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Questions tagged [stochastic-processes]

stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

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55 views

Showing that a market model has arbitrage and describing martingales

This is an exercise which I came upon while studying an introduction to financial mathematics. Exercise : Consider the finite sample space $\Omega = \{\omega_1,\omega_2,\omega_3\}$ and let $\...
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0answers
27 views

How can I estimate the time-varying θ term in the Hull-White one factor model?

I am trying to simulate the prices of bond indexes (e.g. Barclays Aggregate, IBOXX sovereign, IBOXX corporates) using Monte Carlo assuming that they follow the SDE given by the Hull-White model (one-...
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28 views

Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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87 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
53 views

Differential product Correlated processes

I am trying to derive the differential of the product of two processes, but I got stuck. This is what I have until now: We have the following two stochastic processes: $dX_t= \mu_t dt +\sigma_t dW_t$...
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1answer
109 views

Question about quadratic form of f* in the Continuous Kelly Criterion

I am trying to follow the Optimal Kelly derivation on Wikipedia for two continuous assets: one risky and one risk-free. The derivation begins by assuming that the risky assets follows a GBM (a ...
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1answer
75 views

Geometric brownian motion and sudden price drops

Simple question of a curious person: One can say that prices tend to rise "slowly" and drop "all of a sudden". Still, they are a geometric composition upon random returns. As I understand, this is ...
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31 views

Euler discretization with jumps

There is a process $B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$, where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$. ${N_t}$ is a counting process ...
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28 views

Determining the Relationship Between Monte Carlo Breaks and Model Volatility

I'm looking for a statistical test to understand the relationship (if any) between the model volatilities of a stochastic process, and the occurrence of 'break', defined as the instance when an ...
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1answer
71 views

Quadratic variation of an integral of a function of a Brownian motion

I'm asked to find the quadratic variation of the integral $\int_{0}^{t} W_s^2 ds$.
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37 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
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91 views

Need help to interpret the definition of a diffusion process

https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130 In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
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2answers
96 views

How to show that SABR is log-normal for $\beta=1$ and normal for $\beta=0$?

For $\beta = 1$ SABR is log-normally distributed and for for $\beta = 0$ SABR is normally distributed. This is a very common property mentioned in almost every paper about SABR. But I can't find the ...
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72 views

Why is the timings between trades of SPY precisely poised at criticality ? Can this fact be used for prediction?

Let's say we have a point process consisting of the times between trades of SPY for one particular trading day. Empirically, the auto-correlation never dies out to 0 and due to results in Long range ...
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4answers
292 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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42 views

Extension of HJM to multiple factors

The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate- $$df(t,T)=\sigma(t,T)\int_0^T\...
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1answer
76 views

Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ...
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48 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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1answer
96 views

Bond SDE under its own forward measure

I am trying to write the SDE for a forward bond, $dP(t,T_1,T_2)$, under the $T_1$-Forward measure, $Q_{T_1}$. I can easily do this by: Writing the equation of $dP(t,T_1)$ and $dP(t,T_2)$ under the ...
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0answers
45 views

CreditGrades model calibration and initial values

Currently doing a project on structural models, and I want to apply the CreditGrades model. My question is what values are the parameters going to take, in order to update the implied probability of ...
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41 views

Debt per share in CreditGrades model

In order to specify the debt per share in the CreditGrades model one has to specify the liabilities to be included from the firm's balance sheet. I do not have access to the technical document ...
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1answer
198 views

Modelling EUR/USD rate with Ornstein-Uhlenbeck model

I have a data set of daily EUR/USD rate for time period 2000-2018. My goal is to model future behaviour of this financial time series using Ornstein-Uhlenbeck model: $$d X_t = \alpha (\theta - X_t) ...
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38 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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1answer
119 views

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Bernal 2016 says that the solution of $$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$ equals $$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
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46 views

European Call Option Modelling under 2 factor Hull White interest rates

I have modelled the yield curve through the two factor Hull White Model. Now I want to implement in Matlab the price development of a ATM-Call-Option (European). Has someone an idea how to combine ...
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101 views

Calibration of Cox-Ingersoll-Ross process that hits zero (Feller condition violation)

I'm considering a Cox-Ingersoll-Ross (CIR) process $$ dx_{t} = \alpha\left(\theta - x_{t}\right)dt + \sigma \sqrt{x_{t}}\,dW_{t}\,,\qquad \alpha,\beta,\sigma > 0 $$ which by assumption has $2\...
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2answers
199 views

Does numeraire have to be a tradable asset

I thought we create replicating portfolios using underlying and the numeraire i.e. the numeraire has to be a tradable asset (assuming simple binomial model). But I have seen some examples which ...
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0answers
36 views

reconciling arithmetic and geometric compounding

I have just been through 4 papers that make all sorts of clever claims about the 'alternate universes' of arithmetic returns and geometric returns, how thr twain shall never meet, and how they are ...
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0answers
27 views

Squaring lognormal compounding with linear addition of normal returns

Let’s say we start with $100 and invest it for 20 years in stocks and want to predict its terminal value as a random variable (RV). And let’s assume average yearly returns are 10% and volatility is ...
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0answers
36 views

autocorrelation function of the trending OU process

What is the autocorrelation function of the trending Ornstein-Uhlenbeck (OU) process? First, the OU process $dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $ generates coloured noise ...
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31 views

Literature recommendation subordinator models

I'm looking for relevant papers covering subordinator models for stock price modelling. I have alreay read the paper 'A Subordinated Stochastic Process Model with Finite Variance for Speculative ...
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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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1answer
87 views

Random Walk of N Correlated Assets

I am trying to value an option on N assets, say $S^1, S^2,..., S^N$ that expires in $\Delta T$ years using Monte Carlo simulation. I have read many sources that state I should use the following ...
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3answers
235 views

Finding distribution of $\int_0 ^T uW_u du$

I would like to find the distribution of $\int_0 ^T uW_u du$ where $(W_u)_{u\geq0}$ is the Brownian motion. What I have tried: $$\int_0 ^T uW_u du = \int_0 ^T B_udu - \int_0^T \int_0^tB_sdsdt$$ by ...
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1answer
83 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
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154 views

Gyöngy Theorem Proof

Can you please point me to a publicly available text that discusses the proof for the Gyöngy Theorem? Gyöngy, I. (1986), “Mimicking the One-Dimensional Marginal Distributions of Processes Having an ...
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1answer
72 views

Calculate $Cov(e^ {B_t} ,e^{B_s})$

Let $(B_t)_{t \geq0} $ be a Brownian Motion. Calculate $Cov(e^ {B_t} ,e^{B_s})$ I would verify the following solution which the result looks a bit weird. My solution: let $0 \leq s \leq t$. $$Cov(e^ {...
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3answers
162 views

What Process Does the Market Follow in the CAPM?

Consider a multiperiod version of the CAPM $$E_t[r_{i,t+1}-r_{f,t+1}]=\beta_{i,t}E_t[r_{m,t+1}-r_{f,t+1}]$$ where $E_t[r_{i,t+1}-r_{f,t+1}]$ is the time $t$ expectation of the time $t+1$ excess ...
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0answers
39 views

Why is STO (Stochastic Oscillator) typically calculated over 14-periods (hour, day, week)?

I've seen the Stochastic Oscillator calculated over 14-periods everywhere. Whether in hours, days, weeks months. I'm not sure if it's obvious, but why is this?
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59 views

Option pricing with negative strikes

I am valuing corporate securities with call option pricing models. In this scenario, it seems possible to have negative strike prices if we assume that some assets or revenues do not have a diffusion ...
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0answers
57 views

hybrid models with FX

I am working with an hybrid model: $S_f(t)$: is a foreign Equity in a foreign currency f. S follows a BlackScholes model: $dS_f(t) = S_f(t) r_f dt + \sigma_1 S_f(t) dW_1(t) $ $r_f$ follows a hull ...
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47 views

Are there any jump diffusion models(for option pricing) with jump parameters in diffusion coefficient

In Merton model jump parameters located in drift: $$\frac{dS_t}{S_t}= \left(\mu-\lambda k\right)dt + \sigma_tdW_t + \left(y_t-1\right)dN_t $$ where \begin{equation} k=E[y_t-1]=e^{q+\frac{1}{2}\delta^...
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18 views

Difference between Standard VaR and VaR with partial set of Risk Factors

Ciao, I'm working on VaR and Expected Shortfall and this question came out. For a given portfolio VaR can be computed w.r.t. all the risk factors or just for a subset. Infact you can decide to 'freeze'...
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1answer
86 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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58 views

Does pricing contingent claims under the EMM require us to define the distribution?

I am familiar with martingale pricing as primarily a notational abstraction which allows us to price contingent claims on $X_\tau$ by its conditional expectation. Usually, we interpret this to mean ...
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1answer
56 views

How to simulate a path through its solution and conditional expectation / variance

Hi I want to simulate in Matlab the following stochastic integral: $ x(t) = x(s) e^{-a(t-s)} + \sigma \int_s^t e^{-a(t-u)} dW_1(u)$ with $E[x(t) \vert F_s] = x(s) e^{-a(t-s)}$ $Var[x(t) \vert ...
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45 views

Monte Carlo for constructing the Vol smile in SABR

My purpose is to construct the vol smile using Monte Carlo simulation and not market data. When I search for Monte Carlot methods for SABR I often see the Euler scheme as given for instance in these ...
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1answer
220 views

Modelling interest rate

Hi I want to model two stochastic integrals in Matlab, which is given by $ x(t) = x(s) e^{-a(t-s)} + \sigma \int_s^t e^{-a(t-u)} dW_1(u)$ $y(t) = y(s) e^{-b(t-s)} + \eta \int_s^t e^{-b(t-u)} dW_2(...
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2answers
146 views

Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?

The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
2
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1answer
78 views

How to derive the dynamic of the log forward price?

I have the following Schwartz model: $$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$ $$X_t=\ln S_t$$ $$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$ with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$ $$F_t(T)= \exp\...