42
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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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What is 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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15
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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 ...
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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 ...
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14
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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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13
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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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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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13
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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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12
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Arbitragefree Pricing: Q vs. P
In the derivatives context, "arbitrage free" means almost surely for the probability measure under consideration. This is in opposition with statistical arbitrage used at high frequencies for example. ...
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12
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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 ...
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12
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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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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 ...
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12
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What is the Risk Neutral Measure?
Intro:
Great answer given by KeSchn above. I would like to contribute an additional perspective. My experience with and my understanding of the Risk Neutral measure is entirely based on "no ...
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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 ...
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11
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How to choose a risk-neutral measure when the market is incomplete?
A stochastic volatility model for a single risky asset can't be complete because you have two sources of randomness. But you can easily make it complete by adding a derivative whose value depends on ...
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11
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Risk-neutral pricing in incomplete markets
Q: What does the risk-neutral price represent if the option is not replicable?
In an incomplete market, there is no unique martingale measure but instead a set $Q$ of equivalent martingale measures. ...
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10
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What are the main flaws behind Ross Recovery Theorem?
Very simply, Ross' framework assumes a great deal to extract the true pricing kernel. Time homogeneity, additively separable state dependent utility, (discrete time Markovian structure - though these ...
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10
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Why the Black-Scholes formula can be used in the real world?
The short answer is:
As long as a derivative can be perfectly replicated via hedging in the underlying asset then the price of the derivative should be independent of investors' risk aversion and ...
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10
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Intuitive Reasoning for Using Risk-Neutral Measure
this is probably the most asked question in quantitative finance... There are many answers. One nice example to consider is what if the calls were struck at zero.
The call then pays the stock price ...
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What's Risk-Neutral in an Interest Rate Model?
It is a very interesting question. There is a brief explanation in the book Martingale methods in financial modelling. Basically, it says that, the interest short rate $r_t$ can be modeled in any ...
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10
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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 ...
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10
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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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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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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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8
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When to use the real world drift and when the risk neutral one for a Monte-Carlo simulation?
In general these are the two basic approaches to QuantFinance:
Sell side (market maker, risk neutral): You use risk-neutral probabilities ("$\mathbb{Q}$") e.g. in option pricing (to e.g. calculate ...
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8
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Numéraire -- couldn't understand the wiki explanation
If you are interested in the proof of the Baye's Rule for conditional expectations you can find it here
The sake of completeness:
The Baye's rule for conditional expectations states
$$ E^Q[X|\...
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8
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"The drift of stock price becomes the risk-free interest rate" under RNP
Yes, you may as well take this as the definition of the risk-neutral probability $Q$.
I will now try to give you some intuition for that kind of construction.
Assume the risk-free interest rate $r$ ...
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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 ...
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8
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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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8
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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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