# Questions tagged [stochastic]

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### How to calculate mean and variance in Vasicek Model

In the Vasicek model, the short rate of interest under the risk-neutral probability measure is given by: where k, θ, σ > 0 and W is a standard Brownian motion. Consider the related process where ...
63 views

### Distribution of Stochastic Integral Example

I am looking for help on justifying how the integral $$\int_{0}^{t} (t-s) \, dW_{s}$$ is normally distributed. I realize that the general fact that Ito Integrals with deterministic integrands are ...
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### Sum of discretely sampled BM

If an underlying follows lognormal GM with no drift $dS_t = \sigma S_t dW_t$ and $A_N = \Sigma_{i=1}^{N} S_{t_i}$. How to compute variance of $A_N$?
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### 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 ...
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### Inflation Option Modelling Approaches

I am trying to come up with a simplistic inflation option model to get a sense of the materiality of some inflation-indexed contracts containing inflation guarantees. I have a stochastic nominal IR ...
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1 vote
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### Why does an autocall on a linear payoff have vega?

Consider a (stochastic) linear index, say $I(t)$, in that it grows at the risk free rate (with some volatility of course). There exists a maturity date $T$ on which I receive $I(T)$; however there is ...
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### What are the advantages and limitations of predicting future stock prices using stochastic differential equations?

Recently I came across the following stochastic differential equation that "predicts" the value of a given stock: dS_t = \mu S_t dt + \sigma S_tdW_t \\ S_t(0) =S_0 \end{...
101 views

### mixing fractional Brownian motions

Given two Brownian motions $W_t^1, W_t^2$, we can have them correlated by $$W_t^1 = \rho W_t^2+\sqrt{1-\rho^2}Z_t$$ where $W_t^{2}$ and $Z_t$ are independent of each other. My question then: is there ...
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### Asset Pricing and Stochastic Discount Factor: Do well-informed investors only buy efficient portfolios?

I'm currently dealing with the following question: In Asset Pricing, well-informed investors know about the concept of the efficient frontier. Does this mean that they only invest in portfolios that ...
60 views

### difficulty pricing options using stochastic volatility

can someone kindly explain why it was difficult to obtain an explicit formula for pricing options under stochastic volatility. Thanks alot.
58 views

### Find Arithmetic Brownian Motion's transition density

Consider the following stochastic differential equation, an Arithmetic Brownian Motion: 𝑑𝑆(𝑡) = 𝑟 𝑑𝑡 + 𝜎 𝑑𝑊(𝑡) . Find its solution, integrating from t to T, then find its transition density. ...
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### How can I prove that the solution to the Heston SDE is a Markov process?

Consider the Heston model expressed as \begin{align} dS_t &= \mu S_t dt + S_t \sqrt{V_t} \big(\rho dW_t^{(1)}+\sqrt{1-\rho^2}dW_t^{(2)} \big); \tag*{(1)} \\ dV_t &= \kappa(\theta - V_t)dt + \...
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