# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Hagan's explanation of the Local Volatility model

Long story cut short: I am asking why the Local Volatility function can be thought of as a function of the underlying, when in fact it appears to be the function of the strike. Long story: The well-...
• 5,366
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### Reference request: Approximate mapping of a multi-factor stochastic volatility model to single-factor stochastic volatility model

I am looking for approaches to transform a more complicated stochastic volatility model such as the one shown in Section 2.2 of Smile Dynamics II to a single-factor model such as the one shown in ...
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### Feymann Kac pde with correlated process

I have to solve the following PDE: \begin{equation} \begin{cases} \dfrac{\partial F}{\partial t}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial x^2}+\dfrac{1}{2}\dfrac{\partial^2 F}{\partial y^2}+\dfrac{1}{...
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1 vote
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### Conditional Expectation of Integral of Squared Brownian Motion - PDE Approach

I am looking to compute the following using Ito's formula. $$u(t,\beta_t) = \mathbb{E}(\int_t^T\beta_s^2ds|\beta_t)$$ Knowing the properties of brownian motion, it is rather easy to show that the ...
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1 vote
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### Put price characterisation

I am reading Shreve's Stochastic Calculus for Finance II: Continuous-Time Models. I am trying to understand the below two concepts: Topic 8.3.3 Analytical Characterization of the Put price on Page ...
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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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### Backset LIBOR contract

Below is an extract from Steven Shreve’s BK 2, Chapt 10: Term Structure models. LINK I am trying to understand Stochastic Calculus from the above book with the help of a Pure Math PhD student. Despite ...
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### Construction of Itos integral

I am trying to understand the below: Question 1 how can [W(t1) - W(t0)] = [W(t1) - W(0)] =[W(t1) - 0] =some positive number be a profit or loss? In this calculation the purchase price is not taken ...
• 11
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### If $\Delta \log(V_{t})$ behaves like the increments of fractional Brownian motion, why do we model the rough volatility as follows

From Gatheral's paper, Volatility is rough and empirical evidence, it is clear that $\big\{\log(V_{t+1})-\log(V_{t})\big\}_{t}$ behaves like the increments of fractional Brownian motion $B^{H}$ with ...
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### Deriving the variance of G2++ Model

I'm studying G2++ Model in Brigo(2007)'s book. The model constructed as follows, $$r(t) = x(t) + y(t) + φ(t), \quad r(0) = r_0\\$$ with the dynamics of $dx(t)$ and $dy(t)$ described by: \begin{align}...
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1 vote
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### Euler Discretization python code

Write the Euler discretization of the 1-dimensional stochastic equation $dXt = b (t, X_t) \space dt + \sigma (t, X_t) \space dW_t$ For this part I would say all right because it is a purely ...
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### Local volatility implied spot vol correlation

I have a question about local volatility models. In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility ...
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### Transformation of local volatility model

Assume we have an SDE $$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$ where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
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### Maximum entropy probability distribution for $S_T$ implied from discrete market quotes

Consider a maturity $T$, for this maturity I have some implied volatility from market denoted $\sigma^{0}_{i}$. I want to interpolate these volatility using Entropy approach, by using $\sigma^{0}_{i}$...
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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 ...
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1 vote
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### Differential vs. derivative in the Vasicek model [closed]

Can anyone help me in understanding how we get the line I have marked with a red arrow? I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
154 views

### Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?

Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...