# Optimal f :Calculating the expected growth-optimal fraction for discretely-distributed outcomes

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Optimal f :Calculating the expected growth-optimal fraction for discretely-distributed outcomes Ralph Vince November 12, 2015 First version: January, 2012 The president of LSP Partners, LLC, Holbrook Rd, Bentleyville, OH 44022, 1

2 Optimal f : Calculating the expected growth-optimal fraction for discretely-distributed outcomes Abstract Presented is the formulation for determining the exact, expected growth-optimal fraction of equity to risk for all conditions, rather than merely the asymptotic growthoptimal fraction. The formulation presented represents the surface in the leverage space manifold, wherein the loci at the peak of the surface are those fractions for maximizing expected growth. Other criteria can be solved for upon the surface in the leverage space manifold utilizing the equation specified here. Keywords. Growth-optimal portfolio, risk management, Kelly Criterion, finite investment horizon, drawdown. AMS Classifications. 91G10, 91G60, 91G70, 62C, 60E. 2

3 1 Discussion Repeatedly in both the gambling and trading communities the expected growth optimal fraction is employed, often referred to as the Kelly Criterion, and referring to the 1956 paper of John L. Kelly[2]. Though applicable in many gaming situations (though not all, blackjack being a prime example [15]) where the most that can be lost is the amount wagered, the Kelly Criterion solution is not directly applicable to trading situations which are more complicated. Take for example a short sale. Clearly what can be lost is not the same as the price of the stock. Forex transactions incur a different analysis as one is long a currency specified in another currency (and thus vice-versa in effect, short that other currency against the long currency). Commodity futures represent a different problem than simply being long an equity in terms of a floor price (what, specifically, is the lowest, say, wheat can go? Clearly it has value and thus a price of zero may not be realistic, the risk on a long wheat futures transaction therefore actually less than the immediately priced contract amount), and they too are represented in a base currency. CD swaps, complicated strategies involving options, warrants, leaps, trades in volatility where the limiting function of zero as a price and maximum loss is distorted, interest rate products and derivatives all create a situation where the Kelly Criterion solution does not result in the expected growth-optimal fraction to risk, and, in fact, to do so is often to risk more than the actual expected growth-optimal fraction would call for[13]. Capital market situations. given their inherent complexity, are not the same as gambling situations and the Kelly Criterion cannot be applied directly in determining proper expected growth-optimal fractions. Further, as pointed out by Samuelson[8] (though he does not provide a computational solution), the Kelly Criterion solution (in those cases where it is applicable, i.e. wagering situations where what can be lost is equivalent to what is put up) seeks the asymptotic expected growth-optimal fraction as the number of trials approaches infinity. For example, in a single proposition where the probability-weighted expected outcome is greater than 0, the expected growth for a participant whose horizon is one trial is f = 1.0, or to risk his entire equity. If his horizon were longer (but necessarily finite) the expected growth optimal fraction would be greater than the Kelly Criterion solution but less than one. The handicapper who goes to the track for a ten-race card day, seeking to maximize his return for the day employs a horizon, Q, of ten. The portfolio manager, depending on his criteria, has a similarly finite number of periods. The Kelly Criterion solution (when applicable) is only an asymptote; it is never the expected growth-optimal fraction, but rather approached asymptotically as the number of trials approaches infinity. The Kelly Criterion solution, as well as closed-form formulations that seek to solve it, are asymptotic and always less than what is the actual, expected-growth optimal fraction. As Hirashita [1] points out (referring to the asymptotic case), unless the payoff occurs immediately, the cost of assuming the risk is germane to the calculation of the expected growth-optimal fraction. Since we are discussing the finite case here, with a specific horizon, 3

4 we must include the cost of the wager to the specific horizon as is expressed in our equation. Finally, the actual expected growth-optimal fraction formula should incorporate multiple, simultaneous propositions. The Kelly Criterion solution is therefore merely a subset of this more generalized formula, and represents only the asymptote to the special case where it is applicable. Presented here is the formulation for N multiple, simultaneous propositions at a horizon of Q trials, for all potential propositions. The solution is presented in the context of the N + 1 dimensional leverage space manifold where, for each of the N components, a surface bound between f{0, 1} along each axis represents the fraction allocated to that individual proposition. G(f 1,...f N ) Q = n Q ( ((( x=0 Q N Q (1 + ( f i ( (a i,k C i )/w i )))) 1) p k )) (1.1) q=1 i=1 where: Q = the horizon. N = the number of components in the portfolio. n = the number of possible outcomes in the outcomes copula. k = int(x/(n (Q q) ))%n. a N,n = the N x n matrix of outcomes in the outcomes copula. p n = the n-lengthed vector of probabilities associated with each n in the outcomes copula. w N = the worst case outcome of the all discrete outcomes associated with each of the N components. C i = The one period opportunity-cost of the risk in component i, per [13]. This is explained further, below. Note k is a function of the iterators x and q, returning a specific zero th-based row in the outcomes copula. The drawdown constraint or other risk constraint as proposed in [11, 12] for a given Q can be employed upon the surface mapped by this equation in the leverage space manifold. To compute the amount to allocate to replicate the percentage to allocation to a given individual proposition, i for a given f i : q=1 f\$ i = w i /f i (1.2) Then for a given equity amount, the number of units to assume of component i to represent wagering f i percent: NumberOfUnits i = equity/f\$ i (1.3) 4

5 The one period opportunity-cost of the risk in component i, used in (1.1) as C i, is given in [13]: C i = exp(rt)s i S i d i (1.4) where: r = the current risk-free rate. t = the percentage of a year for one period to transpire. d i = dividends, disbursements or costs (negative) assosiated over one period with one unit of the ith component. S i = The maximum of {f\$ i, regulatory (i.e. margin) requirement of the ith component} where: f\$ i = the amount to allocate to replicate the percentage to allocation to a given individual proposition, i for a given f i as given in (1.2) We examine how to collect the data for determining the surface in leverage space at a given Q. Assume we have two separate propositions we wish to engage in simultaneously. For the sake of simplicity, suppose one is a coin toss paying 2:1, where we win with a probability of.5, and the other, a flawed coin paying 1:1 where we win with a probability of.6. We create the copula to use as input from this data: Coin 1 (a 1 ) Coin 2 (a 2 ) Probability (p) We note here that the number of components, N, is two since we have two separate, simultaneous propositions. The number of rows, n representing the space of what can happen for each discrete interval, is 4. w, the worst-case outcome for each N, each column, is 1 for both Coin 1 and Coin 2. To see the equation graphically, consider that for the first interval, the possible outcomes are represented by the table. For the second interval, each node from the first interval now further branches by the number of rows in the table such that at a given period, q toward a horizon, Q, we have n q nodes. The formula can thus account for dependency by permitting the probabilities for the various rows in the copula to change at each subsequent interval based on the previous outcome(s) along a given branch being traversed. The equation represents the surface in the leverage space manifold for given Q, f 1,...f N. This surface represents what one would expect in terms of return, as a multiple on equity, 5

6 after Q trials (hence, G(f 1,...f N ) represents what one would expect in terms of a multiple on equity, on average, per trial). The maximum of this surface (i.e. the greatest G(f 1,...f N ) Q or G(f 1,...f N )), provides the expected growth-optimal fractions. The equation represents the actual (i.e. non-asymptotic) fractions to risk of N components (N >= 1) for all possible propositions; thus, all other expected growth-optimal solutions tend to be subsets of the asymptote (i.e. Q ) of this more generalized equation, which is expressed here as lim G(f 1,...f N ) Q = ( Q + n N (1 + ( f i ( (a i,j C i )/w i ))) p j ) Q (1.5) j=1 Equation (1.5) represents the asymptotic manifestation of equation (1.1). The proof of this is found in [14] and [4]. Since (1.1) yields the surface in the leverge space manifold after Q trials, (1.5) represents what this surface tends to asymptotically as Q +. Since (1.5) is far less computationally expensive than (1.1) we can use (1.5) as a reasonable proxy of (1.1) after even a relatively small Q, thus for many calculations, including the expected risk-adjusted maximizing loci on this surface, ν and ζ as proposed in [4, 14, 15] as well as ψ proposed in [4] can be reasonably determined from (1.5). Acceptance of the underlying proposition in [13], that what the sole player expects to make over Q discrete trials or periods is not the classical expectation, defined as the probability-weighted mean, but rather the median of the ranked cumulative outcomes made along each branch at Q discrete trials or periods, we see a different approach must be undertaken to represent the expectation of the individual over Q discrete trials or periods. Rather what we have thus far considered with Equation (1.1) represents such a campaign performed many times over by the same individual, or, by many individuals over Q discrete trials or periods each. It does not represent what the individual, a sole individual, would expect to occur, for his solitary campaign across Q discrete trials or periods. Ergodicity diminishes as the number of discrete trials or periods approach 1 from the right. I take this to be innately evident, the mathematical proof left for those who are more mathematically adroit than I am. To reject the underlying proposition in [13], is to accept the following. Consider a game with a million-to-one chance of losing one million units, and wins one unit the other 999,999 times. Such a game has a negative expectation in the classic sense, but the individual looking to make one play then leave, would erroneously accept such as his expectation and decline to participate in such a proposition. 1 Equation (1.1) in its raw form suggests an 1 Similarly, this criterion, that of using the classical mathematical expectation for acceptance or rejection of a given proposition, would have the individual accept the reverse of the given proposition. That is, the individual acting on this criterion would accept a one-in-a-million chance of winning one million with a corresponding 999,999 chances in one million of losing one unit. The reader may argue this is precisely what individuals do when they purchase a lottery ticket, but that is an altogether different reasoning process. i=1 6

7 Figure 1: 2:1 coin toss proposition with the possibility of landing on the coins side. f = 0, that is, the player should wager nothing in the classical sense. Clearly though, the median of the sorted outcomes suggest that the individual expects to make one unit profit from such an attempt, hence, his expected growth-optimal wager is to risk 100%, i.e. f = 1. Similarly, rejection of this proposition makes the assumption that a game which pays 1 : 1, 000, 000 with corresponding probabilities of : is the equivalent of a proposition which pays 1 : 1 with corresponding probabilities of : as both have classical (asymptotic) probability-weighted means (mathematical expectations) of , yet they are clearly not the same proposition to someone with a finite horizon of participation. This difference becomes evermore apparent the shorter the horizon. Here we see that Equation (1.1) does, in fact, yield the correct altitude in the leverage space manifold for given f coordinates at Q trials, but what must be amended is the copula used as input to achieve these corrected values. This will be demonstrated by way of a step-by-step example. Consider a proposition as depicted in Figure 1 of three possible outcomes and their associated probabilities. Here, we have a classical coin toss example, wherein the player wins 2 units on heads, loses 1 unit on tails, and in the unlikely (1 in 1,000 chance) of the coin stopping on its side, costs the player 1,000,000 units. The classical, probability-weighted mean outcome calls for a loss of units per play, and hence the individual assessing this proposition should reject participating. However, in this example, we are considering participating only for 2 plays, Q = 2, and we arrive at a dramatically different conclusion. In Figure 2 we see the possible outcomes for two independent trials of this proposition, and their cumulative outcomes. There, one knows the classical expectation is negative when they purchase a lotter ticket. In this sense, we are speaking of a positive expectation lottery purchase, where one continues buying tickets in an attempt to make the positive expectation ultimately manifest. The argument here, is that such manifestation occurs at an unrealistically far horion for the player. 7

8 Figure 2: Possible outcomes for two independent trials and their cumulative outcomes Figure 3: Associated probabilities of the outcomes in Figure 2. The probability product is the product of all probabilities along a given branch. 8

9 Figure 4: Cumulative outcomes and probabilities at the end of each branch Figure 5: Outcomes sorted with their probabilities and median (the individual s expectation) determined. 9

10 Figure 3 is the same as Figure 2 but with the associated probabilities of the outcomes in Figure 2. Here, the probability product is the product of all probabilities along a given branch. In Figure 4 we see the summary of the cumulative outcomes and corresponding probability product at the terminus of each branch. In Figure 5 we sort the values of the outcomes and probabilities at the terminus (Q = 2) of each branch. From here, we can determine the median outcome, representing the expectation the solitary individual expects as outcome at the end of Q trials. We find this value to be a cumulative outcome of 1, characterized by the branch 2, 1( or 1, 2). We select these two rows from the initial copula, and we amend the probabilities of each row to be 1/Q = 1/2 =.5 : Outcome Probability and use this modified copula to determine our altitude in the leverage space manifold (and hence the expected growth-optimal peak) as specified by Equation (1.1). 2 Astonishingly, we find that when we consider the effects of diminishing ergodicity with respect to ever-shorter horizon, for the individual player and his solitary path of outcomes, the shape of leverage space is altered profoundly. Whereas in this example, not only does the classical mathematical expectation indicate a negative proposition that we should therefore not engage in, the Kelly Criterion buttresses this conclusion 3 by its satisfaction with an f = 0, i.e. to risk nothing on this proposition. Yet, we see that for the solitary player, participating to a horizon of Q = 2. he not only expects to profit, but he maximizes his expected profit in this particular proposition (via Equation 1.1, with inputs of 2, -1 and corresponding probabilities of 1/Q, 1/Q) at f =.5. Since all campaigns are ultimately finite, this is the correct calculation for expected 2 When multiple, simultaneous propositions are involved, the process for finding the altitude in leverage space is the same. We amend the input copula by eliminating those rows whose set of outcomes do not lie on the median-sorted-outcome path, and we assign a probability of 1/Q to each of these remaining rows, and performing Equation (1.1) as specified. 3 In fact, the Kelly Criterion, when presented with a one-in-ten chance of winning ten units, and a ninein-ten chance of losing one unit, would specify that.05 as the expected growth optimal wager, regardless of horizon. If one were to play this game, say, one time, the correct expected growth optimal fraction would be to wager nothing as the correct expected growth-optimal wager is a function of horizon always. The Kelly Criterion bettor, to a horizon of one play, should expect to lose.05 of his stake. The characteristic becomes more obvious as the distribution becomes more pronounced. Consider again our one-in-a- million chance of winning one million units, and all other plays losing one unit. The Kelly Criterion solution would require the player to risk one two-milliointh of his stake per play, regardless of horizon. However, for a finite horizon, to a horizon whose length is less than the critical length given in [13], where the positive expectation can be considered to manifest, the expected growth optimal amount to wager is zero, not the positive amount as specified by the asymptotically-dependant, ergodicity-dependant Kelly Criterion. 10

11 optimal growth as well as, more generally, the defined shape of leverage space and any and all other points germane to the individual (e.g. ν, ζ, ψ) within that manifold. This means of assessing the expectation for the individual, experiencing a necessarily finite horizon, as pointed out in [13], is how human beings innately assess a proposition, often assuming what seem to be risky ventures aligned with short-horizon bounty. It is why a commuter will drive to his work as well as why early hominids descended to the savanna despite the risks. It is this innate mechanism of risk-assessment, contrary to our classical understanding of it, that belies our flourishing as a species, absent which, stunted, we would be cowering agoraphobically in the shadows of a primeval world. References [1] Y. Hirashita. Game Pricing and Double Sequence of Random Variables. Journal of Modern Mathematics Frontier, Vol.2 (2013), [2] J. L. Kelly. A new interpretation of information rate. Bell System Technical Journal, 35: , [3] H. A. Latane. Criteria for choice among risky ventures. J. Political Economy, 52:75 81, [4] M. L. de Prado, R. Vince and Q. J. Zhu. Optimal Risk Budgeting Under a Finite Investment Horizon SSRN, , [5] L. C. MacLean, E. O. Thorp, and W. T. Ziemba(Eds.). The Kelly capital growth criterion: theory and practice. World Scientific, [6] L. C. MacLean and W. T. Ziemba. The Kelly criterion: theory and practice. In S. A. Zenios and W. T. Ziemba, editors, Handbook of Asset and Liability Management Volume A: Theory and Methodology. North Holland, [7] Edward O. Thorp and Sheen T. Kassouf. Beat the Market. Random House, New York, [8] P. A. Samuelson. The fallacy of maximizing the geometric mean in long sequences of investing or gambling. Proceedings of the National Academy of Sciences of the United States of America 68: [9] R. Vince. Portfolio Management Formulas. John Wiley and Sons, New York, [10] R. Vince. The New money Management: A Framework for Asset Allocation. John Wiley and Sons, New York, [11] R. Vince. The Handbook of Portfolio Mathematics. John Wiley and Sons, Hoboken, NJ,

12 [12] R. Vince. The Leverage Space Trading Model. John Wiley and Sons, Hoboken, NJ, [13] R. Vince. Risk-Opportunity Analysis. Createspace division of Amazon, [14] R. Vince and Q. J. Zhu. Inflection point significance for the investment size. SSRN, , [15] R. Vince and Q. J. Zhu. Optimal betting sizes for the game of blackjack. SSRN, ,

### Finite Horizon Investment Risk Management

History Collaborative research with Ralph Vince, LSP Partners, LLC, and Marcos Lopez de Prado, Guggenheim Partners. Qiji Zhu Western Michigan University Workshop on Optimization, Nonlinear Analysis, Randomness

### Betting with the Kelly Criterion

Betting with the Kelly Criterion Jane June 2, 2010 Contents 1 Introduction 2 2 Kelly Criterion 2 3 The Stock Market 3 4 Simulations 5 5 Conclusion 8 1 Page 2 of 9 1 Introduction Gambling in all forms,

### Lecture 25: Money Management Steven Skiena. http://www.cs.sunysb.edu/ skiena

Lecture 25: Money Management Steven Skiena Department of Computer Science State University of New York Stony Brook, NY 11794 4400 http://www.cs.sunysb.edu/ skiena Money Management Techniques The trading

### SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION. School of Mathematical Sciences. Monash University, Clayton, Victoria, Australia 3168

SOME ASPECTS OF GAMBLING WITH THE KELLY CRITERION Ravi PHATARFOD School of Mathematical Sciences Monash University, Clayton, Victoria, Australia 3168 In this paper we consider the problem of gambling with

### Pascal is here expressing a kind of skepticism about the ability of human reason to deliver an answer to this question.

Pascal s wager So far we have discussed a number of arguments for or against the existence of God. In the reading for today, Pascal asks not Does God exist? but Should we believe in God? What is distinctive

### Applying the Kelly criterion to lawsuits

Law, Probability and Risk (2010) 9, 139 147 Advance Access publication on April 27, 2010 doi:10.1093/lpr/mgq002 Applying the Kelly criterion to lawsuits TRISTAN BARNETT Faculty of Business and Law, Victoria

### Part I. Gambling and Information Theory. Information Theory and Networks. Section 1. Horse Racing. Lecture 16: Gambling and Information Theory

and Networks Lecture 16: Gambling and Paul Tune http://www.maths.adelaide.edu.au/matthew.roughan/ Lecture_notes/InformationTheory/ Part I Gambling and School of Mathematical

### MONEY MANAGEMENT. Guy Bower delves into a topic every trader should endeavour to master - money management.

MONEY MANAGEMENT Guy Bower delves into a topic every trader should endeavour to master - money management. Many of us have read Jack Schwager s Market Wizards books at least once. As you may recall it

### פרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית

המחלקה למתמטיקה Department of Mathematics פרויקט מסכם לתואר בוגר במדעים )B.Sc( במתמטיקה שימושית הימורים אופטימליים ע"י שימוש בקריטריון קלי אלון תושיה Optimal betting using the Kelly Criterion Alon Tushia

### 6.042/18.062J Mathematics for Computer Science. Expected Value I

6.42/8.62J Mathematics for Computer Science Srini Devadas and Eric Lehman May 3, 25 Lecture otes Expected Value I The expectation or expected value of a random variable is a single number that tells you

### arxiv:1112.0829v1 [math.pr] 5 Dec 2011

How Not to Win a Million Dollars: A Counterexample to a Conjecture of L. Breiman Thomas P. Hayes arxiv:1112.0829v1 [math.pr] 5 Dec 2011 Abstract Consider a gambling game in which we are allowed to repeatedly

### 1 Interest rates, and risk-free investments

Interest rates, and risk-free investments Copyright c 2005 by Karl Sigman. Interest and compounded interest Suppose that you place x 0 (\$) in an account that offers a fixed (never to change over time)

### A New Interpretation of Information Rate

A New Interpretation of Information Rate reproduced with permission of AT&T By J. L. Kelly, jr. (Manuscript received March 2, 956) If the input symbols to a communication channel represent the outcomes

### Probability and Expected Value

Probability and Expected Value This handout provides an introduction to probability and expected value. Some of you may already be familiar with some of these topics. Probability and expected value are

### THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING

THE FUNDAMENTAL THEOREM OF ARBITRAGE PRICING 1. Introduction The Black-Scholes theory, which is the main subject of this course and its sequel, is based on the Efficient Market Hypothesis, that arbitrages

### Review of Basic Options Concepts and Terminology

Review of Basic Options Concepts and Terminology March 24, 2005 1 Introduction The purchase of an options contract gives the buyer the right to buy call options contract or sell put options contract some

### The Kelly criterion for spread bets

IMA Journal of Applied Mathematics 2007 72,43 51 doi:10.1093/imamat/hxl027 Advance Access publication on December 5, 2006 The Kelly criterion for spread bets S. J. CHAPMAN Oxford Centre for Industrial

### The Kelly Criterion. A closer look at how estimation errors affect portfolio performance. by Adrian Halhjem Sælen. Advisor: Professor Steinar Ekern

NORGES HANDELSHØYSKOLE Bergen, Fall 2012 The Kelly Criterion A closer look at how estimation errors affect portfolio performance by Adrian Halhjem Sælen Advisor: Professor Steinar Ekern Master Thesis in

### Betting rules and information theory

Betting rules and information theory Giulio Bottazzi LEM and CAFED Scuola Superiore Sant Anna September, 2013 Outline Simple betting in favorable games The Central Limit Theorem Optimal rules The Game

### .4 120 +.1 80 +.5 100 = 48 + 8 + 50 = 106.

Chapter 16. Risk and Uncertainty Part A 2009, Kwan Choi Expected Value X i = outcome i, p i = probability of X i EV = pix For instance, suppose a person has an idle fund, \$100, for one month, and is considering

### Betting systems: how not to lose your money gambling

Betting systems: how not to lose your money gambling G. Berkolaiko Department of Mathematics Texas A&M University 28 April 2007 / Mini Fair, Math Awareness Month 2007 Gambling and Games of Chance Simple

### 6 Scalar, Stochastic, Discrete Dynamic Systems

47 6 Scalar, Stochastic, Discrete Dynamic Systems Consider modeling a population of sand-hill cranes in year n by the first-order, deterministic recurrence equation y(n + 1) = Ry(n) where R = 1 + r = 1

### A Simple Parrondo Paradox. Michael Stutzer, Professor of Finance. 419 UCB, University of Colorado, Boulder, CO 80309-0419

A Simple Parrondo Paradox Michael Stutzer, Professor of Finance 419 UCB, University of Colorado, Boulder, CO 80309-0419 michael.stutzer@colorado.edu Abstract The Parrondo Paradox is a counterintuitive

### Chapter 1 INTRODUCTION. 1.1 Background

Chapter 1 INTRODUCTION 1.1 Background This thesis attempts to enhance the body of knowledge regarding quantitative equity (stocks) portfolio selection. A major step in quantitative management of investment

### Chapter 7. Sealed-bid Auctions

Chapter 7 Sealed-bid Auctions An auction is a procedure used for selling and buying items by offering them up for bid. Auctions are often used to sell objects that have a variable price (for example oil)

### 9.2 Summation Notation

9. Summation Notation 66 9. Summation Notation In the previous section, we introduced sequences and now we shall present notation and theorems concerning the sum of terms of a sequence. We begin with a

### Session IX: Lecturer: Dr. Jose Olmo. Module: Economics of Financial Markets. MSc. Financial Economics

Session IX: Stock Options: Properties, Mechanics and Valuation Lecturer: Dr. Jose Olmo Module: Economics of Financial Markets MSc. Financial Economics Department of Economics, City University, London Stock

### The Fibonacci Strategy Revisited: Can You Really Make Money by Betting on Soccer Draws?

MPRA Munich Personal RePEc Archive The Fibonacci Strategy Revisited: Can You Really Make Money by Betting on Soccer Draws? Jiri Lahvicka 17. June 2013 Online at http://mpra.ub.uni-muenchen.de/47649/ MPRA

Two-State Options John Norstad j-norstad@northwestern.edu http://www.norstad.org January 12, 1999 Updated: November 3, 2011 Abstract How options are priced when the underlying asset has only two possible

### 4: SINGLE-PERIOD MARKET MODELS

4: SINGLE-PERIOD MARKET MODELS Ben Goldys and Marek Rutkowski School of Mathematics and Statistics University of Sydney Semester 2, 2015 B. Goldys and M. Rutkowski (USydney) Slides 4: Single-Period Market

### The Rational Gambler

The Rational Gambler Sahand Rabbani Introduction In the United States alone, the gaming industry generates some tens of billions of dollars of gross gambling revenue per year. 1 This money is at the expense

### Derivative Users Traders of derivatives can be categorized as hedgers, speculators, or arbitrageurs.

OPTIONS THEORY Introduction The Financial Manager must be knowledgeable about derivatives in order to manage the price risk inherent in financial transactions. Price risk refers to the possibility of loss

### FINANCIAL ECONOMICS OPTION PRICING

OPTION PRICING Options are contingency contracts that specify payoffs if stock prices reach specified levels. A call option is the right to buy a stock at a specified price, X, called the strike price.

### Evaluating Trading Systems By John Ehlers and Ric Way

Evaluating Trading Systems By John Ehlers and Ric Way INTRODUCTION What is the best way to evaluate the performance of a trading system? Conventional wisdom holds that the best way is to examine the system

### Target Strategy: a practical application to ETFs and ETCs

Target Strategy: a practical application to ETFs and ETCs Abstract During the last 20 years, many asset/fund managers proposed different absolute return strategies to gain a positive return in any financial

### 1 Portfolio Selection

COS 5: Theoretical Machine Learning Lecturer: Rob Schapire Lecture # Scribe: Nadia Heninger April 8, 008 Portfolio Selection Last time we discussed our model of the stock market N stocks start on day with

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes consider the way put and call options and the underlying can be combined to create hedges, spreads and combinations. We will consider the

### 1 Introduction to Option Pricing

ESTM 60202: Financial Mathematics Alex Himonas 03 Lecture Notes 1 October 7, 2009 1 Introduction to Option Pricing We begin by defining the needed finance terms. Stock is a certificate of ownership of

### Answers to Concepts in Review

Answers to Concepts in Review 1. Puts and calls are negotiable options issued in bearer form that allow the holder to sell (put) or buy (call) a stipulated amount of a specific security/financial asset,

### Economics 1011a: Intermediate Microeconomics

Lecture 12: More Uncertainty Economics 1011a: Intermediate Microeconomics Lecture 12: More on Uncertainty Thursday, October 23, 2008 Last class we introduced choice under uncertainty. Today we will explore

### Wald s Identity. by Jeffery Hein. Dartmouth College, Math 100

Wald s Identity by Jeffery Hein Dartmouth College, Math 100 1. Introduction Given random variables X 1, X 2, X 3,... with common finite mean and a stopping rule τ which may depend upon the given sequence,

### Black-Scholes-Merton approach merits and shortcomings

Black-Scholes-Merton approach merits and shortcomings Emilia Matei 1005056 EC372 Term Paper. Topic 3 1. Introduction The Black-Scholes and Merton method of modelling derivatives prices was first introduced

### Simplified Option Selection Method

Simplified Option Selection Method Geoffrey VanderPal Webster University Thailand Options traders and investors utilize methods to price and select call and put options. The models and tools range from

### FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008. Options

FIN-40008 FINANCIAL INSTRUMENTS SPRING 2008 Options These notes describe the payoffs to European and American put and call options the so-called plain vanilla options. We consider the payoffs to these

### Example: Find the expected value of the random variable X. X 2 4 6 7 P(X) 0.3 0.2 0.1 0.4

MATH 110 Test Three Outline of Test Material EXPECTED VALUE (8.5) Super easy ones (when the PDF is already given to you as a table and all you need to do is multiply down the columns and add across) Example:

### TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II + III

TPPE17 Corporate Finance 1(5) SOLUTIONS RE-EXAMS 2014 II III Instructions 1. Only one problem should be treated on each sheet of paper and only one side of the sheet should be used. 2. The solutions folder

### Expected Value and the Game of Craps

Expected Value and the Game of Craps Blake Thornton Craps is a gambling game found in most casinos based on rolling two six sided dice. Most players who walk into a casino and try to play craps for the

### Choosing the Wrong Portfolio of Projects. Inability to Find the Efficient Frontier. Lee Merkhofer, Ph.D.

Choosing the Wrong Portfolio of Projects Inability to Find the Efficient Frontier Lee Merkhofer, Ph.D. Inability to find projects on the efficient frontier is one of 5 major reasons organizations have

### Chapter 20 Understanding Options

Chapter 20 Understanding Options Multiple Choice Questions 1. Firms regularly use the following to reduce risk: (I) Currency options (II) Interest-rate options (III) Commodity options D) I, II, and III

### Call and Put. Options. American and European Options. Option Terminology. Payoffs of European Options. Different Types of Options

Call and Put Options A call option gives its holder the right to purchase an asset for a specified price, called the strike price, on or before some specified expiration date. A put option gives its holder

### Options Pricing. This is sometimes referred to as the intrinsic value of the option.

Options Pricing We will use the example of a call option in discussing the pricing issue. Later, we will turn our attention to the Put-Call Parity Relationship. I. Preliminary Material Recall the payoff

### To begin, I want to introduce to you the primary rule upon which our success is based. It s a rule that you can never forget.

Welcome to the casino secret to profitable options trading. I am very excited that we re going to get the chance to discuss this topic today. That s because, in this video course, I am going to teach you

### SAMPLE MID-TERM QUESTIONS

SAMPLE MID-TERM QUESTIONS William L. Silber HOW TO PREPARE FOR THE MID- TERM: 1. Study in a group 2. Review the concept questions in the Before and After book 3. When you review the questions listed below,

### DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II

DIFFERENT PATHS: SCENARIOS FOR n-asset AND PATH-DEPENDENT OPTIONS II David Murphy Market and Liquidity Risk, Banque Paribas This article expresses the personal views of the author and does not necessarily

### INFORMATION FOR OBSERVERS. Gaming Transactions (Agenda Paper 11(i))

30 Cannon Street, London EC4M 6XH, United Kingdom Tel: +44 (0)20 7246 6410 Fax: +44 (0)20 7246 6411 Email: iasb@iasb.org Website: www.iasb.org International Accounting Standards Board This observer note

### Capital budgeting & risk

Capital budgeting & risk A reading prepared by Pamela Peterson Drake O U T L I N E 1. Introduction 2. Measurement of project risk 3. Incorporating risk in the capital budgeting decision 4. Assessment of

### The overall size of these chance errors is measured by their RMS HALF THE NUMBER OF TOSSES NUMBER OF HEADS MINUS 0 400 800 1200 1600 NUMBER OF TOSSES

INTRODUCTION TO CHANCE VARIABILITY WHAT DOES THE LAW OF AVERAGES SAY? 4 coins were tossed 1600 times each, and the chance error number of heads half the number of tosses was plotted against the number

### The Math. P (x) = 5! = 1 2 3 4 5 = 120.

The Math Suppose there are n experiments, and the probability that someone gets the right answer on any given experiment is p. So in the first example above, n = 5 and p = 0.2. Let X be the number of correct

### Algorithmic Trading Session 1 Introduction. Oliver Steinki, CFA, FRM

Algorithmic Trading Session 1 Introduction Oliver Steinki, CFA, FRM Outline An Introduction to Algorithmic Trading Definition, Research Areas, Relevance and Applications General Trading Overview Goals

### MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC

MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC DR. LESZEK GAWARECKI 1. The Cartesian Coordinate System In the Cartesian system points are defined by giving their coordinates. Plot the following points:

### The Kelly Betting System for Favorable Games.

The Kelly Betting System for Favorable Games. Thomas Ferguson, Statistics Department, UCLA A Simple Example. Suppose that each day you are offered a gamble with probability 2/3 of winning and probability

### Introduction to the Rebate on Loss Analyzer Contact: JimKilby@usa.net 702-436-7954

Introduction to the Rebate on Loss Analyzer Contact: JimKilby@usa.net 702-436-7954 One of the hottest marketing tools used to attract the premium table game customer is the "Rebate on Loss." The rebate

### It is more important than ever for

Multicurrency attribution: not as easy as it looks! With super choice looming Australian investors (including superannuation funds) are becoming increasingly aware of the potential benefits of international

### AMS 5 CHANCE VARIABILITY

AMS 5 CHANCE VARIABILITY The Law of Averages When tossing a fair coin the chances of tails and heads are the same: 50% and 50%. So if the coin is tossed a large number of times, the number of heads and

### 3.2 Roulette and Markov Chains

238 CHAPTER 3. DISCRETE DYNAMICAL SYSTEMS WITH MANY VARIABLES 3.2 Roulette and Markov Chains In this section we will be discussing an application of systems of recursion equations called Markov Chains.

### Options pricing in discrete systems

UNIVERZA V LJUBLJANI, FAKULTETA ZA MATEMATIKO IN FIZIKO Options pricing in discrete systems Seminar II Mentor: prof. Dr. Mihael Perman Author: Gorazd Gotovac //2008 Abstract This paper is a basic introduction

### FE670 Algorithmic Trading Strategies. Stevens Institute of Technology

FE670 Algorithmic Trading Strategies Lecture 6. Portfolio Optimization: Basic Theory and Practice Steve Yang Stevens Institute of Technology 10/03/2013 Outline 1 Mean-Variance Analysis: Overview 2 Classical

### Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model

Overview of Violations of the Basic Assumptions in the Classical Normal Linear Regression Model 1 September 004 A. Introduction and assumptions The classical normal linear regression model can be written

### 22Most Common. Mistakes You Must Avoid When Investing in Stocks! FREE e-book

22Most Common Mistakes You Must Avoid When Investing in s Keep the cost of your learning curve down Respect fundamental principles of successful investors. You Must Avoid When Investing in s Mistake No.

### THE WHE TO PLAY. Teacher s Guide Getting Started. Shereen Khan & Fayad Ali Trinidad and Tobago

Teacher s Guide Getting Started Shereen Khan & Fayad Ali Trinidad and Tobago Purpose In this two-day lesson, students develop different strategies to play a game in order to win. In particular, they will

### 1 Limiting distribution for a Markov chain

Copyright c 2009 by Karl Sigman Limiting distribution for a Markov chain In these Lecture Notes, we shall study the limiting behavior of Markov chains as time n In particular, under suitable easy-to-check

### Chapter 4 Lecture Notes

Chapter 4 Lecture Notes Random Variables October 27, 2015 1 Section 4.1 Random Variables A random variable is typically a real-valued function defined on the sample space of some experiment. For instance,

### The Intuition Behind Option Valuation: A Teaching Note

The Intuition Behind Option Valuation: A Teaching Note Thomas Grossman Haskayne School of Business University of Calgary Steve Powell Tuck School of Business Dartmouth College Kent L Womack Tuck School

### LOOKING FOR A GOOD TIME TO BET

LOOKING FOR A GOOD TIME TO BET LAURENT SERLET Abstract. Suppose that the cards of a well shuffled deck of cards are turned up one after another. At any time-but once only- you may bet that the next card

### In the situations that we will encounter, we may generally calculate the probability of an event

What does it mean for something to be random? An event is called random if the process which produces the outcome is sufficiently complicated that we are unable to predict the precise result and are instead

### Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay

Information Theory and Coding Prof. S. N. Merchant Department of Electrical Engineering Indian Institute of Technology, Bombay Lecture - 17 Shannon-Fano-Elias Coding and Introduction to Arithmetic Coding

### K 1 < K 2 = P (K 1 ) P (K 2 ) (6) This holds for both American and European Options.

Slope and Convexity Restrictions and How to implement Arbitrage Opportunities 1 These notes will show how to implement arbitrage opportunities when either the slope or the convexity restriction is violated.

### Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs

Quiz 1 Sample Questions IE406 Introduction to Mathematical Programming Dr. Ralphs These questions are from previous years and should you give you some idea of what to expect on Quiz 1. 1. Consider the

### Betting on Excel to enliven the teaching of probability

Betting on Excel to enliven the teaching of probability Stephen R. Clarke School of Mathematical Sciences Swinburne University of Technology Abstract The study of probability has its roots in gambling

### Lecture 11 Uncertainty

Lecture 11 Uncertainty 1. Contingent Claims and the State-Preference Model 1) Contingent Commodities and Contingent Claims Using the simple two-good model we have developed throughout this course, think

### FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver

FTS Real Time System Project: Portfolio Diversification Note: this project requires use of Excel s Solver Question: How do you create a diversified stock portfolio? Advice given by most financial advisors

### Introduction to Mathematical Finance

Introduction to Mathematical Finance R. J. Williams Mathematics Department, University of California, San Diego, La Jolla, CA 92093-0112 USA Email: williams@math.ucsd.edu DO NOT REPRODUCE WITHOUT PERMISSION

### Probability, statistics and football Franka Miriam Bru ckler Paris, 2015.

Probability, statistics and football Franka Miriam Bru ckler Paris, 2015 Please read this before starting! Although each activity can be performed by one person only, it is suggested that you work in groups

### The power of money management

The power of money management One trader lost (\$3000) during the course of a year trading one contract of system A. Another trader makes \$25,000 trading the same system that year. One trader makes \$24,000

### 1 Portfolio mean and variance

Copyright c 2005 by Karl Sigman Portfolio mean and variance Here we study the performance of a one-period investment X 0 > 0 (dollars) shared among several different assets. Our criterion for measuring

### 1 Introduction. Linear Programming. Questions. A general optimization problem is of the form: choose x to. max f(x) subject to x S. where.

Introduction Linear Programming Neil Laws TT 00 A general optimization problem is of the form: choose x to maximise f(x) subject to x S where x = (x,..., x n ) T, f : R n R is the objective function, S

### Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans

Life Cycle Asset Allocation A Suitable Approach for Defined Contribution Pension Plans Challenges for defined contribution plans While Eastern Europe is a prominent example of the importance of defined

### Numbers 101: Cost and Value Over Time

The Anderson School at UCLA POL 2000-09 Numbers 101: Cost and Value Over Time Copyright 2000 by Richard P. Rumelt. We use the tool called discounting to compare money amounts received or paid at different

### Lecture 13. Understanding Probability and Long-Term Expectations

Lecture 13 Understanding Probability and Long-Term Expectations Thinking Challenge What s the probability of getting a head on the toss of a single fair coin? Use a scale from 0 (no way) to 1 (sure thing).

### Applied Economics For Managers Recitation 5 Tuesday July 6th 2004

Applied Economics For Managers Recitation 5 Tuesday July 6th 2004 Outline 1 Uncertainty and asset prices 2 Informational efficiency - rational expectations, random walks 3 Asymmetric information - lemons,

### Chapter 16. Law of averages. Chance. Example 1: rolling two dice Sum of draws. Setting up a. Example 2: American roulette. Summary.

Overview Box Part V Variability The Averages Box We will look at various chance : Tossing coins, rolling, playing Sampling voters We will use something called s to analyze these. Box s help to translate

### Chapter 15 Introduction to Linear Programming

Chapter 15 Introduction to Linear Programming An Introduction to Optimization Spring, 2014 Wei-Ta Chu 1 Brief History of Linear Programming The goal of linear programming is to determine the values of

### Gambling and Portfolio Selection using Information theory

Gambling and Portfolio Selection using Information theory 1 Luke Vercimak University of Illinois at Chicago E-mail: lverci2@uic.edu Abstract A short survey is given of the applications of information theory

### V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPECTED VALUE

V. RANDOM VARIABLES, PROBABILITY DISTRIBUTIONS, EXPETED VALUE A game of chance featured at an amusement park is played as follows: You pay \$ to play. A penny and a nickel are flipped. You win \$ if either

### Probability and Statistics

CHAPTER 2: RANDOM VARIABLES AND ASSOCIATED FUNCTIONS 2b - 0 Probability and Statistics Kristel Van Steen, PhD 2 Montefiore Institute - Systems and Modeling GIGA - Bioinformatics ULg kristel.vansteen@ulg.ac.be

### Double Deck Blackjack

Double Deck Blackjack Double Deck Blackjack is far more volatile than Multi Deck for the following reasons; The Running & True Card Counts can swing drastically from one Round to the next So few cards