Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

141 questions with no upvoted or accepted answers
Filter by
Sorted by
Tagged with
2
votes
0answers
54 views

How to calculate the multiple integrals where the integral domain is based on the sum of normal distribution random variables?

The integral is shown below: And how to use python to calculate pi (better if we don't need to code for each pi)?
2
votes
0answers
101 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 ...
2
votes
0answers
73 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 ...
2
votes
0answers
190 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 ...
2
votes
0answers
41 views

How does this transformation for Euler Scheme in mean reverting SDEs alleviate instability?

I saw this text in the book - Interest Rate Modelling by Andersen volume 1 on Page 112: I am unable to understand: How does instability arise when we use the Euler scheme on X(t)? What change does ...
2
votes
0answers
68 views

Novikov condition for Vasicek process

Suppose that we have a money account $S^{(0)}$ with dynamics \begin{align} dS^{(0)}_{t} = r_{t} S^{(0)}_{t}\, dt, \end{align} where \begin{align} dr_t = a(b-r_t)\, dt + \sigma_{r} \, dW_t^{(0)}. \...
2
votes
0answers
78 views

Milstein discretization of the CIR process

Given the CIR process $\ dX_t = (a − bX_t ) dt + \sigma \sqrt{X_t}dW_t$ - I want to show that its Milstein scheme is $\ X_{i+1} - X_i = ((a − bX_i) - 0.25\sigma^2)\Delta + \sigma\sqrt{X_i}\sqrt{\...
2
votes
0answers
311 views

For an Ito Process, $d\ln{X} \neq \frac{dX}{X}$ and $(d\ln{X})^2 = (\frac{dX}{X})^2$, but $d\ln{X} \neq \pm \frac{dX}{X}$

In normal calculus we can write $d\ln{x} = \frac{dx}{x}$ since there is no quadratic variation to deal with. This isn't true for stochastic processes, and Ito's Lemma is used to calculate $d\ln{X}$. ...
2
votes
0answers
146 views

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 ...
2
votes
0answers
148 views

SDE of futures price under non-constant interest rate and volatility process

I'm trying to figure out the form of the SDE of futures price under the risk neutral measure, when stock price follows GBM:             &...
2
votes
0answers
61 views

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 ...
2
votes
0answers
296 views

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 : $C_t(T,T+d)$ ...
2
votes
0answers
76 views

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 $X_t^i$...
2
votes
0answers
190 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^{-...
2
votes
0answers
83 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 ...
2
votes
0answers
377 views

Normalized Gains Process is a Q-Martingale - Proof and Intuition

I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked ...
2
votes
0answers
72 views

Laplace Exponent of a Jump-Diffusion Process

I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value: \begin{equation} V_t = ...
2
votes
0answers
208 views

How to understand the integral in the Girsanov theorem?

Let $W^P$ be a $d$-dimesional $P$-wiener procss. Define $L_t = > e^{\int_0^t \phi_s^T dW_s^P - \frac{1}{2} \int_0^t \| \phi_s\|^2 > ds}$.Assuming $E^PL_T = 1$, then the measure given by $dQ = ...
2
votes
0answers
662 views

Stochastic Leibniz rule

We have the following single-factor HJM model $$d_tf(t,T)=\sigma(t,T)dW_t+\alpha(t,T)dt$$ $$f(t,T)=f(0,T)+\int_0^t\sigma(s,T)dW_s+\int_0^t\alpha(s,T)ds$$ The discounted T bond is then \begin{align} Z(...
2
votes
0answers
182 views

Quadratic variation

The following question is more math than quant, but since it arises from a mathematical finance textbook, I've figured the good people in this sub might be able to help me. So here goes. In the 3rd ...
1
vote
0answers
50 views

Replicating portfolio in the Heston model

Given the Heston model $$dS_t=\mu S_tdt+\sqrt{\nu_t}S_tdB_{1,t}\\ d\nu_t=k(\theta-\nu_t)dt+\eta\sqrt\nu_tB_{2,t}$$ how should the replicating portfolio $V_t$ for the derivative $F_t$ be composed? I ...
1
vote
0answers
50 views

Milstein Scheme for Jump-Diffusion models

Hey in this report (Approximation of Jump Diffusions in Finance and Economics by Bruti-Liberati and Platen) is described the Milstein formula (3.5) for simulation SDE with jump component. How it is ...
1
vote
0answers
56 views

How to solve/evaluate an Ito Integral?

I'm given the following Ito integral which I need to evaluate. $Z_t$ is the Brownian motion. My problem is that online resources aren't making much sense because of the notation, so it ends up leaving ...
1
vote
0answers
92 views

A Cauchy Problem: How can I find the following solution?

Suppose that we have the following time-dependent partial differential equation: \begin{equation} \frac{\partial V(t, x)}{\partial t} = \frac{1}{2}\frac{\partial^2 V(t, x)}{\partial x^2} - wxV(t, x), \...
1
vote
0answers
38 views

Do we model stock prices using non-Markovian processes in continuous setting?

In a continuous setting, is it common to model stock prices using non-Markovian processes ? If so, do you have some examples of models ? Or is Markovianity something "embedded" in the ...
1
vote
0answers
98 views

Quanto put hedge\ replication with a brownian motion

Consider $d B_{us}(t)=r_{us} B_{us}(t) dt\\dX(t)=X(t)(r_{us}-r_J)dt+X(t)\sigma^T_J dW(t)\\d B_J(t)=r_{J} B_{J}(t) dt\\dS_J(t)=S_J(t)(r_J-\sigma^T_X\sigma_J)dt+S_J(t)\sigma^T_J dW(t)$ where the $\sigma$...
1
vote
1answer
146 views

Trouble With Applying Ito's Lemma

I am having trouble applying Ito's Formula to the following: Let $Z_t = W_{1t}^2 e^{W_{1t}+ \int_0^t W_{3s}dW_{2s}}$. Find $dZ_t$. $W_1,W_2,W_3$ are independent Brownian motions. I know the formula ...
1
vote
0answers
257 views

Reduced form of credit model

The price for a simple credit bond, where a credit event is modeled as the first jump of a Poisson process $N$, with stochastic hazard rate $\lambda$, is given by $$P_t = P(t, \lambda, N)$$ such that, ...
1
vote
0answers
45 views

Moments of a SDE: a detail on the information set

Very basic questions. Let $(z_t)_{t \geq 0}$ be a standard Brownian motion and let $$dS_t = \mu S_t dt + \sigma S_t dz_t.$$ When we write $E\left( S_t \right)$, do we mean $E\left( S_t \big| F_0 \...
1
vote
0answers
40 views

Change of measure to get a determined drift

let's say I have a real stochastic process $dX_t=dt+\frac{1}{B_t}dB_t$ on $[0,T]$, with $B_t$ Brownian in $\mathbb{P}$ (not centered in 0) in $[0,\tau]$ with $\tau$ some adequate stopping time that ...
1
vote
1answer
158 views

First Hitting Time and Monte Carlo simulation

I am interested in implementing a Monte Carlo simulation in Python of a first hitting time (first passage time) of an Ornstein-Uhlenbeck process (or similar). Specifically interested in fatter tails ...
1
vote
0answers
29 views

Change of numeraire between t1-forward mesure and t2-forward mesure

Let denote $\mathbb{Q}_{t_1}$ the $t_1$-forward mesure associated to zero coupon bond $B(.,t_1)$. Let denote $\mathbb{Q}_{t_2}$ the $t_2$-forward mesure associated to zero coupon bond $B(.,t_2)$. I am ...
1
vote
0answers
76 views

How to solve this particular PDE using Feynman-Kac formula?

I have to solve the PDE $$ \begin{align} \frac{\partial F}{\partial t} + \frac{1}{2}\frac{\partial^2 F}{\partial x^2} + \frac{1}{2}\frac{\partial^2 F}{\partial y^2} + \frac{1}{2}\frac{\partial^2 F}{\...
1
vote
0answers
81 views

Option that never expires

I have been struggling with the problem below for quite some time now. I really don't know how to approach it. All I could think of is to use the Black-Scholes formula with $T \rightarrow \infty$, ...
1
vote
0answers
146 views

Dynamic programming and Bellman equation to obtain the maximum

This is the problem of Marhsall (1992) "Inflation and Asset Returns in a Monetary Economy" and Balvers and Huang (2009) "Money and the C-CAPM" Suppose an endowment economy where the representative ...
1
vote
0answers
53 views

Simulate correlated Brownian motions conditioned on future state(s)

Consider a model defined by 2 geometric Brownian motions $$dY_{1}(t) = \sigma_{2} Y_{1}(t)dW_{1}(t)$$ $$dY_{2}(t) = \sigma_{2} Y_{2}(t)dW_{2}(t)$$ with $Y_{1}(0) = y_{1}$, $Y_{2}=y_{2}$ and $dW_{1}(...
1
vote
0answers
75 views

On Geometric Brownian motion and Itô's formula

Let $S_t$ be a geometric brownian motion such as $$d S(t) = rS(t)dt +\sigma S(t)dW(t),$$ where $W$ is a standard Brownian motion. With Itô's lemma and formulas $(dt)^2=dtdW_t=dW_tdt=0$ and $(dW_t)^2=...
1
vote
0answers
25 views

Order of expectation versus expectation of order (error terms in Taylor expansion)

Given a payoff function $F(X)$ of a random variable $X$, and a Taylor expansion of $F(X)$ around $X=a$, then the expecation of $F(X)$ can be written as $$ E[F(X)] = F(a) + E[ O((X-a))] $$ Under what ...
1
vote
0answers
38 views

Sub replication of contingent claims

I am trying to prove subreplication cost is the infimum of risk neutral expectation of the contingent claim. I wonder if the following equality holds, which is the key to the proof. $$ \lim_n \inf_{Q\...
1
vote
0answers
106 views

How to determine exchange rate dynamics in currency derivatives

I need some guidance regarding exchange rate dynamics in currency derivatives. Following three dynamics are defined below, $\frac{dS(t)}{S(t)}=\alpha dt+\sigma dW(t)$ ; the stock dynamics in the ...
1
vote
0answers
40 views

Realized Variance as an approximation of the Integrated Variance

Realized Variance is written as $RV_{[0,T]}^{n} = \sum_{j = 1}^{n} r_{j,n}^2$, where $r_{j,n}$ is the log return for the $j$th increment, and $n$ is the total number of sample points in the time ...
1
vote
0answers
56 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 ...
1
vote
0answers
118 views

Valuation of Callable Bonds

Is there any way to price American Callable Bonds (those which can be called on any date before expiration) other than basic CRR interest rate trees, since they won't be accurate enough to give ...
1
vote
0answers
35 views

Stochastic process with determinstic frequency of regime changes

Suppose that I have an OU process. For instance, assume that I want to model the interest rates. Suppose that regime change is known ex ante, and is deterministic in terms of frequency (For instance, ...
1
vote
0answers
625 views

Log Contract payoff function

I can’t get where Dr. Rouah gets payoff function of log contract. Could you please take a look at that? https://frouah.com/finance%20notes/Variance%20Swap.pdf It’s on page 2, section 3. I couldn’t ...
1
vote
0answers
156 views

The conditional expectation of a geometric brownian motion

In this question it states that $$\mathbb{E}[e^{\sigma(W_t-W_s)}|\mathcal{F}_s] = \mathbb{E}[e^{\sigma(W_t-W_s)}],$$ and I assume that $0 \leq s \leq t$. The accepted answer states that this step is ...
1
vote
0answers
70 views

Self financing strategy and repo rate

I was wondering how to adjust the self financing condition when cash borrowing cas be secured by the stock. Suppose the risk-free money account is $B_t$ and there is a risky asset $S_t$. One have ...
1
vote
0answers
126 views

On quadratic covariation

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 ...
1
vote
0answers
120 views

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 ...
1
vote
0answers
145 views

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} $ ...