Artificial Intelligence: A Modern Approach
by Stuart Russell and Peter Norvig · 14 Jul 2019 · 2,466pp · 668,761 words
sense of a spoken or written sequence of words. How can dynamic situations like these be modeled? 14.1.1States and observations This chapter discusses discrete-time models, in which the world is viewed as a series of snapshots or time slices.1 We’ll just number the time slices 0, 1
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. On the other hand, in modeling continental drift over geological time, an interval of a million years might be fine. Each time slice in a discrete-time probability model contains a set of random variables, some observable and some not. For simplicity, we will assume that the same subset of variables is
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al. (2008). Significant classified work on filtering was done during World War II by Wiener (1942) for continuous-time processes and by Kolmogorov (1941) for discrete-time processes. Although this work led to important technological developments over the next 20 years, its use of a frequency-domain representation made many calculations quite
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et al., 2006). 1Uncertainty over continuous time can be modeled by stochastic differential equations (SDEs). The models studied in this chapter can be viewed as discrete-time approximations to SDEs. 2The term “filtering” refers to the roots of this problem in early work on signal processing, where the problem is to filter
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. Bertsekas, D. and Tsitsiklis, J. N. (2008). Introduction to Probability (2nd edition). Athena Scientific. Bertsekas, D. and Shreve, S. E. (2007). Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific. Bertsimas, D., Delarue, A., and Martin, S. (2019). Optimizing schools’ start time and bus routes. PNAS, 116 13, 5943–5948. Bertsimas, D
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the magnetospheric multiscale mission and SEXTANT pulsar navigation demonstration. Tech. rep., NASA Goddard Space Flight Center. Witten, I. H. (1977). An adaptive optimal controller for discrete-time Markov environments. Information and Control, 34, 286–295. Witten, I. H. and Bell, T. C. (1991). The zero-frequency problem: Estimating the probabilities of novel
The Concepts and Practice of Mathematical Finance
by Mark S. Joshi · 24 Dec 2003
, we step up another mathematical gear and this is the most mathematically demanding chapter. We introduce the concept of a martingale in both continuous and discrete time, and use martingales to examine the concept of riskneutral pricing. We commence by showing that option prices determine synthetic probabilities in the context of a
Derivatives Markets
by David Goldenberg · 2 Mar 2016 · 819pp · 181,185 words
, PART 1 11. RATIONAL OPTION PRICING 12. OPTION TRADING STRATEGIES, PART 2 13. MODEL-BASED OPTION PRICING IN DISCRETE TIME, PART 1: THE BINOMIAL OPTION PRICING MODEL (BOPM, N=1) 14. OPTION PRICING IN DISCRETE TIME, PART 2: DYNAMIC HEDGING AND THE MULTI-PERIOD BINOMIAL OPTION PRICING MODEL, N>1 15. EQUIVALENT MARTINGALE MEASURES
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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
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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
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.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
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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
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of the more advanced applications. These applications include the Black–Scholes option pricing model. Also, continuous-time derivatives reasoning is so much neater than the discrete-time approach. The more technical aspects of this chapter are in the Appendix, section 4.8, and are well within your reach. I suggest working through
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neither a 100% hedger nor a 100% speculator. He has combined position that represents both hedging and speculating. 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
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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
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simplest model-based, yet still rational (arbitrage-free) option pricing model. This model is called the Binomial Option Pricing Model (BOPM) and it is a discrete time model. The Binomial option pricing model uses a decision tree framework but goes beyond it. In fact, the Binomial option pricing model shows how to
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(specific and unique numbers) for option prices, not just bounds. 13.2 THE BINOMIAL OPTION PRICING MODEL, BASICS 13.2.1 Modeling Time in a Discrete Time Framework The current date is time t and the option’s expiration date is T, which is now called N because the BOPM is a
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discrete time model. For example, N could be the number of days to expiration. Time to expiration is τ=N–t. We divide τ into N sub-
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two independent financial instruments (see Concept Check 7). We already know quite a bit about bond pricing in continuous time. The BOPM is a purely discrete-time model, so we should be using discretely compounded interest rates. Let r′ be the one-period percentage interest rate appropriate to the
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discrete time interval defining the single-period BOPM. r′ represents the de-annualized risk-free rate. In the following, we will also need to define r=1+
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, This should remind you of the derivative of the option price with respect to the stock price because this is essentially what it is in discrete time. Next, we solve for the dollar position in bonds. Substitute (Δ) into (up): But, Therefore, and, Therefore, dividing by r, we find that the bond
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see what is new for the case N>1. ■ KEY CONCEPTS 1. The Objective of Model-Based Option Pricing (MBOP). 2. Modeling Time in a Discrete Time Framework. 3. Modeling the Underlying Stock Price Uncertainty. 4. Path Structure of the Binomial Process, Total Number of Price Paths. 5. Path Structure of the
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European puts and calls that we have effectively demonstrated. The Binomial model is intuitively complete, by our rule of thumb. 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
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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
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financial security to price, and we already did so both in discrete and in continuous time. Here, we price it in discrete time using the AD approach, since the BOPM is a discrete-time, discrete-state space model. We assume that the market permits no-arbitrage. Then, by the first fundamental theorem of asset
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plus the riskless rate. The following set of exercises first shows, in exercise 1, how to use replication to price a unit discount bond in discrete time. Then, the primitive AD securities in the BOPM, N=1 model, ADu(ω) and ADd(ω) will be priced in exercise 2. This is called
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t=0. We now have two results, E(W1(ω)|W0)=W0 and, E(W2(ω)|W1)=W1. 15.3.2 Definition of a Discrete-Time Martingale A discrete-time stochastic process (Xn(ω))n=0,1,2,3,.. is called a martingale if, 1. E(Xn)<∞ and for all n and, 2. E
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violated. Stock prices under risk neutrality are not martingales. However they aren’t very far from martingales. Definition of a Sub (Super) Martingale 1. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a sub-martingale if E(Xn)<∞, and E(Xn+1(ω)|Xn)>Xn for
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all n=0,1,2,3,… 2. A discrete-time stochastic process (Xn(ω))n=0,1,2,3,… is called a super-martingale if E(Xn)<∞, , and E(Xn+1(ω)|Xn)<Xn for
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Prices Not all hope is lost for the martingale representation of stock prices under risk neutrality. We start with defining a bond price process in discrete time, which in this context is a numeraire. (This concept will be discussed in Chapter 16, section 16.4.) Time starts at time t=0. The
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a few of the many properties of martingales that are used in proving results that make martingales useful in applied finance. We restrict attention to discrete-time martingales and sometimes even choose N=2. No attempt at mathematical rigor is claimed. The intuition behind these results is the primary concern. We start
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with a discrete-time stochastic process (Xn(ω)n=0,1,2,3,… with finite first and second moments E(Xn)<∞ and for all n=0,1,2,3
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(ω). 4. Pricing a European Call Option. 5. Pricing any Contingent Claim. 6. Equivalent Martingale Measures (EMMs). 7. Introduction and Examples. 8. Definition of a Discrete-Time Martingale. 9. Martingales and Stock Prices. 10. The Equivalent Martingale Representation of Stock Prices. 11. The Equivalent Martingale Representation of Option Prices. 12. Discounted Option
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time, we want to work through some important examples to get the flavor of the continuous-time framework. Many of the same ideas from the discrete-time framework we have been discussing carry over to the continuous-time case. Risk-neutral valuation carries over, and dynamic hedging which we discussed in the
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scaled Brownian motion, which by assumption is Gaussian (see property 3 in the definition of an ABM). Note that all of these calculations are in discrete time. When we move to continuous time, things get more complicated and we will need Itô’s lemma, which provides a stochastic calculus for the continuous
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manner that μ remains constant. In section 16.5.2, we will see that this is the defining characteristic of the continuous analogue of the discrete-time GBM process in (μ equation discrete, risk-adjusted). That process is called the log-normal process or geometric Brownian motion (GBM). The reason for calling
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process (N=2) 440; for N-period binomial process, summary of 444–5; stock price tree (N=2) 488; stock price uncertainty 439; time in discrete time framework, modeling of 437–8; trinomial model (three stock outcomes) 464 Black-Scholes option pricing model: option pricing in continuous time 566–85, 588–9
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prices and 26; underlying securities and 66 directional trades 371–2 discounted option prices 527–8 discounted stock price process 524–5, 527–8, 530 discrete-time martingale, definition of 521 diversifiable risk 225 diversification, maximum effect of 419–20 dividend-adjusted geometric mean (for S&P 500) 227 dividend payments, effect
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515–17; current price as predictor of future stock prices 531; discounted option prices 527–8; discounted stock price process 524–5, 527–8, 530; discrete-time martingale, definition of 521; double expectations (DE) 534–5; efficient market hypotheses (EMH) basis for modeling 517; features of 532; guide to modeling prices 529
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293 terminological navigation 596–7; interest-rate swaps 278–81, 293–4 tick size 228, 229 ticker symbol 214, 215, 228, 229, 261 time in discrete time framework, modeling of 437–8 time lines 20, 23 time premia 326, 330–1, 333, 337 total stock process with dividends (before dividends are paid
Mathematical Finance: Theory, Modeling, Implementation
by Christian Fries · 9 Sep 2007
four Prices . . . 6.2.2. Example (2): Interpolation of two Prices . . . . 6.3. Arbitrage Free Interpolation of European Option Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. Hedging in Continuous and Discrete Time and the Greeks English version of this part not available 7.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2. Deriving the Replications Strategy from Pricing Theory . . . . . . . . 7.2.1
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PDE under a BlackScholes Model . . . . . . . . . . . . . . . . . . . . . . . . . 7.3. Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1. Greeks einer europäischen Call-Option unter dem BlackScholes Modell . . . . . . . . . . . . . . . . . . . . . . . . . 7.4. Hedging in Discrete Time: Delta- and Delta-Gamma-Hedging . . . . 7.4.1. Delta Hedging . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2. Error Propagation . . . . . . . . . . . . . . . . . . . . . . . . 7.4.3. Delta-Gamma Hedging . . . . . . . . . . . . . . . . . . . . . 12 This work is
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Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ Contents 7.4.4. Vega Hedging . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5. Hedging in discrete Time: Minimizing the Residual Error (BouchaudSornette Method) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1. Minimizing the Residual Error at Maturity T . . . . . . . . . 7.5.2. Minimizing the Residual Error in each
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gives a construction of the Brownian motion. Tip (time discrete realizations): In the following we will often consider the realizations of a stochastic process at discrete times 0 = T 0 < T 1 < . . . < T N only (e.g. this will be the case when we consider the implementation). If we need only the
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changes are independent from the position and time at which they occur. To motivate the class of Itô processes we consider the Brownian motion at discrete times 0 = T 0 < T 1 < . . . < T N . The random variable W(T i ) (position of the particle) may be expressed through the increments ∆W(T
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particle movement: First we assume that the particle may lose energy over time 11 In a (one dimensional) random walk a particle changes position at discrete time steps by a (constant) distance (say 1) in either direction with equal probability. In other words we have binomial distributed Yi in Theorem 25. 40
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7 Hedging in Continuous and Discrete Time and the Greeks An English version of this part is currently not available. What follows it the German version. 7.1. Introduction Die vorstehende Bewertungstheorie
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Beide Modellierungen (kontinuierliche oder diskrete Replikation) entsprechen nicht der Realität: Zum einen ist es nicht möglich Finanzprodukte kontinuierlich und in
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS 7.2.1. Deriving the Replication Strategy under the Assumption of a Locally Riskless Product Der obige Ansatz V(t) = V(t
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Dies und dS i dS j = 0 falls i = 0 oder j = 0 eingesetzt in (7.4) liefert n ∂V(t)n
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Für die Funktion V(t, n, s) = sΦ(d+ ) − n K Φ(d− ) N(T ) mit σ̄2 (T − t) 1
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Die Definition von Rho wird lediglich für Aktienderivate verwendet. Für Zinsderivate, wie wir sie in den folgenden Kapitel betrachten werden
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Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING 7.4. Hedging in Discrete Time: Delta- and Delta-Gamma-Hedging Wird ein Delta-Hedge kontinuierlich angewendet, so liefert er eine exakte Replikation. Das Portfolio ist gegenu
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS zum Beispiel durch die Wahl von φ0 1 φ0 (tk ) = φ0 (tk−1 ) − S 0 (tk ) n X · φi (tk ) − φi (tk
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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING 7.4.2. Error Propagation Die Wahl von φ0 bestimmt sich zum Zeitpunkt t = 0 aus der Anfangsbedingung Π(0
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Wir benutzen diese Wahl um an diskreten Zeitpunkten T i , i = 0, 1, . . . in einem Replikationsportfolio zu handeln. Die Größe der
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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING Hedge Portfolio versus Payoff 1,0 Wöchentlicher Delta Hedge Wöchentlicher Delta Hedge mit falscher Volatilität Portfolio Value 0,8 0
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Der durch die nur lineare Approximation des Derivates durch das Replikationsportfolio gemachte Fehler lässt sich Reduzieren, in dem Produkte in das
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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING so bleibt ein Restrisiko |∆V(t) − ∆Π(t)| = O(|∆t|2 , |∆t∆S i |, |∆S i |3 ). Um das Gamma
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Hedge Portfolio versus Payoff 1,0 Monatlicher Delta Hedge Monatlicher Delta-Gamma Hedge Portfolio Value 0,8 0,5 0,2 0
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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.4. HEDGING IN DISCRETE TIME: DELTA- AND DELTA-GAMMA-HEDGING Hedge Portfolio versus Payoff 1,0 Portfolio Value 0,8 Wöchentlicher Delta Hedge mit falscher Zinsrate Wöchentlicher Delta-Gamma Hedge
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Hedge Portfolio versus Payoff 1,0 Statischer Delta Hedge Statischer Delta-Gamma Hedge Portfolio Value 0,8 0,5 0,2 0
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Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.5. HEDGING IN DISCRETE TIME: MINIMIZING THE RESIDUAL ERROR (BOUCHAUD-SORNETTE METHOD) 7.5. Hedging in discrete Time: Minimizing the Residual Error (Bouchaud-Sornette Method) Der Delta-Hedge überträgt die optimale Handelstrategie für kontinuierliches Handeln
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Proof: Der Beweis ist elementar. Mit E(X · Y|C) = Y · E(X|C) und E(Y 2 |C) = Y 2 folgt
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welcome. ©2004, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ 7.5. HEDGING IN DISCRETE TIME: MINIMIZING THE RESIDUAL ERROR (BOUCHAUD-SORNETTE METHOD) gilt – man beachte, dass φ0 in der letzten Summe nicht mehr vorkommt. Somit lässt sich mit φ0
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, 2005, 2006 Christian Fries Version 1.3.19 [build 20061210]- 13th December 2006 http://www.christian-fries.de/finmath/ CHAPTER 7. HEDGING IN CONTINUOUS AND DISCRETE TIME AND THE GREEKS Handelsstrategie entsteht aus der Forderung im Zeitschritt T k das Residualrisiko zur Zeit T k+1 durch Wahl des Portfolios φ(T
Tools for Computational Finance
by Rüdiger Seydel · 2 Jan 2002 · 313pp · 34,042 words
of the (S, t) half strip is semi-discretized in that it is replaced by parallel straight lines with distance ∆t apart, leading to a discrete-time model. The next step of discretization replaces the continuous values Si along the parallel t = ti by discrete values Sji , for all i and appropriate
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of Wtj and (Wtj+1 − Wtj ), but not of Wtj+1 and (Wtj+1 − Wtj ). Wiener processes are examples of martingales — there is no drift. Discrete-Time Model Let ∆t > 0 be a constant time increment. For the discrete instances tj := j∆t the value Wt can be written as a sum
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number of units of the asset held in a portfolio at time t. We start with the simplifying assumption that trading is only possible at discrete time instances tj , which define a partition of the interval 0 ≤ t ≤ T . Then the trading strategy b is piecewise constant, b(t) = b(tj−1
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the Assumption 1.2(c) of a geometric Brownian motion see also the notes and comments on Sections 1.7/1.8. For reference on discrete-time models, see [Pli97], [FöS02]. on Section 1.3: References on specific numerical methods are given where appropriate. As computational finance is concerned, most quotations
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. A Bermudan option restricts early exercise to specified dates during its life. In our context, the “dates” are created artificially by a finite set of discrete time instances ti . This resembles the time discretization of the binomial method of Section 1.4, see Figure 1.8. In this semi-discretized setting the
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). Decide whether an exact calculation of the hitting point makes sense. (Run experiments comparing such a strategy to implementing the hitting time restricted to the discrete time grid.) Think about how to implement upper bounds. Chapter 4 Standard Methods for Standard Options We now enter the part of the book that is
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.1 The Payoff There are several ways how an average of past values of St can be formed. If the price St is observed at discrete time instances ti , say equidistantly with time interval h := T /n, one obtains a times series St1 , St2 , . . . , Stn . An obvious choice of average is the
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Discrete Monitoring Instead of defining a continuous averaging as in (6.3), a realistic scenario is to assume that the average is monitored only at discrete time instances t1 , t2 , . . . , tM . These time instances are not to be confused with the grid times of the numerical discretization. The discretely sampled arithmetic average
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) riskless. As will be shown next, there is a simple analytic formula for ∆ in case of European options. (In reality, hedging must be done in discrete time.) The Black-Scholes equation has a closed-form solution. For a European call with continuous dividend yield δ as in (4.1) (in Section 4
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] [BBG97] [BTT00] [Br91] [BrS77] [BrS02] [Br94] [BrD97] [BrG97] [BrG04] [BH98] [BuJ92] [CaMO97] [CaF95] [CaM99] [Cash84] [CDG00] G.I. Bischi, V. Valori: Nonlinear effects in a discrete-time dynamic model of a stock market. Chaos, Solitons and Fractals 11 (2000) 21032121. F. Black, M. Scholes: The pricing of options and corporate liabilities. J
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). [Fisz63] M. Fisz: Probability Theory and Mathematical Statistics. John Wiley, New York (1963). [FöS02] H. Föllmer, A. Schied: Stochastic Finance: An Introduction to Discrete Time. de Gruyter, Berlin (2002). [FV02] P.A. Forsyth, K.R. Vetzal: Quadratic convergence of a penalty method for valuing American options. SIAM J. Sci. Comp
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York (1983). E. Platen: An introduction to numerical methods for stochastic differential equations. Acta Numerica (1999) 197-246. S.R. Pliska: Introduction to Mathematical Finance. Discrete Time Models. Blackwell, Malden (1997). D. Pooley, P.A. Forsyth, K. Vetzal, R.B. Simpson: Unstructured meshing for two asset barrier options. Appl. Mathematical Finance 7
Frequently Asked Questions in Quantitative Finance
by Paul Wilmott · 3 Jan 2007 · 345pp · 86,394 words
of either economists or applied mathematicians. Mike Harrison and David Kreps, in 1979, showed the relationship between option prices and advanced probability theory, originally in discrete time. Harrison and Stan Pliska in 1981 used the same ideas but in continuous time. From that moment until the mid 1990s applied mathematicians hardly got
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hedging There are quite a few problems with delta hedging, on both the practical and the theoretical side. In practice, hedging must be done at discrete times and is costly. Sometimes one has to buy or sell a prohibitively large number of the underlying in order to follow the theory. This is
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to have a method for forecasting future volatility. There is one slight problem with these econometric models, however. The econometrician develops his volatility models in discrete time, whereas the option-pricing quant would ideally like a continuous-time stochastic differential equation model. Fortunately, in many cases the
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discrete-time model can be reinterpreted as a continuous-time model (there is weak convergence as the time step gets smaller), and so both the econometrician and
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in an otherwise perfect Black-Scholes world. The only reason why this is not exactly a Black-Scholes world is because we are hedging at discrete time intervals. The Black-Scholes models prices in the expected value of this expression. You will recognize the from the Black-Scholes equation. So the hedging
Data Mining: Concepts, Models, Methods, and Algorithms
by Mehmed Kantardzić · 2 Jan 2003 · 721pp · 197,134 words
, shown in Figure 7.1, constituting the only computational node of the network. Neuron k is driven by input vector X(n), where n denotes discrete time, or, more precisely, the time step of the iterative process involved in adjusting the input weights wki. Every data sample for ANN training (learning) consists
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as a filter to perform three basic information-processing tasks: 1. Filtering. This task refers to the extraction of information about a particular quantity at discrete time n by using data measured up to and including time n. 2. Smoothing. This task differs from filtering in that data need not be available
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time n can also be used to obtain the required information. This means that in smoothing there is a delay in producing the result at discrete time n. 3. Prediction. The task of prediction is to forecast data in the future. The aim is to derive information about what the quantity of
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a word C″ is represented as C″ = (bcccccbaaaaabbccccbb). The main advantage of the SAX method is that 100 different discrete numerical values in an initial discrete time series C is first reduced to 20 different (average) values using PAA, and then they are transformed into only three different categorical values using SAX
Handbook of Modeling High-Frequency Data in Finance
by Frederi G. Viens, Maria C. Mariani and Ionut Florescu · 20 Dec 2011 · 443pp · 51,804 words
long-memory property, which intuitively means that observations that are far apart are highly correlated. Harvey (1998) and Breidt et al. (1998) independently introduced a discrete time model under which the log-volatility is modeled as a fractional ARIMA(p,d,q) process, while at the same time Comte and Renault (1998
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the LMSV model (Eq. 8.1). The model is in continuous-time and the volatility process is not observed, but we only have access to discrete time observations of historical stock prices. However, we assume that we are able to obtain high frequency (intraday) data, for example, tick-by-tick observations. 8
…
the case of Gaussian observations has been studied by Robinson (1995a,b) and Hurvich, Deo and Brodsky (1998). However, the log squared returns in the discrete time LMSV model are not Gaussian and asymptotic properties of the estimator in this case have been established by Deo and Hurvich (2001). Our model is
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to determine the implied value of H . As we also mentioned in the introduction, although the model is in continuous time, we only have available discrete time observations, the historical stock prices. The option pricing algorithm consists of two main parts. In the first part, our goal is to compute the empirical
…
variance of stock returns. Econ Lett 1994;45(3):281–285. Del Moral P, Jacod J, Protter P. The Monte-Carlo method for filtering with discrete time observations. Probab Theor Relat Field 2001;120:346–368. Deo RS, Hurvich CM. On the log periodogram regression estimator of the memory parameter in long
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continuously has to be relaxed. The first model in that direction was initiated by Leland (1985)[2]. He assumes that the portfolio is rebalanced at discrete time δt fixed and transaction costs are proportional to the value of the underlying; that is, the costs incurred at each step is κ|ν|S
…
C(S, t) be the value of the option and be the value of the hedge portfolio. The asset follows a geometric Brownian motion. Using discrete time, we assume the underlying asset follows the process √ δS = μSδt + σ S δt, (14.1) where is drawn from a standard normal distribution, μ is
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coefficients, 384 measuring, 243–244 pathwise computing of, 251 Director compensation, 59. See also CEO compensation Direct reinforcement learning, 65 Dirichlet (DIR) kernel, 255, 261 Discrete time model, 220 DIS data series, DFA and Hurst methods applied to, 153 Disjoint union, 314, 317 Distributional partial derivative, 386 Distribution distortions, 93 Distribution family
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, 120 Terminal condition, 348, 353 TH2 -type kernels, 261, 263 Tick-by-tick data, 29, 244 Time. See also Calendar time sampling; Continuous-time entries; Discrete time model; Exit time; Fixed stopping time; Fixed time interval; Infinite time horizon; Lunch-time trader activity; Optimal stopping time; Prespecified terminal time; Refresh time entries
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of Interest Rates: a New Methodology, Cornell University, 1987. Heath D., Jarrow R. and Morton A., Bond pricing and the term structure of interest rates: discrete time approximation, Journal of Financial and Quantitative Analysis, Vol. 25, 1990, pp. 419–40. Bonds 139 • the absence of arbitrage opportunity; • hypotheses relating to stochastic processes
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independent of time and follow a Gaussian distribution. The variation must also be finished. Volumes will be observed at equidistant moments (case of process in discrete time). We will take as an example the floating-demand savings accounts in LUF/BEF Techniques for Measuring Structural Risks in Balance Sheets 319 from 1996
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of Interest Rates: a New Methodology, Cornell University, 1987. Heath D., Jarrow R., and Morton A., Bond pricing and the term structure of interest rates: discrete time approximation, Journal of Financial and Quantitative Analysis, Vol. 25, 1990, pp. 419–40. Ho T. and Lee S., Term structure movement and pricing interest rate
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