Questions tagged [stochastic]
The stochastic tag has no usage guidance.
64
questions
4
votes
1
answer
135
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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-...
- 41
0
votes
0
answers
45
views
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.
...
- 1
1
vote
1
answer
160
views
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
votes
0
answers
37
views
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
203
views
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 ...
- 41
3
votes
1
answer
139
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
$$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
0
votes
1
answer
103
views
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 $A_N$?
- 89
3
votes
2
answers
193
views
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+\...
- 373
1
vote
2
answers
232
views
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. ...
- 183
10
votes
2
answers
615
views
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 $...
- 690
2
votes
0
answers
141
views
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 ...
- 33
1
vote
0
answers
81
views
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 ...
- 111
1
vote
1
answer
117
views
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 ...
- 1,634
2
votes
3
answers
910
views
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
104
views
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 ...
- 288
0
votes
0
answers
105
views
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
votes
0
answers
63
views
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
votes
1
answer
76
views
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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1
vote
3
answers
578
views
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
241
views
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(...
- 311
2
votes
1
answer
110
views
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 ...
- 65
1
vote
1
answer
293
views
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 ...
- 11
2
votes
1
answer
184
views
Problem of stochastic differential equation (SDE)
Please help to answer this stochastic differential equation (SDE). Thank you very much.
- 29
5
votes
1
answer
219
views
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
123
views
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 ...
- 493
4
votes
1
answer
194
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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 ...
- 53
1
vote
1
answer
663
views
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.
- 53
2
votes
0
answers
175
views
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 ...
- 179
5
votes
1
answer
567
views
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
109
views
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 ...
- 617
-1
votes
2
answers
483
views
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....
- 57
1
vote
1
answer
158
views
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 ...
- 11
2
votes
2
answers
630
views
Application of Itô's lemma - Forward process
How would be applied the itô's lemma in the following equation:
And we know that:
- 98
2
votes
0
answers
65
views
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 ...
- 51
2
votes
0
answers
46
views
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 ...
- 53
1
vote
1
answer
380
views
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?
- 53
1
vote
1
answer
328
views
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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0
votes
2
answers
3k
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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
390
views
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 ...
- 11
1
vote
0
answers
45
views
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
1
vote
3
answers
392
views
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
$$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]...
- 73
2
votes
0
answers
69
views
$\int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2}$ normally-distributed?
I have came across the following stochastic integrals:
$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$
which was claimed to be standard normally distributed ($W_x$ and $W_y$ are ...
- 21
4
votes
2
answers
1k
views
What is meant by innovations in volatility?
I am currently reading about stocks with "high sensitivity to innovations in aggregate volatility". I am not a native speaker so this might be a trivial question, but I truly cannot find an answer ...
- 41
2
votes
1
answer
3k
views
Expected Value of Stochastic Process
Given the following stochastic process:
$$ 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 ...
- 83
2
votes
1
answer
242
views
Calibration of stochastic volatility models
Which are good references to know about different calibration methods for stochastic volatility models such as Heston? I know that there are a lot of way of carrying this task out and I was just ...
- 61
2
votes
2
answers
632
views
Hawkes process intensity solution
Hail to all,
I am struggling to solve the following SDE for intensity:
$d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t $
I know to expect the solution in the form of
$\lambda_t = c(0)e^{-...
- 746
3
votes
1
answer
309
views
FX options pricing exchange rate regimes
how can we estimate the impact of a exchange rate regime switch ( from fixed to float) on the options prices i'm talking about the moroccan case (EUR/MAD USD/MAD) options OTC , is there any stochastic ...
8
votes
2
answers
319
views
Why won't Bjork ever show that the integrability condition is satisfied?
A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0.
This is based on a result presented ...
- 81
1
vote
1
answer
307
views
Question on implied vol (surface) and strikes
there have been loads of papers on skews ATM / OTM, volatility premium and such. Lots of explanations for why iv is different on same stock with different strikes focused on preference of informed ...
- 11
1
vote
1
answer
263
views
Piecewise Ito formula
Usually Ito's lemma is stated for $C^{1,2}(\mathbb{R}^{d+1},\mathbb{R})$ functions.
My question is does Ito still hold if the domain is restricted. That is if the semi-martingale $Z_t$ is only ...
- 353