Important Mathematical Topics for Quants | Street Of Walls
Gamma and theta have an inverse correlation. A positive gamma position will lose money every day due to time decay, while a negative. could anyone outline the inter-relationships between theta and vega and THETA is also basically the inverse of GAMMA so, positive VEGA. Do you know the difference in meaning between “stochastic” and “statistic”? in- the-money option increases as expiration nears; the opposite is true for an out-of- the-money option. Daily P&L = Gamma P&L + Theta P&L + Vega P&L + Other.
So would the profit from the puts completely neutralize the loss on the stock. Actually, you would do better. This results because delta itself changed. Gamma is the change in delta for each unit change in the price of the underlying. The absolute magnitude of delta increases as the time to expiration of the option decreases, and as its intrinsic value increases.
Gamma changes in predictable ways. As an option goes more into the money, delta will increase until it tracks the underlying dollar for dollar; however, delta can never be greater than 1 or less than When delta is close to 1 or -1, then gamma is near zero, because delta doesn't change much with the price of the underlying. Gamma and delta are greatest when an option is at the money—when the strike price is equal to the price of the underlying. The change in delta is greatest for options at the money, and decreases as the option goes more into the money or out of the money.
Both gamma and delta tend to zero as the option moves further out of the money. The total gamma of a portfolio is called the position gamma.
Theta Options are a wasting asset. The option premium consists of a time value that continuously declines as time to expiration nears, with most of the decline occurring near expiration. Theta is a measure of this time decay, and is expressed as the loss of time value per day.
relationship between delta, theta, and gamma
Thus, a theta of. Theta is minimal for a long-term option because the time value decays only slowly, but increases as expiration nears, since each day represents a greater percentage of the remaining time. Theta is also greatest when the option is at the money, because this is the price where the time value is greatest, and, thus, has a greater potential to decay.
For the same reason, theta is greater for more volatile assets, because volatility increases the option premium by increasing the time value of the premium. With higher volatility, an option has a greater probability of going into the money for any given unit of time. For the option writer, theta is positive, because options are more likely to expire worthless with less time until expiration.
Theta measures changes in value of options or a portfolio that is due to the passage of time. The holding of options has a negative position theta because the value of options continuously declines with time. Because time decay favors the option writer, a short position in options is said to have positive position theta. The net of the positive and negative position thetas is the total position theta of the portfolio.
- Inverse-gamma distribution
Vega aka Tau Volatility is the variability in the price of the underlying over a given unit of time. The Black-Scholes equation includes volatility as a variable because it affects the probability of the option going into the money: Historical volatility is easily measured, but current volatility cannot be measured because the unit of time is reduced to now. On the other hand, the price of the underlying, the option premium, time until expiration, and the other factors, except volatility, are known.
Therefore, volatility can be measured by rearranging the Black-Scholes equation to solve for volatility in terms of the other known factors. This is referred to as implied volatilitybecause the volatility is implied by the other known variables to the Black-Scholes equation. Consequently, vega is often used to measure the change in implied volatility. Vega measures the change in the option premium due to changes in the volatility of the underlying, and is always expressed as a positive number.
Because volatility only affects time value, vega tends to vary like the time value of an option—greatest when the option is at the money and least when the option is far out of the money or in the money. For any given time until expiration, the time value of an option is greatest when the option is at the money, and diminishes as it moves farther either out of the money or in the money.
Because theta and vega only measure the effect of time passage and volatility on the time value of an option, both theta and vega are greatest when the time value is greatest, and declines with time value when the price of the underlying moves away from the strike price.Understanding Theta - Time Decay Of Options
The position vega measures the change in option or portfolio values with changes in the volatility of the underlying. Rho Higher interest rates generally result in higher call premiums, according to option pricing modelsbecause the present value of the strike price is subtracted in these models.
Hence, higher interest rates correspond to lower present values, so less is subtracted, leading to higher call prices. Prevailing Interest Rates, Put Premiums A more intuitive way to understand why higher interest rates increases call prices is to understand that a call is like a forward contract, in that it allows the holder to buy the stock at a specified price before the expiration date, so the money that would have been used to otherwise buy the stock can, instead, be invested in Treasuries to earn a risk-free interest rate until the date in which the stock is purchased.
Because the stockholder incurs a cost of holding the stock, which is the forfeited interest that could otherwise be earned, a higher price is charged for the call to compensate the stockholder for the forfeited interest. By the same reasoning, dividends decrease the price of calls because only the stockholder is entitled to receive the dividends, not the call holder.
Variables that are distributed lognormally include stock prices and typically interest rates. What is a correlation coefficient? How is it calculated? Note that an equity Beta for stock x is the covariance of the stock x with the market divided by the variance of the market. What is the difference between a permutation and combination? When items may only be selected once i. This is a permutation. This combination is typically written using the following shorthand: Combinations are symmetric, so choosing 3 balls out of 16 or choosing 13 balls out of 16 results in the same number of combinations.
What is the probability that I flip this penny 5 times, it will come up heads at least 2 times?
The Greeks: Delta, Gamma, Theta, Vega, and Rho
Think of this as the inverse to the problem: The probability out of 5 flips of getting 0 heads: What is a Monte Carlo simulation? A Monte Carlo simulation is a computerized mathematical procedure for sampling random outcomes for a given process.
It provides a range of possible outcomes and their associated probabilities rather than a discrete point estimate of a given outcome. Monte Carlo simulation performs risk analysis by building models of possible results by substituting a range of values—a probability distribution—for any factor that has inherent uncertainty.
It calculates results over and over, each time using a different set of random values chosen from the probability functions. During a Monte Carlo simulation, values are sampled at random from the input probability distributions.
Each set of results from that sample is recorded; the values comprise a probability distribution of possible outcomes. Depending upon the number of uncertainties and the ranges specified for them, a Monte Carlo simulation could involve thousands or tens of thousands of recalculations. Results show not only what could happen, but also how likely each outcome is. Assume that I tell you that a prize is behind one of three doors.
If you pick a door say Door 2and I tell you that the prize is not behind another door Door 3, for exampleand I give you the option of remaining with your first pick Door 2 versus switching to Door 1, should you switch?
Inverse-gamma distribution - Wikipedia
This question has been asked for decades; the answer is much simpler than many think. In order for at least 2 people to have the same birthday, this means that 2 or 3 or 4…. This latter event is a permutation; if John is born on January 1 and Jeff is born on December 31, this is a different outcome than John being born on December 31 and Jeff being born on January 1 i. The total number of ways to choose k different birthdays from elements with no repetitions is !
So for a room full of k people, the probability that at least 2 have the same birthday is: Econometrics What is an R2 statistic? An R2 statistic is a measure of goodness-of-fit, also known as the coefficient of determination.
It is the proportion of variability in a data set that is accounted for by the chosen model. What is a random walk? Is it stationary or non-stationary? A random walk is a path that consists of taking successive random steps.
Random walks are not stationary, i. Non-stationary behaviors can include trends, cycles, random walks or combinations of the three. Note that while non-stationary data cannot be modeled or forecasted, the data can usually be transformed so that they can be modeled. The random walk is a non-mean reverting process that can move away from the mean either in a positive or negative direction. Put another way, the means, variances and co-variances of the walk change over time. Another characteristic of a random walk is that the variance evolves over time and goes to infinity as time goes to infinity; therefore, a random walk cannot be predicted.
What happens if you create a regression based on 2 variables that each are continuously increasing with time? If you attempt to model two series that are both time-dependent say, consumption and incomeyou will get a spurious regression, i. This is because both series are non-stationary.
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With non-stationary variables, one needs to transform them into stationary series; the easiest way to do this is by differencing, or looking at changes in the series.
Changes in a non-stationary series are usually stationary. What to do if two series are non-stationary: Rather than differencing each series, one may be able to create a better model by finding a cointegrating relationship.
With cointegration, the aim is to detect any common stochastic trends in the underlying data; whereas the two series may not be stationary, the difference between two non-stationary series may itself, be stationary. If I have a regression between x and y, what test statistics should I look at to determine whether I have a good model? Explains how much of the movement in the independent variable can be explained by the dependent variables chosen.