All Questions
Tagged with finance stochastic-differential-equations
54
questions
1
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0
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30
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For $A_t= B_t^l Y_t \mathbb{E}(\frac{A_T}{B_T^l Y_T} \mid \mathcal{F}_t)$, find the values of $l$ to replicate $A_t$ by a self financing portfolio $X$
Background:
In attempting to resolve the below problem, I have arrived at an answer that appears to counter intuition (and therefore, I suspect that it is wrong). I would appreciate assistance in ...
- 4,331
1
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0
answers
37
views
ARCH-Vasicek model closed-form solution
I understand how we can obtain the solution of Vasicek model $dr_t=\alpha(\mu-r_t)dt+\sigma dW_t$:
$$
r_t=r_0e^{-\alpha t}+\mu(1-e^{-\alpha t})+\sigma\int_0^te^{-\alpha(t-s)dW_{s}}
$$
This easily ...
- 11
3
votes
1
answer
72
views
Characteristic function of a random variable by Fourier transform
this is character function in probability theory
$$\phi(u)=\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}ux}f(x)\mathrm{d}x$$
Let an asset price $S_t$ (e.g. a stock) be modeled with a Geometric ...
- 33
1
vote
1
answer
41
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Returns of an asset in risk-neutral measure and its PDE
I am a bit confused regarding how an asset returns in a risk-neutral measure (say $\mathbb{Q}$), and subsequently its Black-Scholes-esque PDE.
In class, we learned about the approach to take when ...
1
vote
1
answer
126
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Is Heston model an affine jump-diffusion?
In Duffie, Pan and Singleton's paper "Transform Analysis and Asset Pricing for Affine Jump-diffusions" (2000) they define affine jump-diffusion (AJD) a process of the following form:
$$dX_t=\...
- 392
2
votes
0
answers
328
views
Are the Black-Scholes equation models still considered accurate predictors for asset pricing? [closed]
Sorry if this equation is not phrase in precise mathematical form. I am open to suggestions to improve the explanation, and I have tried to formulate the problem as precisely as I could.
I was talking ...
- 2,531
1
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1
answer
87
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Covariance of Black-Scholes
Let us assume that $W^1, W^2$ are 1-dimensional Brownian motions such that $(W^1, W^2)$ has a jointly normal distribution, and denote $c_t:=\operatorname{cov}(W^1_t,W^2_t)$ their covariance at time $t$...
- 398
1
vote
1
answer
148
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Delta and Gamma computation using the likelihood ratio method in Black Scholes
Context:
We consider a geometric brownian motion: $$dS_t = rS_t dt + \sigma S_t dW_t$$ with $S_0 = x$.
And we are interested in the function $$ u(x) = E(g(S_T)) = \int_{\mathbb{R}^+} g(s) p(s,x) ds $$
...
- 62
3
votes
0
answers
87
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Understanding proof of stability in an stochastic differential equation
I am currently doing my thesis on stochastic models applied to interest rates. I am partly basing myself on the article "Stability Behavior of Some Well-Known
Stochastic Financial Models" of ...
0
votes
1
answer
61
views
Application of Ito's lemma to consol price process
I have a question related to this paper https://www.jstor.org/stable/2245302.
Given a process for the short rate $r$, the authors consider the price process $Y$ for a consol bond that satisfies
\begin{...
- 1
0
votes
1
answer
370
views
How to derive the following in Ho Lee Model?
I am trying to understand the proof of the zero bond price $Z(t)$ of the Ho-Lee model which is the unique solution of the following SDE:
$$ dZ(t) = -Z(t) [ \sigma(T-t)dW(t) + [ \int_t^T \alpha(t,u)du -...
- 322
2
votes
1
answer
200
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How to solve SDE: $ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$
Problem:
I would like to find a solution to the following SDE:
$$ dX_t = \bigg(\sqrt{1+X_t^2}+\frac{1}{2}X_t \bigg)dt + \sqrt{1+X_t^2} dW_t$$
Calculations:
by Ito's lemma: $$df(t,X_t) = \Bigg(f_t(t,...
- 85
1
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0
answers
400
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Prove that the stochastic process $s_t$ follows a normal distribution where the mean and the variance are functions of time in each case.
The two basic models of finance are the following:
$\textbf{The Samuelson SDE (aka Black - Scholes - Merton model):}$ Suppose that $Z=\left(Z_t, t\in\mathbb{R}^{+}\right)$ is a Wiener process (aka ...
- 257
0
votes
1
answer
88
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How to derive the HJM drift condition?
I'm trying to derive the Heath Jarrow Morton drift condition (from Björk, page 298) and this equation is the part that I'm not able to derive:
$$ A(t,T) + \frac{1}{2} ||S(t,T)||^2 = \sum_{i=0}^d S_i(t,...
- 322
3
votes
1
answer
198
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Technical difficulties with degenerate PDEs
Crossposted at Quantitative Finance SE
I have seen lot of discussions in this Math. S.E. platform about 'degenerate partial differential equations'. But I still unclear about the 'technical ...
- 775