Questions tagged [stochastic]

The tag has no usage guidance.

Filter by
Sorted by
Tagged with
4 votes
1 answer
135 views

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 ...
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 $A_N$?
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+\...
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 $...
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 ...
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 ...
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. ...
  • 1
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 + \...
user avatar
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 ...
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 ...
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.
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 ...
4 votes
1 answer
194 views

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....
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 ...
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 ...
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 ...
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?
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 ...
  • 11
0 votes
2 answers
3k views

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)]...
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 ...
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^{-...
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