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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Standard deviation of the difference between a time series and its EMA?

I have a time series $Q={\{q_t\}}$ of known standard deviation $\sigma$, and its EMA of parameter $\alpha$ : $\{EMA_t(\alpha)\}$. My question is : I'm looking for a formula that would give the ...
1 vote
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34 views

Can I extend the private information model of Kyle in in a continuous analogue, e.g. the Ornstein–Uhlenbeck process?

Taking into account an old post of maths.stackexchange, I recall the following: On the one hand, we know that the Ornstein–Uhlenbeck process can also be considered as the continuous-time analogue of ...
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Boundary conditions for put option seems wrong when solving three-dimensional PDE

I am solving the the Heston-Hull-White and Heston-CIR hybrid models using a finite difference approach. The hybrid models is given by $$dS=(r-q) S dt + \sqrt{v} S dZ_1$$ $$dv = \kappa_v (\theta_v - v) ...
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Calibrating Hull-White model using historical data

I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates. $$dr(t) = (\theta(t) - \alpha ...
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41 views

Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model

I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
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Feymann Kac pde with correlated process

I have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
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2 votes
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Optimal consumption process [Munk (2011)]

I'm trying to solve problem 4.4 in Munk (2011). The problem is as follows: Assume the market is complete and $\xi = (\xi_{t})$ is the unique state-price deflator. Present value of any consumption ...
1 vote
1 answer
53 views

Dynamics of discounted prices (multi-dimensional)

My objective is to find the dynamics of the discounted prices, given by $\mathbf{y}_{t} = \mathbf{P}_{t}\mathrm{e}^{-\int^{t}_{0} r_{s} ds}$. I know the dynamics should be $d\mathbf{y}_{t} = \mathrm{...
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Pricing Leveraged ETF option based on base ETF

I am following along with the paper linked here: https://math.nyu.edu/~avellane/thesis_Zhang.pdf . In section 4.4, equation (4.4.2) makes the claim: $$\sigma(k) = |\beta|\sigma_s(S_0k^*)$$ where: $$...
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Any innovations in mathematical processes behind option pricing models?

I am working on my thesis about option pricing models beyond classical Black-Scholes Model by looking for some recent innovations on mathematical processes behind the pricing structures. By that I ...
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Munk (2011) exercise 3.6

I'm trying to solve the exercise in Munk (2011). The exercise reads: "Find the dynamics of the process: $\xi^{\lambda}_{t} = \exp\left\{-\int^{t}_{0} \lambda_{s} dz_{s} - \frac{1}{2}\int^{t}_{0} \...
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Analytical expression for SDE

I'm trying to find an analytical expression for the following. Suppose $X$ is a geometric Brownian motion, such that: $dX_{t} = \mu X_{t} dt + \sigma X_{t} dW_{t}$. Suppose furthermore, that the ...
2 votes
1 answer
106 views

Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)

I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following $$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
3 votes
1 answer
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Derivations of the pricing PDE for the Heston-Hull-White or Heston-CIR models

Consider the hybrid model given by $$dS=(r-q) S dt + \sqrt{v} S dZ_1$$ $$dv = \kappa_v (\theta_v - v) dt + \sigma_v \sqrt{v} dZ_2$$ $$dr = \kappa_r (\theta_r - r) dt + \sigma_r r^p dZ_3$$ with ...
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1 vote
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Quantile function for fractional Brownian motion (fBm)

If anyone could help me to understand if it is possible calculate the quantile function for fBm? I’ve checked several papers([1],[2],[3]), and although several works stated that it is centralised ...
2 votes
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151 views

If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows

From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
4 votes
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134 views

optimal stopping time problem

I'm currently reading a paper (The Optimal Stopping Time for Selling an Asset When It Is Uncertain Whether the Price Process Is Increasing or Decreasing, American Journal of Operations Research, March ...
4 votes
1 answer
243 views

Simulating Iterated Brownian Motions

I was going through an interesting article (https://arxiv.org/pdf/1112.3776.pdf) while I was trying to read about subordinated processes. I wanted to simulate subordinated processes (in R or python) ...
0 votes
1 answer
187 views

Deriving the stochastic process for a dividend-yielding stock (under Black-Scholes assumptions)

In order to derive the Black-Scholes equation for a stock $S(t)$ yielding dividends at the continuous rate $d$ $$ S(t) = S_0 e^{(\mu - d - \frac{\sigma^2}{2})t + \sigma \sqrt{t} N(0,1)} \text{,} $$ M. ...
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1 answer
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European option with payoff $X_T^2$ [closed]

I have been ask to price a European option with payoff $H(X_T,T) = X_T^2$ using the equivalent martingale measure (EMM). For this I used the process: \begin{equation} dX_t = r X_t dt + \sigma X_t d\...
2 votes
1 answer
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Deriving the variance of G2++ Model

I'm studying G2++ Model in Brigo(2007)'s book. The model constructed as follows, $$ r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\ $$ with the dynamics of $dx(t)$ and $dy(t)$ described by: \begin{align}...
2 votes
1 answer
400 views

Heston Model python MC simulation

I have this exercise. $\\\\$ Look for realistic values ​​of the parameters and calculate the price of a European Call with maturity $T = 0.5$ and $S_0 = 1$ for the strike values $​​K = 0.5,0.6, ......,...
1 vote
1 answer
243 views

Euler Discretization python code

Write the Euler discretization of the 1-dimensional stochastic equation $dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$ For this part I would say all right because it is a purely ...
8 votes
1 answer
200 views

If the spread between two assets is an OU process, what processes do the two assets follow?

Let $(\Omega,\mathcal{F}, \mathbb{P}, (\mathcal{F}_{t})_{t\geq0})$ be a filtered probability space. Furthemore, let $(S_{t}^{1},S_{t}^{2})_{t\geq0}$ be two assets (adapted to filtration, etc). Define $...
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95 views

Hitting time of Brownian motion with drift using Feynman-Kac

I was studying this question from "A Practical Guide to Quantitative Finance Interviews" and was having some trouble understanding one solution. Please advise if misunderstood anything or if ...
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42 views

Does the Lévy characterization imply that the price process of any asset is a Brownian motion?

While studying Brownian motion applied to mathematical finance, I came across these lecture notes by prof Steve Lalley. In the prologue, he gives this explanation for the occurrence of Brownian motion ...
2 votes
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58 views

NFT Floor Price

I'm interested in modeling NFT Floor Price. Specifically, I'm trying to answer the question: Given current bid-ask info on an NFT collection, what is the probability distribution of the lowest ask ...
0 votes
1 answer
92 views

Implication of unique risk neutral measure

I'm reading Shreve Stochastic Calculus II, theorem 5.4.9 (Second fundamental theorem of asset pricing), This is the part that confuses me : suppose there is only one risk-neutral measure. This ...
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153 views

Pricing a put-option in the Heston Model

Assume the Heston Model with dynamics under the martingale measure $Q$ given by \begin{align} dS_t &= (r-q)S_t dt + \sqrt{v_t}S_tdW_{1,t}^Q\\ dv_t &= \kappa(\theta-v_t)dt + \sigma\sqrt{v_t}dW_{...
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2 answers
405 views

Transformation of local volatility model

Assume we have an SDE $$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$ where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
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GARCH option pricing

I have been trying to implement GARCH(1,1) model for pricing call options. Suppose I have calibrated Garch(1,1) model for modelling the conditional volatility using the historical data of an equity ...
0 votes
1 answer
95 views

Why we introduce correlations between Wiener processes? [closed]

Wiener processes are used to model various assets, and I wonder why we are introducing correlations between the Wiener processes and what is the interpretation? Because when the correlations between ...
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1 answer
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Pairs trading using dynamic hedge ratio - how to tell if stationarity of spread is due to genuine cointegration or shifting of hedge ratio?

I'm very new to pairs trading, and am trying it out on a few dozen pairs. It seems very natural to me to use a dynamic hedge ratio, as it seems likely that the ratio will move over time. To accomplish ...
1 vote
0 answers
94 views

Pairs Trading - isn't any spread stationary if your rolling lin-reg window is small enough?

I have a set of 7 assets, and I have run an ADF test on all possible pair-spreads to find possible pair strategies. I am creating the spreads using a rolling window in which I run linear regression to ...
-1 votes
1 answer
99 views

Integration of exponential raised with Brownian Motion wrt the Brownian Motion

I have to derive several things for my thesis, however, I have the following expression: $$ \int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t} $$ Does anyone know what the solution for this is? Kind regards.
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1 vote
1 answer
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Calculating Expectation of Stochastic Volatility

I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM STRUCTURE OF OPTION IMPLIED VOLATILITY AND VOLATILITY COMPONENTS by Guo, Han, and Zhao. I don't understand why the above equations ...
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1 vote
1 answer
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What kind of interpolation is this?

I have Wiener process $W_t=\int_0^t\sigma(t)dB(t)$ where $B(t)$ - Brownian Motion and $\sigma(t)$ - piecewise constant function. I also take $t_k<t<t_{k+1}$ where I know the values of $W_{t_k}$ ...
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Simulating sum of squared brownian motions process

I'm trying to simulate the following stochastic process: \begin{equation} R_t = \sum_{i=1}^nB_{i,t}^2 \end{equation} which has the following dynamics: \begin{equation} \begin{aligned} dR_t = \sum_{...
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63 views

Optimal Entry, Exit, And Stop Loss From Historical Stock Data

I'm trying to build a system that recommends stock trades. My goal is calculate optimal values for the following: Entry Parameter: expressed as a percentage change downwards from the opening price. ...
1 vote
0 answers
106 views

Differential vs. derivative in the Vasicek model [closed]

Can anyone help me in understanding how we get the line I have marked with a red arrow? I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
0 votes
1 answer
110 views

Separating jumps and diffusion

I want to model energy prices. I have two markets, lets say market 1 and 2. Market 1 is continuously traded, and I will assume it follows brownian motion. So the value of the asset could be defined ...
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Asset pricing using a two period binomial tree

Recall that in the binomial model the risk-neutral probability for the price going up is given by p = 1+r−d / u−d where u > 1 and d < 1 specify the possible price jumps in the risky asset and r ...
7 votes
3 answers
694 views

Why does the diffusion term remain the same when we change pricing measure?

Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion In plenty of interest rate examples, I have ...
2 votes
0 answers
195 views

How do you hedge volatility risk?

Suppose I model an asset $S_1(t)$ under a stochastic volatility model. To price an option on $S_1$, I must assume the existence of an asset $S_2$ that is used to hedge against changes in the ...
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2 answers
181 views

Expectation of functions with Brownian Motion embedded

Trying to solve a problem set with: Let $W_t$ be a Brownian Motion and $X_t = e^{izW_t}$ where $z$ is real, $i = \sqrt{-1}$. I need to find $\mathbb{E}\left(X_t\right)$... I am a bit stuck. I have ...
2 votes
1 answer
203 views

Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
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3 votes
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94 views

How to merge ML-based $\alpha$-signal with stochastic control approach?

I'm having a hypothetical situation where I have a set of ML-based alpha signals $\{\alpha_i\}_{i=1}^{N}$ that describe a different states of order book - imbalances, order flow, spread properties etc....
1 vote
1 answer
289 views

How to deal with negative intercept terms on GJR-GARCH(1,1) model?

Recently, I have been studying the relationship between COVID-19 and stock returns using a GJR form of threshold ARCH model. However, I got some unusual estimation results I can't figure out whether ...
1 vote
2 answers
288 views

On moving Linear Correlation (rolling correlation)

Let's say I have two random variables $X$ and $Y$ which each represents the daily returns of two given stocks. I can easy calculate their (total) correlation by finding their covariance matrix $\Sigma[...
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1 answer
249 views

Jump Diffusion Process question

I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
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