# All Questions

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### Is there any method/module/library to directly solve an SDE in python? Especially if it's just geometric brownian motion

Now, I'm given an SDE $$dS_t = 2S_t\,dt + 4 S_t\,dW_t$$ which I need to find the solution of. I have the solution on paper, but I want to know if there's any way I can solve this directly in python. ...
• 101
1 vote
94 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 ...
• 183
346 views

### Testing the fit of an Ornstein-Uhlenbeck process

I would like to check if a time-series follows an Ornstein-Uhlenbeck process defined by an SDE: $$dX_t - \lambda (\mu - X_t) dt = \sigma dW_t$$ where $\lambda > 0$ is the mean-reversion ...
• 135
49 views

### Why does it hold true that $\theta_{t} d\overline{X}_{t}$ is a local $Q$ martingale if $\overline{X}$ is a local $Q$ martingale

I am learning from Bernt Oksendal's Stochastic Differential Equations and on page 276 Lemma 12.1.6, it is stated that: The existence of an equivalent martingale measure $Q$ on the discounted price ...
• 459
157 views

### COS Method and existence of density

Hey in the COS method we use characteristic function of $\ln{S_T}$ to price european options (by recovering density from characteristic function). But how do we know that density exists? For example I ...
117 views

### Explicit form for forwards Feynman-Kac formula

This might be a simple question, but I'm having trouble with it. Consider the Cauchy problem with final condition. \begin{cases} \frac{\partial u}{\partial t}(t,x) + \mathcal{L}u(t,x) ...
619 views

### How can I learn stochastic process & stochastic calculus in two weeks? [closed]

I am going for an interview for a quant job. The interview will focus on my mathematical knowledge about stochastic process & stochastic calculus, and I believe I will definitely be asked to solve ...
• 11
128 views

### SDF derivation by a stochastic process

I have a stochastic process to model the stochastic discount factor (SDF) with M: $$dM_t = aM_tdt + bM_t d Z_t$$ where, $Z_t$ is a standard brownian motion. How do I show ...
1 vote
232 views

### What is the interpretation if the real world measure $\mathbb P$ is equal to the martingale measure $\mathbb Q$

Out of interest, is there anything noteworthy about a market when its real world measure $\mathbb P$ is actually also its martingale measure. In other words the real world measure $\mathbb P$ is equal ...
• 459
523 views

### Calibrating the Ornstein-Uhlenbeck process with an additional parameter

Firstly I find the spread between two cointegrated time-series $Y_t$ and $Z_t$ by finding the best slope parameter $\beta$ in the equation $spread_t = Y_t - \beta Z_t$ (via Cointegrated Dickey-Fuller ...
• 135
118 views

### What is the difference between the geometric brownian motion and cumulative product of percentage returns?

I wonder why the following code: one using GBM and the other using cumulative product of normally distributed percentage returns slightly different values. ...
185 views

### Summary of Stochastic Derivatives, Integrals, Expectations, and Variances

I wanted to make a summary table of stochastic functions to improve my understanding. Maybe the following should be a wiki page on this site so others can add functions and examples? Does the ...
• 113
141 views

### Is it possibile to use Ito Formula here?

I have this process: $dY_s^y=\alpha(s,Y_s^y)ds + \frac{1}{2}\beta^2(Y_s^y)^2dW_s$ with inital value $Y_s^y=y$. Moreover $\alpha(s,y)$ is a linear function in $y$ and bounded is $s$. I was wondering if ...
• 33
1k views

### Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
• 73
190 views

### Pairs trading by transforming two cointegrated series into a mean-reverting process?

I am slightly confused about the following. Let us assume I have two cointegrated time-series. I would like to model their 'cointegration' by a mean-reverting Ornstein-Uhlenbeck process since if they ...
• 135
250 views

### Expectation of $\int_0^t \frac{1}{1+W_s^2} \text dW_s$ [duplicate]

I am trying to calculate the expectation of $$\int\limits_0^t \frac{1}{1+W_s^2} \text dW_s,$$ where $(W_t)$ is a Wiener process. I was told that the value of this expectation is zero. Can someone ...
• 347
1 vote
54 views

• 499
1 vote
99 views

• 133
37 views

• 21
1 vote
89 views

### Properties of integrated GBM

(I asked this question on MSE but I think it might have more success here) Good day, I was going over some exercises and I stumbled upon a question that, for its solution, requires me to find/...
• 157
269 views

### Stochastic Interest Rates in Option pricing

My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
• 411
732 views

### Idea of using logarithm for solving SDE in Black-Scholes model

In the Black-Scholes model they consider that the stock follows this stochastic differential equation: $$dS = \mu S dt + \sigma S\ dW$$ I was wondering, was it common at the time they work on this ...
• 385
440 views

### Difference between $W_t$ and $X_t= \sqrt{t}Z$

$W_t$ is a brownian motion and $X_t:= \sqrt{t}Z$, where: $Z\sim N(0,1)$. How to show that for a bounded continuous $f$ process, $$U_t = \int_0^t (f(W_s))ds$$ and $$V_t = \int_0^t (f(X_s))ds$$ have the ...
• 63
1 vote
It says assuming a no-uncertainty Weiner process that models stock price: $$\Delta S = \mu S\Delta t$$ Can be rearranged to (after taking the limit of $\Delta t \to 0$... $$\frac{dS}{S}=\mu dt$$ ...