hyman Minsky, implied volatility, income inequality, incomplete markets, index fund, inflation targeting, information asymmetry, interest rate swap, invisible hand, Kenneth Rogoff, laissez-faire capitalism, law of one price, London Interbank Offered Rate, Long Term Capital Management, loss aversion, margin call, market bubble, market clearing, market friction, market fundamentalism, market microstructure, mental accounting, merger arbitrage, mittelstand, moral hazard, Myron Scholes, negative equity, New Journalism, oil shock, p-value, passive investing, Paul Samuelson, performance metric, Ponzi scheme, prediction markets, price anchoring, price stability, principal–agent problem, private sector deleveraging, purchasing power parity, quantitative easing, quantitative trading / quantitative finance, random walk, reserve currency, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, riskless arbitrage, Robert Shiller, Robert Shiller, savings glut, selection bias, Sharpe ratio, short selling, sovereign wealth fund, statistical arbitrage, statistical model, stochastic volatility, stocks for the long run, survivorship bias, systematic trading, The Great Moderation, The Myth of the Rational Market, too big to fail, transaction costs, tulip mania, value at risk, volatility arbitrage, volatility smile, working-age population, Y2K, yield curve, zero-coupon bond, zero-sum game

In conditional models, βs (factor sensitivities) or λs (factor risk premia) can be specified as linear functions of any set of predictors (conditioners) that are assumed to be in investors’ information sets. Time-varying risks Rolling estimates of second moments (historical volatilities, correlations, and factor sensitivities) are simple proxies of ex ante risks. In the past 20 years, academics have tried to improve on them by developing theoretical models of stochastic volatilities, simple linear models that predict future volatilities with a set of conditioners, more complex regime-switching models, and most importantly, a variety of so-called GARCH models. Opening up the acronym GARCH, to read “generalized autoregressive conditional heteroskedasticity”, scares many investors, but the main idea is simply that volatility is not constant but varies predictably over time.

They regress single-stock return variances on S&P 500 index variance to estimate variance betas and find that stocks with higher variance betas are associated with higher variance risk premia (a wider gap between implied and realized variances). Apparently, investors are willing to accept some losses in exchange for hedging away market volatility spikes. The variance risk premium may reflect (1) the empirical linkage between stock return and variance and/or (2) stochastic volatility as a distinct risk factor. Stock market variance tends to be higher during down markets; thus long (short) variance positions have a negative (positive) stock market beta. Because this beta risk can explain empirically only a small portion of observed profits from variance selling (variance risk premia), an independent variance risk factor seems to be the key contributor. This evidence is relatively weak; stronger evidence for a volatility premium comes from the finding that volatility selling is more profitable for stocks with high volatility.

Assessing irrational and rational explanations Irrational explanations rely on (1) the demand from irrational investors (end-users) distorting option prices and (2) the inability of rational risk-neutral arbitrageurs or market-makers to offset these distortions. In contrast to the ideal BSM world, in the real world market-makers cannot perfectly hedge their option positions. Noncontinuous pricing, jumps, stochastic volatility, and trading costs as well as capital constraints make market-makers sensitive to unhedgeable option risks and cause them to require compensation for bearing them. Since market-maker actions no longer offset end-user demand biases, it is important to assess what those biases are. New academic studies document many demand regularities that are consistent with observed pricing regularities:• End-users tend to be net long index options and net short single-stock options.

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Mathematics of the Financial Markets: Financial Instruments and Derivatives Modelling, Valuation and Risk Issues by Alain Ruttiens

algorithmic trading, asset allocation, asset-backed security, backtesting, banking crisis, Black Swan, Black-Scholes formula, Brownian motion, capital asset pricing model, collateralized debt obligation, correlation coefficient, Credit Default Swap, credit default swaps / collateralized debt obligations, delta neutral, discounted cash flows, discrete time, diversification, fixed income, implied volatility, interest rate derivative, interest rate swap, margin call, market microstructure, martingale, p-value, passive investing, quantitative trading / quantitative finance, random walk, risk/return, Satyajit Das, Sharpe ratio, short selling, statistical model, stochastic process, stochastic volatility, time value of money, transaction costs, value at risk, volatility smile, Wiener process, yield curve, zero-coupon bond

Coming back to Eq. 9.2 the first term of this relationship, E({rt |t - 1), is the conditional mean of rt, and is predictable. It values art−1 if it is modeled by an AR(1) process. The second term, t, is the innovation term, that is unpredictable. To go a step further, it makes thus sense to now model t, to (try to) reduce this forecast error. There are two ways: either, by a stochastic equation (stochastic volatility model, cf. Chapter 12, Section 12.2), or, similarly as for the conditional mean, by a linear (auto)regression, that is, by an ARCH model: Autoregressive conditional heteroskedastic or ARCH processes – developed by R. Engle6 – aim to model the error term t, responsible for the volatility of the returns, by considering that the variance – that is, the squared volatility – of the process is also conditional to the available information, through a specific function ht.

Figure 12.17 Correlation between Nasdaq and S&P 500 data (2000–2009) The standard deviation of these 50-day series of volatilities, correlations and covariances shows a much bigger dispersion of the correlation than of the related volatilities in the case of the uncorrelated EUR/USD versus S&P 500, while it is of the same order of magnitude in the case of the well-correlated S&P 500 versus NASDAQ 100. Needless to say, the problem of a correlation model, or process, becomes even harder with respect to more than two assets, via correlation matrixes. Currently, the main trails followed by researchers consist of looking for multivariate GARCH models11 or for a multivariate stochastic volatility model, generalizing the Heston model (cf. Section 12.2) in a matrix process of n Wiener processes, leading to a (complex) stochastic correlation model that still allows for analytic tractability.12 12.5 VOLATILITY AND VARIANCE SWAPS Volatility and variance swaps belong to the family of performance swaps, presented in Chapter 6, Section 6.7.6, but are developed here, given the particular nature of the swapped commodity.

TAYLOR, Asset Price Dynamics, Volatility and Prediction, Princeton University Press, 2007, 544 p. 1 The end of Section 12.1.5 will contribute to justify this precision. 2 See Andrew LO, “The statistics of Sharpe Ratios”, Financial Analysts Journal, vol. 58, no. 4, 2002, pp. 36–45. 3 Provided the corresponding random variables are independent. 4 In absolute value of the delta, this allows for a common representation for both calls and puts. 5 For an annualized value, also in the following formulae, 1/n has to be replaced by 250/n. 6 For more details about these processes, see for example D. YANG, Q. ZHANG, “Drift-independent volatility estimation based on high, low, open, and close prices”, Journal of Business, 2000, vol. 73, no. 3, pp. 477–491. 7 For further details about this model, see P.S. HAGAN, D. KUMAR, A.S. LESNIEWSKI, D.E. WOODWARD, “Managing smile risk”, Wilmott Magazine, July 2002, pp. 84–108. 8 See A. LEWIS, The mixing approach to stochastic volatility and jump models,, March 2002. Let us also mention the dynamic model developed by A. SEPP, which involves the VIX spot, the underlying S&P 500, and the VIX futures and options: A. SEPP, “VIX option pricing in a jump-diffusion model”, RISK, April 2008, pp. 84–89. 9 A detailed example of such a GARCH model to volatilities exceeds the framework of this book, both in size and in calculations volume: see, for example, Amit GOYAL, Predictability of stock return volatility from GARCH models, Anderson Graduate School of Management, UCLA, May 2000 (working paper). 10 For further reading, see, for example, T.G.

pages: 246 words: 16,997

Financial Modelling in Python by Shayne Fletcher, Christopher Gardner

Brownian motion, discrete time, interest rate derivative, London Interbank Offered Rate, stochastic volatility, yield curve, zero day, zero-coupon bond

There are many reasons for wanting to fi the discretisation of the time axis but the main reason is that many stochastic differential equations do not have analytic solutions, which means that the equations have to be discretised (e.g. using the Euler’s scheme) in order to solve them. Naturally any discretisation scheme is approximate and the writer of the model will want to control the discretisation error by fixin the size of the evolution step. The authors have found that the aforementioned abstractions of the model into core components works extremely well, even for sophisticated models like the Libor Market Model with stochastic volatility, which requires a non-trivial discretisation scheme in order to evolve the state variables of the model forwards in time. The following snippet illustrates a typical application of the evolve component. >>> import >>> from numpy import zeros 114 >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> >>> Financial Modelling in Python expiries = [0.0, 0.5, 1.0, 1.5, 2.0, 3.0, 4.0, 5.0] tenors = [0, 90] values = zeros((8, 2)) values.fill(0.001) surf =, tenors, values) env = key = "ve.term.eur.hw" env.add surface(key, surf) key = "" env.add constant(key, 0.01) r = ppf.model.hull white.requestor() s = ppf.model.hull white.monte carlo.state(10000) e = evolve("eur") e.evolve(0.0,0.5,s,r,env) Once again unit tests are provided in the module ppf.test.test hull white.

.; mathematics; NumPy; ppf package basics 193–205 batch interpreter mode 193–4 benefit 1–4 built-in data types 1, 195–7 C++/Python ‘Hybrid Systems’ 4, 159–63 C API routines 19–26, 161–3 C/C++ interoperability benefit 2, 3–4, 7–9, 11–26, 157 class basics 2–3, 201–3 COM servers 4, 5–6, 98, 165–89 concepts 1–4, 193–205 control fl w statements 197–200 dictionaries 119–22, 181, 196–7, 215–16 dynamic type system 2–3 encapsulation support 2–3, 58–61, 209 expressiveness aspects 1 extensibility aspects 1–4, 7–9, 11–26 financia engineering 1–4, 11–26 function basics 2–3, 200–1 functional programming idioms 1 GUI toolkits 2 high-level aspects 1 indented code 198–200 inheritance basics 122, 202–3, 209–11 interactive interpreter mode 193–4 interoperability aspects 1–4, 7–9, 11–26, 157 interpreters 2, 24–6, 45–6, 193–4, 208–9 list basics 3, 196–7, 215 Microsoft Excel 4, 165–89 misconceptions 2–3 module basics 203–5 overview of the book 3–4 package basics 1, 203–5 productivity benefit 1 simple expressions 194–5 Index standard libraries 1 structure misconceptions 2–3 tuples 131, 195–7, 200–1, 215 visualisation software integration 2 whirlwind tour 193–205 white space uses 198–200 Python Distutils package 9 Python Programming on Win32 (Hammond & Robinson) 165 Python Scripting for Computational Science (Langtangen) 2 python.hpp 214–15 python -i command 193 PyUnit testing module, concepts 6–7, 9 quadratic roots, concepts 3, 46–9 quadratic fo 132–42 quadratic roots 46–9 quantitative analysis 1–4, 27–61, 123–43, 165–89 raise 15–16, 30–1, 35–6, 40–1, 43–5, 50–8, 66–7, 78, 81–3, 85–7, 88, 89, 91, 98, 102, 105–6, 113, 118–19, 131, 133, 134–6, 166–9, 170–6 random 27–8, 45–6 random number generation, concepts 3, 27–8, 45–6, 112–22 random variables, expectation calculations 3, 49–61 range function, Python basics 197–8 ratio 48–9 rcv flows 89–91, 96–8, 120–2 redemption cap 153–6 redemption floor 153–6 reference counts 20–6 references, C++ 212–14 reg clsid 169–76, 177–87, 188–9 register com class 169–76 register date... 11–16, 160–3 register date more.cpp 160 register numpy.cpp 23 register special functions 18–19 reg progid 169–76, 177–89 regression schemes 4, 132–42, 150–2, 219–20 regression model 136–42, 150–2 regressions 132–42 regrid 55–61 regridder 58–61 regrid fs 55–7 regrid xT 58–61 regrid yT 58–61 relative date 66–7, 105–22, 125–8, 146–57 233 requestor 100–22, 124–8, 129–42, 146–57 requestor component, pricing models 100–22, 124–8, 129–42 reset currencies, concepts 70–9, 96–8, 105–22 reset dates 69–79, 95–8 see also observables reset basis 72–9, 89–91, 96–8, 120–2, 178–87 reset ccy 69–79, 90 reset currency 71–9, 89–90, 96–8, 105–22, 177–87 reset date 69–79, 95–8 reset duration 73–9, 89–91, 96–8, 120–2 reset holiday centres 72–9 reset id 69–79, 93–8, 146–57 reset lag 72–9 reset period 73–9, 89–91, 96–8, 120–2 reset shift method 72–9, 89–91, 96–8, 120–2, 178–87 retrieve 66–7, 97–8, 124–8, 169–76, 179–87, 188–9 retrieve constant 67, 100–22 retrieve curve 66–7, 100–22, 148–57 retrieve surface 67, 100–22 retrieve symbol... 97–8, 124–8, 131–42, 149–52 return statements, Python basics 200–5 risk the Greeks 142–3 management systems 4 Robinson, Andy 165 rollback 57–61, 108–22, 124–8 rollback component, pricing models 108–22, 124–8, 177–87 rollback max 57–61, 108–22, 124–8 rollback tests 109–22 roll duration 72–9, 84–5, 89–91, 96–8, 120–2, 177–87 roll end 72–9, 81–3, 84–5, 86–91 roll period 72–9, 84–5, 89–91, 96–8, 120–2, 177–87 rolls 14–16, 72–3 roll start 77–8, 81–3, 86–91 root-findin algorithms bisection method 35–6, 37 concepts 3, 35–7 Newton–Raphson method 36–7 root finding 35–7 roots 46–9, 53–7 RuntimeError 30–1, 35–6, 40–1, 43–5, 50–8, 66–7, 77, 81–3, 85–7, 88, 89, 90, 98, 102, 104–5, 113, 118–19, 131, 133, 134–6, 147–8, 169, 170–6, 177–8, 202–3 sausage Monte-Carlo method 143 Schwartz, E.S. 219 234 Index SciPy 1, 3, 8 see also NumPy scope guard techniques 20 SDEs 218 second axis 64–5 seed 112–22, 150–2 self 31–4, 45–6, 51–61, 63–7, 69–79, 93–122, 124–8, 130–42, 146–57, 178–89 semi-analytic conditional expectations, concepts 57–61 semi analytic domain integrator 57–61 server 166–89 set event 126–8, 130–42 set last cfs 135–42 sgn 46–9 shape 43–6, 50–1, 58–61, 81–3, 103–22, 133–42 shared ptr hpp 20–4 shift 14–16, 72–9, 86–8, 111–22 shift convention 14–16, 73–9, 80–2, 83–91, 96–8, 120–2, 151–2 shift method 14–16, 73–9, 80–2, 83–91, 120–2 short rates 101–2 sig 42–6, 65 sign 35–6 simple expressions, Python basics 194–5 sin 205 singular value decomposition of a matrix see also linear algebra concepts 42–6 singular value decomposition back substitution 42–6 solve tridiagonal system 2–3, 34, 39–40 solve upper diagonal system 17–19, 40–4, 50–1 solving linear systems see also linear algebra concepts 39–40 solving tridiagonal systems see also linear algebra concepts 2–3, 34, 39–40 solving upper diagonal systems see also linear algebra concepts 17–19, 40–4, 49–51 sort 48–9 special functions 17–18, 27–61 spread 70–9, 154–6 sqrt 48–9, 52–7, 59–61, 100–22 square tridiagonal matrices 33–4, 40–1 standard deviations 44–6, 51–7, 102–22, 133–42, 188–9 standard libraries 1 standard normal cumulative distributions see also N concepts 3, 27–9, 31, 51–7, 102–22 start 80–3, 83–91, 96–8, 120–2, 177–87, 198–200 start date 83–4 start of to year 16 state 59–61, 102–22, 124–8, 129–42, 146–57 state component, pricing models 101–22, 124–8, 129–42, 145–52 stddev 53–7, 102–22 step, Python basics 198–200 STL functions, C++ 29 stochastic volatility 113–14 stop, Python basics 198–200 str 71–9, 80–3, 86–7, 94, 166–8, 170–6 string literals, Python basics 194–6 structure misconceptions, Python 2–3 sum array 23–6 surf 101–22 surface 64–7, 101–22 surfaces see also environment concepts 3, 63, 64–7, 100–22, 170–6 definitio 64 volatility surfaces 3, 6, 63–7, 100–22 surface tests 64–7 svd 42–4 see also singular value decomposition of a matrix swap rates 70–9, 104–5, 115–22 swap obs 105–22 swap rate 74–9, 116–22 swaps 4, 70–9, 101–2, 104–5, 115–22, 123–8, 132–42, 145–52, 157 swap tests 149–52 swaptions 4, 101–2, 115–16, 126–8, 132–42, 145–52, 157 symbol table 97–8, 124–8, 129–42 symbol table listener 125–8, 129–42 symbol value pair 130–42, 155–6 symbol value pairs to add 130–42, 155–6 sys 27–8 table 82–4, 169 tables, adjuvants 82–4, 147–52, 153–6, 177–87 tag 169–76, 177–89 target redemption notes (TARNs) 4, 101–2, 145, 152–7 concepts 152–7 definitio 152 pricing models 4, 101–2, 145, 152–7 target indicator 153–6 tarn coupon leg payoff 152–6 tarn funding leg payoff 154–6 Index TARNs see target redemption notes tarn tests 155–6 templates 18–26, 159–63 tenor duration 72–9 tenor period 72–9 tenors 67, 84–5, 101–22, 170–6 term 28–9, 103–22 term structure of interest rates see yield curves term volatility, Hull–White model 100–22 terminal T 104–22 term var 100–22 term vol 100–22 test 6–7, 9, 17–19, 59–61, 64–7, 109–22, 148–57 test bond 115–22 test bond option 111–22 test bound 30–1 test bound ci 31 test constant 111–22 test discounted libor rollback 109–22 test explanatory variables 117–22 test hull white 67, 109–22 testing concepts 6–7, 9, 17–19 test lattice pricer 148–57 test market 64–7 test math 59–61 test mean and variance 114–22 test monte carlo pricer 154–6 test value 149–52 theta 48–9, 205 throw error already set 21–6 timeline 94–8, 125–8, 129–42 see also events Tk 2 tline 96–8 to ppf date 168–9, 178–87 tower law 60 tower law test 60–1 trace 23–6 Traceback 195–7, 202–3 trade 87–91, 94–8, 125–8, 129–42, 150–7, 177–87 trade representations, concepts 3, 69–91, 93–8 trade server, COM servers 176–87 trade utilities, concepts 88–91 trade VBA client 181 trade id 188–9 trades see also exercise...; fl ws; legs concepts 3, 69, 87–91, 93–8, 123–43, 176–87 definitio 69, 88 TradeServer 176–87, 188–9 trade server 176–87, 188–9 235 trade utils 89–91, 129–42, 153–6, 176–87 transpose 41–4 tridiagonal systems 2–3, 33–4, 39–40 try 27–8, 171–6, 177–87 Trying 6–7 tuples, Python basics 131, 195–7, 200–1, 215 TypeError 195–7 Ubuntu... 8 underlying 127–8, 130–42 unicode 172 unit fo 132–42 update indicator 134–42 update symbol 97–8, 126–8, 131–42 upper bound 29–31 USD 70–83, 152 utility 6–7, 29–61, 64–7 utility functions 17–26, 29–61 utils 168–76, 187–9 values 101–22 vanilla financia instruments, pricing approaches 99, 123–8, 145–57 var 102–22 variance 51–61, 102–22 variates 103–22 varT 102–22 Vasicek models 217–18 see also Hull–White model VB... see Microsoft... vector 41, 44–6, 133–42, 212 vectorize 133–42 visualisation software integration, Python benefit 2 vol 51–7, 59–61, 114–22 volatility Hull–White model 100–22 piecewise polynomial integration 51–61 surfaces 3, 6, 63–7, 100–22 vols 65 volt 59–61, 59–61 weekdays 15–16, 159–63 while statements, Python basics 199–200 white space, Python basics 198–200 win32 165–89 Win32 Python extensions 165–89 xh 52–7 xl 52–7 xprev 53–7 xs 56–61 xsT 60–1 xT 58–61, 108–22 xtT 58–61, 108–22 xt 58–61, 108–22

pages: 209 words: 13,138

Empirical Market Microstructure: The Institutions, Economics and Econometrics of Securities Trading by Joel Hasbrouck

Alvin Roth, barriers to entry, business cycle, conceptual framework, correlation coefficient, discrete time, disintermediation, distributed generation, experimental economics, financial intermediation, index arbitrage, information asymmetry, interest rate swap, inventory management, market clearing, market design, market friction, market microstructure, martingale, price discovery process, price discrimination, quantitative trading / quantitative finance, random walk, Richard Thaler, second-price auction, selection bias, short selling, statistical model, stochastic process, stochastic volatility, transaction costs, two-sided market, ultimatum game, zero-sum game

Lyons (2001) discusses the market microstructure of the foreign exchange market, providing a useful alternative to the present treatment, which is based more on equity markets. Survey articles include Hasbrouck (1996a), Madhavan (2000), and Biais, Glosten, and Spatt (2005). Amihud, Mendelson, and Pedersen (2005) survey the rapidly growing field that links microstructure and asset pricing. Shepard (2005) is a useful collection of key readings in stochastic volatility. This research increasingly relies on high-frequency data and therefore more deeply involves microstructure issues. Some characteristics of security price dynamics are best discussed in context of the larger environment in which the security market operates. Cochrane (2005) is a comprehensive and highly comprehensible synthesis of the economics of asset pricing. Related background readings on financial economics include Ingersoll (1987), Huang and Litzenberger (1998), and Duffie (2001).

Sandas, Patrik, 2001, Adverse selection and competitive market making: evidence from a pure limit order book, Review of Financial Studies 14, 705–34. Sargent, Thomas J., 1979, Macroeconomic Theory (Academic Press, New York). Seppi, Duane J., 1990, Equilibrium block trading and asymmetric information, Journal of Finance 45, 73–94. Seppi, Duane J., 1997, Liquidity provision with limit orders and a strategic specialist, Review of Financial Studies 10, 103–50. Shephard, Neil, 2005, Stochastic Volatility (Oxford University Press, Oxford); esp. General introduction. 193 194 REFERENCES Smith, Jeffrey W., 2000, Market vs. limit order submission behavior at a Nasdaq market maker, (National Association of Securities Dealers (NASD), NASD Economic Research). Smith, Jeffrey W., James P. Selway III, and D. Timothy McCormick, 1998, The Nasdaq stock market: historical background and current operation (Nasdaq working paper 98-01).

Mathematical Finance: Core Theory, Problems and Statistical Algorithms by Nikolai Dokuchaev

Black-Scholes formula, Brownian motion, buy and hold, buy low sell high, discrete time, fixed income, implied volatility, incomplete markets, martingale, random walk, short selling, stochastic process, stochastic volatility, transaction costs, volatility smile, Wiener process, zero-coupon bond

J. of Financial Economics 7, 229–263. [5] Föllmer, H. and Schied, A. (2002). Stochastic Finance: An Introduction in Discrete Time 2. De Gruyter Studies in Mathematics, Berlin. [6] Gujarati, D. (1995). Basic Econometrics. New York: McGraw-Hill. [7] Higham, D.J. (2004). An Introduction to Financial Option Valuation. Cambridge: Cambridge University Press. [8] Hull, J. and White, A. (1987). The pricing of options on assets with stochastic volatilities. Journal of Finance 42, 281–300. [9] Karatzas, I. and Shreve, S.E. (1998). Methods of Mathematical Finance. New York: SpringerVerlag. [10] Korn, R. (2001). Option Pricing and Portfolio Optimization: Modern Methods of Financial Math. Providence, R.I.: American Mathematical Society. [11] Lambertone, D. and Lapeyre, B. (1996). Introduction to Stochastic Calculus Applied to Finance. London: Chapman & Hall. [12] Li, D. and Ng, W.L. (2000).

pages: 240 words: 60,660

Models. Behaving. Badly.: Why Confusing Illusion With Reality Can Lead to Disaster, on Wall Street and in Life by Emanuel Derman

Albert Einstein, Asian financial crisis, Augustin-Louis Cauchy, Black-Scholes formula, British Empire, Brownian motion, capital asset pricing model, Cepheid variable, creative destruction, crony capitalism, diversified portfolio, Douglas Hofstadter, Emanuel Derman, Eugene Fama: efficient market hypothesis, fixed income, Henri Poincaré, I will remember that I didn’t make the world, and it doesn’t satisfy my equations, Isaac Newton, Johannes Kepler, law of one price, Mikhail Gorbachev, Myron Scholes, quantitative trading / quantitative finance, random walk, Richard Feynman, riskless arbitrage, savings glut, Schrödinger's Cat, Sharpe ratio, stochastic volatility, the scientific method, washing machines reduced drudgery, yield curve

The Black-Scholes formula, which translate estimates of volatility into option prices, seemed so arcane when it burst upon the world that Black and Scholes had great difficulty getting their paper accepted for publication. Then, as users of the model grew more experienced, volatility became common currency. Nowadays traders and quants have grown so sophisticated that they talk fluently about models with stochastic volatility, a volatility that is itself volatile. Sweep Dirt Under the Rug, but Let Users Know About It One should be humble in applying mathematics to markets, and be wary of overly ambitious theories. Whenever we make a model of something involving human beings, we are trying to force the ugly stepsister’s foot into Cinderella’s pretty glass slipper. It doesn’t fit without cutting off some essential parts.

pages: 923 words: 163,556

Advanced Stochastic Models, Risk Assessment, and Portfolio Optimization: The Ideal Risk, Uncertainty, and Performance Measures by Frank J. Fabozzi

algorithmic trading, Benoit Mandelbrot, capital asset pricing model, collateralized debt obligation, correlation coefficient, distributed generation, diversified portfolio, fixed income, index fund, Louis Bachelier, Myron Scholes, p-value, quantitative trading / quantitative finance, random walk, risk-adjusted returns, short selling, stochastic volatility, Thomas Bayes, transaction costs, value at risk

This was displayed, for example, in the second equation of equation (D.6) where K was the P(Sτ ≤ K)-quantile of Sτ, which translated into the P(Sτ ≤ K)-quantile K/S0 of the new random variable Y = Sτ/S0. References Bachelier, Louis. [1900] 2006. Louis Bachelier’s Theory of Speculation: The Origins of Modern Finance. Translated by Mark Davis and Alison Etheridge. Princeton, N.J.: Princeton university Press. Barndorff-Nielsen, Ole E. 1997. “Normal Inverse Gaussian Distributions and Stochastic Volatility Modelling.” Scandinavian Journal of Statistics 24 (1): 1-13. Black, Fischer, and Myron Scholes. 1973. “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81 (3): 637-654. ———. 1987. “A Conditionally Heteroscedastic Time-Series Model for Security Prices and Rates of Return Data.” Review of Economics and Statistics 69: 542-547. Bollerslev, Tim. 2001. “Financial Econometrics: Past Developments and Future Challenges.”

After the i-th independent variable has been removed, the sum of square residuals of the regression with the remaining k − i variables is given by SSEk-i. 263 See Chapter 17 for what is meant by best linear unbiased estimators. 264 Standardized residuals would be compared with the standard normal distribution, N(0,1). 265 The chi-square distribution is covered in Chapter 11. 266 The p-value is explained in Chapter 19. 267 The term “conditional” in the title of the two models means that the variance depends on or is conditional on the value of the random variable. 268 This pattern in the volatility of asset returns was first reported by Mandelbrot (1963). 269 See Bollerslev (2001). 270 In addition to ARCH/GARCH models, there are other models that deal with time-varying volatility, such as stochastic-volatility models, which are beyond the scope of this introductory chapter. 271 For convenience, we have dropped the error term subscript. 272 This simplest ARCH model is commonly referred to as ARCH(1) since it incorporates exactly one lag-term (i.e., the squared deviation from the mean at time t - 1). 273 Since here we use the squared deviations from three immediately prior periods (i.e., lag terms), that is commonly denoted as ARCH(3). 274 See Savin and White (1977). 275 The function f(x) = ln (x) is the natural logarithm.

Monte Carlo Simulation and Finance by Don L. McLeish

Black-Scholes formula, Brownian motion, capital asset pricing model, compound rate of return, discrete time, distributed generation, finite state, frictionless, frictionless market, implied volatility, incomplete markets, invention of the printing press, martingale, p-value, random walk, Sharpe ratio, short selling, stochastic process, stochastic volatility, survivorship bias, the market place, transaction costs, value at risk, Wiener process, zero-coupon bond, zero-sum game

These uniform random variables determine such quantities as the normally distributed increments of the logarithm of the stock price. In summary, the simulation is used simply to estimate a multidimensional integral of the form E(g(U1 , ..., Ud )) = Z Z .. Z g(u1 , u2 , ...ud )du1 du2 . . . dud (4.1) over the unit cube in d dimensions where often d is large. As an example in finance, suppose that we wish to price a European option on a stock price under the following stochastic volatility model. INTRODUCTION 205 Example 33 Suppose the daily asset returns under a risk-neutral distribution is assumed to be a variance mixture of the Normal distribution, by which we mean that the variance itself is random, independent of the normal variable and follows a distribution with moment generating function s(s). More specifically assume under the Q measure that the stock price at time n∆t is determined from S(n+1)∆t = Sn∆t exp{r∆t + σn+1 Zn+1 } m( 12 ) where, under the risk-neutral distribution, the positive random variables σi2 are assumed to have a distribution with moment generating function m(s) = E{exp(sσi )}, Zi is standard normal independent of σi2 and both (Zi , σi2 ) are independent of the process up to time n∆t.

pages: 364 words: 101,286

The Misbehavior of Markets: A Fractal View of Financial Turbulence by Benoit Mandelbrot, Richard L. Hudson

Albert Einstein, asset allocation, Augustin-Louis Cauchy, Benoit Mandelbrot, Big bang: deregulation of the City of London, Black-Scholes formula, British Empire, Brownian motion, business cycle, buy and hold, buy low sell high, capital asset pricing model, carbon-based life, discounted cash flows, diversification, double helix, Edward Lorenz: Chaos theory, Elliott wave, equity premium, Eugene Fama: efficient market hypothesis, Fellow of the Royal Society, full employment, Georg Cantor, Henri Poincaré, implied volatility, index fund, informal economy, invisible hand, John Meriwether, John von Neumann, Long Term Capital Management, Louis Bachelier, mandelbrot fractal, market bubble, market microstructure, Myron Scholes, new economy, paper trading, passive investing, Paul Lévy, Paul Samuelson, plutocrats, Plutocrats, price mechanism, quantitative trading / quantitative finance, Ralph Nelson Elliott, RAND corporation, random walk, risk tolerance, Robert Shiller, Robert Shiller, short selling, statistical arbitrage, statistical model, Steve Ballmer, stochastic volatility, transfer pricing, value at risk, Vilfredo Pareto, volatility smile

. • Reprint: Beyond Efficiency and Equilibrium. Edited by Doyne Farmer and John Geanakoplos. Oxford, UK: The University Press, 2004. Mandelbrot, Benoit B. 2001c. Scaling in financial prices, III: Cartoon Brownian motions in multifractal time. Quantitative Finance 1: 427-440. Mandelbrot, Benoit B. 2001d. Scaling in financial prices, IV: Multifractal concentration. Quantitative Finance 1: 641-649. Mandelbrot, Benoit B. 2001e. Stochastic volatility, power-laws and long memory. Quantitative Finance 1: 558-559. Mandelbrot, Benoit B. 2002. Gaussian Self-Affinity and Fractals: Globality, the Earth, 1/f Noise, and R/S. New York: Springer Verlag. Mandelbrot, Benoit B. 2003. Heavy tails in finance for independent or multifractal price increments. Handbook on Heavy Tailed Distributions in Finance. Edited by Svetlozar T. Rachev (Handbooks in Finance: 30, Senior Editor: William T.

pages: 313 words: 101,403

My Life as a Quant: Reflections on Physics and Finance by Emanuel Derman

Berlin Wall, bioinformatics, Black-Scholes formula, Brownian motion, buy and hold, capital asset pricing model, Claude Shannon: information theory, Donald Knuth, Emanuel Derman, fixed income, Gödel, Escher, Bach, haute couture, hiring and firing, implied volatility, interest rate derivative, Jeff Bezos, John Meriwether, John von Neumann, law of one price, linked data, Long Term Capital Management, moral hazard, Murray Gell-Mann, Myron Scholes, Paul Samuelson, pre–internet, publish or perish, quantitative trading / quantitative finance, Sharpe ratio, statistical arbitrage, statistical model, Stephen Hawking, Steve Jobs, stochastic volatility, technology bubble, the new new thing, transaction costs, volatility smile, Y2K, yield curve, zero-coupon bond, zero-sum game

Years ago, when I first became aware of the smile and hoped to find the "right" model, I used to ask colleagues at other firms which model they thought was correct. But now there is such a profusion of models that I ask more practical questions-not "What do you believe?" but rather "When you hedge a standard S&P 500 option, do you use the Black-Scholes hedge ratio, something larger, or something smaller?" Local volatility models produce smaller hedge ratios, while stochastic volatility models tend to produce larger ones. The differences between the models are even more dramatic for exotic options. In 2003, at a derivatives meeting in Barcelona, I led a small roundtable discussion group on the smile. There were fifteen of us, traders and quants from derivatives desks all over the world. I asked everyone my simple question: When you hedge an S&P 500 option, would you use the Black-Scholes hedge ratio, something larger, or something smaller?

pages: 483 words: 141,836

Red-Blooded Risk: The Secret History of Wall Street by Aaron Brown, Eric Kim

activist fund / activist shareholder / activist investor, Albert Einstein, algorithmic trading, Asian financial crisis, Atul Gawande, backtesting, Basel III, Bayesian statistics, beat the dealer, Benoit Mandelbrot, Bernie Madoff, Black Swan, business cycle, capital asset pricing model, central bank independence, Checklist Manifesto, corporate governance, creative destruction, credit crunch, Credit Default Swap, disintermediation, distributed generation, diversification, diversified portfolio, Edward Thorp, Emanuel Derman, Eugene Fama: efficient market hypothesis, experimental subject, financial innovation, illegal immigration, implied volatility, index fund, Long Term Capital Management, loss aversion, margin call, market clearing, market fundamentalism, market microstructure, money market fund, money: store of value / unit of account / medium of exchange, moral hazard, Myron Scholes, natural language processing, open economy, Pierre-Simon Laplace, pre–internet, quantitative trading / quantitative finance, random walk, Richard Thaler, risk tolerance, risk-adjusted returns, risk/return, road to serfdom, Robert Shiller, Robert Shiller, shareholder value, Sharpe ratio, special drawing rights, statistical arbitrage, stochastic volatility, stocks for the long run, The Myth of the Rational Market, Thomas Bayes, too big to fail, transaction costs, value at risk, yield curve

But by 1987 the terminology was so entrenched that the definition of derivative was changed rather than change what people called options. This kind of redefinition is another reason many people didn’t notice the shift. Smile and skew blew many quant strategies out of the water, but also created a host of new ones. We got two major models to account for smile and skew pretty quickly—called local volatility and stochastic volatility—but they gave opposite advice on what to do about the phenomena, and no one has resolved the two models yet or come up with anything better. The most amazing part is that this didn’t happen just in the stock market. You might expect it there. The same change happened simultaneously in option markets for foreign exchange (FX) and interest rates, even in commodity markets. There had been no major disruptions in these markets, and there weren’t a lot of people who traded multiple types of options.

pages: 651 words: 180,162

Antifragile: Things That Gain From Disorder by Nassim Nicholas Taleb

Air France Flight 447, Andrei Shleifer, banking crisis, Benoit Mandelbrot, Berlin Wall, Black Swan, business cycle, Chuck Templeton: OpenTable:, commoditize, creative destruction, credit crunch, Daniel Kahneman / Amos Tversky, David Ricardo: comparative advantage, discrete time, double entry bookkeeping, Emanuel Derman, epigenetics, financial independence, Flash crash, Gary Taubes, George Santayana, Gini coefficient, Henri Poincaré, high net worth, hygiene hypothesis, Ignaz Semmelweis: hand washing, informal economy, invention of the wheel, invisible hand, Isaac Newton, James Hargreaves, Jane Jacobs, joint-stock company, joint-stock limited liability company, Joseph Schumpeter, Kenneth Arrow, knowledge economy, Lao Tzu, Long Term Capital Management, loss aversion, Louis Pasteur, mandelbrot fractal, Marc Andreessen, meta analysis, meta-analysis, microbiome, money market fund, moral hazard, mouse model, Myron Scholes, Norbert Wiener, pattern recognition, Paul Samuelson, placebo effect, Ponzi scheme, principal–agent problem, purchasing power parity, quantitative trading / quantitative finance, Ralph Nader, random walk, Ray Kurzweil, rent control, Republic of Letters, Ronald Reagan, Rory Sutherland, selection bias, Silicon Valley, six sigma, spinning jenny, statistical model, Steve Jobs, Steven Pinker, Stewart Brand, stochastic process, stochastic volatility, Thales and the olive presses, Thales of Miletus, The Great Moderation, the new new thing, The Wealth of Nations by Adam Smith, Thomas Bayes, Thomas Malthus, too big to fail, transaction costs, urban planning, Vilfredo Pareto, Yogi Berra, Zipf's Law

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