Girsanov theorem
In probability theory, the Girsanov theorem describes how the dynamics of stochastic processes change when the original measure is changed to an equivalent probability measure. The theorem is important in the theory of financial mathematics as it tells how to convert from the physical measure, which describes the probability that an underlying instrument will take a particular value or values, to the risk-neutral measure, a useful tool for pricing derivatives on the underlying instrument. Results of this type were first proved by Cameron–Martin in the 1940s and by Girsanov in 1960, they have been subsequently extended to more general classes of process culminating in the general form of Lenglart. Girsanov's theorem is important in the general theory of stochastic processes since it enables the key result that if Q is a measure continuous with respect to P every P-semimartingale is a Q-semimartingale. We state the theorem first for the special case when the underlying stochastic process is a Wiener process.
This special case is sufficient for risk-neutral pricing in the Black–Scholes model and in many other models. Let be a Wiener process on the Wiener probability space. Let X t be a measurable process adapted to the natural filtration of the Wiener process with X 0 = 0. Define the Doléans-Dade exponential E t of X with respect to W E t = exp , where t is the quadratic variation of X t. If E t is a positive martingale, a probability measure Q can be defined on such that we have Radon–Nikodym derivative d Q d P | F t = E t Then for each t the measure Q restricted to the unaugmented sigma fields F t W is equivalent to P restricted to F t W. Furthermore, if Y is a local martingale under P the process Y ~ t = Y t − t is a Q local martingale on the filtered probability space. If X is a continuous process and W is Brownian motion under measure P W ~ t = W t − t is Brownian motion under Q; the fact that W ~ t is continuous is trivial. In many common applications, the process X is defined by X t = ∫ 0 t Y s d.
For X of this form a sufficient condition for E {\displaystyle
Normal distribution
In probability theory, the normal distribution is a common continuous probability distribution. Normal distributions are important in statistics and are used in the natural and social sciences to represent real-valued random variables whose distributions are not known. A random variable with a Gaussian distribution is said to be distributed and is called a normal deviate; the normal distribution is useful because of the central limit theorem. In its most general form, under some conditions, it states that averages of samples of observations of random variables independently drawn from independent distributions converge in distribution to the normal, that is, they become distributed when the number of observations is sufficiently large. Physical quantities that are expected to be the sum of many independent processes have distributions that are nearly normal. Moreover, many results and methods can be derived analytically in explicit form when the relevant variables are distributed; the normal distribution is sometimes informally called the bell curve.
However, many other distributions are bell-shaped. The probability density of the normal distribution is f = 1 2 π σ 2 e − 2 2 σ 2 where μ is the mean or expectation of the distribution, σ is the standard deviation, σ 2 is the variance; the simplest case of a normal distribution is known as the standard normal distribution. This is a special case when μ = 0 and σ = 1, it is described by this probability density function: φ = 1 2 π e − 1 2 x 2 The factor 1 / 2 π in this expression ensures that the total area under the curve φ is equal to one; the factor 1 / 2 in the exponent ensures that the distribution has unit variance, therefore unit standard deviation. This function is symmetric around x = 0, where it attains its maximum value 1 / 2 π and has inflection points at x = + 1 and x = − 1. Authors may differ on which normal distribution should be called the "standard" one. Gauss defined the standard normal as having variance σ 2 = 1 / 2, φ = e − x 2 π Stigler goes further, defining the standard normal with variance σ 2 = 1 /: φ = e − π x 2 Every normal distribution is a version of the standard normal distribution whose domain has been stretched by a factor σ and translated by μ: f = 1 σ φ.
The probability density must be scaled by 1 / σ so that the integral is still 1. If Z is a standard normal deviate X = σ Z + μ will have a normal distribution with expected value μ and standard deviation σ. Conversely, if X is a normal deviate with parameters μ and σ 2 Z = / σ
Black–Scholes model
The Black–Scholes or Black–Scholes–Merton model is a mathematical model for the dynamics of a financial market containing derivative investment instruments. From the partial differential equation in the model, known as the Black–Scholes equation, one can deduce the Black–Scholes formula, which gives a theoretical estimate of the price of European-style options and shows that the option has a unique price regardless of the risk of the security and its expected return; the formula led to a boom in options trading and provided mathematical legitimacy to the activities of the Chicago Board Options Exchange and other options markets around the world. It is used, although with adjustments and corrections, by options market participants. Based on works developed by market researchers and practitioners, such as Louis Bachelier, Sheen Kassouf and Ed Thorp among others, Fischer Black and Myron Scholes demonstrated in the late 1960s that a dynamic revision of a portfolio removes the expected return of the security, thus inventing the risk neutral argument.
In 1970, after they attempted to apply the formula to the markets and incurred financial losses due to lack of risk management in their trades, they decided to focus in their domain area, the academic environment. After three years of efforts, the formula named in honor of them for making it public, was published in 1973 in an article entitled "The Pricing of Options and Corporate Liabilities", in the Journal of Political Economy. Robert C. Merton was the first to publish a paper expanding the mathematical understanding of the options pricing model, coined the term "Black–Scholes options pricing model". Merton and Scholes received the 1997 Nobel Memorial Prize in Economic Sciences for their work, the committee citing their discovery of the risk neutral dynamic revision as a breakthrough that separates the option from the risk of the underlying security. Although ineligible for the prize because of his death in 1995, Black was mentioned as a contributor by the Swedish Academy; the key idea behind the model is to hedge the option by buying and selling the underlying asset in just the right way and, as a consequence, to eliminate risk.
This type of hedging is called "continuously revised delta hedging" and is the basis of more complicated hedging strategies such as those engaged in by investment banks and hedge funds. The model's assumptions have been relaxed and generalized in many directions, leading to a plethora of models that are used in derivative pricing and risk management, it is the insights of the model, as exemplified in the Black–Scholes formula, that are used by market participants, as distinguished from the actual prices. These insights include risk-neutral pricing. Further, the Black–Scholes equation, a partial differential equation that governs the price of the option, enables pricing using numerical methods when an explicit formula is not possible; the Black–Scholes formula has only one parameter that cannot be directly observed in the market: the average future volatility of the underlying asset, though it can be found from the price of other options. Since the option value is increasing in this parameter, it can be inverted to produce a "volatility surface", used to calibrate other models, e.g. for OTC derivatives.
The Black–Scholes model assumes that the market consists of at least one risky asset called the stock, one riskless asset called the money market, cash, or bond. Now we make assumptions on the assets: The rate of return on the riskless asset is constant and thus called the risk-free interest rate; the instantaneous log return of stock price is an infinitesimal random walk with drift. The stock does not pay a dividend. Assumptions on the market: There is no arbitrage opportunity, it is possible to borrow and lend any amount fractional, of cash at the riskless rate. It is possible to buy and sell any amount fractional, of the stock; the above transactions do not incur any costs. With these assumptions holding, suppose there is a derivative security trading in this market. We specify that this security will have a certain payoff at a specified date in the future, depending on the value taken by the stock up to that date, it is a surprising fact that the derivative's price is determined at the current time though we do not know what path the stock price will take in the future.
For the special case of a European call or put option and Scholes showed that "it is possible to create a hedged position, consisting of a long position in the stock and a short position in the option, whose value will not depend on the price of the stock". Their dynamic hedging strategy led to a partial differential equation which governed the price of the option, its solution is given by the Black–Scholes formula. Several of these assumptions of the original model have been removed in subsequent extensions of the model. Modern versions account for dynamic interest rates, transaction costs and taxes, dividend payout; the notation used throughout this page will be defined as follows: S, the price of the underlying asset at time t..
Option (finance)
In finance, an option is a contract which gives the buyer the right, but not the obligation, to buy or sell an underlying asset or instrument at a specified strike price prior to or on a specified date, depending on the form of the option. The strike price may be set by reference to the spot price of the underlying security or commodity on the day an option is taken out, or it may be fixed at a discount or at a premium; the seller has the corresponding obligation to fulfill the transaction – to sell or buy – if the buyer "exercises" the option. An option that conveys to the owner the right to buy at a specific price is referred to as a call. Both are traded, but the call option is more discussed; the seller may grant an option to a buyer as part of another transaction, such as a share issue or as part of an employee incentive scheme, otherwise a buyer would pay a premium to the seller for the option. A call option would be exercised only when the strike price is below the market value of the underlying asset, while a put option would be exercised only when the strike price is above the market value.
When an option is exercised, the cost to the buyer of the asset acquired is the strike price plus the premium, if any. When the option expiration date passes without the option being exercised, the option expires and the buyer would forfeit the premium to the seller. In any case, the premium is income to the seller, a capital loss to the buyer; the owner of an option may on-sell the option to a third party in a secondary market, in either an over-the-counter transaction or on an options exchange, depending on the option. The market price of an American-style option closely follows that of the underlying stock being the difference between the market price of the stock and the strike price of the option; the actual market price of the option may vary depending on a number of factors, such as a significant option holder may need to sell the option as the expiry date is approaching and does not have the financial resources to exercise the option, or a buyer in the market is trying to amass a large option holding.
The ownership of an option does not entitle the holder to any rights associated with the underlying asset, such as voting rights or any income from the underlying asset, such as a dividend. Contracts similar to options have been used since ancient times; the first reputed option buyer was the ancient Greek mathematician and philosopher Thales of Miletus. On a certain occasion, it was predicted that the season's olive harvest would be larger than usual, during the off-season, he acquired the right to use a number of olive presses the following spring; when spring came and the olive harvest was larger than expected he exercised his options and rented the presses out at a much higher price than he paid for his'option'. In London, puts and "refusals" first became well-known trading instruments in the 1690s during the reign of William and Mary. Privileges were options sold over the counter in nineteenth century America, with both puts and calls on shares offered by specialized dealers, their exercise price was fixed at a rounded-off market price on the day or week that the option was bought, the expiry date was three months after purchase.
They were not traded in secondary markets. In the real estate market, call options have long been used to assemble large parcels of land from separate owners. Film or theatrical producers buy the right — but not the obligation — to dramatize a specific book or script. Lines of credit give the potential borrower the right — but not the obligation — to borrow within a specified time period. Many choices, or embedded options, have traditionally been included in bond contracts. For example, many bonds are convertible into common stock at the buyer's option, or may be called at specified prices at the issuer's option. Mortgage borrowers have long had the option to repay the loan early, which corresponds to a callable bond option. Options contracts have been known for decades; the Chicago Board Options Exchange was established in 1973, which set up a regime using standardized forms and terms and trade through a guaranteed clearing house. Trading activity and academic interest has increased since then.
Today, many options are created in a standardized form and traded through clearing houses on regulated options exchanges, while other over-the-counter options are written as bilateral, customized contracts between a single buyer and seller, one or both of which may be a dealer or market-maker. Options are part of a larger class of financial instruments known as derivative products, or derivatives. A financial option is a contract between two counterparties with the terms of the option specified in a term sheet. Option contracts may be quite complicated.
Mathematical proof
In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other established statements, such as theorems, can be used. In principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions. Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is always true, rather than enumerate many confirmatory cases. An unproved proposition, believed to be true is known as a conjecture. Proofs employ logic but include some amount of natural language which admits some ambiguity. In fact, the vast majority of proofs in written mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in symbolic language instead of natural language, are considered in proof theory; the distinction between formal and informal proofs has led to much examination of current and historical mathematical practice, quasi-empiricism in mathematics, so-called folk mathematics.
The philosophy of mathematics is concerned with the role of language and logic in proofs, mathematics as a language. The word "proof" comes from the Latin probare meaning "to test". Related modern words are the English "probe", "probation", "probability", the Spanish probar, Italian provare, the German probieren; the early use of "probity" was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, which outweighed empirical testimony. Plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof, it is that the idea of demonstrating a conclusion first arose in connection with geometry, which meant the same as "land measurement". The development of mathematical proof is the product of ancient Greek mathematics, one of the greatest achievements thereof. Thales and Hippocrates of Chios proved some theorems in geometry. Eudoxus and Theaetetus formulated did not prove them.
Aristotle said definitions should describe the concept being defined in terms of other concepts known. Mathematical proofs were revolutionized by Euclid, who introduced the axiomatic method still in use today, starting with undefined terms and axioms, used these to prove theorems using deductive logic, his book, the Elements, was read by anyone, considered educated in the West until the middle of the 20th century. In addition to theorems of geometry, such as the Pythagorean theorem, the Elements covers number theory, including a proof that the square root of two is irrational and that there are infinitely many prime numbers. Further advances took place in medieval Islamic mathematics. While earlier Greek proofs were geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, etc. for "lines."
He used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, who used it to prove the binomial theorem and properties of Pascal's triangle. Alhazen developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures. There is no longer an assumption; as practiced, a proof is expressed in natural language and is a rigorous argument intended to convince the audience of the truth of a statement. The standard of rigor has varied throughout history. A proof can be presented differently depending on the intended audience. In order to gain acceptance, a proof has to meet communal statements of rigor; the concept of a proof is formalized in the field of mathematical logic. A formal proof is written in a formal language instead of a natural language. A formal proof is defined as sequence of formulas in a formal language, in which each formula is a logical consequence of preceding formulas.
Having a definition of formal proof makes the concept of proof amenable to study. Indeed, the field of proof theory studies formal proofs and their properties, for example, the property that a statement has a formal proof. An application of proof theory is to show; the definition of a formal proof is intended to capture the concept of proofs as written in the practice of mathematics. The soundness of this definition amounts to the belief that a published proof can, in principle, be converted into a formal proof. However, outside the field of automated proof assistants, this is done in practice. A classic question in philosophy a
Greeks (finance)
In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of derivatives such as options to a change in underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used. Collectively these have been called the risk sensitivities, risk measures or hedge parameters; the Greeks are vital tools in risk management. Each Greek measures the sensitivity of the value of a portfolio to a small change in a given underlying parameter, so that component risks may be treated in isolation, the portfolio rebalanced accordingly to achieve a desired exposure; the Greeks in the Black–Scholes model are easy to calculate, a desirable property of financial models, are useful for derivatives traders those who seek to hedge their portfolios from adverse changes in market conditions. For this reason, those Greeks which are useful for hedging—such as delta and vega—are well-defined for measuring changes in Price and Volatility.
Although rho is a primary input into the Black–Scholes model, the overall impact on the value of an option corresponding to changes in the risk-free interest rate is insignificant and therefore higher-order derivatives involving the risk-free interest rate are not common. The most common of the Greeks are the first order derivatives: delta, vega and rho as well as gamma, a second-order derivative of the value function; the remaining sensitivities in this list are common enough that they have common names, but this list is by no means exhaustive. The use of Greek letter names is by extension from the common finance terms alpha and beta, the use of sigma and tau in the Black–Scholes option pricing model. Several names such as ` vega' and ` zomma' sound similar to Greek letters; the names'color' and'charm' derive from the use of these terms for exotic properties of quarks in particle physics. Delta, Δ, measures the rate of change of the theoretical option value with respect to changes in the underlying asset's price.
Delta is the first derivative of the value V of the option with respect to the underlying instrument's price S. For a vanilla option, delta will be a number between 0.0 and 1.0 for a long call and 0.0 and −1.0 for a long put. The difference between the delta of a call and the delta of a put at the same strike is close to but not in general equal to one, but instead is equal to the inverse of the discount factor. By put–call parity, long a call and short a put is equivalent to a forward F, linear in the spot S, with factor the inverse of the discount factor, so the derivative dF/dS is this factor; these numbers are presented as a percentage of the total number of shares represented by the option contract. This is convenient. For example, if a portfolio of 100 American call options on XYZ each have a delta of 0.25, it will gain or lose value just like 2,500 shares of XYZ as the price changes for small price movements. The sign and percentage are dropped – the sign is implicit in the option type and the percentage is understood.
The most quoted are 25 delta put, 50 delta put/50 delta call, 25 delta call. 50 Delta put and 50 Delta call are not quite identical, due to spot and forward differing by the discount factor, but they are conflated. Delta is always negative for long puts; the total delta of a complex portfolio of positions on the same underlying asset can be calculated by taking the sum of the deltas for each individual position – delta of a portfolio is linear in the constituents. Since the delta of underlying asset is always 1.0, the trader could delta-hedge his entire position in the underlying by buying or shorting the number of shares indicated by the total delta. For example, if the delta of a portfolio of options in XYZ is +2.75, the trader would be able to delta-hedge the portfolio by selling short 2.75 shares of the underlying. This portfolio will retain its total value regardless of which direction the price of XYZ moves.. The Delta is close to, but not identical with, the percent moneyness of an option, i.e. the implied probability that the option will expire in-the-money.
For this reason some option traders use the absolute value of delta as an approximation for percent moneyness. For example, if an out-of-the-money call option has a delta of 0.15, the trader might estimate that the option has a 15% chance of expiring in-the-money. If a put contract has a delta of −0.25, the trader might expect the option to have a 25% probability of expiring in-the-money. At-the-money calls and puts have a delta of 0.5 and −0.5 wi
Straddle
In finance, a straddle strategy refers to two transactions that share the same security, with positions that offset one another. One holds the other short; as a result, it involves the purchase or sale of particular option derivatives that allow the holder to profit based on how much the price of the underlying security moves, regardless of the direction of price movement. A straddle involves put with same strike price and expiration date. If the stock price is close to the strike price at expiration of the options, the straddle leads to a loss. However, if there is a sufficiently large move in either direction, a significant profit will result. A straddle is appropriate when an investor is expecting a large move in a stock price but does not know in which direction the move will be; the purchase of particular option derivatives is known as a long straddle, while the sale of the option derivatives is known as a short straddle. A long straddle involves "going long," in other words, purchasing both a call option and a put option on some stock, interest rate, index or other underlying.
The two options expire at the same time. The owner of a long straddle makes a profit if the underlying price moves a long way from the strike price, either above or below. Thus, an investor may take a long straddle position if he thinks the market is volatile, but does not know in which direction it is going to move; this position is a limited risk. At the same time, there is unlimited profit potential. For example, company XYZ is set to release its quarterly financial results in two weeks. A trader believes that the release of these results will cause a large movement in the price of XYZ's stock, but does not know whether the price will go up or down, he can enter into a long straddle, where he gets a profit no matter which way the price of XYZ stock moves, if the price changes enough either way. If the price goes up enough, he ignores the put option. If the price goes down, he ignores the call option. If the price does not change enough, he loses money, up to the total amount paid for the two options.
The risk is limited by the total premium paid for the options, as opposed to the short straddle where the risk is unlimited. If the stock is sufficiently volatile and option duration is long, the trader could profit from both options; this would require the stock to move both below the put option's strike price and above the call option's strike price at different times before the option expiration date. This is an at-the-money Straddle with 1 year to expiry: After 50 days, the P/L graph of the straddle will look as follows: The P/L blue graph is negative at prices from 84 to 107 dollars, which means that in order for the strategy to be profitable after 50 days, the stock price should be either higher than 107 dollars or lower than 84 dollars; as time goes by, due to time decay, the straddle P/L graph goes down, until it reaches the orange line. The distance between the break-even points increases. A short straddle is a non-directional options trading strategy that involves selling a put and a call of the same underlying security, strike price and expiration date.
The profit is limited to the premium received from the sale of call. The risk is unlimited as large moves of the underlying security's price either up or down will cause losses proportional to the magnitude of the price move. A maximum profit upon expiration is achieved if the underlying security trades at the strike price of the straddle. In that case both puts and calls comprising the straddle expire worthless allowing straddle owner to keep full credit received as their profit; this strategy is called "nondirectional" because the short straddle profits when the underlying security changes little in price before the expiration of the straddle. The short straddle can be classified as a credit spread because the sale of the short straddle results in a credit of the premiums of the put and call. A risk for holder of a short straddle position is unlimited due to the sale of the call and the put options which expose the investor to unlimited losses or losses limited to the strike price, whereas maximum profit is limited to the premium gained by the initial sale of the options.
A tax straddle is straddling applied to taxes used in futures and options to create a tax shelter. For example, an investor with a capital gain manipulates investments to create an artificial loss from an unrelated transaction to offset their gain in a current year, postpone the gain till the following tax year. One position accumulates the other a loss; the position with the loss is closed prior to the completion of the tax year, countering the gain. When the new year for tax begins, a replacement position is created to offset the risk from the retained position. Through repeated straddling, gains can be postponed indefinitely over many years. Publication 17 Your Federal Income Tax Form 1040 series of forms and instructions Social Security's booklet "Medicare Premiums: Rules for Higher-Income Beneficiaries" and the calculation of the Social Security MAGI. McMillan, Lawrence G.. Options as a Strategic Investment. New York: New York Institute of Finance. ISBN 978-0-7352-0197-2. McMillan, Lawrence G..
Options as a Strategic Investment. Prentice Hall Press. ISBN 978-0-7352-0465-2. Specific
