45
votes
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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 ...
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36
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 $M$...
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21
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&...
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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 ...
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16
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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 ...
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16
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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 ...
- 15.3k
15
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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 ...
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14
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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{...
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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 ...
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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 ...
- 561
13
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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 ...
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12
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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 ...
- 7,999
11
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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 ...
- 14.5k
11
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 ...
- 14.5k
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 ...
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9
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Why Drifts are not in the Black Scholes Formula
"Intuitively, everything else being equal, if a stock has higher drift, shouldn't it have higher probability of finishing in-the-money (and higher probability of having higher payoff), and the call ...
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9
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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 ...
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9
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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
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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 ...
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9
votes
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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 ...
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8
votes
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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 ...
- 14.5k
8
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 ...
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8
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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 ...
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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.
$Q$ ...
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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, ...
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7
votes
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Risk neutral drift vs real world
The risk neutral drift is the risk free rate for an asset with no dividends, no cost of carry, no repo cost, etc. Otherwise the drift has to be adjusted to take these into account, and the easiest way ...
- 5,632
7
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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 ...
- 7,999
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, ...
- 5,891
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-\...
- 1,096
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 ...
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