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Probability
The probability of a certain event is a number expressing the likelihood that a specific event will occur, expressed as the ratio of the number of actual occurrences to the number of possible occurrences. In mathematics, it is a measure of how often an event will happen and is the basis of statistics.
14,643 Questions
What is Risk neutral probability measure?
A probability measure allocates a non-negative probability to each possible outcome. All individual probabilities together add up to 1.
The "risk-neutral probability measure" is used in mathematical finance. Generally, risk-neutral probabilities are used for the arbitrage-free pricing of assets for which replication strategies exist. This is about relative pricing, based on possible replication strategies. The first argument is that a complete and arbitrage-free market setting is characterised by unique state prices. A state price is the price of a security which has a payoff of 1 unit only if a particular state is reached (these securities are called Arrow securities). In a complete market, every conceivable Arrow security can be traded. It is more easy to visualise these securities in terms of discrete scenarios. (On a continuous range of scenarios we would have to argue in terms of state price density.) The arbitrage-free price of every asset is the sum (over all scenarios) of the scenario-payoff weighted with its state price. Any pricing discrepancy with regards to an implicit state price would enable arbitrage in a complete market. The assumption is that the pursuit of such opportunities drives the prices towards the arbitrage-free levels. Hence the state prices are unique.
Since the whole set of Arrow securities is the same as a risk-free bond (sure payoff of 1 unit at maturity), the price of the whole set of Arrow securities must be e^(-rt) (assuming we are now at maturity minus t). Risk-neutral probabilities can then be defined in terms of state prices, or vice versa. A probability measure has to fulfil the condition that the sum of all individual probabilities adds up to 1. Therefore, if we want to create an artificial probability distribution based on the state price distribution, we have to multiply each state price with e^(rt) in order to obtain its probability equivalent. It is not surprising then that any expectation taken under the risk-neutral probability measure grows at the risk-free rate. This is an artificial probability measure, why should we create such a construct? This connection allows us to exploit mathematical tools in probability theory for the purpose of arbitrage-free pricing. The main difficulty about risk-neutral probabilities is that the probability concepts used have not initially been developped for the purpose of financial pricing, therefore, two different languages are used, which can easily be confusing. The economic interpretation of a risk-neutral probability is a state price compounded at the risk-free rate. Anything that has an effect on a state price (preferences, real probability, ...), has an effect on the risk-neutral probability.
So now we have a bridge to go from state prices to risk-neutral probabilities and back again. What is this good for? According to the second argument, we can, under certain conditions, specify the unique risk-neutral probability distribution of an underlying asset price with the help of an only incomplete specification of its real probability distribution, thanks to the Girsanov Theorem. If the innovation in the price of the underlying asset is driven by a Brownian motion, then all we need to obtain the risk-neutral probability distribution is the volatility parameter. What can we now do with this risk-neutral probability distribution? We can use the first argument to convert the obtained risk-neutral probability distribution back to a state price distribution, and the state price distribution applied to the payoff distribution (i.e. taking the sum over all scenarios) leads to the arbitrage-free price. These arguments save us a lot of trouble when trying to calculate the arbitrage-free price of an asset. They allow us to avoid the estimation of risk premia, by implicitly using those incorporated in the underlying asset price. The arbitrage-free price is, however, NOT independent of risk-premia. The price of the underlying asset is part of the pricing equation, and the risk-premia are inherent in this price, but because the price of the underlying asset is known to us, we obviously do not need estimate it.
It is important to emphasise that the risk-neutral valuation approach only works if the asset to be priced can be perfectly replicated. This is often not true in reality, especially when dynamic replication strategies are involved.
Paper explaining risk-neutral probabilities:
http://ssrn.com/abstract=1395390
It is 156/2652 = 0.0588
Almost 12 %
It's the same as flip a penny ten times.
There are 2^10 = 1024 different outcomes (each penny being either head or tail).
The ones we can 'use' are those with 3 heads.
Let '1' be head and '0' be tail:
0001010100 is a valid outcome - and 1110000000 as well.
So how many way can we arrange three 1's in ten spaces?
In 120 ways.
120 ways out of 1024 is the answer.
120/1024=0.1171875 = 11.71875 %
5/18
It doesn't matter what number the first roll produces. The chance that the second throw will be the same is 1/6, so the chance of throwing one of the five different remaining numbers with the second dice is 5/6. Now there are only 4/6 ways of scoring with the third throw, and finally only 3/6 possibilities for the last throw.
So the probabiliy of throwing 4 different numbers = 1 * 5/6 * 4/6 * 3/6, w
hich is 5/6 * 2/3 * 1/2 = 5/18 = 0.2777777777777777or 27.77777777%
which is almost 28%
What is a probability based upon an event that has already occurred?
The probability based on an event that has already occurred is 100%. If the event has occurred, it has occurred.
The likelihood of an event occurring?
The likelihood of an event occurring is known as the probability of occurrence. This can be calculated based on previous patterns and other factors.
How do you Find the mean with a random variable and a probability?
say you have 3,45,5 you do 3+45+5 then divide it by 3 because there is 3 numbers
25%
ANSWER: One in four. Each ant can only move in two directions. Multiply that to get the total number of possibilities (2×2x2=8 ways). There are only two ways the ants can avoid running into each other. Either they all travel clockwise, or they all travel counterclockwise. Otherwise, there has to be a collision.
Pick one ant and call him "Bill." Once Bill decides which way to go (clockwise or counterclockwise), the other ants have to go in the same direction to avoid a collision. Since the ants choose randomly, there is a one-in-two chance the second ant will move in the same rotational direction as Bill, and a one-in-two chance the third ant will. That means there is a one in four chance of avoiding a collision.
Let's Say each ant can move left or right, here are the possible directions:
LLR, LRL, RLL,
RRL, RLR, LRR,
LLL, RRR
Using 2 dice how many different combinations can you roll?
you cam make 36 different combinations.
Ans 2.If you roll one dice twice, there are 36 combinations, if and only if you count 1,2 as different from 2,1.
If you are rolling two dice (at once) there are only 21 different combinations !
A dice is rolled three times in succession what is the probability of rolling three sixes?
The probability of rolling a six is one in six. The probability of rolling three consecutive sixes is one in 216. (1/6 x 1/6 x 1/6 = 1/216)
What is an addition expression?
An addition expression is an equation without an equals sign (=)
An example of an addition expression is 2+3. And that's it! Very simple!
more examples:
2+8
2+6
6+1
4+1
9+1239759736493856
7824501287565240564929384658834765+7569276520972809742659258215
Get it now?
Where can you buy 20 sided dice?
On the Deep Internet, Black Market Reloaded.
You need bitcoins to buy them though.
Probability of getting a multiple of 2 on a die?
2/6
It should be 3/6. Because the sample space is S=(1,2,3,4,5,6) As 2,4 and 6 is a multiple of 2, it should be 3/6
What is the next number in the sequence?
In fact, what are the next 2 numbers:
7, 14, 17, 21, 27, 28, 35, 37, ?, ?
The next two numbers are 42 and 47. It's a set of numbers that contain or can be divided by 7.
7 women in a group of 11 people
7/11
Can a joint probability have a value greater than 1?
A joint probability can have a value greater than one. It can only have a value larger than 1 over a region that measures less than 1.
The probability of a woman being chosen in this situation is 7/11.
Example of stratified random sampling?
In stratified sampling, the population to be sampled is divided into groups (strata), and then a simple random sample from each strata is selected. For example, a state could be separated into counties, a school could be separated into grades. These would be the 'strata'.
There are infinitely many numbers and so the probability of the second event is 0. As a result the overall probability is 0.
Example of theoretical probability?
Roll a fair cubic die. The theoretical probability of it ending up with any particular face on top is 1/6
A dice is rolled six times in succession what is the probability of rolling six sixes?
The chance that one dice falls on 6 is 1 out of 6. The probability that dice falls twice in a row is 1 out of 6 and 1 out of 6, that is (1/6)(1/6)=(1/6)^2=1/36.
Hence the probability of six sixes in a row is(1/6)^6=1 out of 46,656.
What is the probability of heads on one flip of a coin?
you would think there is a 50% chnce of it landing on heads. However due to the weight of the metal on the picture on the coin it is actually about 5009/10,000 chance it will land on heads.
