All Questions
Tagged with finance stochastic-integrals
46
questions
1
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0
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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
0
votes
1
answer
372
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Expected value of Ornstein-Uhlenbeck process
In the paper "The Impact of Jumps in Volatility and Returns" by Nicholas Polson, Bjorn Eraker, and Michael Johannes (2003), the authors state in footnote 6 on page 1273 that, given an ...
- 392
1
vote
1
answer
40
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Extracting a conditional density from a formula involving 2 stochastic integrals
I have a problem coming from a financial maths application, that involves trying to
extract the conditional density of a variable expressed as an integral over a Brownian motion, conditioned on ...
- 51
1
vote
1
answer
106
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Expectation of stock price that follows a stochastic differential equation
Let the price $S_t$ of an asset satisfy $dS_t = \alpha (\mu - \ln{S_t})S_tdt + \sigma S_t dW_t$, where $W_t$ is a Brownian motion.
I managed to show that $x_T = x_te^{-b(T-t)} + \frac{a}{b}(1 - e^{-b(...
- 167
0
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0
answers
122
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Pathwise differentiation under integrated measurement (stochastic volatility) models.
Setup
Consider the (general) stochastic volatility model
\begin{align}
\mathrm{d}X_t &= \mu^X(X_t)\mathrm{d}t + \sigma^X(X_t)\mathrm{d}W^X_t \\
\mathrm{d}Y_t &= \mu^Y(X_t)\mathrm{d}t + ...
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
0
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1
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170
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Pricing a call option in the Black Scholes Market - calculation steps
I am working on computing the price of a standard European call option under a Black-Scholes market.
Using knowledge of the payoff, I can split the calculation into:
$
e^{-rT}(E[S_t] \mathbb{1}_{S_T &...
- 145
2
votes
1
answer
1k
views
How to solve the simple Ito (stochastic) integral over Brownian motion via the Ito formula?
Question:
Let $X(t)=\int_0^t W(s)dW(s)$ where $W(t)$ is the Brownian motion (Wiener process). What function $f(X(t))$ of $X(t)$ can be used with the Itô formula to get the explicit result $X(t)=\frac{...
- 153
0
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0
answers
33
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Stochastic differential notation in Standard Brownian Market Models
I am trying to become familiar with stochastic integration and stochastic differential notation. I tried to do the following little exercise. In my lecture notes the risky Asset is defined in the ...
- 63
3
votes
1
answer
282
views
Quadratic variation of a stochastic integral w.r.t. a local martingale
I am trying to prove the (seemingly simple) property: for a continuous local martingale $M$ and an $M$-integrable process $H$, the quadratic variation $\langle\int H\,dM\rangle$ of $\int H\,dM$ is ...
- 249
4
votes
2
answers
348
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Finding Option Probability Density Using Local Volatility from Dupire Model
This question is different than https://quant.stackexchange.com/questions/31050/pricing-using-dupire-local-volatility-model
I am reading about the Dupire local volatility model. I have found ways to ...
2
votes
1
answer
227
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False proof: Every continuous-time financial market has arbitrage.
We know that many continuous-time financial markets do not have arbitrage. But I have created a proof which shows a very large class of them has arbitrage, so there must be a mistake in the proof. ...
- 1,262
0
votes
1
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160
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Zero coupon bond price dynamics under HJM Model
I am reading this article: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=3330240
and am trying to figure out what is written in the appendix, pages 22 and 23
They wrote:
\begin{align*}
R_j(t) &...
- 133
6
votes
1
answer
343
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Advantages of pathwise stochastic integrals over standard Itô integrals
I am currently reading about Föllmer's construction of a stochastic Integral that is defined in a pathwise sense. But I am not sure what exactly the purpose of such a construction is. The main ...
- 335
3
votes
1
answer
78
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Compute $\mathbb{E}(\int_{0}^{T} |B_{t}|^{2} dt)$
For $(B_t)_{t\geq0}$ a standard brownian motion and $T$ the first time the standard brownian motion hits $1$ or $-1$ I have to compute $\mathbb{E}(\int_{0}^{T} |B_{t}|^{2} dt)$. My lecturer has given ...
- 141