# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

54 questions
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### Integral of Brownian motion w.r.t. time

Let $$X_t = \int_0^t W_s \,\mathrm d s$$ where $W_s$ is our usual Brownian motion. My questions are the following: Expectation? Variance? Is it a martingale? Is it an Ito process or a Riemann ...
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### How were these SDE derived?

Can anyone give me a detailed explanation of how below equations (3) and (4) are derived from (1) and (2)? \begin{align*} \frac{dF_{t,T}}{F_{t,T}} &=\sigma e^{-\lambda(T-t)}dB_t, \tag{1}\\ \ln(F_{...
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### 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$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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### Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ ...
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### 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 ...
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### 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 ...
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### Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ...
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### Show that $E[B_t|\mathscr{F}_s] = B_s$ for $B_t = W_t^3 - 3 t W_t$

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 $B_t = W_t^3 - 3tW_t$. ...
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Can someone point me into the right direction to calculate this one: $E(B^4_t)=3t^2$ I had tried using the following property with no luck: $E(B^4_t)=E(B^2_tB^2_t)=E(\int B^2 dt )E(\int B^2 dt )=[E(\... 1answer 471 views ### Why is the value of an adaptive stochastic process known at time t? I am having a hard time to understand the concept of an adapted stochastic process. Using an analogy to finance, I have been told we can think of adaptiveness of a stock price process as having an ... 1answer 2k views ### Correlation coeffitiont between two stochastic processes I want to find correlation coeffitiont between$W_t$and$\int_{0}^{t}W_s ds$. I think that these are uncorrelated. But Why? So thanks 1answer 196 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 ... 2answers 410 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 ... 2answers 2k 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 ... 1answer 1k views ### CIR Process from Ornstein–Uhlenbeck process The wikipedia entry on the CIR Model states that "this process can be defined as a sum of squared Ornstein–Uhlenbeck process" but provides no derivation or reference. Can any one do that? I could only ... 1answer 189 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: $$... 1answer 474 views ### How to express the volatility of two correlated Ito processes Wt_1, Wt_2 expressed in terms of W_t? Having two correlated Ito processes (W_t^1 and W_t^2 are correlated Brownian motions with correlation \rho) dX_{t} =\mu_{1} dt + \sigma_1 dWt_1 dY_{t} = \mu_{2} dt + \sigma_2 dWt_2 ... 1answer 262 views ### Simple question about stochastic differential What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let X_t be a process and \alpha(t) a real function. What would be d(\alpha(t)X_t)? 2answers 729 views ### Why does the short rate in the Hull White model follow a normal distribution? Consider Hull White model dr(t)=[\theta(t)-\alpha(t)r(t)]dt+\sigma(t)dW(t) when we solve the SDE above we have r(t)=e^{-\alpha t}r(0)+\frac{\theta}{\alpha}(1-e^{-\alpha t})+\sigma e^{-\alpha t}\... 1answer 274 views ### “Expectation” of a FX Forward I have an FX process X_t = X_0 \exp((r_d-r_f)t+ \sigma W_t). Now clearly E[X_t] = F_{0,t}^X. i.e. a forward contract of the process X starting at time 0 and maturing at time t. What if I ... 2answers 249 views ### Ho Lee model in Baxter&Rennie I am currentyl reading Baxter&Rennie and I have a difficulty with understanding a derivation of formula for one function, g(x,t,T) (this can be found on page 152 in the book). I know that there ... 1answer 469 views ### Baxter & Rennie HJM: differentiating Ito integral From Baxter and Rennie, page 138:$$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)dsZ_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)dZ_t=... 0answers 417 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 (... 2answers 260 views ### Application of Ito's lemma Let$X_t$be some stochastic process driven by wiener process ($W_t)$so it can be expressed as: $$dX_t=(...)dt+(...)dW_t$$ Let$f(t,x)$be some$C^2$function. Define the process$Z_s=f(t-s,X_s)$... 1answer 167 views ### How to derive an option price for an asset with these dynamics? Assuming my underline asset price follows the process: $$d\ln (F_{t,T})=-(1/2)\sigma ^2e^{-2\lambda(T-t)}dt+\sigma e^{-\lambda(T-t)}dB_t$$ How should I derive an option price formula? 1answer 2k views ### How to get Geometric Brownian Motion's closed-form solution in Black-Scholes model? The Black Scholes model assumes the following dynamics for the underlying, well known as the Geometric Brownian Motion: $$dS_t=S_t(\mu dt+\sigma dW_t)$$ Then the solution is given:$$S_t=S_0\,e^{\... 2answers 255 views ### Ito lemma of Convertible Bond under Two-factor Model Interest Rate @Behrouz Maleki has provided the PDE of two factor model in other post so could anyone please provide Ito lemma of this equation and how this PDE was derived from Vasicek model. as far as I know it ... 1answer 157 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 ...
The Black Scholes model assumes the following form for the Wiener process describing the evolution of the stock price S: $dS=\mu S dt + \sigma S dX$ Clearly $S$ ...