Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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7
votes
2answers
246 views

Stochastic Integral Graph

As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below, I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
3
votes
1answer
112 views

stochastic dominance displaced diffusions

Suppose I have two processes both satisfying a displace lognormal diffusion: $$ dX(t) = \alpha(t)[X(t) - a] dW(t) $$ $$ dY(t) = \beta(t)[Y(t) - b] dW(t) $$ Note that the processes are perfectly ...
-3
votes
0answers
63 views

Integrals with respect to Brownian motion

I have two questions: Let $(B_t)$ be a Brownian motion. Find all constants $a$ and $b$ such that $X_t =\int_a^t (a +b\frac{u}{t}) \mathrm{d}B_u$ is also a Brownian motion. Find all constants $a$, $b$...
3
votes
2answers
496 views

Finding price of the power option

Let's assume a market with $d=1$ and $X=X^1$ satisfying $dX_t=\sigma X_t\,dW_t,\: \: X_0=1,$ where $(W_t)$ is a standard Brownian motion. Assume that $\mathbb{F}$ is the natural filtration of $X$ ...
-5
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0answers
64 views

The basic knowledge for financial mathematics? [closed]

I was a objective trader in Asian Stocks and Forex, and study master degree in the major of quantitative finance currently. Well, it feels not good about learning experiences without strong math ...
4
votes
1answer
99 views

How do we calcualte $E[W_sW_t|W_s]$

$W_t$ is a Brownian motion. How do we calculate this expectation? there are two cases: $s < t$ $t < s$ Do we have to distinguish the two cases or there is a unified way of calculating it
1
vote
2answers
88 views

Instantaneous change in value of portfolio

I am trying to figure out an intuitive explanation for the instantaneous change for the value of a portfolio (essentially I'm creating a self-financing portfolio to replicate a derivative payoff). ...
1
vote
1answer
79 views

Stochastic solution (mean, variance) to lognormal drift and normal volatility

I have trouble deriving the state equations for a mixture of normal/lognormal stochastic differential, namely for its a) expected mean, (b) variance, and (c) drift adjustment for LMM - libor model I ...
3
votes
1answer
92 views

What the expectation of S^2 is from GBM? [closed]

I was at an interview and was asked to write down the SDE for GBM. $$ dS = S\mu dt + S\sigma dX $$ Then I was asked how I would compute the expectation of S^2. I didn't know where to start. Any ...
4
votes
1answer
159 views

Evaluating the SDE $dX_t = t\,dS_t$

The process $S$ is a geometric Brownian motion with an SDE: $dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating $E(X_t)$ and $V(X_t)$, where $dX_t = t\,dS_t$.
6
votes
1answer
195 views

Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$

I want to solve the following SDE: $$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$ Integrating, I get: $$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$ $$ ...
-1
votes
1answer
66 views

Stochastic Vol Mathematical derivation [closed]

I want to understand the mathematical steps done. Can someone please simplify the derivation of d(pi) from Pi? Thanks in advance.
28
votes
4answers
19k views

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 ...
2
votes
0answers
48 views

Interchange Expectation and Supremum in Snell Envelope/American Options

I had a question about the properties of a snell envelope, $\sup_{t\le\tau\le T} \Bbb E\left(Z_\tau\mid \mathcal F_t\right)$, which came to me while studying American options. I know that in general,...
4
votes
1answer
114 views

Invariance Scaling of Brownian Motion

Prove $\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$ converges to $\sup\limits_{t\in [0,1]}B_t$ in distribution as $t\to\infty$. I have a sense to use scaling invariance, but no ...
1
vote
1answer
77 views

Are the Ito's Lemma given in Mark Joshi's Concept and Practice in Mathematical Finance same as what I learn?

In Joshi's Concepts and Practice in Mathematical Finance, page $110,$ he stated the Ito's Lemma: Theorem $5.1$ (Ito's Lemma) Let $X_t$ be an Ito process satisfying $$dX_t = \mu(X_t,t)dt + \sigma(...
0
votes
1answer
57 views

integration of squared brownian motion w.r.t time

How to prove $\int_0^1 B_s^2ds$ is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion.
0
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0answers
33 views

Change of numeraire/probability when asset pays dividends

So I was looking at Margrabe's formula for exchange call options in the book 'Mathematical Methods for Financial Markets' (Jeanblanc, Chesney, Yor), and I was having trouble justifying their change of ...
1
vote
1answer
73 views

Integrating Brownian Motion [closed]

I just wonder how to integrate standard Brownian motion on time interval $(t, T)$. Let $Z$ be a standard Brownian motion with mean $0$ and standard deviation $1$, with $dZ^2 = dt$. How to derive the ...
2
votes
0answers
38 views

Volatility of a perpetuity $E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\vert\mathcal{F}_t\Big]$

Let $z$ be a brownian motion, let $\mathcal{F}$ be the filtration it generates. For $k>0$ and $m\in\mathbb{R}$, I define the process $Y$ as $$Y_t=E\Big[\Big(\int_0^\infty e^{-ks+mz_s}ds\Big)^\eta\...
0
votes
1answer
54 views

Event Occurs Almost Surely

Consider an uncountably infinite space, an infinite coin-tossing. Let $(\Omega,\mathcal{F},\mathbb{P})$ be the probability space. If a set $A\in\mathcal{F}$ satisfies $\mathbb{P(A)=1},$ then we say ...
30
votes
4answers
8k views

What is a stationary process?

How do you explain what a stationary process is? In the first place, what is meant by process, and then what does the process have to be like so it can be called stationary?
1
vote
1answer
73 views

Calculating the value of Beta - Martingales

Assume a risk free bond $B_t$and the stock St follow the dynamics of the Black & Scholes model. (with interest rate r, stock drift $\mu$ and volatility $\sigma$). Find $\beta$ such that the ...
11
votes
5answers
945 views

Free or Relatively Less Pricey Quant Finance courses online

I am trying to figure out what all online Quant Finance courses are out there which are free or relatively less pricey? CQF is not less pricey Financial Engineering course on Coursera - Not so great ...
3
votes
0answers
57 views

Stochastic differential of a time integral

Suppose that $S$ follows a geometric brownian motion: $$ dS(u) = r S(u)du + S(u)\sigma(u,S(u))dW(u) , $$ with $r$ a deterministic constant, and let the process $Z$ be defined by: $$ Z(t) = \int_0^t ...
3
votes
1answer
116 views

Bond Option Hedging

(My question) Please show me how to solve from (2) to (4) with computation processes. These are too difficult to solve. Thank you for your help in advance. (Cross-link) I have posted the same ...
3
votes
1answer
78 views

Why is the value of the Brownian motion bounded by the maximum value of this square difference?

This comes from Taleb and Madeka's paper (https://www.academia.edu/39998351/All_Roads_Lead_to_Quantitative_Finance_Response_to_Clayton_?auto=download) regarding arbitrage restrictions on binary ...
2
votes
2answers
57 views

Cumulative Integration with regard to Vasicek Model's Bond Price and its Forward Price

(My Question) Please show me how to compute the following expectation with its computation process. Besides, $B_t$ is S.B.M. $$E\left[ \exp \left( - \int^T_t \int^u_0 \sigma e^{-b(u-s)} d B_s du \...
2
votes
1answer
68 views

The Riccatti equation for The Cox-Ingerson-Ross Model

(My Question) I went through the calculations halfway, but I cannot find out how to calculate the following Riccatti equation. Please tell me how to calculate this The Riccatti equation with its ...
2
votes
0answers
58 views

The Ho-Lee Model (1986)'s Bond Call Option Pricing [closed]

(My Question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (the details in this ...
2
votes
0answers
39 views
2
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0answers
50 views

The Ho-Lee Model (1986)

(My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides, $W_t$ is a S.B.M. (Thank you for your ...
3
votes
1answer
82 views

HJM in infinite dimensions

I recently started reading Filipovic's Consistency problems for HJM interest rate models and came across the Musiela reparametrization $$r_t(x)=f(t,x+t)$$ so the forward curve can be thought of as a ...
12
votes
1answer
472 views

Transformation of Volatility - BS

I have recently seen a paper about the Boeing approach that replaces the "normal" Stdev in the BS formula with the Stdev \begin{equation} \sigma'=\sqrt{\frac{ln(1+\frac{\sigma}{\mu})^{2}}{t}} \end{...
3
votes
0answers
54 views

Discretisation of OU (mean reverting) process with a jump process

I have a question about how to apply the Euler approximation on OU process with a jump process. The stochastic process $X_t$ has dynamic $$dX_t=\alpha(\beta-X_t)dt+\sigma dW_t+dY_t$$ where $dY_t=...
0
votes
1answer
62 views

Not clear on an SDE solution example on YouTube [closed]

This video, from about 6 to 12 minutes: https://youtu.be/qdbkvD4N-us I feel like I’m following him ok, but then at the end his f(t,B(t)) has become an f(t,x) and there is no B(t) in his result, so it ...
3
votes
1answer
203 views

Bayes Theorem with change of measure

Tomas bjork- arbitrage theory in continuous time. Appendix B, proposition B41 says: The proof is not clear to me. Thanks to Gordon's comment below of $E^Q (X/G)$ being $G$ measurable, I think the ...
3
votes
2answers
63 views

For Ito Integrals with respect to a Brownian motion, why would the amount of stock held be a stochastic process?

Suppose that $B$ is a Wiener process and suppose $H$ is a right-continuous, adapted, and locally bounded process. Suppose $$\int_0^t H dB$$ is the Ito integral of $H$ with respect to the Wiener ...
2
votes
1answer
56 views

$\beta = 1$: Simulation of SABR and whether a solution is *exact*

Quick question regarding the conditional distributions (SABR is just an example here) Consider $$dS_t = \sigma_tS_tdW_t$$ $$d\sigma_t = \alpha\sigma_tdV $$ $$dW_tdV_t=\rho dt$$ Hence a SABR process ...
0
votes
2answers
83 views

Advantage of continuous time stochastic calculus over discrete version?

I'm new to the stochastic calculus, and I keep converting the continuous stochastic differential equation to its counterpart in discrete time, such as the autoregressive models. I wonder in practice, ...
2
votes
2answers
489 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 ...
1
vote
0answers
69 views

Taylor expansion of stochastic variables with dynamics of the form $dX_t=b(\sigma_t,X_t)dW_t$

https://www.math.nyu.edu/~cai/Courses/Derivatives/compfin_lecture_5.pdf In the above document stochastic taylor expansions are nicely explained. Let us now consider a typical SDE model in finance ...
4
votes
1answer
121 views

Bond-price dynamics in the Vasicek model

Hello I am studying about interest rate modeling There is one good source about Vasicek (link: https://web.mst.edu/~bohner/fim-10/fim-chap4.pdf). However there is one equation that I try but unable ...
1
vote
2answers
114 views

Why is Delta Hedging a Hedge Against Short Position? [closed]

Consider the usual one-period binomial model. The delta-hedging formula, following Shreve's convention, is: $$\Delta_0=\frac{V_1(H)-V_1(T)}{S_1(H)-S_1(T)}$$ Shreve states: "The agent has ...
1
vote
1answer
79 views

Example of complex structured products on FX market?

Lately I have been working a lot with the vol smile and different stochastic volatility models with FX forwards data. Now I want to work with pricing examples through simulations. Can you suggest some ...
3
votes
1answer
90 views

Problem at deriving Bachelier formula with interest rates

In the Bachelier model, I have difficulties with a certain step. I want to figure out the distribution of $S_T$, which is the price process in the Bachelier model. So far I could state that ($\mathbb{...
1
vote
0answers
48 views

Symbol “.” in the derive of Quanto Adjustment

I am reading "Analysis, Geometry and Modeling in Finance". In section 2.10.2 which derives the quanto adjustment, it states that (in page 46) by definition the process $S_t^{d/f}S_t^f$ is the foreign ...
3
votes
1answer
128 views

What is the easiest way to learn Option pricing with PDE?

I was reading about Ito's formula and Girsanov theorem, but I am still struggling to grasp how in reality these are combined to compute the price of an option. What are the main source to understand ...
1
vote
1answer
41 views

Why the variance of a process is $\left( \frac{dS_T^2}{dt}\right)^2$?

Consider an Ito process $dS_t = f(t,S_t) dt + g(t,S_t)dW_t $ What is the reason that we can compute the variance as: $\sqrt{VaR(S_t)} = \frac{(dS_t)^2}{dt}$
2
votes
0answers
105 views

Term structure equation in the Vasicek model

Consider the SDE $$dr_t = (b-ar_t)dt +\sigma dW_t, \text{with } a; b > 0.$$ Let $$F(t; r) = E(\exp(-\int_{t}^{T}r_sds)| r_t = r).$$ (F can be interpreted as price of a zero coupon bond with ...