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4
votes
3answers
262 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
579 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
1answer
137 views

Stochastic Differential

Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because $$ \mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta. $$ But am I allowed to actually write ...
4
votes
2answers
292 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
6answers
590 views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
4
votes
1answer
82 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
224 views

Closed form solution of PDE of Option Price

Let $V=V(S_t,t)$ be the option price and \begin{align} V_t+\mu\,S\,V_S+\frac{1}{2}\sigma^2\,S^2\,V_{SS}=0\\ V(S_T,T)=\ln (S_T)^{2}. \end{align} My question: How can I obtain a closed form solution of ...
4
votes
1answer
134 views

How can I calculate $Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right)$

How can I calculate? \begin{align} Cov\left(\int_{0}^{s}W_u\,du\,\,\,,\,\int_{0}^{t}W_v\,dv\right) \end{align} Thank you for your attention.
4
votes
1answer
166 views

Lipschitz condition in mathematical finance

I am interested in a rigorous explanation on why the Lipschitz condition plays a major part in stochastic calculus, most significantly in mathematical finance. To be specific, suppose we want to ...
4
votes
1answer
150 views

Derivation of the Stochastic Vol PDE

I'm trying to follow the derivation of the stochastic vol pde for an option price - as given in Gatheral (The vol surface), Wilmott on Quant Finance and many other places. As usual one starts off with ...
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 ...
4
votes
1answer
132 views

Computation of Expectation

This question has so long preoccupied my mind.Please help me to solve it. Question: Assume $X_t$ described by the following stochastic differential equation $$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
4
votes
1answer
89 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?
4
votes
1answer
238 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
4
votes
1answer
105 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
137 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
284 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
2answers
244 views

Uniqueness of equivalent martingale measure in Black Scholes-Model

Let's consider standard Black-Scholes model with price process $S_t$ satisfying SDE $$dS_t = S_t(bdt + \sigma dB_t)$$, where $B_t$ is standard Brownian Motion for probability $\mathbb{P}$. I ...
4
votes
1answer
263 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
2answers
736 views

Regime switching in mean reverting stochastic process

Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$. This time series could be ...
4
votes
1answer
125 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 ...
4
votes
0answers
136 views

Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]

Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is $$ \nu \left( dx\right) = A ...
3
votes
3answers
152 views

For $B_t$ a Brownian motion what is the probability that $B_1>0$ and $B_2<0$?

Let $B_t$ be a Brownian Motion. What's the probability that $B_1>0$ and $B_2<0$?
3
votes
3answers
277 views

Show that $E[B_t|\mathscr{F}_s] = B_s$

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 $(B_t)_{t \geq 0}$ where $B_t = W_t^3 - 3tW_t$. ...
3
votes
2answers
780 views

Financial Mathematics - Martingales example

Was hoping somebody could help me with the following question. Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. ...
3
votes
1answer
156 views

How to get Black Scholes' Geometric Brownian Motion differential form form the closed form?

My instructor has mostly self contained notes, where our textbook is mostly a reference. She has it written that: $$S_t = S_0e^{(\mu - \frac{\sigma^2}{2})t + \sigma W_t} \iff dS_t = S_t(\mu dt + ...
3
votes
2answers
257 views

CVaR/VaR Ratio as alpha goes to 1

I am having trouble taking the following limit of CVaR/VaR for a normal distribution as alpha approaches 1: $\lim_{\alpha \to 1} \frac{\mu + \sigma \frac{\phi^{-1}(\alpha)}{1-\alpha}}{\mu + \sigma ...
3
votes
1answer
183 views

Ho and lee derivation for short rates model

A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
3
votes
1answer
303 views

What is augmented data when simulating stochastic differential equations using Gibbs Sampler?

I am reading this paper on Bayesian Estimation of CIR Model. Basically, it is about estimating parameters using Bayesian inference. It estimates this stochastic differential equation: $$dy(t)=\{ ...
3
votes
2answers
67 views

Conditional expectation of a non stochastic process

In an example I was working through it was shown that $W_{t}^{2} - t$ was a martingale with respect to the Brownian motion filtration $\mathcal{F}_{s}^{W}$ with $t>s$. Everything was fine except a ...
3
votes
1answer
456 views

Covariance matrix and Cholesky decomposition

I am simulating a spread option with stochastic volatility using Monte Carlo simulation. I have the positive-definite covariance matrix $$ \rho = \left( \begin{array}{cccc} 1 & \rho_{1,2} & ...
3
votes
1answer
367 views

Integrating log-normal

The usual log normal model in differential form is: $dS = \mu S dt + \sigma S dX$ where $dX$ is the stochastic part, so $\frac{dS}{S} = \mu dt + \sigma dX$ (1) and we normally solve this by ...
3
votes
1answer
120 views

What is the correlation between these two functions of GBMs?

Let's say that I have two correlated GBMs: $$dA_t = A_t \sigma^A dW^A_t$$ $$dR_t = R_t \sigma^R dW^R_t$$ $$dW^R_t dW^A_t = \rho dt$$ I am trying to price a derivative which payoff at time $T$ is: ...
3
votes
1answer
100 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 $X_{t}=c*\beta_{t}$ ...
3
votes
2answers
136 views

Stochastic Differentials - Ito's formula for a self-financing portfolio

Suppose I have a portfolio of stocks $(S)$ and savings account ($\beta_t$) then, the value is $$V = a_t S_t + b_t \beta_t$$ and for this portfolio to be self replicating, we need by Ito's lemma $$dV ...
3
votes
2answers
244 views

Libor Market Model: numeraire change

I am currently studying the Libor forward market model, and although I get the mechanics behind the main arguments, I still do not have an intuitive idea of what's exactly the objective behind ...
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.
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
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 ...
3
votes
1answer
127 views

Distribution of Black Scholes call option price at time 0<t <T

Does anyone know how to find the probability law (distribution) under P* of a Black Scholes Call Option price $C_t$ for $0 < t < T $? (Under P*, $ dC_t = \frac{\partial c}{\partial s}\sigma S_t ...
3
votes
3answers
383 views

Why is the CAPM securities market line straight?

Let $\gamma$ be the expected return, in terms of its exponential growth rate, of the market asset. If we set $\gamma=\mu-\sigma^2/2$ as explained by the Doléans-Dade exponential, then the expected ...
3
votes
0answers
106 views

How to compute the stochastic integral of log-normal process?

How do you compute the following integral: $$\int_0^t e^{\mu s + \sigma W_s} ds$$ or $$\int_0^t e^{\mu s + \sigma W_s} dW_s$$ ? Are those integrals stochastic processes of some well-know type ...
3
votes
0answers
368 views

Test for stationarity and make use of non-stationary points in financial market?

I have two questions to ask: What are the best methods to determine stationarity in a financial market (such as stocks) using MATLAB? What methods would you recommend to use in order to change from ...
2
votes
2answers
152 views

Is the Brownian motion multiplication rule a definition or is it a theorem?

Is the Brownian motion multiplication rule a definition or is it a theorem? Refer to the highlight part of http://i.stack.imgur.com/doQuT.png where $dw_1(t)dw_1(t)=dt$
2
votes
1answer
212 views

Simple question about stochastic differential

What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let $X_t$ be a process and $\alpha(t)$ a real function. What would be $d(\alpha(t)X_t)$?
2
votes
2answers
69 views

Problem with deriving the dynamics of a process

I'm trying to solve the following problem. Given a process $X_t$ and a process $Z_t$, with the dynamics of $X_t$ as $$ dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t $$ and $Z_t$ defined ...
2
votes
3answers
57 views

Is a bond expiring at $T$ clean or dirty price a martingale under the $T$-Forward measure?

When we say Bond prices are martingale under T-Forward measure, do we mean their Clean Price is a martingale or is it their dirty price. I guess it should be dirty price, as clean price is just a ...
2
votes
2answers
116 views

Using Black-Scholes to price a geometric average price call

Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway. I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
2
votes
2answers
449 views

SVCJ (SVJJ) Duffie et. al Model implementation in Matlab

I'm attempting to implement aforementioned SVCJ model by Duffie et al in MATLAB. so far without success. It's supposed to price vanilla (european) calls . parameters provided, the expected price is: ...
2
votes
3answers
281 views

Quadratic variation question

Here I have this question (i) state Ito's formula (ii) hence or otherwise show that $\int^t_0B_s dB_s = \dfrac{1}{2}B^2_t -\dfrac{1}{2} t$ (iii) define the quadratic variation $Q(t)$ of Brownian ...