Questions tagged [stochastic]
The stochastic tag has no usage guidance.
70
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Can Heston volatility model be used to calculate VaR or CVaR?
I'm just a beginner and third-year statistics major student. Based on what I read in some journal, most common model that used to calculate VaR or CVAR is GARCH. Is there any possibility that I can ...
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vote
0
answers
144
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Expectation of the realized volatility
I was reading Zhang and Wang 2023 and I have some doubts regarding it. The realized Stochastic Volatility Model is expressed as follows:
$$\begin{matrix}
y_t = \exp \big( \frac{h_t}{2} \big) \...
- 37
3
votes
1
answer
215
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How does the inclusion of stochastic volatility in option pricing models impact the valuation of exotic options?
Been lurking this forum for quite some time and there’s this concept I can’t wrap my head around:
How does the inclusion of stochastic volatility in option pricing models impact the valuation of ...
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0
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43
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Transition probabilities of a stochastic volatility model
I have a stochastic volatility model for commodity price which follows an AR(1) process:
ln(pt ) − m = ρ (ln(pt−1) − m) + exp(σt)ut ut ∼ IID(0, 1)
σt − μ = ρσ(...
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0
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22
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Estimating expectation from a Markov chain process with AR(1) framework and stochastic volatility
I have a stochastic volatility model of a commodity price as follows:
...
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1
answer
160
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Necessary conditions to ensure that stochastic integral is a normal variable
Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
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4
votes
1
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227
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Calibration of Local or Stochastic Volatility Models to Prices vs Implied Volatilities
As the title suggests, what is the difference between calibrating an option pricing model (say the Heston model) to market option prices instead of computing their implied volatilities using Black-...
- 559
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Derive the Probability of Default (PD) of private companies with Merton Model
Do you know a well used method how to calculate the PD of private companies using the Merton Model.
The main challenges I am facing is to get the appropriate volatility of the assets and the drift.
...
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1
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1
answer
276
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4th Order Brownian Motion Martingale [closed]
I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
2
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What does a non-stochastic limiting shrinkage function mean?
I'm reading the paper "The Power of (Non-)Linear Shrinking: A Review and Guide to Covariance Matrix Estimation" by Ledoit and Wolf (2020). When a function that is used to transform the ...
2
votes
1
answer
253
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Simple Black-Scholes alternatives
I work at an accountancy firm and we use Black-Scholes to value equity in private companies that has option like features. The equity we typically value is akin to deeply out of the money European ...
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3
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1
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156
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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$$...
0
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1
answer
103
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Sum of discretely sampled BM
If an underlying follows lognormal GM with no drift $dS_t = \sigma S_t dW_t $ and $A_N = \Sigma_{i=1}^{N} S_{t_i}$. How to compute variance of $A_N$?
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203
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Change of measure for a stochastic process to be a martingale
$\text { Give a measure change so that } X_{t}=e^{B_{t}}\left(B_{t}-t / 2\right) \text { is a martingale, } 0 \leq t \leq T$
My attempt
Using Ito's lemma on $X_{t}$ we get:
$-\frac{e^{B t}}{2} d t+\...
- 383
1
vote
2
answers
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Taleb's Black-Swan: interpretation of the exponent
I am reading Taleb's "Black Swan" (revised 2020th edition). In chapter 16 "The Aesthetics of Randomness" he describes the meaning of the exponent in the context of extrapolation. ...
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votes
2
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853
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conditional expectation of stochastic integral
let $M_t$ be the following stochastic integral
$$
M_t = \int_0^t \sigma_s dW_s
$$
where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
- 700
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143
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Is it possibile to use Ito Formula here?
I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$.
Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
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vote
0
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Inflation Option Modelling Approaches
I am trying to come up with a simplistic inflation option model to get a sense of the materiality of some inflation-indexed contracts containing inflation guarantees. I have a stochastic nominal IR ...
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1
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1
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143
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Why does an autocall on a linear payoff have vega?
Consider a (stochastic) linear index, say $I(t)$, in that it grows at the risk free rate (with some volatility of course). There exists a maturity date $T$ on which I receive $I(T)$; however there is ...
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3
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What are the advantages and limitations of predicting future stock prices using stochastic differential equations?
Recently I came across the following stochastic differential equation that "predicts" the value of a given stock:
\begin{equation}
dS_t = \mu S_t dt + \sigma S_tdW_t \\
S_t(0) =S_0
\end{...
4
votes
0
answers
110
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mixing fractional Brownian motions
Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by
$$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$
where $W_t^{2}$ and $Z_t$ are independent of each other.
My question then: is there ...
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131
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Asset Pricing and Stochastic Discount Factor: Do well-informed investors only buy efficient portfolios?
I'm currently dealing with the following question:
In Asset Pricing, well-informed investors know about the concept of the efficient frontier. Does this mean that they only invest in portfolios that ...
0
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0
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64
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difficulty pricing options using stochastic volatility
can someone kindly explain why it was difficult to obtain an explicit formula for pricing options under stochastic volatility. Thanks alot.
0
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1
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113
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Find Arithmetic Brownian Motion's transition density
Consider the following stochastic differential equation, an Arithmetic Brownian Motion: 𝑑𝑆(𝑡) = 𝑟 𝑑𝑡 + 𝜎 𝑑𝑊(𝑡) . Find its solution, integrating from t to T, then find its transition density. ...
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3
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How can I prove that the solution to the Heston SDE is a Markov process?
Consider the Heston model expressed as
\begin{align}
dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\
dV_t &= \kappa(\theta - V_t)dt + \...
user39119
5
votes
2
answers
264
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Stochastic Calculus problem with three processes? (Itô calculus)
Can someone help me solve this following Itô Calculus problem?
Let $Z(t):= [B(t)*X(t)]/S(t)$
We have the following dynamics of B(t), X(t) and S(t):
$dS(t)=\alpha S(t)dt+\sigma S(t)dW(t)$
$dB(t)=rB(...
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1
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Independence of increments of the stochastic process $\frac{1}{t}\int_0^t u dW_u $
Let $X_t$ be a stochastic process such that
$$X_{t} =\frac{1}{t}\int_0^t u dW_u $$
I know that for
$$Y_{t} =\int_0^t u dW_u$$
$Y_t-Y_s$ is independent of $Y_s$ where $t>s$.
But is this also true ...
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1
vote
1
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325
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Heston model with jumps in both variance and underlying dynamic
How can I build on Matlab a Heston model using characteristic function adding jumps in both variance and underlying dynamic ? Suppose that the number of jumps is Poisson-distributed but the jump size ...
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1
answer
209
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Problem of stochastic differential equation (SDE)
Please help to answer this stochastic differential equation (SDE). Thank you very much.
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1
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231
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Evaluating the SDE $dX_t = t\,dS_t$
The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
1
vote
1
answer
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SDE Parameter Estimation
Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?"
Let's say $X_t$ follows the process:
$dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2 $
I think I've checked ...
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4
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1
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Invariance Scaling of Brownian Motion
Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
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1
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integration of squared brownian motion w.r.t time
How to prove $\int_0^1 B_s^2ds$ is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion.
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0
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The Ho-Lee Model (1986)
(My question)
I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M.
(Thank you for your ...
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1
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Expected value of exponential of hitting time of GBM
We have a stopping time
$$
\tau=\inf\{t\geq 0: S_0e^{\sigma B_t+(r-\sigma^2/2)t}=S^* \}
$$
where $S_0,\sigma,r,S^*$ are constants and $S^*<S_0$, and $B_t$ is a brownian motion. I wish to compute ...
- 53
2
votes
0
answers
120
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stochastic volatility and smile
Can we say that the volatility smile contain for sure stochastic volatility information ?
If yes why ?
Saying that BlackScholes does not explain the smile does not necessary mean there is an ...
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2
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529
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Detect trend of an index
My question is about determining the trend and it can break down to 3 parts.
To clarify, a trend in my point of view, and in simple form, is the last close at time t relative to its time reference, i....
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change of measure expectation
How to find expectation of this stochastic process? Also, to show that the expectation of a stochastic process expression [Xt - St] in one measure is equal to expectation of another expression (of the ...
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2
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Application of Itô's lemma - Forward process
How would be applied the itô's lemma in the following equation:
And we know that:
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0
answers
69
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Negative theta in Log-linear stochastic volatility model
I was asked to simulate the following geometric Brownian motion to get paths for the SPX stock price. the process follows a Log-Linear stochastic volatility.
$dS_t = \mu S_tdt+e^VS_tdW_1 $
where ...
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0
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A Soft Problem: Application of Stochastic Differential Equations in Hilbert Space Beyond HJM Interest Rate Model
I am reading books on stochastic differential equations (SDE) in Hilbert spaces. It seems that every book just discusses HJM interest rate model as an application when discussing financial ...
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1
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Good references on Heston Model?
I am looking for good bibliographic references on Heston Model and Stochastic volatility models in general.
Does anyone know any good introductory/intermediate references on this topic?
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1
answer
347
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CIR calibration
I'm using a CIR short rate model to forecast interest rate paths.
I've been thinking and also searching online about different ways of estimating its parameters (a, b and sigma). While there are a ...
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2
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4k
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Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]
My question is about a stochastic integral of brownian motion w.r.t time.
Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this:
\begin{eqnarray}
X(t)=\int_{0}^t dt' W(t').
\...
1
vote
0
answers
409
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Forward implied vol vs Instantaneous vol
In the Discrete Stochastic Implied Volatility Model which is from the standard Heston Model, the model shows the evolution of forward implied volatilities with time.
I thought forward implied ...
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vote
0
answers
47
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Total Variance of an asset in case of stochastic rates
Let's suppose the underlying S follows a BS dynamic with the drift being the short rate that follows a short dynamic model.
the "local volatility" of the equity should be the implied volatility from ...
- 173
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vote
3
answers
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Heging against stochastic interest rate
I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash.
I use the following Black-Scholes formula
$$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]...
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0
answers
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$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
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votes
2
answers
1k
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What is meant by innovations in volatility?
I am currently reading about stocks with "high sensitivity to innovations in aggregate volatility". I am not a native speaker so this might be a trivial question, but I truly cannot find an answer ...
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1
answer
4k
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Expected Value of Stochastic Process
Given the following stochastic process:
$$ dX = a(X,t)dt + b(X,t)dz $$
where:
$$ dz = A \sqrt{dt}$$
and $A$ is a random variable with mean zero and variance $1$.
Is there a way to calculate the ...
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