The stochastic-calculus tag has no wiki summary.
21
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1answer
2k views
What is the role of stochastic calculus in day-to-day trading?
I work with practical, day-to-day trading: just making money. One of my small clients recently hired a smart, new MFE. We discussed potential trading strategies for a long time. Finally, he expressed ...
16
votes
3answers
2k 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?
10
votes
2answers
556 views
Missing step in stock price movement equations
Assuming a naive stochastic process for modelling movements in stock prices we have:
$dS = \mu S dt + \sigma S \sqrt{dt}$
where S = Stock Price, t = time, mu is a drift constant and sigma is a ...
10
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3answers
686 views
Solving Path Integral Problem in Quantitative Finance using Computer
I've asked this question here at Physics SE, but I figured that some parts would be more appropriate to ask here. So I'm rephrasing the question again.
We know that for option value calculation, path ...
9
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0answers
321 views
Law of an integrated CIR Process as sum of Independent Random Variables
It is known (see for example Joshi-Chan "Fast and Accureate Long Stepping Simulation of the Heston SV Model" available at SSRN) that for a CIR process defined as :
$$dY_t= \kappa(\theta -Y_t)dt+ ...
8
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3answers
584 views
Deterministic interpretation of stochastic differential equation
In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by ...
6
votes
2answers
253 views
Obtaining characteristics of stochastic model solution
I want to use the following stochastic model
$$\frac{\mathrm{d}S_{t}}{ S_{t}} = k(\theta - \ln S_{t}) \mathrm{d}t + \sigma\mathrm{d}W_{t}\quad (1)$$
using the change in variable $Z_t=ln(S_t)$
we ...
6
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1answer
354 views
How to perform basic integrations with the Ito integral?
From the text book Quantitative Finance for Physicists: An Introduction (Academic Press Advanced Finance) I have this excercise:
Prove that
$$
...
6
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1answer
171 views
Upper bound concerning Snell envelope
Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
6
votes
2answers
580 views
How do practitioners use the Malliavin calculus (if at all)?
This question is inspired by the remark due to Vladimir Piterbarg made in a related thread on Wilmott back in 2004:
Not to be a party-pooper, but Malliavin calculus is essentially useless in ...
6
votes
1answer
187 views
Why does the price of a derivative not depend on the derivative with which you hedge volatility risk?
I'm trying to derive the valuation equation under a general stochastic volatility model.
What one can read in the literature is the following reasoning:
One considers a replicating self-financing ...
6
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0answers
263 views
Probability distribution of maximum value of binary option?
A binary option with payout \$0/\$100 is trading at \$30 with 12 hours to
expiration.
Assuming the underlying follows a geometric Brownian motion (hence volatility remains constant), what ...
6
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0answers
211 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}}
...
5
votes
2answers
339 views
What is the average stock price under the Bachelier model?
Let's say stock price follows following process:
$$dS(t) = \sigma dW(t)$$
where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price ...
5
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3answers
435 views
How to use Itô's formula to deduce that a stochastic process is a martingale?
I'm working through different books about financial mathematics and solving some problems I get stuck.
Suppose you define an arbitrary stochastic process, for example
$ X_t := W_t^8-8t $ where $ W_t ...
5
votes
2answers
588 views
How does one go from measure P to Q(risk-neutral) when modeling an asset paying dividends?
I am really having a terrible time applying Girsanov's theorem to go from the real-world measure $P$ to the risk-neutral measure $Q$. I want to determine the payoff of a derivative based an asset ...
5
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2answers
287 views
What is the mean and the standard deviation for Geometric Ornstein-Uhlenbeck Process?
I am uncertain as to how to calculate the mean and variance of the following Geometric Ornstein-Uhlenbeck process.
$$d X(t) = a ( L - X_t ) dt + V X_t dW_t$$
Is anyone able to calculate the mean ...
5
votes
1answer
308 views
What is the forward rate for a Black-Karasinski interest rate model?
I was wondering if anyone could help me with the instantaneous forward rate equation for a Black-Karasinski interest rate model?
I was also after the Black-Karasinski Bond Option Pricing Formula.
4
votes
1answer
197 views
Derivation of Ito's Lemma
My question is rather intuitive than formal and circles around the derivation of Ito's Lemma. I have seen in a variety of textbooks that by applying Ito's Lemma, one can derive the exact solution of a ...
4
votes
1answer
142 views
Non-arbitrage theory and existence of a risk premium
Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
4
votes
1answer
276 views
Change of measure discrete time
Suppose I have a random walk $X_{n+1} = X_n+A_n$ where $A_n$ is an iid sequence, $\mathsf EA_n = A>0$. How to construct a martingale measure for this case?
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0answers
80 views
Simple question concerning Jump process (Lévy process) model for a risky actif price process [closed]
Consider $X= \left( X_t \right)_{t\geq 0}$ is a Lévy process whose characteristic triplet is $\left( \gamma, \sigma ^2, \nu \right)$ and where its Lévy measure is
$$ \nu \left( dx\right) = A ...
3
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1answer
191 views
What is augmented data when simulating stochastic differential equations using Gibbs Sampler?
I am reading this paper on Bayesian Estimation of CIR Model.
Basically, it is about estimating parameters using Bayesian inference.
It estimates this stochastic differential equation:
$$dy(t)=\{ ...
3
votes
1answer
159 views
Integrating log-normal
The usual log normal model in differential form is:
$dS = \mu S dt + \sigma S dX$
where $dX$ is the stochastic part, so
$\frac{dS}{S} = \mu dt + \sigma dX$ (1)
and we normally solve this by ...
3
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1answer
170 views
How to measure a non-normal stochastic process?
If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
2
votes
1answer
155 views
Simple question about stochastic differential
What is the equivalent of product rule for stochastic differentials? I need it in the following case: Let $X_t$ be a process and $\alpha(t)$ a real function. What would be $d(\alpha(t)X_t)$?
2
votes
2answers
424 views
Financial Mathematics - Martingales example
Was hoping somebody could help me with the following question.
Prove that under the risk-neutral probability $\tilde{\mathsf P}$ the stock and the bank account have the same average rate of growth. ...
2
votes
2answers
110 views
Quadratic variation quesiton
Here I have this question
(i) state Ito's formula
(ii) hence or otherwise show that
$\int^t_0B_s dB_s = \dfrac{1}{2}B^2_t -\dfrac{1}{2} t$
(iii) define the quadratic variation $Q(t)$ of Brownian ...
2
votes
1answer
237 views
Regime switching in mean reverting stochastic process
Let you have a mean reverting stochastic process with a statistically significant autocorrelation coefficient; let it looks like you can well model it using an $ARMA(p,q)$.
This time series could be ...
2
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0answers
100 views
Measure change in a bond option problem
This is not a homework or assignment exercise.
I'm trying to evaluate $\displaystyle \ \ I := E_\beta \big[\frac{1}{\beta(T_0)} K \mathbf{1}_{\{B(T_0,T_1) > K\}}\big]$, where $\beta$ is the ...
2
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0answers
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Stochastic discount factor (aka deflator or pricing kernel) and class D processes
When (under what assumptions on the model) does a Stochastic Discount Factor need to be of Class D? What would be the implications if it was not? Is it connected to one of the no-arbitrage notions?
1
vote
1answer
218 views
Does Ito/Malliavin calculus have any applications helpful for direction based trading?
I'm an aspiring computer scientist who want to move into algorithmic trading at some point.
At the moment I'm mostly focusing on courses in machine learning/data analysis etc. but I've noticed that ...
1
vote
3answers
171 views
Why is the CAPM securities market line straight?
Let $\gamma$ be the expected return, in terms of its exponential growth rate, of the market asset. If we set $\gamma=\mu-\sigma^2/2$ as explained by the Doléans-Dade exponential, then the expected ...
