# Questions tagged [stochastic]

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### Financial Time-Series: Stochastic or Dynamic?

I have learned how some methods of constructing predictive models of financial time-series involves assumptions of stochasticity. For example, reinforcement learning utilizes the Markov Decision ...
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### Confusion about how price of a contingent claim at time 1 could give arbitrage

I have been reading the book Tomas Bjork's Arbitrage Theory in Continuous Time and could not understand how there could be arbitrage if the price of a contingent claim is not $X$. To give some ...
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### How does the inclusion of stochastic volatility in option pricing models impact the valuation of exotic options?

Been lurking this forum for quite some time and there’s this concept I can’t wrap my head around: How does the inclusion of stochastic volatility in option pricing models impact the valuation of ...
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### Necessary conditions to ensure that stochastic integral is a normal variable

Let $\left(W_t\right)_{t\geq 0}$ be a Brownian motion with respect to filtration $\mathbb{F}=\left(\mathcal{F}_t\right)_{t\geq 0}$. Let $\left(\alpha_t\right)_{t\geq 0}$ be an $\mathbb{F}$-adapted ...
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### Calibration of Local or Stochastic Volatility Models to Prices vs Implied Volatilities

As the title suggests, what is the difference between calibrating an option pricing model (say the Heston model) to market option prices instead of computing their implied volatilities using Black-...
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### Derive the Probability of Default (PD) of private companies with Merton Model

Do you know a well used method how to calculate the PD of private companies using the Merton Model. The main challenges I am facing is to get the appropriate volatility of the assets and the drift. ...
1 vote
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### 4th Order Brownian Motion Martingale [closed]

I understand the first order MG of brownian motion is Bt.. the second order is Bt^2 - t and the third order is bt^3 - 3tBt. How can I find the fourth and beyond order of a Brownian Motion Martingale?
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### What does a non-stochastic limiting shrinkage function mean?

I'm reading the paper "The Power of (Non-)Linear Shrinking: A Review and Guide to Covariance Matrix Estimation" by Ledoit and Wolf (2020). When a function that is used to transform the ...
334 views

### Simple Black-Scholes alternatives

I work at an accountancy firm and we use Black-Scholes to value equity in private companies that has option like features. The equity we typically value is akin to deeply out of the money European ...
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### Integral of Function of Brownian Motion w.r.t Time (Context: Computing Quadratic Variation)

I am looking to compute the quadratic variation of $$S_t = S_0e^{\sigma B_t}$$ where $B_t$ is Brownian Motion. Applying Itô's lemma, I having the following $$(dS_t)^2 = S_0^2\sigma^2e^{2\sigma B_t}dt$$...
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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{...
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### 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 ...
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### 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.
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### 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. ...
1 vote
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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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