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Questions tagged [probability]

The tag has no usage guidance.

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Probability ITM formula for options

Given a stock of price price and annual volatility annual_volatility, and given an option with strike price ...
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1answer
53 views

In search of nice (approx) function forms of the volatility of cumulative simple returns

Let's consider a period $t\in[0,T]$, and let the simple return over year $t$ ($1\le t\le T$) be $r_t$. Assume $r_t$ are iid normal. The cumualative simple return over the whole period $[0,T]$ is $$R_T=...
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70 views

How to calculate number of round trips given volatility?

Suppose we know stock price volatility is normally distributed with mean = 0 and annual volatility say 20%. Let's assume markets never close and we can trade at 1 second intervals. Let's assume stock ...
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1answer
60 views

Bayesian trade probability with factors

I have a strategy Y which is influenced by some factors X1, ..., Xn (for example asset volatility, distribution of macroeconomic factors). At moment t0 I have historical distribution(prior) of X1, ...,...
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1answer
57 views

Expectation of the product of two Brownian motions [closed]

Could you please let me know the steps to follow to get to the solution?
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1answer
78 views

Hedging Value-Financial Mathematics

EXERCISE We consider a free from arbitrage financial market $(Ω,F,P,S_0,S_1)$ with $α<S_0^{1}\cdot(1+r)<β$,where $$0<α:=min_{ω \in Ω} S_1^{1}(ω), β:=max_{ω \in Ω}S_1^{1}, α<β$$ Let ...
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1answer
73 views

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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1answer
137 views

What is the Probability Distribution of Max-Drawdown?

How to obtain the probability distribution of Maximum Drawdown, starting from the probability distribution of Daily Returns? Here the details: Suppose I have a time serie of N=1000 daily returns. ...
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1answer
79 views

The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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1answer
136 views

Conditional Probability - Geometric Brownian Motion

Background I am trying to find a way to price a variant of a gap option by using closed-end expressions. What makes this option a bit tricky is that it can be exercised at four predetermined dates (t=...
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1answer
39 views

How skew in vertical put spreads change the payoff?

An spx four strikes wide Put Spread from at the money has a payoff ratio of 1 to 2 meaning if the Premium on the spread is \$10 your reward is \$20; yet the corresponding Call Spread with the same ...
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0answers
25 views

About a Trivariate Density of Brownian motion, its local and occupation times

I need help on a technical detail in the article by Ioannis Karatzas and Steven E. Shreve (1984) titled: " Trivial density of Brownian motion, its local time and occupation,with application to ...
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3answers
230 views

From Butterfly Price to Probability of $S_T$ Falling within a Range

If a butterfly in the limit represents a probability (by the Breeden-Litzenberger result), what can be said about the relative likelihood of a random variable $S_0$ from the price of a vanilla-option ...
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0answers
25 views

About a formula concerning the occupation time of a Brownian motion (The arc-sine formula)

Let the process $Z_t \buildrel\textstyle\over={W_t}+ {\lambda}t $, $t\geq 0$ a brownian Motion with Drift and $A_{T}^{+k, Z}$ his occupation time above the barrier $k$ defined by $$A_{T}^{+k, Z}=\...
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0answers
83 views

Probability of Implied Volatility Move [closed]

I want to see the probability of Implied Volatility of an underlying moving up or down from its current position. Would it just be 50% probability of going up and 50% of it going down? Because I've ...
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50 views

Square Integrable Process Implication

In Sergii Kuchuk and Yuliya Mishura paper, Pricing the European Call Option in the Model with Stochastic Volatility Driven by Ornstein-Uhlenbeck Process, Exact Formulas, the model can be represented ...
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0answers
21 views

Informal definition of the almost complete convergence

I'm searching for the informal definition of the almost complete convergence (P. L. Hsu and H. Robbins (1947), Dugué (1955)), which means a definition without any mathematics formula. Thank you for ...
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0answers
44 views

Jump diffusion model and Firm probability of default

I want to examine whether corporate events affect firm's probability of default. My initial thought was a jump diffusion model, although in the literature, the only work I found, involved CDS market ...
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0answers
115 views

Detecting butterfly spread arbitrage for American options through European option prices

It's easy to demonstrate that if European option prices are concave with strike, then an arbitrage exists. For example, the risk-neutral probability density is the second derivative of European put ...
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0answers
59 views

How to determine the default probability of a county in a bond that is not in its native currency?

Disclaimer: This post is cross posted in here also. Consider the following case: Country P uses the currency Euro and gives p percent interest on a one year bond issued in Euro. Country Q uses the ...
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0answers
41 views

Prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$

Using the Dynkin's formula, prove that $F(s,x_0)=0$, $F(t,x)=1$ and $\frac{\partial F}{\partial t}+\frac{1}{2}\frac{\partial^2 F}{\partial x^2}=0$ where $F(s,t)=2\int_{x-x_0}^{\infty}\frac{1}{\sqrt{2\...
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How are Risk indices linked to Physical Trading returns?

Ref to my previous question here: Physical trading spot transaction analysis-Quantified I have been able to narrow down my aim to defining a physical trading strategy P&L. My question is, how ...
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1answer
68 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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1answer
230 views

credit risk - marginal default probability

I have been working on an assignment trying to calculate marginal/conditional probability of default. Using a logistic regression framework, I was able to compute the 12-month unconditional PD for ...
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0answers
33 views

Uniqueness of data metric [closed]

Is there a metric that calculates "uniqueness of data"? For example if i have two sets of 200 observations, DataSet 1 has 70 unique values but 4 values take up the next 130 observations. DataSet 2 ...
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1answer
90 views

Distribution in Heston

$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$ $W_t$ is wiener process and the rest is just some parameters. For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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0answers
95 views

How to compute SABR's probability density function

I am trying to compute the probability density function of the forward rate implied by the SABR formula approximation in order to see how the density implied by the approximation has negative ...
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1answer
61 views

Can you determine USD swap rate movement probability from OTM swaption premiums?

E.g., the USD 1y x 4y swap rate is currently 2.84%. ATM receiver swaption , European exercise is currently at ATM premium of 1.15% while swaption premium at strike 1.5% is 0.15% or about 90% lower ...
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1answer
201 views

conditional probability of default

I would like to ask the following question. I would appreciate if someone could help me out. On what argument is based that states that conditional default rates ( loans of corporate borrowers) ...
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0answers
50 views

Computing the Expectation to a Max function

if $X_T$ is log-normally distributed and $k$ is a constant, how do I compute: $$E[\max(X_T-k,0)]$$ I can compute $E[X_T-k]$ and $P(X_T-k>0)$. I was thinking that an approach will be compute to $$E[...
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1answer
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Stochastic Calculus: How to test for dependency of random variables

If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as: $$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process. For $s<t$, Will $X_s$ and $X_s-X_t$ then be ...
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2answers
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Is a wiener proces measurable? (exercise from Bjork)

I will claim $$E[W(T) \vert F_t] = 0$$ for $t<T$. Anyway, in an exercise in Bjork the results requires that $$E[W(t) \vert F_t] = 0$$ But why? Isn't $W(t)$ measurable at time $t$ and hence not ...
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0answers
182 views

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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2answers
320 views

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 ...
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How does HMM apply to Forex for example, when any imaginable state you can think of involving price is observable?

Intuitively it seems like you can add states to the transition probability matrix $A$ and use a learning process to figure out the new transitions. If that's correct then it answers my question as ...
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1answer
75 views

When predicting Forex price using HMM what, typically, are the states and what are the observations?

I understand their abstract definition but having trouble applying the HMM method to Forex prices. What should the observations be? Then what should the states be (like "hot", "cold", etc.)?
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1answer
78 views

Spot Interest Rate at time $t$

I know that the general model for the dynamics of the spot interest rate is $$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$ My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?
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1answer
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Deriving $dR(t)$ For Reverse Exchange Rate

Say $Q(t)$ is the exchange rate at time $t$. It's the price in domestic currency of one unit of foreign currency and converts foreign currency into domestic currency. The model for the dynamics of ...
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1answer
181 views

R Calculate future price range and plot the result

First I want to say that I've read this post (How to calculate future distribution of price using volatility?) but it doesn't help much. Here is what I'm trying to do (values are not real) Let's ...
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0answers
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Computing the PDF of the sum of N moves of an empirical PDF for USDJPY 1-minute moves

Per-minute tick data for USDJPY is available here. Suppose we download this file to usdjpy.txt and then save it into a Numpy array in Python 3 as follows: ...
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2answers
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Am I calculating my Kelly Criterion correctly?

I'm taking a look at my trading history over a particular time period and have 500 trades on with an win rate of 82%. My average win is $W$. My average loss is $L$. So am I correct in assuming the ...
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1answer
112 views

Girsanov Transform and Likelihood Process Domestic to Foreign

Working two exercises relating to $Q^d$ and $Q^f$. I'm comfortable working with transforms and likelihood processes on a risky asset between $Q$ and $Q^s$, and also on an exchange rate $X$ between $Q$ ...
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1answer
70 views

What's the relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$ if the probability distribution function is not symmetric?

If the probability distribution function $f(x)$ is not symmetric, is there any relationship between $VaR_{\alpha}(X)$ and $VaR_{1-\alpha}(X)$? Here, $VaR$ is defined as $$ VaR_{\alpha}(X) := \inf\...
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1answer
222 views

Probability in different measures

I'm having some troubles understanding a problem. The problem: "Show how a measure change can be used to estimate the probability for $Y > 100$ when $Y \sim \mathcal{N}(0, 1)$. The book I'm using ...
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1answer
238 views

Can a Kelly Criterion Percent be very high?

This is my personal record trading options (selling spreads) over a certain time period: Win Rate: 83.94% Average Win: $299 Average Loss: $1,181.40 The formula for the Kelly Criterion is: $$ f=\frac{...
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1answer
368 views

Subadditivity of Expected Shortfall

I am able to see why Expected Shortfall will be subadditive for normal distribution or a uniform distribution. I am trying to prove the result for any generic distribution. I came across many proofs ...
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1answer
399 views

Probability of exercise in the Black-Scholes Model

What's the intuition behind the fact that the limit of $\mathcal{N}(d_2)$, i.e. the (risk-neutral) probability of exercise, in the Black-Scholes Model tends to $0$ when the volatility tends to ...
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1answer
57 views

Quantile with periodic investing

Short Version Can I get a quantile of such an expression? \begin{equation} \sum_{k=1}^{n} A_k\exp(\mathcal{N}(t_k\mu-\sigma\sqrt{t_k}/2,\sigma))) \end{equation} I know I can do it for one part of ...
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1answer
344 views

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

I'd like this question to definitively guide a practitioner to using both $\mathbb{P}$ vs $\mathbb{Q}$ probabilities in trading and research. Let's take only one fact as given: if I have a risk-...
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1answer
193 views

Quantile normal and lognormal

Let's assume we have a normal distribution $X\sim \mathcal{N}(\mu,\sigma^2)$. In a normal distribution the quantile can be calculated as follows: \begin{equation} \Phi_X ^{-1}(p)=\mu +\sigma {\sqrt {...