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Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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70 views

Derive a mathematical equation for Eurodollar future rate

If we suppose that r(t) follows a Vasicek model, which is: $$dr(t) = (\mu - \kappa r(t))dt + \sqrt\sigma dW(t)$$ How to derive an expression for Eurodollar future rate?
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32 views

Levy process and random measure

I am wondering if random measures are used under a Levy process and how this connects to finance (particularly pricing). Any paper or books for suggestions is welcomed.
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Solving BSDE in R

I was wondering how to implement a BSDE approximation in R. For example, if I have the toy BSDE $$ dX_t = \mu dt + \sigma dW_t ; X_T\sim N(\mu_1,\sigma_1), $$ for fixed real numbers $\mu,\mu_1,\sigma,...
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35 views

Views on Functional Ito Calculus [closed]

Setting up a thread to see your views and opinions on the applications of Functional Ito Calculus. Would like to get an overview intuition of it before delving deep into the mathematics. Bruno Dupire ...
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54 views

Question about Stochastic Calculus,(change of measure)?

Can any one give some hint for this question? Let $\{S_t\}_{t=0}^\infty$ be an asset price process defined on the probability space $(\Omega,\mathcal{F},\mathbb{P})$. Assume that the log-return of $...
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35 views

about stochastic calculus. and for the Gaussian process

I don’t know how to integrate the u function about Brownian motion in this picture, so I can’t calculate.
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38 views

Characteristic function and distribution of a random variable

This is exercise 4.3 in Bjork, Arbitrage Theory in Continous Time. $$ X_t = \int^t_0 \sigma(s)dW_s $$ $\sigma$ is a deterministic function and $W_t$ is brownian motion. I am asked to find the ...
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1answer
138 views

For the Brownian motion integrate

I want to calculate $$\operatorname{E} \left[ \int_0^1{W(t)dt \cdot \int_0^1{t^2W(t)dt}} \right].$$ I discovered that the first integral is $\operatorname{N}(0, \frac{1}{3})$ but I don't know how to ...
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71 views

Statistical estimation vs Stochastic calibration of models

I have never been able to deduce the precise differences between model building from the statistical perspective and the stochastic processes/calibration perspective. I can only infer that these are ...
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24 views

Variance of integrated dynamical system

Define time increment $\mu:=t_{k+1}-t_{k}$. Consider the signal $x(\mu)-\mathbb{E}[x(\mu)]$ defined as $x(\mu)-\mathbb{E}[x(\mu)]=\frac{1}{\mu}\int_{t_{k}}^{t_{k+1}}\int_{0}^{\tau}e^{A(\tau-\delta)}...
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42 views

Black and Normal Model for Caplet using Python

I am able to Price Caplet using Black 76 model in Python. However, I am unable to price the same with Normal Model. Can anyone suggest what is missing ? I am valuing caplet that caps interest rate on ...
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111 views

How to calculate the covariance between two stochastic integrals?

How to calculate the covariance between the integral of a Brownian motion at different times: $$\text{Cov}\left(\int^{t_1}_0\sigma(t)dW_t,\int^{t_2}_0\sigma(t)dW_t\right)\ ?$$ I know the answer is: $$\...
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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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93 views

What is the Expectation of price S with respect to probability measure P and risk neutral measure Q?

If I have a stock with price process $S_t$, and we have that $W$ is a Brownian motion under the probability measure $\mathbb{P}$, and $\mathbb{Q}$ is the risk neutral/equivalent martingale measure. ...
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35 views

Euler discretization with jumps

There is a process $B_t = B_0\prod_{i=1}^{N_t}(1-Z_n)$, where $Z_n=e^{-ξ_n}$ for i.i.d exponentially distributed random variables $(ξn)_{n≥1}$ with rate $ρ=20$. ${N_t}$ is a counting process ...
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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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43 views

Correlated GBM and OU processes

I want to model two different stochastic processes, such that: $X_t , V_t$ are correlated with coefficient $\rho$. Where: $\frac{dX_t}{X_t}=\mu_1dt+\sigma_1 dW_{1,t}$ and $dV_t=\theta(\mu_2-V_t)dt+\...
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3answers
95 views

Need help to interpret the definition of a diffusion process

https://studentportalen.uu.se/uusp-filearea-tool/download.action?nodeId=1134155&toolAttachmentId=218130 In these lecture notes at page 15 and 16 I am looking at the definition of diffusion ...
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50 views

Price of a stochastic game between an agent and the market

In the article Pricing via utility maximization and entropy from Richard Rouge and Nicole El Karoui, they define the value function of the optimization problem as \begin{align} V(x,C) = \dfrac{1}{\...
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2answers
181 views

Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate]

My question is about a stochastic integral of brownian motion w.r.t time. Let $W(t)$ the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \...
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1answer
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The duality of the free energy and relative entropy used to deduce deduce the stochastic game between an agent and the market

I am reading the article Pricing via utility maximization and entropy by Richard Rouge and Nicole El Karoui. They talk about the relative entropy of a probability measure $Q$ with respect to the ...
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1answer
58 views

How to deduce the formula of the wealth process of a stochastic volatility model?

I am reading the paper Solution of the HJB Equations Involved in Utility-Based Pricing from Daniel Hernandez and Shuenn Jyi Sheu. The authors consider the utility function $U: \mathbb{R} \to \...
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1answer
42 views

standard brownian vs brownian motion

We say Xt with paramters (µ,σ) is brownian process if (Xt-s - X t) ~N (µs,σ2 s) AMONG other conditons . Here we don't speak about any particular distribution for X t. We only say it is a brownian ...
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Pricing caplet with Bachelier (normal dynamic) using forward measure

I'm trying to price caplet with Bachelier under forward measure, but I can't find any solution. Remind that Bachelier assumed rates follow a normal dynamic. So here what I was doing : $C_t(T,T+d)$ ...
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2answers
121 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 ...
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88 views

Compo Feature in Asian Option on Futures

I'm pricing an Asian option on futures using Turnbull–Wakeman (other suggestions welcome) where the average is defined as $A _ { t _ { 1 } , t _ { n } } ^ { A , f } = \frac { 1 } { n } \sum _ { i = 1 }...
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31 views

If you have normally distributed returns, shouldn't you have the same adjustment factor as lognormally distributed?

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 $\...
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1answer
59 views

What does $\int dS \phi (S - K)$ mean in Gatheral's book?

In Gatheral's book on stochastic volatility, he writes the price of an option as $$\int_K^\infty dS \phi (S - K)$$ where $\phi$ is a density. Where does this come from? I have multiple questions: ...
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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\...
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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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62 views

Discretizing a Continuous Time Stochastic Volatility Model

Forgive me for cross-posting. I have the following continuous time SDE for a stochastic volatility model. $S_t$ is the price, and $v_t$ is a variance process. $$ dS_t = \mu S_tdt + \sqrt{v_t}S_t dB_{...
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Impact analysis of parameters in the 2 Factor Hull White Model

Through the 2-Factor-Hull White Model you can model the yield curve if you have the parameters $a, b, \sigma, \eta$ given. Is there any way to measure the impact of these parameters on the yield ...
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81 views

Dynamics of an option on a future

I have trouble understanding why $$V_s=exp(\int_s^t r_u du) \int_s^t exp(−\int_t^v r_u du)\theta_v dW_v$$ is the solution to the following SDE $dV_s=\theta_s dW_s+r_s V_s ds$. I tried of course with ...
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Prove the given stochastic integral are equally distributed

Let $W^i_t$ and $W_t$ be pairwise independent Brownian motions for $i \in \{1, \dots , d\}$. Let $X_t^i$ be $d$ independent Ornstein–Uhlenbeck processes for $i \in \{1, \dots , d\}$, i.e. each $X_t^i$...
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122 views

Two papers - two different solutions of the Ornstein-Uhlenbeck process

Bernal 2016 says that the solution of $$ dr_{t}=\lambda*(\mu-r_{t})*dt+\sigma dW_{t} \qquad (eq.1) $$ equals $$ r_{t}=r_0*exp(-\lambda t)+\mu(1-exp(-\lambda t))+\sigma \int_{0}^{t} exp(-\lambda t)...
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48 views

Bond prices at future times under Vasick one-factor model

In Vasicek one-factor model (and in other affine models), the price of a zero-coupon bond at time $t$ conditional on the information at this time is $$P(t,T) = E[e^{-\int^T_tr(u)du}|F_t] = A(t,T)e^{-...
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1answer
127 views

Calculating the stochastic integral of $\exp(-rt)S_t$

I am currently reading lecture notes which aim to show that if $$ S_t = S_0 \exp (\mu t + \sigma W_t) $$ then, under the probability measure $\tilde{\mathbb{P}}$ with density $$ \gamma_T = \exp (c W_T ...
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143 views

Applying Ito's formula to complex functions

Within my lecture notes, the following definition is given: We say that the stochastic process $X_t$ has stochastic differential $$ dX_t = b_t dt + \sigma_t dW_t $$ if and only if $$ X_t = ...
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autocorrelation function of the trending OU process

What is the autocorrelation function of the trending Ornstein-Uhlenbeck (OU) process? First, the OU process $dX_t = -\frac{1}{\mu} X_t + \sqrt{\frac{2\sigma^2}{\mu}} dW_t $ generates coloured noise ...
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1answer
66 views

Girsanov's Theorem for Multiple Risky Assets

Girsanov's theorem provides the measure transformation from probability measure P to Q such that- $dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
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1answer
106 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
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1answer
169 views

Show that the Ito integral is Gaussian

Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
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squaring stochastic calculus and other solutions [closed]

It is well-known that the solution to the stochastic SDE $$ dS = S_0(\mu dt + \sigma dWt) $$ is $$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$ Were $\sigma=0$, this is simply the formula for ...
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1answer
87 views

Distribution in Heston

$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$ $W_t$ is wiener process and the rest is just some parameters. For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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1answer
101 views

Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
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Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?

The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
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1answer
78 views

How to derive the dynamic of the log forward price?

I have the following Schwartz model: $$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$ $$X_t=\ln S_t$$ $$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$ with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$ $$F_t(T)= \exp\...
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1answer
105 views

Properties of Stochastic Exponential

Let $\{X_t\}_{t \ge 0},\{Y_t\}_{t \ge 0}$ be a continuous semi-martingale with $X_0 = Y_0 = 0$, let ${\cal E}(X)$ to be the unique solution of: $dZ_t = Z_t dX_t$ with $Z_0=1$. We can show that ${\cal ...
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1answer
700 views

Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
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0answers
52 views

Computing Malliavin Derivative for European Call Payoff

Let $X_t$ be a continuous local-martingale modeling the stock price given by $$ X_t = \int_0^t \sigma_t(T,K)dW_t , $$ and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...