# Tagged Questions

stochastic processes is a collection of random variables representing the evolution of some system of random values over time.

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### How to deal with negative ARCH terms?

Lately I have been trying to fit a GJR-GARCH(1,1) model to fit against the S&P 500 returns over 1985-2015 but I have ran into some problems I can't quite figure out. The GJR-GARCH(1,1) model I am ...
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### ARMA-GARCH model, bset model selection and confidence levels calculations

I'm a newbie in GARCH models. I tried to realize ARMA(p, q)-GARCH(u, v) model via fGarch. So, 2 main questions. 1) Can I use BIC/AIC for selection best model for all (p, q)-(u, v) models? So, is it ...
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### Spread Return and Mean Reversion Model

http://papers.ssrn.com/sol3/papers.cfm?abstract_id=2399915 The above paper proposes an interesting method for modeling credit spreads. I have tried to implement it in R but keep obtaining unrealistic ...
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### Lookback option to find stock price

Consider the payoff equation for the lookback option $\psi(T)= max(S_t-S_T)$, where $t\in[0,T]$ and $S_t$ is modeled by the geometric Brownian motion with constant parameters. Find the price of stock ...
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### Stochastic Integration

I have the following derivation question: A small company is investing resources in a risky project that it hopes will be profitable. The project could, for example, represent the manufacturing and ...
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### Processes used in quant finance

What are the main stochastic processes (and their SDE) used in quant finance? For example to model currency prices, stock prices, etc.
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### Stochastic process with non-independent increments

All stochastic process I see always have independent increments. It is true for: standard brownian motion geometric brownian motion (?) Ornstein Uhlenbeck (?) in general, Levy process etc. What ...
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### Black-Scholes formula proof, without stochastic integration

I've looked into many books at my academic library, and very often it goes like this: Brownian motion Then, stochastic integration (ItÃ´'s formula etc.) Application: Black-Scholes formula for price ...
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### Avellaneda/Cont model Order Book Model

The model given in the following paper by Avellaneda et al http://people.stern.nyu.edu/jreed/Papers/limitorder.pdf On page 7 he explains that the initial Bid and Ask size should be normalised by ...
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### Prove that $E[g(X_T)|\mathscr F_t] = E[g(X_T)|X_t]$

Let $T > 0$. Let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathscr F_t = \sigma(W_u, u \in [0,t])$ where $W_t$ is standard Brownian ...
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### How do one solve $\int_t^T \exp[\int_0^u-( r-\delta_s)ds] dW_u$? Double integral with general deterministic function $\delta(t)$

How do one solve $\int_t^T \exp[\int_0^u-\left( r-\delta_s\right)ds] dW_u$ ? $\delta(t)$ is a general deterministic function. $r$ is constant.
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### Motivation: Stochastic Interest rate model

what is a reason that someone might be interested in a stochastic-interest model such as the Chen model? Also can you provide me with a link to an easy to read motivational paper/part of a paper on ...
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### How to change to risk neutral measure in a mean reversion process?

For example, in the Ornstein-Uhlenbeck process do I just replace the drift term with the risk free rate, like in the GBM case?
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### $\mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace$ ? Exp. of an indicator funct and a diffusion with non-proportional vol

How to compute $\mathop{\mathbb{E^{}}}\left\lbrace 1_{S_T > K} \; S_T \right\rbrace$ ? where $dS_t = S_t r dt + \sigma dW_t$ and $1_{S_T > K}$ is the indicator function being one when ...
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### Any idea of compound Poisson processes in betting? [closed]

Any suggestions on compound poisson processes in bets of a customer?
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### Modelling commodity price uncertainty with brownian motion - time period impacts

background I have two separate models of a metals resources company. Each model produces a series of accounting and cashflows forecast for different assets, and consolidates these to a overall ...
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### Feynman Kac Formula for path-dependent options

Consier geometric Brownian motion: $dS_t/S_t=\mu dt+\sigma dW_t$ Feynman Kac theorem tells us that the conditional expectation $v(t,x)=E[ e^{-rT}\Psi(S_T) | S_t=x]$ can be computed by solving the ...
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### Specifying integration level of time series [closed]

Following model was estimated on 200 observations. How to specify the level of integration of $X_t?$ In brackets there are standard errors and p-value of Breusch-Godfrey test is also shown. \$X_t=0,02+...