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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 ...
Quantuple's user avatar
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37 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 $M$...
Kevin's user avatar
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20 votes
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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&...
Jan Stuller's user avatar
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$\mathbb{P}$ 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 ...
Matthew Gunn's user avatar
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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 ...
fni's user avatar
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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 ...
Kevin's user avatar
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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 ...
Matthew Gunn's user avatar
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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{...
Kevin's user avatar
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13 votes
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Obtaining risk-neutral probability from option prices

The risk-neutral probability density function $q(.)$ is indeed given by $$ q(S_T=s) = \frac{1}{P(0,T)} \frac{ \partial^2 C }{\partial K^2} (K=s,T) $$ where $P(0,T)$ figures the relevant discount ...
Quantuple's user avatar
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13 votes
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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 ...
peter carr's user avatar
13 votes
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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 ...
ragoragino's user avatar
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 ...
Dave Harris's user avatar
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How do we determine the "correct measure"?

Recall that any traded asset divided by a numéraire is a martingale under the measure associated to that numéraire. For the 3 interest rates you mention, the natural measure (namely the one that makes ...
Daneel Olivaw's user avatar
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 ...
Quantuple's user avatar
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11 votes
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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 ...
Chris Taylor's user avatar
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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 ...
Kevin's user avatar
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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 ...
lehalle's user avatar
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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 ...
Nicolas Gutierrez's user avatar
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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 ...
Quantuple's user avatar
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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 ...
Quantuple's user avatar
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8 votes
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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 ...
LocalVolatility's user avatar
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. $Q$ ...
Matthew Gunn's user avatar
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8 votes
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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, ...
Kevin's user avatar
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7 votes
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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 ...
Forgottenscience's user avatar
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 ...
Daneel Olivaw's user avatar
7 votes
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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, ...
Chris Taylor's user avatar
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7 votes
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Equivalent Martingale Measure(EMM) of Inverse of Stock Price

Let $dB_t = rB_t dt$. Now \begin{equation} 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-\...
Freelunch's user avatar
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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 ...
Antoine Savine's user avatar
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 ...
alexbougias's user avatar
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7 votes
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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., ...
NBF's user avatar
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