All Questions
Tagged with finance stochastic-calculus
214
questions
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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
4
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43
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Question regarding a value in a one-period model
There is a script at my university (can't post it for copyright reasons) for a course on discrete time financial mathematics. I decided to give it a try and found this problem:
Consider a financial ...
- 51
1
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37
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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
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19
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Boundary hitting and arbitrage in Reflecting Brownian Motion approximation
I am facing the following quantitative finance problem.
Suppose that $z$ follows a Brownian Motion centered at $0$. There are two boundaries that define a no-arbitrage region: $[-b,b]$. Trades take ...
- 21
1
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38
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Hedging a long position, multiple periods (Steven E. Shreve, Stochastic Calculus for Finance I)
I have attempted to answer this question, but I'm unsure if I'm on the right track. I've started with setting the value of the portfolio at time 3 to the desired value:
$$
X_3(HHH) = X_3(HHT) = X_3(...
3
votes
1
answer
72
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Characteristic function of a random variable by Fourier transform
this is character function in probability theory
$$\phi(u)=\int_{-\infty}^{\infty}\mathrm{e}^{\mathrm{i}ux}f(x)\mathrm{d}x$$
Let an asset price $S_t$ (e.g. a stock) be modeled with a Geometric ...
- 33
1
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1
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63
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Conditional distribution of Brownian motion given the passage time?
I am stuck with this question. Suppose a stock price follow Brownian motion starting at price $p$. Denote by $\tau_r$ the passage time reaching level $r$ with $r<p$. What would be the distribution ...
- 197
-1
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1
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94
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The partial derivative of a call option with respect to $t$ [closed]
In Black-Scholes related computations, why do we not treat the stock price $S$ as a function of $t$ when taking partial derivatives with respect to $t$? For example, if
$$c(t,T)=SN(d_1)-Ke^{-r(T-t)}N(...
- 285
0
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111
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Is this the correct proof of Proposition 10.23 in Björk's ''Arbitrage theory in continuous time''?
I want to prove Proposition 10.23 from Tomas Björk's ''Arbitrage theory in continuous time'' in the snippet below.
My attempt: For simplicity, assume everything is one-dimensional, with one risky ...
- 135
2
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2
answers
134
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An exercise about replicable random variables
The following question is an exercise which I have in my course for Financial Mathematics:
Let $h:[0,\infty) \to [0,\infty)$ be twice differentiable with $h'' \geq 0$. Establish, with the help of ...
- 109
0
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1
answer
57
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HJM model forward rate explosion
In Steven Shreve's excellent book, page 436, it says the forward rate $f(t,T)$ of the Heath-Jarrow-Morton (HJM) model explodes as $t\to T-$. The attached screenshot shows the calculation where the HJM ...
- 462
0
votes
1
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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
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1
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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:
$$dX_t=\...
- 392
1
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1
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87
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Covariance of Black-Scholes
Let us assume that $W^1, W^2$ are 1-dimensional Brownian motions such that $(W^1, W^2)$ has a jointly normal distribution, and denote $c_t:=\operatorname{cov}(W^1_t,W^2_t)$ their covariance at time $t$...
- 398
1
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1
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148
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Delta and Gamma computation using the likelihood ratio method in Black Scholes
Context:
We consider a geometric brownian motion: $$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 $$
...
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