This resulted in the so-called Farmer Curve of acceptable probability of an event versus its consequence. The technique as a whole is usually referred to as probabilistic risk assessment (PRA) (or probabilistic safety assessment, PSA). See WASH-1400 for an example of this approach. In finance, risk is the chance that the return achieved on an investment will be different from that expected, and also takes into account the size of the difference. This includes the possibility of losing some or all of the original investment. In a view advocated by Damodaran, risk includes not only "downside risk" but also "upside risk" (returns that exceed expectations).

Catalog of articles in probability theory

Gambler's fallacy. Gambler's ruin / (L:D). Game of chance. Inverse gambler's fallacy. Lottery. Lottery machine. Luck. Martingale. Odds. Pachinko. Parimutuel betting. Parrondo's paradox. Pascal's wager. Poker probability. Poker probability (Omaha). Poker probability (Texas hold 'em). Pot odds. Proebsting's paradox. Roulette. Spread betting. The man who broke the bank at Monte Carlo. Bible code. Birthday paradox. Birthday problem. Index of coincidence. Spurious relationship. Algorithmic Lovász local lemma. Box–Muller transform. Gibbs sampling. Inverse transform sampling method. Las Vegas algorithm. Metropolis algorithm. Monte Carlo method. Panjer recursion. Probabilistic Turing machine.

Gerolamo Cardano

CardanoCardanCardano, Gerolamo
Cardano was notoriously short of money and kept himself solvent by being an accomplished gambler and chess player. His book about games of chance, Liber de ludo aleae ("Book on Games of Chance"), written around 1564, but not published until 1663, contains the first systematic treatment of probability, as well as a section on effective cheating methods. He used the game of throwing dice to understand the basic concepts of probability. He demonstrated the efficacy of defining odds as the ratio of favourable to unfavourable outcomes (which implies that the probability of an event is given by the ratio of favourable outcomes to the total number of possible outcomes ).


(Other risk types include foreign exchange, shape, volatility, sector, liquidity, inflation risks, etc.) It focuses on when and how to hedge using financial instruments; in this sense it overlaps with financial engineering. Similar to general risk management, financial risk management requires identifying its sources, measuring it (see: Risk measure#Examples), and formulating plans to address these, and can be qualitative and quantitative. In the banking sector worldwide, the Basel Accords are generally adopted by internationally active banks for tracking, reporting and exposing operational, credit and market risks.

Event (probability theory)

eventeventsrandom event
For the standard tools of probability theory, such as joint and conditional probabilities, to work, it is necessary to use a σ-algebra, that is, a family closed under complementation and countable unions of its members. The most natural choice is the Borel measurable set derived from unions and intersections of intervals. However, the larger class of Lebesgue measurable sets proves more useful in practice. In the general measure-theoretic description of probability spaces, an event may be defined as an element of a selected σ-algebra of subsets of the sample space.

Derivative (finance)

derivativesderivativefinancial derivatives
Options valuation is a topic of ongoing research in academic and practical finance. In basic terms, the value of an option is commonly decomposed into two parts: Although options valuation has been studied since the 19th century, the contemporary approach is based on the Black–Scholes model, which was first published in 1973. Options contracts have been known for many centuries. However, both trading activity and academic interest increased when, as from 1973, options were issued with standardized terms and traded through a guaranteed clearing house at the Chicago Board Options Exchange.

Gambling mathematics

mathematics of gamblingadvantage to the dealergaming mathematician
Poker probability. Provably fair. Statistical association football predictions. The Mathematics of Gambling, by Edward Thorp, ISBN: 0-89746-019-7. The Theory of Gambling and Statistical Logic, Revised Edition, by Richard Epstein, ISBN: 0-12-240761-X. The Mathematics of Games and Gambling, Second Edition, by Edward Packel, ISBN: 0-88385-646-8. Probability Guide to Gambling: The Mathematics of Dice, Slots, Roulette, Baccarat, Blackjack, Poker, Lottery and Sport Bets, by Catalin Barboianu, ISBN: 973-87520-3-5 excerpts. Luck, Logic, and White Lies: The Mathematics of Games, by Jörg Bewersdorff, ISBN: 1-56881-210-8 introduction. Probability and gambling math discussion from the Wizard of Odds.

Butterfly (options)

In finance, a butterfly is a limited risk, non-directional options strategy that is designed to have a high probability of earning a limited profit when the future volatility of the underlying asset is expected to be lower or higher than the implied volatility when long or short respectively. A long butterfly position will make profit if the future volatility is lower than the implied volatility. A long butterfly options strategy consists of the following options: where X = the spot price (i.e. current market price of underlying) and a > 0. Using put–call parity a long butterfly can also be created as follows: where X = the spot price and a > 0.

Expected value

*The roulette game consists of a small ball and a wheel with 38 numbered pockets around the edge. As the wheel is spun, the ball bounces around randomly until it settles down in one of the pockets. Suppose random variable X represents the (monetary) outcome of a $1 bet on a single number ("straight up" bet). If the bet wins (which happens with probability in American roulette), the payoff is $35; otherwise the player loses the bet. The expected profit from such a bet will be Let X be a random variable with a countable set of finite outcomes x_1, x_2, ..., occurring with probabilities p_1, p_2, ..., respectively, such that the infinite sum converges.

Put option

putput optionsputs
A put option is said to have intrinsic value when the underlying instrument has a spot price (S) below the option's strike price (K). Upon exercise, a put option is valued at K-S if it is "in-the-money", otherwise its value is zero. Prior to exercise, an option has time value apart from its intrinsic value. The following factors reduce the time value of a put option: shortening of the time to expire, decrease in the volatility of the underlying, and increase of interest rates. Option pricing is a central problem of financial mathematics. Trading options involves a constant monitoring of the option value, which is affected by changes in the base asset price, volatility and time decay.


uncertainuncertaintiesstandard uncertainty
Type B, those evaluated by other means, e.g., by assigning a probability distribution. measured value ± uncertainty. measured value. measured value (uncertainty). Uncertainty is designed into games, most notably in gambling, where chance is central to play. In scientific modelling, in which the prediction of future events should be understood to have a range of expected values. In optimization, uncertainty permits one to describe situations where the user does not have full control on the final outcome of the optimization procedure, see scenario optimization and stochastic optimization.

Nassim Nicholas Taleb

Nassim TalebTaleb, Nassim NAntifragility
Taleb's writings discuss the error of comparing real-world randomness with the "structured randomness" in quantum physics where probabilities are remarkably computable and games of chance like casinos where probabilities are artificially built. Taleb calls this the "ludic fallacy".

Parlay (gambling)

parlayaccumulatoraccumulator bet
Since the probability of all possible events will add up to 1 this can also be looked at as the weighted average of the event. The table below represents odds. Column 1 = number of individual bets in the parlay Column 2 = correct odds of winning with 50% chance of winning each individual bet Column 3 = odds payout of parlay at the sportsbook Column 4 = correct odds of winning parlay with 55% chance of winning each individual bet The table illustrates that with even a 55% chance of winning each individual bet parlays are profitable in the long term.

Financial economics

financial economistfinancial economistsfinance
Hertz in 1964, allow financial analysts to construct "stochastic" or probabilistic corporate finance models, as opposed to the traditional static and deterministic models; see. Relatedly, Real Options theory allows for owner—i.e. managerial—actions that impact underlying value: by incorporating option pricing logic, these actions are then applied to a distribution of future outcomes, changing with time, which then determine the "project's" valuation today.

Dutch book

However, if Horse 4 was withdrawn and the bookmaker does not adjust the other odds, the implied probabilities would add up to 0.95. In such a case, a gambler could always reap a profit of $10 by betting $100, $50 and $40 on the remaining three horses, respectively, and not having to stake $20 on the withdrawn horse, which now cannot win. Other forms of Dutch books can exist when incoherent odds are offered on exotic bets such as forecasting the order in which horses will finish. With competitive fixed-odds gambling being offered electronically, gamblers can sometimes create a Dutch book by selecting the best odds from different bookmakers, in effect undertaking an arbitrage operation.

Outline of finance

List of valuation topicsFinanceList of insurance topics
Monte Carlo methods for option pricing. Monte Carlo methods in finance. Quasi-Monte Carlo methods in finance. Least Square Monte Carlo for American options. Trinomial tree. Volatility. Implied volatility. Historical volatility. Local volatility. Stochastic volatility. Constant elasticity of variance model. Heston model. SABR volatility model. Volatility smile (& Volatility surface). Implied binomial tree. Implied trinomial tree. Edgeworth binomial tree. Swaps. Swap valuation. Currency swap #Valuation and Pricing. Interest rate swap #Valuation and pricing. Variance swap #Pricing and valuation. Interest rate derivatives (bond options, swaptions, caps and floors, and others).

Monte Carlo methods in finance

Monte Carlo simulationsimulationsimulation techniques
Then, these results are combined in a histogram of NPV (i.e. the project’s probability distribution), and the average NPV of the potential investment - as well as its volatility and other sensitivities - is observed. This distribution allows, for example, for an estimate of the probability that the project has a net present value greater than zero (or any other value). See further under Corporate finance. *In valuing an option on equity, the simulation generates several thousand possible (but random) price paths for the underlying share, with the associated exercise value (i.e. "payoff") of the option for each path.

Financial transaction tax

financial transactions taxtax on financial transactionstransaction taxes
However, excess speculation is often deemed not only a source of volatility but a distraction of talent and dangerous shift of focus for a developed economy. By contrast, hedging is necessary for the stability of enterprises. Tax schemes in general seek to tax speculation - seen as akin to gambling - while trying not to interfere with hedging (a form of insurance).

Binary option

binary optionsdigital optionasset-or-nothing call
The Isle of Man, a self-governing Crown dependency for which the UK is responsible, has issued licenses to companies offering binary options as "games of skill" licensed and regulated under fixed odds betting by the Isle of Man Gambling Supervision Commission (GSC). This positions binary options as a form of gambling, and the administrator of the trading as something akin to a casino, as opposed to an exchange or brokerage house. On October 19, 2017, London police raided 20 binary options firms in London. On January 3, 2018, Financial Conduct Authority (FCA) took over regulation of binary options from the Gambling Commission.

Mathematical finance

financial mathematicsquantitative financequantitative trading
Put–call parity (Arbitrage relationships for options). Intrinsic value, Time value. Moneyness. Pricing models. Black–Scholes model. Black model. Binomial options model. Implied binomial tree. Edgeworth binomial tree. Monte Carlo option model. Implied volatility, Volatility smile. Local volatility. Stochastic volatility. Constant elasticity of variance model. Heston model. SABR volatility model. Markov switching multifractal. The Greeks. Finite difference methods for option pricing. Vanna–Volga pricing. Trinomial tree. Implied trinomial tree. Garman-Kohlhagen model. Lattice model (finance). Margrabe's formula. Pricing of American options. Barone-Adesi and Whaley. Bjerksund and Stensland.

Corporate finance

financial managementbusiness financefinance
An application of this methodology is to determine an "unbiased" NPV, where management determines a (subjective) probability for each scenario – the NPV for the project is then the probability-weighted average of the various scenarios; see First Chicago Method. (See also rNPV, where cash flows, as opposed to scenarios, are probability-weighted.) A further advancement which "overcomes the limitations of sensitivity and scenario analyses by examining the effects of all possible combinations of variables and their realizations" is to construct stochastic or probabilistic financial models – as opposed to the traditional static and deterministic models as above.

Glossary of Australian and New Zealand punting

betting plungeAustralian and New Zealand EnglishAustralian and New Zealand punting
Unbackable: A horse which is quoted at such extremely short odds that investors decide it is too short to return a reasonable profit for the risk involved. Under double wraps: An expression indicating that a horse won very easily without being fully extended. Unders: Odds about a horse which are considered to be bad value because they are shorter than its estimated winning probability. Undertaker: A bookmaker said to only be interested in laying "dead 'uns". Urger: see coat-tugger. Via the cape: The horse ran wide on the home turn and covered too much ground.

List of probability topics

Game of chance. Odds. Gambler's fallacy. Inverse gambler's fallacy. Parrondo's paradox. Pascal's wager. Gambler's ruin. Poker probability. Poker probability (Omaha). Poker probability (Texas hold 'em). Pot odds. Roulette. Martingale (betting system). The man who broke the bank at Monte Carlo. Lottery. Lottery machine. Pachinko. Coherence (philosophical gambling strategy). Coupon collector's problem. Birthday paradox. Birthday problem. Index of coincidence. Bible code. Spurious relationship. Monty Hall problem. Probable prime. Probabilistic algorithm = Randomised algorithm. Monte Carlo method. Las Vegas algorithm. Probabilistic Turing machine. Stochastic programming.

List of statistics articles

list of statistical topicslist of statistics topics
Prior probability. Prior probability distribution redirects to Prior probability. Probabilistic causation. Probabilistic design. Probabilistic forecasting. Probabilistic latent semantic analysis. Probabilistic metric space. Probabilistic proposition. Probabilistic relational model. Probability. Probability and statistics. Probability bounds analysis. Probability box. Probability density function. Probability distribution. Probability distribution function (disambiguation). Probability integral transform. Probability interpretations. Probability mass function. Probability matching. Probability metric. Probability of error. Probability of precipitation. Probability plot.

Frank P. Ramsey

Frank RamseyRamseyRamsey, F.P.
Thus personal beliefs that are formulated by this individual knowledge govern probabilities, leading to the notions of subjective probability and Bayesian probability. Consequently, subjective probabilities can be inferred by observing actions that reflect individuals' personal beliefs. Ramsey argued that the degree of probability that an individual attaches to a particular outcome can be measured by finding what odds the individual would accept when betting on that outcome. Ramsey suggested a way of deriving a consistent theory of choice under uncertainty that could isolate beliefs from preferences while still maintaining subjective probabilities.