The tag has no usage guidance.

learn more… | top users | synonyms

1
vote
0answers
11 views

Analytical Bond Price under Rendlemen-Bartter?

Assuming the short rate $r_t$ follows the risk-neutral (so $W_t$ is a $Q$-Brownian motion) process $$ dr_t = ar_t dt + \sigma r_t dW_t, $$ does anyone know of an analytical bond price formula? We ...
0
votes
0answers
30 views

For a square-root process (CIR), how to verify the characteristic function of the transition density?

I am trying to solve a financial mathematical question. I derived PDE (a) for the characteristic function as follows. But, I don't know how to verify the following characteristic function of the ...
5
votes
1answer
81 views

Modelling EUR/USD with Ornstein-Uhlenbeck + jumps?

I'm trying to simulate a process as close as possible to EUR/USD of the ten past years. I've used a Ornstein-Uhlenbeck process: $$d X_t = -\theta (X_t - \mu) d t + \sigma d B_t$$ with the ...
3
votes
2answers
63 views

Bounded Stochastic discrete process

I just came across this stochastic process (link): $dY_t = (a-bY_t)dt + c \sqrt{Y_t(1-Y_t)}dW_t$, where $dW_t$ is a Wiener Process. According to the author under certain conditions this process is ...
3
votes
2answers
120 views

What's the name of this nearly-brownian stochastic process?

1) Does the following algorithm (my question is math, not programming-related): ...
0
votes
0answers
48 views

Analytical solution to the Black-Scholes equation with time-dependent volatility

I am stuck with the following exercise and I would appreciate any help with it. I have to calculate the analytical function for the price of a call option given the following process for the ...
1
vote
2answers
86 views

Geometric brownian motion vs. Ornstein Uhlenbeck

I'm looking at the SDE of Geometric brownian motion(*): $$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$ (with analytic solution $X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}$) and the SDE of ...
0
votes
0answers
74 views

Black-Scholes formula proof, without stochastic integration

I've looked into many books at my academic library, and very often it goes like this: Brownian motion Then, stochastic integration (Itô's formula etc.) Application: Black-Scholes formula for price ...
1
vote
0answers
30 views

Avellaneda/Cont model Order Book Model

The model given in the following paper by Avellaneda et al http://people.stern.nyu.edu/jreed/Papers/limitorder.pdf On page 7 he explains that the initial Bid and Ask size should be normalised by ...
5
votes
0answers
74 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends ...
4
votes
1answer
123 views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
0
votes
0answers
48 views

Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
3
votes
1answer
88 views

How do one solve $ \int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u $? Double integral with general deterministic function $\delta(t)$

How do one solve $ \int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u $ ? $\delta(t)$ is a general deterministic function. $r$ is constant.
0
votes
1answer
36 views

Motivation: Stochastic Interest rate model

what is a reason that someone might be interested in a stochastic-interest model such as the Chen model? Also can you provide me with a link to an easy to read motivational paper/part of a paper on ...
1
vote
1answer
28 views

How to change to risk neutral measure in a mean reversion process?

For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
2
votes
2answers
79 views

$ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $ ? Exp. of an indicator funct and a diffusion with non-proportional vol

How to compute $ \mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace $ ? where $ dS_t = S_t r dt + \sigma dW_t $ and $ 1_{S_T > K} $ is the indicator function being one when ...
2
votes
1answer
34 views

Can the differential operator be removed to get the mean/variance of an Ito process?

If $X_t$ is an Ito process, such that: $dX_t = \mu(t, X_t)dt + \sigma(t, Xt)dW_t$ where $W_t$ is a standard brownian motion. Then we can say that: $E(dX_t) = \mu(t, X_t)dt$ and that $Var(dX_t) = ...
1
vote
1answer
39 views

Asymmetric Random Walk / Prove that $T:= \inf\{n: X_n = b\}$ is a $\{\mathscr F_n\}_{n \in \mathbb N}$-stopping time

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
3
votes
2answers
99 views

Asymmetric Random Walk / Prove that $E[T:= \inf\{n: X_n = b\}] < \infty$

Given random variables $Y_1, Y_2, ... \stackrel{iid}{\sim} P(Y_i = 1) = p = 1 - q = 1 - P(Y_i = -1)$ where $p > q$ in a filtered probability space $(\Omega, \mathscr F, \{\mathscr F_n\}_{n \in ...
3
votes
1answer
101 views

How to apply the Feynman-Kac formula?

I've been learning about Feynman-Kac recently and I understand the underlying ideas. I am stuck however in actually computing explicit solutions for specific problems. For example, suppose I have the ...
3
votes
1answer
199 views

open problems in mathematical finance

What are open problems in mathematical finance that use fundamental concepts of mathematics (functional analysis, geometry and topology, algebra and number theory etc.) and not data-driven. I have ...
3
votes
1answer
63 views

Expected value of log-GARCH process

Is there a way to analitycally compute expectation of log-GARCH process? The GARCH(1,1) process: $dU_t = \theta(\omega - U_t) dt + \xi U_t d W_t$ The log-GARCH(1,1) process: $e^{U_t}$ The ...
1
vote
0answers
23 views

Soft: Interpretation Fractional BM in finance

Suppose we are in the BS framework. If we replace the Brownian Motion with a more general fractional Brownian motion therein, how can it be interpreted? That is what is a financial interpretation of ...
2
votes
1answer
59 views

How to use Euler discretization for this interest rate model?

How can I perform Euler discretization on this model where $\delta t=1$ and $\delta x_t = x_t-x_{t-1}$
5
votes
2answers
76 views

Constructing a Brownian motion from a Simple Random Walk

I'm trying to get my head around how a Brownian motion is formed from a simple random walk. I've seen two similar methods used: Why has one approach used $\frac{1}{\sqrt{k}}$ and the other ...
1
vote
1answer
63 views

Obtaining the drift of a Wiener process formed from a random walk

I'm trying to understand how the equation for Geometric Brownian Motion is formed from a random walk. I'm following the book 'Statistics of Financial Markets' but I'm struggling to follow how the ...
6
votes
1answer
94 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $P(\tau < t) \geq (|W_t|>2)\\$ ...
5
votes
1answer
152 views

Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
1
vote
0answers
64 views

State of Art - Nelson Siegel Modeling

My idea is to work with dynamic Nelson Siegel models(DNS) on my master's thesis. As I am finishing undergraduation this year I started researching on the subject. I wonder what is being discussed in ...
5
votes
0answers
104 views

Why is it useless to model stochastic volatility when pricing Vanilla style derivatives?

With respect to the answer by user AFK in Ideas about Stochastic volatility models. I am specifically interested in interest rate options (IR Caps/Floors and Swaptions).
2
votes
1answer
79 views

Why is my Euler discretization error increasing with number of steps?

I'm trying to see how the Euler discretization error behaves with respect to the number of steps. To do this I'm simulating a geometric brownian motion and comparing it with it's 'exact' solution. ...
2
votes
0answers
56 views

Risk Neutral Variance Gamma

In the risk neutral version of the Variance Gamma model the stock dynamics are $S_T=S_0 e^{ (r-q+\omega)t + X(t;\sigma,\nu,\theta)}$ with $\omega=\frac{1}{\nu}ln(1-\theta \nu - \frac{\sigma^2 \nu ...
1
vote
0answers
42 views

Any idea of compound Poisson processes in betting? [closed]

Any suggestions on compound poisson processes in bets of a customer?
0
votes
0answers
38 views

Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...
1
vote
1answer
161 views

Feynman Kac Formula for path-dependent options

Consier geometric Brownian motion: $dS_t/S_t=\mu dt+\sigma dW_t$ Feynman Kac theorem tells us that the conditional expectation $v(t,x)=E[ e^{-rT}\Psi(S_T) | S_t=x]$ can be computed by solving the ...
2
votes
0answers
31 views

Specifying integration level of time series [closed]

Following model was estimated on 200 observations. How to specify the level of integration of $X_t?$ In brackets there are standard errors and p-value of Breusch-Godfrey test is also shown. ...
3
votes
2answers
207 views

Relationships between white noise and random walk

I would like to ask 5 questions about relations between these processes. 1) Could white noise be also a random walk? 2) Could random walk be also a white noise? 3) Could white noise be stationary? ...
1
vote
2answers
86 views

Question about find no arbitrage trading strategy

We got the stochastic process for stock price of n stocks at continues time. We can find if there is a arbitrage trading strategy or dominant trading strategy. I wonder if we cannot find such ...
4
votes
2answers
177 views

Does No arbitrage(NA) imply efficient markets (EMH)?

The EMH states that stocks are traded at its fair values. This means there is no arbitrage strategy in efficient markets. However, if the market is no arbitrage, can we conclude the market is ...
2
votes
2answers
387 views

Ideas about Stochastic volatility models

I am currently working on comparing different models for modelling the volatility and then pricing vanilla options (I use option prices on real stocks in order to calibrate my models and then I ...
0
votes
1answer
53 views

Interpreting and scaling of Realized Variance with sample data

I have some question about realized variance(RV) and I have some sample prices below to work with. You can run the R code below to build a vector of log returns. Three are 78 5-minute buckets in a ...
3
votes
1answer
144 views

Derivation using Ito's Lemma of price process

Define $q(t)$ as the log price minus a linear trend $$ q(t) = \ln P(t) - \mu t $$ Assume the log price process = Equation 1: $$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$ Can you show that the ...
5
votes
1answer
111 views

justification of square root process

In finance, many stochastic processes $X(t)$ are defined via \begin{equation} dX = \text{(some drift term)} dt + \sigma X^\gamma dW_t \end{equation} with $\gamma = 1/2$ (for instance the Heston model ...
1
vote
1answer
96 views

Calibration of non-mean-reverting OU process

I'm looking for some reference on how to calibrate a non-mean-reverting Ornstein-Uhlenbeck process to historical data using MLE or OLS. The model has the following SDE: ...
3
votes
0answers
118 views

Is there a countably infinite Sigma-Algebra? Why?

Assume $\,\mathcal{F}$ be a nonempty collection of subsets of $\Omega$. $\,\mathcal{F}$ is called a $\sigma$-Algebra whenever if $A\in\mathcal{F}$ then $A^c\in\mathcal{F}$, and if ...
2
votes
2answers
762 views

Speed of mean reversion of an interest rate model

I would like to have a bit more of intuition about the concept of "speed of mean reversion" for an interest rate model, e.g. Vasicek or CIR. In particular, is a negative speed of mean reversion ...
5
votes
2answers
180 views

Reflection Principle

Let $(\Omega,\mathcal{F},P)$ be a probability space and $\{W_t ∶ t ≥ 0\}$ be a standard Wiener process. By setting $\tau$ as a stopping time and defining \begin{align} ...
3
votes
2answers
258 views

Ito's formula for Jump process

Let $\{N_t\,|\,0\leq t\leq T\}$ be a Poisson process with intensity $\lambda>0$ defined on the probability space $(\Omega,\mathcal{F}_t,P)$ with respect to the filtration $\mathcal{F}_t$ and ...
3
votes
1answer
125 views

Extended CIR and discretization

Did someone know how to discretize this process efficiently : $dX(t) = \kappa [\theta(t)-X(t)]dt + \sigma \sqrt{X(t)}dW(t)$ I am looking for something more sophisticated than the trivial Euler ...
2
votes
1answer
67 views

Underlying Sample Space in Continuous Market Model

E.g., a model for $N$ stocks might have each follow a GBM $dS_i = \mu_i S_i dt + \sigma_i S_i dW_i$, where each $W_i$ is independent of the others. Letting $(\Omega, \mathcal{F}, P)$ be the ...