Skip to main content

# Tag Info

## Hot answers tagged risk-neutral-measure

49 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 ...
• 14.7k
38 votes

### Explaining the Risk Neutral Measure

Life Without a Risk-Neutral Measure How would we price assets without the measure $\mathbb Q$? Well, we would start with some version of the Euler equation $P_t=\mathbb{E}_t[M_{t+1}P_{t+1}]$, where $M$...
• 16.1k
21 votes
Accepted

### Explaining the Risk Neutral Measure

Intro: Great answer given by Kevin. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no arbitrage&...
• 6,213
18 votes

### $\mathbb{P}$ vs $\mathbb{Q}$ Probabilities - Transitioning Between Measures

$\mathbb{P}$ is the true probability measure. Measure $\mathbb{Q}$ is a measure of convenience that allows risk neutral pricing. Stochastic discount factor $M$ takes you between the two. If you care ...
• 6,974
16 votes
Accepted

### Risk-neutral vs. physical measures: Real-world example

You can price an asset paying $X_{t+1}$ in two ways: $$P_t=\frac{1}{R_f}\sum_{\omega} Q(\omega)X_{t+1}(\omega)$$ $$P_t=\sum_{\omega} P(\omega)M_{t+1}(\omega)X_{t+1}(\omega)$$ As you can see, the price ...
• 1,896
16 votes
Accepted

### What is the connection between the risk neutral implied density and the real world density?

I'll outline how you can estimate the (implied) real-world density function from (observed) option prices. Having found this real-world density, you can then compute all sorts of probabilities and ...
• 16.1k
15 votes

### What is the difference between risk neutral probabilities and stochastic discount factor?

The risk neutral probability measure $Q$ is the true probability measure $P$ times the stochastic discount factor $M$ but rescaled so $Q$ sums to 1. Simple derivation For maximum simplicity, I'll ...
• 6,974
15 votes
Accepted

### Path-dependent options valuation

Risk-neutral pricing A time-$T$ payoff is an integrable, $\mathcal{F}_T$-measurable random variable $\xi$. The value process of the discounted payoff is then a $\mathbb{Q}$-martingale, i.e., \begin{...
• 16.1k
13 votes
Accepted

### What are the main flaws behind Ross Recovery Theorem?

This is a loaded question. Ross' recovery theorem has both flaws and insights. The single answer thus far did a great job of addressing the flaws from an economics perspective. No one questions that ...
• 146
13 votes
Accepted

### Probability in different measures

In order to estimate a probability of an event with small probability, you might want to try to estimate the probability for a changed random variable that allocates a larger probability mass for the ...
• 561
13 votes

### Why quants think that the risk-neutral measure should not be used for financial forecasting?

There is a deeper issue. Frequentist distributions are not probability distributions because they are designed to be minimax distributions rather than actual distributions. This ignores all of the ...
• 4,309
12 votes
Accepted

### How do we determine the "correct measure"?

Recall that any traded asset divided by a numÃ©raire is a martingale under the measure associated to that numÃ©raire. For the 3 interest rates you mention, the natural measure (namely the one that makes ...
• 8,139
11 votes

### Risk neutral measure for jump processes

Assume a constant risk-free rate $r$ and no dividends. Generalisation is straightforward. To preclude arbitrage opportunities, under the risk-neutral measure $\Bbb{Q}$, the discounted asset price ...
• 14.7k
11 votes
Accepted

### Would it be possible to combine long butterfly with long straddle, achieving profit no matter the outcome?

Your butterfly is short a straddle and long a strangle. If you add a long straddle with the same strike/notional you are now just long a strangle. The payoff for a strangle is zero if the terminal ...
• 5,931
10 votes
Accepted

### What is a Brownian motion "under the risk-neutral measure"?

A Brownian motion is always defined with repect to a given probability space. Let $(\Omega,\mathcal{F},\mathbb{P})$ be a probability space and $X_t=W_t^\mathbb{P}$ a Brownian motion, i.e. a stochastic ...
• 16.1k
9 votes

### Risk-neutral vs. physical measures: Real-world example

First of all, you need to understand risk-neutral measures are not meant to make predictions of future prices, but they are meant to allow hedging (ie risk replication). Historical measures on their ...
• 12.4k
9 votes

### Why quants think that the risk-neutral measure should not be used for financial forecasting?

In their book "Counterparty Credit Risk, Collateral and Funding" D. Brigo, M. Morini and A. Pallavicini start with a dialogue between a Physics PhD graduate and an experienced practitioner of ...
9 votes
Accepted

### Girsanov Theorem, Radon-Nikodym Derivative backward

The result you're looking for is $$\left. \frac{d\Bbb{P}}{d\Bbb{Q}}\right\vert_{\mathcal{F}_t} = \left( \left. \frac{d\Bbb{Q}}{d\Bbb{P}}\right\vert_{\mathcal{F}_t} \right)^{-1}$$ This is a result ...
• 14.7k
8 votes
Accepted

### How to price this basket option?

No offense but it will be much more complicated than what you think... I'm not even sure that you are familiar with risk-neutral pricing in the first place? I'll try to give you some clues. This ...
• 14.7k
8 votes
Accepted

### Why must a riskless portfolio earn the risk-free rate?

If you imagine you have two risk-less assets that have a unit payoff at maturity $V_1(T) = V_2(T) = 1$ but their present value is not equal, e.g. $V_1(t) < V_2(t)$. You buy the cheaper, sell the ...
• 6,064
8 votes

### Why quants think that the risk-neutral measure should not be used for financial forecasting?

Perhaps a case of views based upon theoretical possibilities rather than empirical realities? In theory, $P$ and $Q$ can be extremely different $P$ is the real world, actual probability measure. $Q$ ...
• 6,974
8 votes
Accepted

### Risk Neutral Valuation, Drifts and Calibration

There are two parts to your question which I try to answer separately. The first one is about what calibration actually is whereas the second question deals with risk-neutral pricing. As an example, ...
• 16.1k
7 votes
Accepted

### Risk-adjusted returns ratio that does not reward high risk for negative returns

Yes, you are correct on both terms - it doesn't make much sense, and there exists a well-cited solution by C. Israelsen: "A refinement to the Sharpe ratio and information ratio." Journal of Asset ...
7 votes

### Why discounted derivative price is a martingale?

Under a Black-Scholes framework, the dynamics of the stock price under the risk-neutral measure $\mathbb{Q}$ are given by ... $$S_t = r S_tdt +\sigma S_tdW^{\mathbb{Q}}_t$$ ... and those of the ...
• 8,139
7 votes
Accepted

### Drift rate vs. Riskless rate in the Black-Scholes model

Let's take your questions in turn - If volatility isn't a concern, an investor is concerned with $$E[\log S_T] = \log S_0 + (\mu - \tfrac{1}{2}\sigma^2)T$$ when deciding to purchase a stock, ...
• 5,931
7 votes
Accepted

### Equivalent Martingale Measure(EMM) of Inverse of Stock Price

Let $dB_t = rB_t dt$. Now d\Big(\frac{1}{B_t S_t}\Big) = -\frac{dS_t}{B_t S_t^2} -\frac{dB_t}{B_t^2S_t} +\frac{2}{2}\frac{(dS_t)^2}{B_t S_t^3} = (-\mu-r+\sigma^2)\frac{1}{B_tS_t}dt-\...
• 1,086
7 votes

### Machine Learning usage in Q part of Quant Finance

There is at least one clear area of application of ML in Q quant finance, it is the LSM algorithm invented by Longstaff, Schwartz and Carriere in the late 1990s for the valuation of callable exotics ...
7 votes

### Measure theory in quantitative finance

Measure theory helps us overcome some of the drawbacks of constructing measures (measure of probability when ranged at $[0,1]$). Classic probability theory is effective for probability models whose ...
• 1,426
7 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., ...
• 1,098
7 votes

### Is "risk-neutral probability" a misnomer?

Originally "risk-neutral" is a term from economics describing the attitude of investors towards risk: if they are risk-neutral they only factor in the expected value of a decision and not the level of ...
• 27.5k

Only top scored, non community-wiki answers of a minimum length are eligible