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Questions tagged [risk-neutral-measure]

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Equivalent martingale measure exists if and only if $a < S_0^1(1+r)< b$

Exercise : We consider a market of one period $(\Omega, \mathcal{F}, \mathbb P, S^0, S^1)$, where the sample space $\Omega$ has a finite number of elements and the $\sigma-$algebra $\mathcal{F} = 2^...
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Describing all the equivalent martingale measures

Exercise : Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\mathbb P$ be a probability measure such that $\mathbb P[\{\omega_1\}] > 0$ for all $i=1,2,3$. We ...
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1answer
88 views

Risk neutral valuation formula

I am totally new to Finance and Arbitrage theory and I have started reading Björk (2018) Arbitrage theory in continuous time. Can anyone please explain to me what is the risk-neutral valuation formula ...
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88 views

What is the Expectation of price S with respect to probability measure P and risk neutral measure Q?

If I have a stock with price process $S_t$, and we have that $W$ is a Brownian motion under the probability measure $\mathbb{P}$, and $\mathbb{Q}$ is the risk neutral/equivalent martingale measure. ...
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1answer
45 views

Suggestions of papers for computing market implied probability distribution function

I need suggestions of papers that propose simple and fast methods (not heavily dependent on simulations, nut can depend on simulation) to derive the market implicit probability distribution function ...
3
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1answer
87 views

condition of risk neutral pricing

The theorem says if $U$ is a numeraire and let $\mathbb{Q}^U$ be the corresponding measure. Then for every tradable asset $S$, the relative price $S_t/U_t$ is a martingale under $\mathbb{Q}^U$. But I ...
3
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1answer
88 views

Change of numeraire between T-forward and Bank Account

I follow a course, and get to the point that one bond price discounted by another one is a martingale: $$ \frac{P(t,T_0)}{P(t,T_1)} - \text{ is a } \mathbb{Q}^{T_1} \text{ martingale } $$ I can not ...
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53 views

Dividend paying asset, why can't be taken as numéraire?

Why when considering numéraires, one cannot use a dividend paying asset to define a risk neutral measure? Here's where I got my question : (Shreve - Stochastic Calculus For Finance II)
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1answer
107 views

Libor Market Model (LMM) under risk neutral measure

I would like to establish the equations of forward libors under risk neutral measure. Here is how I do it, and what I get : Under the $P_{T_j} $ measure, forward Libor $L_j$ is martingale. Thus: $$ ...
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0answers
30 views

Checking arbitrage for the SABR model - analytical vs numerical approach

I wish to check if the fitted volatility smile/surface from the SABR model for a fixed time period is arbitrage free. Through my research, I've learnt the following need to be checked: The RND (risk ...
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0answers
59 views

Why a currency is not considerend as a numéraire for a risk neutral measure

We often say that "A risk neutral measure is associated with the money market account, not the currency. Currency pays a dividend because it can be invested in the money market." How is a currency ...
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1answer
88 views

Confusion regarding the risk neutral and physical measures

I may be confused. I am looking at the risk neutral vs. physical measures. We know that knowing the short interest rate stochastic process $r$, a bond maturing at time $T$ can be considered as a ...
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28 views

Joint risk neutral distribution

If I have stock index option and individual stock option data, where individual stocks are components of this index, can I estimate joint risk neutral distribution of underlying stocks
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39 views

Unique risk neutral measure in Black Scholes vs Merton Model

I was going through a question on the unique risk neutral measure in the Black Scholes model : Unique risk neutral measure for Brownian Motion One of the answers said it is essentially because there ...
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1answer
89 views

Vasicek short rate: Risk-neutral measure into real-world measure

I consider the Vasicek model under the risk-neutral measure $\mathbb{Q}$: $$ dr_t=\kappa(\theta−r_t) dt+\sigma dW^{\mathbb{Q}}_t.$$ I have already determined $$\mathbb{E}^{\mathbb{Q}}\left[e^{−\int\...
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Equivalent martingale measure in time changed Levy models

I am investigating time changed Levy models. As far as I have seen, these models are usually directly described under the risk neutral measure $\mathbb{Q}$. However, I'm interested in first modelling ...
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1answer
30 views

calibrating two (or X) equity diffusion trees

I have two equities S1 and S2. Each one follows the following tree evolution : $$S_1 \rightarrow \left \{ \begin{matrix} S_1 (1+u_1) & \text{with probability } p_1 \\ S_1 (1-d_1)...
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What is the risk neutral density and how is it estimated? [closed]

I don't understand the words "risk-neutral density". Please explain what it is, and how it can be estimated in practice. My guess would be that we have an underlying probability space $(\Omega, P)$. ...
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4answers
293 views

Measure theory in quantitative finance

When I read up on stochastic modeling, the use of "measure" comes up a lot. So far I just read the word "measure" as "probabilities" or "distribution" and was able to get away with it when trying to ...
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135 views

Is there a relationship between Risk Neutral Pricing framework and Nash Equilibria?

Based on the Fundamental Theorem of Asset Pricing, the risk neutral price of a contingent claim on an asset in a liquid, arbitrage free market can be determined by switching to an equivalent $Q-$ ...
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1answer
141 views

Stock forward price argument

Hi I am strangling to understand where is the mistake with the following strategy. Can anyone help me with the following argument? Assuming a stock price follows geometric Brownian motion then the ...
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30 views

Going from Stochastic Discount Factor / Risk Neutral Density -> Hedge Ratio

Assuming a probability distribution function is known in its entirety, what methods are available to construct a hedge ratio? For guidance, I went to the canonical Empirical Pricing Kernels and found ...
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1answer
102 views

Calculating the stochastic integral of $\exp(-rt)S_t$

I am currently reading lecture notes which aim to show that if $$ S_t = S_0 \exp (\mu t + \sigma W_t) $$ then, under the probability measure $\tilde{\mathbb{P}}$ with density $$ \gamma_T = \exp (c W_T ...
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1answer
118 views

Question about the Cameron-Martin-Girsanov (CMG) theorem

Within my lecture notes, the following definition of the CMG theorem is given: Under the probability measure $\mathbb{\tilde{P}}$ with density $\gamma_T = \exp(cW_T - \frac{c^2}{2}T)$, the process $...
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2answers
207 views

Monte-Carlo simulation Hull-White process: physical and risk-neutral measure

From Monte-Carlo simulation Hull-White process I get paths in risk-neutal measure. How can I get paths in physical measure?
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1answer
63 views

Girsanov's Theorem for Multiple Risky Assets

Girsanov's theorem provides the measure transformation from probability measure P to Q such that- $dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
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Machine Learning usage in Q part of Quant Finance

Machine Learning algorithms is broadly used in trading strategies and in general when it comes to working with financial time series. The webpage Quantopian is a platform to see some of the ...
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Tail estimation of the risk neutral density with GEV

I am a little bit unfamiliar with GEV distribution fitting and wanted to see how to implement it in the particular case of risk neutral density estimation. I use a common approach to estimate my risk ...
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0answers
65 views

Hull Martingales and measures problem 27.16 7e?

Here's a question from Hull's Options Futures and Other derivatives which I'd appreciate if someone helped me to clarify. The question is from the chapter "Martingales and Measures" Suppose that the ...
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0answers
52 views

Martingale Measure in Fundamental Asset Prcing Formula

I want to know when we use FAPF to price the Value of an Asset according to the formula, $$V_{asset} = \mathbb{E}^{P} [\sum_{\tau=0}^T (PV[C_\tau])]$$, where: $P$ is some equivalent probability ...
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1answer
100 views

Equivalent Martingale Measure result Hull?

I've been reading Hull's chapter about Martingales and measures where he states that if you have the dynamics of two securities as follows: \begin{align} \frac{df}{f} = (r + \lambda \sigma_f) dt + \...
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0answers
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How to Calculate the Value of a Growing Perpetuity Using a State Price Matrix?

Summary I wish to value perpetual cash flows through state contingent claims on real consumption, where the state of the economy is assumed to follow a finite markov chain (Similar to Banz and Miller ...
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1answer
302 views

Hull-White Extension of Vasicek Model

I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process ...
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1answer
103 views

Term structure used in Geometric Brownian Motions under Risk Neutral Measure?

When using a GBM under a risk-neutral measure to simulate stock prices, we have to use the risk-free interest rate, but how exactly do you determine what interest rate to use? I have used the Vasicek ...
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1answer
230 views

Change of Numeraire to price European swaptions

In the pricing of a European swaption, it is common to use the annuity factor $A(t)$ as the Numeraire. I was trying to write down the pricing formula via the bank account as numeraire to see if they ...
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1answer
388 views

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

My question is regarding the difference between risk neutral probabilities and stochastic discount factor? I am confused as to how are they related?
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3answers
238 views

Risk neutral measure doubt

For a derivative in a complete market, we can say that: $h_0 = E(h_t)$ assuming 0 risk free rate. Is the above relation also valid for a stock/ non derivative i.e. $s_0 = E(s_t)$ under the same risk ...
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1answer
364 views

Equivalent Martingale Measure(EMM) of Inverse of Stock Price

I met this question says how to price a vanilla call option $C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock $V_{t} = \frac{1}{S_{t}}$ at maturity if the stock price follows a geometric ...
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Complete Financial Market: Integrability condition for Contingent Claims

Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
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69 views

Bivariate risk neutral distribution through copula

I want to build a bivariate risk-neutral distribution from two liquid assets (A and B) through the use of a copula. As A and B are liquid, I have the marginal distributions from the market. All I have ...
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Constant volatility and risk-free rate assumptions of Black Scholes

I'm studying the risk-neutral derivation of Black-Scholes formula and feel confused about the requirement for the volatility of the underlying asset and the risk-free rate to be constant. It seems ...
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Overview of pricing using e.g. forward measure numeraire change

I am taking my first steps into numeraire change and change of probability measure. My overall struggle is relating numeraire change and all its aspects to e.g. the easy intuition behind risk-neutral ...
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2answers
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The two fundamental theorems of Finance, as they relate to the martingale measure

I RECENTLY read this in an article by Battig and Jarrow, "the first fundamental theorem relates the notion of no arbitrage to the existence of an equivalent martingale measure, while the second ...
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1answer
483 views

Girsanov Theorem, Radon-Nikodym Derivative backward

Given a filtered probablity space $(\Omega,\mathcal{F},{\mathcal{F}}_t,\mathbb{P})$ and a standard Brownian motion $W_t$. Normally, in Girsanov Theorem, we use the exponential martingale $Z_t=\exp(-\...
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1answer
82 views

Convexity adjustment when payment if after interest natural term?

I've been working with a convexity adjustment for an interest rate payoff and the next question came to me: The usual problem that gives rise to the convexity adjustment I'm referring to is as ...
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1answer
197 views

How to infer real world measure from risk neutral measure

Assume we have inferred risk neutral density of stock price at time T from option prices. Assume we have obtained a parameterized density p(S). How can we infer real world measure? I know about ...
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Binomial model's Radon-Nikodym derivative

Related: Dumb question: is risk-neutral pricing taking conditional expectation? In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, I think it's $\frac{d \mathbb Q}{d \mathbb P}...
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How are the two concepts No arbitrage & Risk neutral probability related?

The title, and might I add, that this question is in relation to the Black-Scholes model and why the concepts are important for option pricing in general.
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2answers
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Dumb question: is risk-neutral pricing taking conditional expectation?

Dumb question: is risk-neutral pricing taking conditional expectation? $\tag{1}$ In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally ...