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22
votes
3answers
6k views

What is a stationary process?

How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?
22
votes
1answer
4k views

What is the role of stochastic calculus in day-to-day trading?

I work with practical, day-to-day trading: just making money. One of my small clients recently hired a smart, new MFE. We discussed potential trading strategies for a long time. Finally, he expressed ...
13
votes
0answers
631 views

Law of an integrated CIR Process as sum of Independent Random Variables

It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as : $$dY_t= \kappa(\theta -Y_t)dt+ ...
11
votes
4answers
834 views

Solving Path Integral Problem in Quantitative Finance using Computer

I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again. We know that for option value calculation, path ...
10
votes
3answers
774 views

Deterministic interpretation of stochastic differential equation

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by ...
10
votes
2answers
727 views

Missing step in stock price movement equations

Assuming a naive stochastic process for modelling movements in stock prices we have: $dS = \mu S dt + \sigma S \sqrt{dt}$ where S = Stock Price, t = time, mu is a drift constant and sigma is a ...
9
votes
2answers
1k views

Why Ito calculus?

Coming from physics, I am used to the fact that the Ito interpretation of most natural stochastic equations is wrong, and one should be using Stratonovich calculus instead (of course they are ...
9
votes
2answers
4k views

Worked examples of applying Ito's lemma

In most textbooks Ito's lemma is derived (on different levels of technicality depending on the intended audience) and then only the classic examples of Geometric Brownian motion and the Black-Scholes ...
9
votes
0answers
482 views

Probability distribution of maximum value of binary option?

A binary option with payout \$0/\$100 is trading at \$30 with 12 hours to expiration. Assuming the underlying follows a geometric Brownian motion (hence volatility remains constant), what ...
8
votes
2answers
2k views

What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?

I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process. $$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$ Is anyone able to calculate the mean ...
8
votes
2answers
334 views

Why does Black-Scholes equation hold on continuation region of American Option?

Explanation for Put Option: $ \frac{\partial V}{\partial t}+ \mathcal{L}_{BS} (V) = 0 $, where $\mathcal{L}_{BS} (V) = \frac{1}{2} \sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r-q) S ...
8
votes
2answers
510 views

Change of measure discrete time

Suppose I have a random walk $X_{n+1} = X_n+A_n$ where $A_n$ is an iid sequence, $\mathsf EA_n = A>0$. How to construct a martingale measure for this case?
7
votes
2answers
327 views

Obtaining characteristics of stochastic model solution

I want to use the following stochastic model $$\frac{\mathrm{d}S_{t}}{ S_{t}} = k(\theta - \ln S_{t}) \mathrm{d}t + \sigma\mathrm{d}W_{t}\quad (1)$$ using the change in variable $Z_t=ln(S_t)$ we ...
7
votes
1answer
308 views

Girsanov Theorem and Quadratic Variation

Girsanov theorem seems to have many different forms. I've got a problem matching the form in wiki to the one in Shreve's book, due to the difficulty of quadratic variation calculation. Below is the ...
7
votes
6answers
954 views

Self-financing and Black-Scholes-Merton formula

Self-financing is an important concept in financial product replicating, normally used in pricing. I read about several ways to derive Black-Scholes-Merton (BSM) formula. Seems some approaches ...
7
votes
2answers
286 views

Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?

I'm trying to derive the valuation equation under a general stochastic volatility model. What one can read in the literature is the following reasoning: One considers a replicating self-financing ...
6
votes
2answers
879 views

Derivation of Ito's Lemma

My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
6
votes
2answers
965 views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
6
votes
2answers
282 views

Why is this stochastic integral a martingale?

Suppose that: $W^*_t$ is a Wiener process under probability measure $\mathbb{P}^*$ and; $\tilde{S}_t=S_0+\sigma\int_{0}^{t}S(u)dW^*_s$. In my lecture notes, it says that $\tilde{S}_t$ is a ...
6
votes
2answers
2k views

How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?

I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
6
votes
1answer
842 views

How to perform basic integrations with the Ito integral?

From the text book Quantitative Finance for Physicists: An Introduction (Academic Press Advanced Finance) I have this excercise: Prove that $$ ...
6
votes
2answers
242 views

Filtration and measure change

I asked this question in math stackexchange but to no avail. So i'm trying the luck here. I'm reading Steven E. Shreve's "Stochastic calculus for finance II", and find myself not really understand ...
6
votes
1answer
265 views

Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
6
votes
2answers
866 views

How do practitioners use the Malliavin calculus (if at all)?

This question is inspired by the remark due to Vladimir Piterbarg made in a related thread on Wilmott back in 2004: Not to be a party-pooper, but Malliavin calculus is essentially useless in ...
6
votes
1answer
214 views

What is a good Computer Algebra System for financial engineering?

I would like to know if there exists some computer algebra systems adapted to calculate pricing based on particular models, i.e. pricing YoY Inflation Swap under Jarrow Yildirim Model. I know that ...
6
votes
2answers
132 views

Deriving the definition of stochastic integrals with respect to Ito processes from first principles

When I first encountered the definition of integrals with respect to Ito processes (Shreve's Stochastic Calculus for Finance Vol II), I didn't think twice. However, I wanted to see if the definition ...
6
votes
0answers
278 views

Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \begin{equation} \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} ...
5
votes
4answers
465 views

Geometric Brownian motion - Volatility Interpretation (in the drift term)

A Geometric Brownian motion satisfying the SDE $dS_t = rS_t dt+\sigma S_t dW_t$ has the analytic solution $$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$ Recently ...
5
votes
2answers
168 views

A question on Ito

If we know the dynamics of $S$, then we can estimate the value of $S$ at a time point, $t$. Here, I have a question concerning how to solve for $S_t$ by Itô because I obtained different results by ...
5
votes
3answers
2k views

How to use Itô's formula to deduce that a stochastic process is a martingale?

I'm working through different books about financial mathematics and solving some problems I get stuck. Suppose you define an arbitrary stochastic process, for example $ X_t := W_t^8-8t $ where $ W_t ...
5
votes
1answer
173 views

unique equivalent martingale measure in incomplete markets

Do you have any idea about how we can prove, and under which conditions, that an equivalent martingale measure (EMM) in an incomplete market is unique? The assumptions we have made are: 1) that the ...
5
votes
1answer
638 views

Multi Fractals Models

From a quant point of view, how would you explain Multi Fractals Models in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am ...
5
votes
1answer
519 views

What is the forward rate for a Black-Karasinski interest rate model?

I was wondering if anyone could help me with the instantaneous forward rate equation for a Black-Karasinski interest rate model? I was also after the Black-Karasinski Bond Option Pricing Formula.
5
votes
2answers
123 views

how we can derive $PIDE$ of double exponential Jump-diffusion model (we know as kou model)?

I'm working in double exponential Jump-diffusion model (we know as kou model) with following form , under the physical probability measure $P$: \begin{equation} ‎\frac{dS(t)}{S(t-)}=\mu‎‏ ‎dt+\sigma ...
5
votes
1answer
195 views

Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ ...
5
votes
1answer
160 views

What is the stochastic differential of a general semimartingale?

By using the canonical representation of a semimartingale in Eberlein, Glau and Papapantoleon's "Analysis of Fourier Transform Valuation Formulas and Applications", on page 3: $$H = B + H^c + h(x) ...
5
votes
2answers
209 views

Ito, Stochastic Exponential and Girsanov

This is a two-part question relating to the change of measure density used in Girsanov and secondly to the Stochastic Exponential. Whilst reading notes relating to Girsanov it is stated that the ...
5
votes
1answer
185 views

Distribution of hitting time of the integrated CIR process

If an increasing process $X_t$ has a known Laplace transform $\mathbb{E} e^{-s X_t} = m_t(s)$, define its hitting time $\tau$ to some level $B$ to be $$ \tau = \inf\{ u > 0 : X_u \geq B \}. $$ Can ...
4
votes
3answers
215 views

Determine $E[W_p W_q W_r]$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let 0 < p < q < r. Determine $E[W_p W_q W_r]$. ...
4
votes
2answers
503 views

Differential equation for log-returns

I have a question that might be trivial to most of you, but somehow I'm not able to solve it by myself. I have a disagreement with my colleague on the distributional properties of a Geometric Brownian ...
4
votes
2answers
480 views

Malliavin Calculus

From a quant point of view, how would you explain Malliavin calculus in few words ? I have the level to take these courses, but won't be able to do it next year, so I want to know what I am missing. ...
4
votes
2answers
247 views

question on Leif Andersen's “Interest Rate Modeling, vol 2 Term Structure Models”

I'm reading Leif Andersen's "Interest Rate Modeling, vol 2 Term Structure Models" and met a problem on Chapter 14 LM Dynamics and Measures, $\S$ 14.2.5 Stochastic Volatility, Lemma 14.2.6, on page ...
4
votes
1answer
77 views

Discounted risky asset stochastic process problem

$S_t$ is the random variable representing the risky asset price at time $t$. M_t is the riskless asset. They are governed by the equations $\frac{dS_t}{dt}=\mu dt + \sigma dZ_t$ and $dM_t = rM_t ...
4
votes
1answer
71 views

Multidimensional Ito's Lemma for Vector-Valued functions

Consider the vector of $n$ Ito processes $$ d \mathbf{X}_t = \mathbf{\mu}(\mathbf{X}_t,t)dt + \Sigma(\mathbf{X}_t,t)d\mathbf{W}_t $$ where $\mathbf{\mu} \in \mathbb{R}^n$ and $\Sigma \in ...
4
votes
2answers
203 views

Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
4
votes
1answer
89 views

backward Kolmogorov equations - Markov properties

I'm a physicist who's research has lead him into the theory of stochastic differential equations. If this question is not appropriate for this forum, please feel free to delete it. So I've been ...
4
votes
1answer
115 views

The distribution of jump gaps for Levy processes

Assume $X_{t}$ is a Levy process with triplet $(\sigma^{2}, \lambda, \nu)$, here $\nu$ is the Levy measure of $X_{t}$. Define $\tau_{1},\tau_{2},\dots$ be the time gap between the successive jumps ...
4
votes
1answer
270 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
4
votes
1answer
255 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
4
votes
1answer
116 views

Explicit solution SDE

I have the following SDE: $$dY_{t}=A\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{1}+B\left(\frac{W_{t}^{1}}{\sqrt{t}},\frac{Y_{t}}{\sqrt{t}}\right)dW_{t}^{2}$$ where ...