# Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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### Price a contingent claim with payoff $(S_{1T}-S_{2T})^+$ at time $T$

Two stocks are modelled as follows: $$dS_{1t}=S_{1t}(\mu_1dt+\sigma_{11}dW_{1t}+\sigma_{12}dW_{2t})$$ $$dS_{2t}=S_{2t}(\mu_2dt+\sigma_{21}dW_{1t}+\sigma_{22}dW_{2t})$$ with $dW_{1t}dW_{2t}=\rho dt$....
1 vote
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### Why is Feynman-Kac formula applicable in Burgard-Kjaers PDE paper?

In the paper Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs by Burgard and Kjaer, they say we may formally apply the Feynman-Kac theorem ...
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### State space equation of CARMA(p,q) processes

Thanks for visting my question:) I am currently working on Carma(p,q) processes and do not understand how to derive the state equation. So the CARMA(p,q) process is defined by: for $p>q$ the ...
52 views

### How to understand Short Gamma and Long Volatility for Leveraged ETFs?

In the book Leveraged Exchange-Traded Funds: Price Dynamics and Options Valuation, it describes a static delta-hedged long volatility position by simultaneously shorting regular/inverse leveraged ETFs ...
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### Ito formula and confusion with the differential operator $d$

Thanks for visiting my question. Im am currently working on this paper (https://arxiv.org/abs/2305.02523) and I am stuck at page 21 (Theorem 14 proof). First these SDE's were defined: \begin{align*} ...
50 views

### Kalman Filtering to estimate parameters of G2++ Model

I'm trying to use Kalman Filtering to estimate the parameters of the G2++ short rate model. For this, I've been using Implementing Short Rate Models: A Practical Guide by F.C. Park. For reference, he ...
176 views

### Balland - SABR goes normal

To summarise this very long post : please help me understand the undetailed proof of the quoted paper. I am not comfortable using a result I do not fully understand. I am reading Balland & Tran ...
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### Solving Equation for estimation risk averse parameter

Let the portfolio value follow the SDE: $$V_t=(\mu w+r(1-w))\cdot V_t\cdot dt +\sigma \cdot w\cdot V_t \cdot dB_t$$ where $\mu$ = drift of the portfolio, $\sigma$=standard deviation of the portfolio, ...
98 views

### Is homogeneity preserved under change of measure?

In a paper, Joshi proves that the call (or put) price function is homogeneous of degree 1 if the density of the terminal stock price is a function of $S_T/S_t$. In the paper I think Joshi is silently ...
310 views

### Integrated Brownian motion

I occasionally see a post here: Integral of brownian motion wrt. time over [t;T]. This post has the conclusion that $\int_t^T W_s ds = \int_t^T (T-s)dB_s$. However, here is my derivation which is ...
142 views

### Time-shifted power law in path dependent volatility

I can't understand a function which is part of a volatility model. This is all explained in an open access paper titled "Volatility is (mostly) path-dependent" by Guyon and Lekeufack. My ...
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### Can the PDE of Black and Scholes really be derived from the CAPM?

Black and Scholes (1973) argue that their option pricing formula can directly be derived from the CAPM. Apparently, this was the original approach through which Fischer Black derived the PDE, although ...
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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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### Volatility swaps hedging

I have heard that traders use a straddle to hedge volatility swaps (in the FX context), although I could not figure out the specifics. Is this type of hedge used in practice? And if yes, how does it ...
208 views

### Smile wings and varswap pricing

Is it true that far wings of the volatility smile have an outsized influence on the price of a variance swap? Is there a mathematical argument demonstrating this idea? What do we generally refer as ...
130 views

### how to calculate pdf and cdf for an Ornstein-Uhlenbeck process

I have the Task. For Ornstein-Uhlenbeck process generate a path and plot a) cumulative distribution (cdf), b) density function (pdf), c) calculate the 95%-quantile. My solution. From the literature we ...
1 vote
60 views

### Transform non-linear HJB PDE into system of linear ODEs [closed]

I am reading this market making paper, and am trying to understand the transformation presented on page 6. A good resource for background relevant to the transformation is this other market-making ...
1 vote