金融工程 Quantitative Finance

 

At- In- Out- the Money

Options are said to be at the money when the underlying asset price is equal to the strike price. A call option is said to be in of the money if the price of the underlying is above the strike price. An in of the money call option has positive intrinsic value. Conversely, a put option is in of the money if the price of the underlying is below the strike price. A call option is said to be out of the money if the price of the underlying is below the strike price. An out of the money call option has no intrinsic value. Conversely, a put option is out of the money if the price of the underlying is above the strike price.

Amortized Options

Amortized options, sometimes referred to as Whale options, where the payoff is typically given as the difference between the average asset price over the averaging period and the strike price (typically given as a percentage of the underlying price) as a ratio over the average asset price.

Binomial Distribution

One of the more popular probability distributions, the Binomial model shows the number of occurrences of an event within a number of observations. The binomial concept can be illustrated via the use of a coin, in that the model represents the number of times that the outcome of flipping a coin is either a head, or a tail.

Main characteristics of the distribution / model: 1. Underlying variable is discrete (in that it is not continuous) 2. There can only be 2 outcomes 3. The event must be mutually exclusive 4. The variable is randomly selected。

It can also be applied within option pricing via the aptly named Binomial Pricing Model which gives the price of an asset (an option for example) given a rise or fall in the underlying asset.

Brownian Motion

One of the important applications of physics within finance is the concept of Brownian Motion. It is a stochastic process which has stationary independent increments which follows a Normal distribution and also has continuous sample paths. In physics, Brownian motion has been used to show the movement of atomic particles which are subject to many "shocks". It is a Markovian process, and is also known as a Wiener process.

It is often applied to stock returns as many conclude that log of the returns follow a normal distribution. A variable z follows this process if 1. The change in z (dz) is = . Where is a random pick from a normal distribution and dt is a small change in time. 2. The values of dz for two separate time intervals of dt are independent. In other words, the change in the variable z is not dependent on any 2 individual intervals of time (dt).

Copula

A copula of two variables x and y is a cumulative probability function defined directly as a function of the marginal cumulative probabilities of x and y. A copula is thus a way to specify the co-dependence between variates entirely independently on their individual marginal distribution.

A sensible copula must be a function of all the cumulative marginal probabilities, and some kind of control parameters that determine the strength of co-dependence of the individual variables.

Fade-in Option

A fade-in option is the product of a standard plain vanilla option and a fade-in factor , this factor increases with the time spent by the asset within a given rage [L, H]. if the asset never leaves the range, the payoff is a plain vanilla payoff. More formally, the payoff of the fade-in call at maturity T is given by

, with , where (ti, i=1,2,...,N) is a set of fade-in dates within the lifetime of the option.

Improving Option

An improving option has a maturity date T and a strike K like a plain vanilla option. in addition, a refixing date t prior to the maturity date as well as a factor alpha are stipulated at the start of the option. At the refixing date, the strike of the option is changed to alpha times the current spot if this is profitable for the holder of the option.

Pearson's Correlation

The correlation between two variables reflects the degree to which the variables are related. The most common measure of correlation is the Pearson Product Moment Correlation (called Pearson's correlation for short). When measured in a population the Pearson Product Moment correlation is designated by the Greek letter rho (ρ). When computed in a sample, it is designated by the letter "r" and is sometimes called "Pearson's r." Pearson's correlation reflects the degree of linear relationship between two variables. It ranges from +1 to -1. A correlation of +1 means that there is a perfect positive linear relationship between variables. The scatterplot shown on this page depicts such a relationship. It is a positive relationship because high scores on the X-axis are associated with high scores on the Y-axis.

Option Delta

The ratio comparing the change in the price of the underlying asset to the corresponding change in the price of a derivative. Sometimes referred to as the "hedge ratio". For example, with respect to call options, a delta of 0.7 means that for every $1 the underlying stock increases, the call option will increase by $0.70. Put option deltas, on the other hand, will be negative, because as the underlying security increases, the value of the option will decrease. So a put option with a delta of -0.7 will decrease by $0.70 for every $1 the underlying increases in price. As an in-the-money call option nears expiration, it will approach a delta of 1.00, and as an in-the-money put option nears expiration, it will approach a delta of -1.00.

Prolongation Option

A prolongation option consists of a plain vanilla option and a prolongation feature. The prolongation feature only comes into life if the plain vanilla option expires worthless, in which case the holder of the option will receive a new plain vanilla option. there are two types of prolongation option: Straight prolongation: the strike of the new option is fixed at time 0; Refixed prolongation: the strike of the new option will be fixed at t at a certain percentage in- or out-of-the-money. this percentage has to be stipulated at time 0.