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Integration in the Hull-White SDE

I'm stuck in solving the SDE in Hull-White interest rate model. I do not have a thorough background in math (only Real Analysis during my blissful undergrad years), so I am having trouble ...
93 views

How to express the volatility of two correlated Ito processes $Wt_1, Wt_2$ expressed in terms of $W_t$?

Having two correlated Ito processes ($W_t^1$ and $W_t^2$ are correlated Brownian motions with correlation $\rho$) $dX_{t} =\mu_{1} dt + \sigma_1 dWt_1$ $dY_{t} = \mu_{2} dt + \sigma_2 dWt_2$ ...
981 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 ...
I am analysing a problem where I have two correlated stocks described by Brownian motions $$\frac{dS^{1}_{t}}{S^{1}_{t}}=\mu_{1} dt + \sigma_{1} dW^{1}_{t} \quad \quad (1)$$ $$\frac{dS^{2}_{t}}{S^{... 1answer 127 views pdf of simple equation, compound Poisson noise I would like to find the probability density function (at stationarity) of the random variable X_t, where: \begin{equation*} dX_t = -aX_t dt + d N_t, \end{equation*} a is a constant and N_t is a ... 1answer 98 views Square of arithmetic brownian motion process We have an arithmetic Brownian motion process X_t that follows dX_t=\mu dt + \sigma dZ_t and we define the asset price S_t=X_t^2 and we are asked to find the stochastic differential equation ... 5answers 861 views Geometric Brownian motion - Volatility Interpretation (in the drift term) A Geometric Brownian motion satisfying the SDE dS_t = rS_t dt+\sigma S_t dW_t has the analytic solution$$S_t = S_0\exp\left\{\left(r-\frac{\sigma^2}{2}\right)t\right\}\exp\{\sigma W_t\}$$Recently ... 0answers 201 views Real world application of stochastic portfolio theory There is a branche of stochastic portfolio theory (see also this question). Fernholz and Karatzas have published research in this field (e.g. "Diversity and relative arbitrage in equity markets") and ... 2answers 125 views Intergral of Brownian motion w.r.t. Brownian motion I don't understand why S (highlight on picture), I learned$$\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 2answers 90 views ARMA-GARCH model, bset model selection and confidence levels calculations I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch. So, 2 main questions. 1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it ... 1answer 85 views Lookback option to find stock price Consider the payoff equation for the lookback option \psi(T)= max(S_t-S_T), where t\in[0,T] and S_t is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ... 2answers 55 views Asymptotic behavior property of geometric Brownian Motion proof Online I found the asymptotic behavior property of geometric Brownian Motion X_tas: If \mu (drift parameter) is \ge \sigma^2/2 where \sigma is the volatility parameter, then X_t \... 0answers 62 views Stochastic Integration I have the following derivation question: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and ... 1answer 220 views Implications of shifting the lognormal model for forward rates from a probability perspective I have a question regarding the application of a shift to the Black-Scholes formula for negative forward rates. I am reading in the Brigo book that "increasing the shift \alpha shifts the ... 1answer 32 views prove the normality, with given moments, of this process: I have this process: dx_t = -\frac{k}{2}x_tdt + \frac{\beta}{2}dz_t and must prove it's normally distributed with first two moments: \mu = e^{-\frac{1}{2}kt}x_0 \sigma^2 = \frac{\beta^2}{4k}(... 0answers 40 views Regularity requirement for convergence of Euler scheme for stochastic integral? Let S_t be follow Black Scholes, then I am interesting in simulating the process \int ^t _0 e^{-rt}1_{\{S_t\leq K\}}dS_t which is like a naive hedge of a European put, which does not work in ... 0answers 39 views How to solve this system of ODEs? Im trying to replicate the procedure of the Hackbarth et al. 2006 paper. Im trying to solve the ODEs (12) and (13) on page 525 in the paper, following the solution by the authors given in appendix A. ... 1answer 86 views Prove uniqueness, and prove Y_t is a martingale by considering dZ_t and dL_t Suppose we are given a filtered probability space (\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P}), where \{\mathscr{F}_t\}_{t \in [0,T]} is the filtration generated by standard ... 0answers 56 views What is the maximum of a brownian motion with drift over the interval [t_1,t_2] I am having a problem deriving the equation:$$P(max_{(t_1 \leq t \leq t_2)} S(t) > B | S(t_1),S(t_2))= e^{-\frac{2}{T}ln\bigg{(}\frac{B}{S(t_1)}\bigg{)} ln\bigg{(}\frac{B}{S(t_2)}\bigg{)}}$$... 2answers 182 views Ito Formula for Stochastic Integral Suppose I have$$dS_t = \mu(S_t,t) dt + \sigma(S_t,t)dW_t$$What would be the process satisfying the following process of y_t?$$y_t = \int_0^t S_u du + \int_0^t S_u dW_u$$I'm not quite sure ... 2answers 106 views Stochastic process theory question *S follows a process dS= mSdt + oSdz where m and o are constant. What is the probability followed by  Y=(Se)^{(r-t)} . If S follows a process  dS= k (b-S) dt + oSdz  where k, b, o are ... 0answers 50 views For a square-root process (CIR), how to verify the characteristic function of the transition density? I am trying to solve a financial mathematical question. I derived PDE (a) for the characteristic function as follows. But, I don't know how to verify the following characteristic function of the ... 2answers 231 views Geometric brownian motion vs. Ornstein Uhlenbeck I'm looking at the SDE of Geometric brownian motion(*):$$d X(t) = \sigma X(t) d B(t) + \mu X(t) d t$$(with analytic solution X(t) = X(0) e^{(\mu - \sigma^2 / 2) t + \sigma B(t)}) and the SDE of ... 2answers 105 views Question about the martingale property of stochastic integral Let W_{t} be a Wiener process, and let$$X_{t} = \int^{t}_{0}W_{\tau}d\tau$$Is X_{t} a martingale? We can rewrite in differential form as$$dX_{t} = W_{t}dt$$,which means X_{t} is a diffusion ... 0answers 25 views self financing property vs. unlimited borrowing How the self financing property of a portfolio should be understood in the problems where the unlimited access to the borrowing is assumed? 1answer 137 views Methods of SDE Calibration There is somewhere summary of methods that can be used to estimate parameters of SDE? I currently using MLE and regression due to linear dependence between samples. I searching for something ... 2answers 79 views Problem with deriving the dynamics of a process I'm trying to solve the following problem. Given a process X_t and a process Z_t, with the dynamics of X_t as$$ dX_t = (\alpha + \beta X_t)dt + (\gamma + \sigma X_t)dW_t $$and Z_t defined ... 1answer 149 views How to use the Girsanov theorem to prove \hat{W_t} is a \hat{\mathbb P}-Brownian motion? Let T > 0, and let (\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P) be a filtered probability space where \mathbb P = \tilde{\mathbb P} (risk-neutral measure) and \mathscr F_t ... 0answers 87 views Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus) I am doing a research on communication protocol design. A file of n blocks is transferred in several rounds and R_i denotes the number of blocks received in the i-th round. The sender sends n-... 1answer 85 views How to prove \int_0^t W_s^2dWs = \frac{1}{3}W_s^3 - \int_0^t W_s ds using Ito's formula? [closed] Please help me with this problem. 1answer 46 views Please help me with this problem of double exponential distribution please help me with this problem of double exponential distribution 0answers 63 views Prove that E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t] Let T > 0. Let (\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P) be a filtered probability space where \mathscr F_t = \sigma(W_u, u \in [0,t]) where W_t is standard Brownian ... 2answers 77 views zero coupon problem calculus I encounter a problem: do we have the following equality : B(0,T_{i})e^{\int_{0}^{t}r_{s}ds}=B(t,T_{i}) and if yes why because I am stuck with this ... I try to use that : B(t,T_{i}) = B(0,T_{i})e^... 2answers 114 views Distribution of stochastic integral Suppose that f(t) is a deterministic square integrable function. I want to show$$\int_{0}^{t}f(\tau)dW_{\tau}\sim N(0,\int_{0}^{t}|f(\tau)|^{2}d\tau)$$. I want to know if the following approach is ... 1answer 93 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. 1answer 143 views Differential of stochastic term Question 1: How does one come up with the equation in the red box below? It looks like some kind product rule, but I'm not sure how to apply Ito's lemma here. Bjork doesn't seem to explain it ... 1answer 69 views Prove E_{\mathbb Q}[X_t | \mathscr F_u] = X_u given Y_t is a martingale We are given a filtered probability space (\Omega, \mathscr{F}, \{\mathscr{F}_t\}_{t \in [0,T]}, \mathbb{P}), where \{\mathscr{F}_t\}_{t \in [0,T]} is the filtration generated by standard \mathbb ... 2answers 86 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 ... 1answer 45 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 ... 1answer 78 views Brownian motion. Solve stoc. integral by using Ito's lemma I want to show that following statement is true by using Ito's lemma to solve stochastic integrals: I define the functions in Ito's model: a()=0, b()= (2wt-2)^2. f(t)=Integrate[(2wt-2)^2] Then df=(b^... 1answer 55 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? 1answer 147 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 ... 1answer 701 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 ... 2answers 629 views How to use the stock as a numeraire to price a derivative with payoff of the form (S_T f(S_T))^+? I have \frac{dS_t}{S_t} = rdt + \sigma dW_t as usual under the money-market numÃ©raire and I need to price options with payoffs$$(S_T f(S_T))^+$$How do I express the stock dynamics using the ... 2answers 1k views How to express the Black Derman & Toy Model in a dr=A\,dt+B\, dW form? The Black Derman & Toy (BDT) model is given by$$d(\ln\,r)=\left(\theta(t)-\frac {d(\ln\sigma(t))}{dt}\ln r\right)\,dt+\sigma(t) \, dW.$$How can one rewrite the BDT model as dr=A\,dt+B\, dW, ... 1answer 161 views Derivation using Ito's Lemma of price process Define q(t) as the log price minus a linear trend$$ q(t) = \ln P(t) - \mu t $$Assume the log price process = Equation 1:$$ dq(t) = - \Theta q(t) dt + \sigma dW(t) $$Can you show that the ... 2answers 795 views Regime switching in mean reverting stochastic process Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an ARMA(p,q). This time series could be ... 1answer 176 views Extended Hull White Interest Rate Model for Zero Coupon Bond Please taking the following SDE dr = u (r; t) dt + w (r; t) dX: u (r; t) = a(t)-br; w (r; t) = c; b&c are constants and a(t) arbitrary function of time. If Zero Coupon Bond Z (r; T; T) = 1 ... 0answers 146 views How to compute the stochastic integral of log-normal process? How do you compute the following integral:$$\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 (...
I have a question about the solution of the Black-Scholes PDE for the European call option when I read the book Stochastic Calculus for Finance II of Steven E.Shreve. Let $c(t,x)$ be the value of the ...