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# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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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 ...
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### Proving Flow Property of Stochastic Differential Equation

I am trying to show that $X_t^{s,x} = X_t^{r, X_r^{s,x}}$ for $0 \leq s \leq r \leq t$, $x \in \mathbb{R}^n$ is a given initial condition for time $s$, for some SDE: \begin{equation*} d X(u)=b(X(u))d ...
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I ran through an equality in a paper I was reading but couldn't check if it is correct. Let $W^1_t$, $W^2_t$ and $W^3_t$ be three brownian motions, not necessarily independent, is it true that the ...
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### Ultra Powerfull Vibrato Montecarlo for delta sensitivities of a not regular payoff

Ciao, I am working on a derivative with the following payoff at time $T$: $$\sqrt{(S_T - K)^+}$$ where $S_T$ is the value of the stock at the expiring date. As usual we will assume $S_t$ to be a ...
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### kolmogorov backward equation intuition

The kolmogorov backward equation equation states that the probability density of a random variable $x$ which follows $dx= \mu dt + \sigma dw$ is given by $-p_t = \mu p_x + 0.5\sigma^2 p_{xx}$ ...
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### Model of asset substitution/risk shifting in continuous time

Consider a firm with cash flows $X_t$, which under a risk-neutral probability measure, follows a geometric brownian motion: $$dX_t = X_t[(r-\beta)dt + \sigma dZ_t]$$ where $r>0$ is the risk-free ...
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### From one period to multi period risk neutral pricing

For a one period economy, we have the price of an asset as: $p_0 = E^Q [p_1 * \frac {B0}{B1}]$ where $B0 = e^{-r_0}$ = time 0 price of risk free bond maturing at time =1 and $r_0$ is known at t0. ...
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We know that when using lognormal returns, the number you need to plug in is not the apparent return, but $\mu-\sigma^2/2$ because what you really have is, in essence, (1) a deterministic growth of $\... 0answers 56 views ### Extension of HJM to multiple factors The HJM model calibrates the entire forward curve using the existing yield curve data and this results in the following expression for its instantaneous forward rate- $$df(t,T)=\sigma(t,T)\int_0^T\... 0answers 189 views ### Pricing a structured note instrument I am trying to work out the following fixed income problem, where I am asked to price a structured note in Excel, which seems to me to be a reverse collar. My purpose was replicating this structured ... 0answers 98 views ### Geometric Brownian Motion: Drawdown as a function of time Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration? 1answer 349 views ### Stochastic integrals wrt to independent Wiener processes are uncorrelated, but potentially dependent? In Proof of Proposition 1.2.20 in the following lectures notes http://math.uni-heidelberg.de/studinfo/reiss/sode-lecture.pdf I found following quote " stochastic integrals with respect to ... 0answers 47 views ### Is the 'constant weight in the risky asset' portfolio-strategy self-financing? My question concerns a topic in quantitative finance that I feel is often brushed under the table: is a given strategy self-financing. We have two assets, one risky and one riskless, defined by the ... 1answer 51 views ### Discretizing the conditional variance in the Arbitrage Free Dynamic Nelson Siegel model for my thesis I am trying to fit the correlated factor arbitrage free dynamic Nelson Siegel model to yield data. I use the Kalman filter to model this but since the model is in continuous time, I need ... 0answers 140 views ### Characteristic function of SDE with coefficients depending upon second coupled SDE Say we have the following two SDEs driven by the same single Brownian:$$ dx_t = -0.5\sigma^2g(\psi)^2dt + \sigma g(\psi)dW_t \quad\quad d\psi_t = -(H\psi_t+0.5\sigma^2)dt + \sigma dW_t$$where$...
Consider the following setup: Let $S=\left(S_1,\ldots,S_n\right)$ be a $n$-dimensional price process and denote by $V$ its value process defined by $V_t=\phi_t\dot\ S_t$ for $t=0,\ldots,T$. In "...