42
votes
Accepted
How to estimate real-world probabilities
The risk-neutral measure $\mathbb{Q}$ is a mathematical construct which stems from the law of one price, also known as the principle of no riskless arbitrage and which you may already have heard of in ...
- 13.9k
6
votes
Accepted
Vasicek short rate: Risk-neutral measure into real-world measure
Vasnicek by itself does not specify what form the change of measure should be and how you should parameterise the market price of risk.
A very natural parameterisation is affine in the factor, i.e., ...
- 1,008
6
votes
Accepted
Are all change of measure operations between equivalent probability measures Doléans-Dade exponentials?
The answer is yes.
Proof:
Theorem (Radon-Nikodym)
Let $(\Omega, \mathcal{F})$ be a measurable space. Let $\mathbb{P}$ and $\widetilde{\mathbb{P}}$ be two $\sigma$-finite measures. Let $\widetilde{\...
- 510
6
votes
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Risk Neutral and Real World Valuations using Monte Carlo
Just to add to the answer by @Kevin :
There are at least two things going on here. First of all let $\{Q_i \}$ denote a set of equivalent probability measures, which includes your $P$ and $Q$ above.
...
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6
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Risk Neutral and Real World Valuations using Monte Carlo
You probably wonder whether $\mathbb{E}^\mathbb{P}[P_T\mid\mathcal{F}_t]= \mathbb{E}^\mathbb{Q}[P_T\mid\mathcal{F}_t]$. Note the $T$ as index, i.e. the future unknown payoff and not the current price $...
- 13.8k
5
votes
What is the numeraire for the real world measure $\mathbb{P}$?
Let $N_t$ be the numeraire for the real world measure $P$. Let $r$ be the risk free rate and $P^*$ be the risk neutral measure. Then from the standard change of measure results one must have
$
\frac{...
- 5,532
5
votes
Accepted
Real world probabilities from option implied risk neutral density?
Great question! Unfortunately, it's not easy. We can use option prices to get the $\mathbb{Q}$-distribution. However, the probability measure $\mathbb{Q}$ merges the stochastic discount factor (SDF) $...
- 13.8k
4
votes
What is the numeraire for the real world measure $\mathbb{P}$?
To build on Antoine's answer (which covers the case where the market consists only of a stock $S$ and a risk free asset $r$). In the general case, if the real world measure $\mathbb{P}$ numéraire ...
- 2,110
4
votes
Accepted
Objective probability of default from CDS spread
(Bloomberg and Reuters News are fond is reporting that some name is trading at some such CDS spread, "which implies N% probability of default". They neglect to mention what recovery ...
- 9,229
4
votes
Accepted
Variance-Covariance Matrix under $\mathbb{P}$ and $\mathbb{Q}$
Just to expand on Alex answer.
Empirically it is simply not true. Focusing on the diagonal of the variance-covariance matrix, we know that there is a large variance risk premium. Take a look at table ...
- 6,685
3
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How to estimate real-world probabilities
Two remarkably simple solutions have been missed. Let us go a completely different route. Let's assume that the standard models don't work sufficiently well, for whatever reason, and that we need a ...
- 4,064
2
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Stock forward price argument
You ask where the mistake is, but there isn't one. If you buy stocks using money borrowed at the risk free rate you will expect to make money, but there is risk. There's no contradiction.
- 13.7k
2
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Are all changes of measures for continuous diffusion processes given by the change of drift?
I have read that for diffusion processes, indeed the volatility must be preserved under a change of measure. This old question appears to be relevant :
Version of Girsanov theorem with changing ...
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2
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How to estimate real-world probabilities
You can definitely calculate the real-world probabilities. For instance, just think log-returns are normally distributed, take the mean and standard deviation of the past log-returns and ta-da... You ...
- 1,021
1
vote
Estimating risk aversion from option bid-ask spreads
Bid Ask spreads should reflect the willingness of parties to exchange at a certain price, where market makers are the sellers it represents the risks they are prepared to take in order to make the the ...
- 174
1
vote
Are all changes of measures for continuous diffusion processes given by the change of drift?
Change of measure and change of variable are two separate things. In measure change, you keep the same variable and redistribute the probability. Keeping the variable the same is the key to the ...
- 6,044
1
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Estimation of Radon–Nikodym derivative from historical returns and option price data
Since Girsanov changes the drift but keeps the volatility unchanged, it would be hard to reconcile say a simple exponential brownian motion under $\mathbb{P}$ with a skew/smile structure under $\...
- 5,532
1
vote
Accepted
Uniqueness of Risk-neutral measure: Probabilistic view
Basically the argument is that we have arrow-debreu securities (instrument that pays 1 if you arrive in a certain state). In the absence of arbitrage the price of this arrow-debreu security should be ...
- 1,401
1
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Confusion regarding the risk neutral and physical measures
The equivalent martingale measure (EMM) $\mathbb{Q}$ is a measure under which all the asset prices discounted using a risk-free bond are martingales, i.e. given the bond price $B(t)$ and the asset ...
- 1,856
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