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Questions tagged [stochastic]

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5
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1answer
89 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 ...
2
votes
0answers
45 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 ...
-1
votes
2answers
235 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
0answers
64 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
2answers
186 views

Application of Itô's lemma - Forward process

How would be applied the itô's lemma in the following equation: And we know that:
1
vote
0answers
35 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
0answers
31 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
1answer
109 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
1answer
121 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 ...
0
votes
2answers
601 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
0answers
130 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 ...
1
vote
0answers
40 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 ...
1
vote
2answers
184 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
0answers
64 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 ...
2
votes
2answers
316 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 ...
0
votes
1answer
1k 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 ...
2
votes
1answer
160 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 ...
3
votes
2answers
383 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
1answer
280 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
2answers
252 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 ...
1
vote
1answer
170 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 ...
1
vote
1answer
133 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 ...
1
vote
0answers
198 views

Is there a way I could find a matlab or R code to estimate a regime switching stochastic volatility model (discrete)?

Sorry to bother you with this request but, does anyone know where I could find a matlab or R code to estimate a regime switching stochastic volatility model (discrete)? Thank you very much.
2
votes
1answer
102 views

Merton portfolio allocation problem proportions/weights >1 or <0?

In the classical Merton portfolio problem, lets assume: $$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t) = \pi_t X_t (\mu dt +\sigma dW_t) $$ ie: zero interest rates for simplicity. ...
2
votes
1answer
148 views

Problem with derivating integral

I have a doubt : I know that if $x_{t}=\int_{0}^{t}\gamma(s)dW_{s}$ (with $W_{s}$ a brownian motion), we have : $dx_{t}=\gamma(t)dW_{t}$ What about if $x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}$. Do I have ...
3
votes
3answers
187 views

existence of implied volatility

I read a book where it was written : 1/ "implied volatility is the market's consensus on the volatility of the asset between now and the maturity of the option". -> Could someone explain me this ...
11
votes
4answers
4k views

Is it really possible to create a robust algorithmic trading strategy for intraday trading?

I'm an engineer doing academic research for my master thesis in the area of quantitative finance, basically the purpose is to study the possibility to create an intraday-trading algorithm. I've tried ...
4
votes
2answers
877 views

Intergral of Brownian motion w.r.t. Brownian motion

I don't understand why $S$ (highlight on picture), I learned $$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t $$ everyone please explain for me. Thank you
5
votes
1answer
116 views

Why $W_{t}^3$ is not a martigale?(by Definition)

If $W_t$ be a wiener process then,how can i show that $W_{t}^{3}$ is not a martingale by definition?
3
votes
1answer
138 views

stochastic calculus - brownian motion

I don't know how to prove this : let be $X_t = \int_{0}^{t}\sigma_{u}dW_{u}$ where $\sigma_{t}$ is a predictable process. If $|\sigma_{t}| = c$ a.s. how can I prove that $X_{t}=c*\beta_{t}$ (...
1
vote
1answer
185 views

stochastic calculus - Itô formula?

I encounter a problem in the proof below: I don't know how to proove the first line in yellow (cf below): it makes me think about the Itô formula a lot I don't undertand the deduction (ok $\gamma^{\...
2
votes
1answer
60 views

equality in distribution

I encounter the following problem : I have the equality in distribution: for all $\lambda >0, ((1/\lambda)*\int_{0}^{\lambda t}\sigma_{u}^{2}du,t\geq0)=(\int_{0}^{t}\sigma_{u}^{2}du,t\geq0)$ ...
2
votes
1answer
88 views

forward option, stochastic calculus

I encounter a problem to understand this: The price of a forward option is : $C(K,t,T)=\mathbb{E}[((S_{T}/S_{t})-K)+]$ OK The option should only depend on $T-t$ because the yield randomness (for a ...
1
vote
3answers
4k views

How to differentiate a brownian motion?

By definition a wiener process cannot be differentiated. But when we use Ito's lemma on $F = X^2$, where X is wiener process we have total change in $$dF = 2XdX + dt$$ How can we calculate $\...
0
votes
0answers
137 views

Stochastic Volatility for Stocks, FTSE

Can someone help me with calculating Stochastic Volatility (of stocks and options) using SAS or R or Matlab please? I am new to SAS and I am trying to use Heston model, White-Hull model or any other ...
1
vote
0answers
62 views

What are the estimation methods for SV models?

I want to know about some methods like Methods-of-Moments, Quasi-Maximum Likelihood method, Baysian methods using Markov Chain Monte Carlo methods. Is there any reference to have an idea of these ...