Questions tagged [stochastic-processes]

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

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Asset pricing using a two period binomial tree

Recall that in the binomial model the risk-neutral probability for the price going up is given by p = 1+r−d / u−d where u > 1 and d < 1 specify the possible price jumps in the risky asset and r ...
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473 views

Why does the diffusion term remain the same when we change pricing measure?

Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion In plenty of interest rate examples, I have ...
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168 views

How do you hedge volatility risk?

Suppose I model an asset $S_1(t)$ under a stochastic volatility model. To price an option on $S_1$, I must assume the existence of an asset $S_2$ that is used to hedge against changes in the ...
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2answers
152 views

Expectation of functions with Brownian Motion embedded

Trying to solve a problem set with: Let $W_t$ be a Brownian Motion and $X_t = e^{izW_t}$ where $z$ is real, $i = \sqrt{-1}$. I need to find $\mathbb{E}\left(X_t\right)$... I am a bit stuck. I have ...
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1answer
156 views

Obtaining the dynamics of the Vasicek model using Itô

Consider the following expression for the short-term interest rate $$r_t=r_0 e^{\beta t}+\frac{b}{\beta}\left(e^{\beta t}-1\right)+\sigma e^{\beta t}\int_0^te^{-\beta s}dW_s \tag{1},$$ which is ...
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57 views

How to merge ML-based $\alpha$-signal with stochastic control approach?

I'm having a hypothetical situation where I have a set of ML-based alpha signals $\{\alpha_i\}_{i=1}^{N}$ that describe a different states of order book - imbalances, order flow, spread properties etc....
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1answer
178 views

How to deal with negative intercept terms on GJR-GARCH(1,1) model?

Recently, I have been studying the relationship between COVID-19 and stock returns using a GJR form of threshold ARCH model. However, I got some unusual estimation results I can't figure out whether ...
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2answers
108 views

On moving Linear Correlation (rolling correlation)

Let's say I have two random variables $X$ and $Y$ which each represents the daily returns of two given stocks. I can easy calculate their (total) correlation by finding their covariance matrix $\Sigma[...
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1answer
200 views

Jump Diffusion Process question

I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
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187 views

Pricing of European options on two underlying assets

Is anybody able to give the solution to the following problem? Suppose we have two assets, each of which follows a GBM process, and where $dW_S$ and $dW_X$ are correlated $(dW_SdW_X=\rho)$. $dS=\mu_s ...
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1answer
136 views

Interpolation of $\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$

Let's assume that we have SDE $$dX(t)=\mu(t,X(t))dt+\sigma(t,X(t))dW(t)$$ and we simulate it on a time grid which contains points $t_k$ and $t_{k+1}$. How can we then calculate value of $X$ at time $...
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124 views

Solving SDE using integration factor and Ito's lemma

I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
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1answer
77 views

Ito's lemma for option pricing with Levy-alpha stable drift

Consider $$dS=\omega\left(\Lambda-S\right)dt+\sigma_S S dW_t,$$ such that such that $W_t$ is a Wiener process, $\sigma_S$ is constant, $\omega: t\rightarrow\mathbb{R}$ represents anticipated drift and ...
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Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?

I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous ...
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Is it true that interest rates options with different maturities are free of calendar arbitrage because of the different underlying rates dynamics?

The title says it all - is it true that European style interest rates options (lets say on LIBOR 3M for the sake of simplicity) with different maturities are free of calendar arbitrage because ...
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Why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire?

why is the price of any asset divided by a reference asset(numeraire) is a martingale under the measure associated with that numeraire? For example, if I have the price of a forward price $f_t$ and a ...
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82 views

Stochastic process as integral over window function

Consider the following stochastic integral of a deterministic function $f(t,s)$ with respect to the Wiener process $W_s$: $$\int_0^\infty f(t,s) d W_s$$ My questions are: Is such an integral ...
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89 views

Black Scholes derivation: Why treat Delta as a constant?

In the derivation of the Black-Scholes equation, it is argued (e.g. in the original paper and in Hull) that $$dV(S_t, t)=(…)dt + \frac{\partial V}{\partial S} dS_t,$$ where $V(S_t, t)$ is the value at ...
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Change of Numeraire technique (Cross-currency models)

Hey I have problem with understanding change of numeraire technique. For example we have $dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank ...
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Instantaneous correlations in multi-currency G2++ model

Hey in "Interest Rate Models - Theory and Practice With Smile, Inflation and Credit" by Damiano Brigo, Fabio Mercurio we have dynamics for two interest rates and FX rate between them: $$r_1(...
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91 views

Is this process log normally distributed?

I came across a question that I guess $P$ is lognormally distributed. where $y_n$ is log-normally distributed. Am I right on the guessing? Here is the full solution if interested.( my guessing comes ...
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Existence of the solution for SDE with Gaussian Process

I'm interested in the existence of the solution for a non-Ito SDE. Sloppy notation but assume a SDE given by $\dot{x}=f(x),\quad f(x)\sim GP(0,k(x,x')),$ where $f$ is a Gaussian Process with kernel $k$...
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55 views

How to compute this current value using no arbitrage condition?

Suppose $X_t$ is a geometric Brownian motion with drift $\mu$ and volatility $\sigma$. $X_0$ is known. You have a machine that produces something worth $X_t$ at random times $t$ generated by a Poisson ...
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56 views

A question in information strucutres and probability measures - How are they connected?

Suppose that $\mathcal{I}=(X,\sigma^{\mathcal{X}},\mu)$ is an information strucutre, which is a probability space, where $X=X^1\times X^2$ is the cartesian product of the individual finite sets of ...
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60 views

Infill lower frequency data: Brownian Bridge

Given monthly returns data, I would like to infill those to get daily returns. Roughly estimates imply that annual volatility is about 1.5x of SPY. One option that came up in my initial research was ...
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1answer
225 views

Expected value and Variance of a stopped random process

Consider the random process where you keep drawing samples from [0,1] uniformly at random as long as the current sample is larger than the previous sample. What are the Expected value and the Variance ...
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1answer
50 views

The non-negativity condition of a discretized mean-reverting Heston model with stochastic violatilities

I happened to encounter the following discretized mean-reverting Heston model with stochastic volatilities in a paper $$ P(t) = P(t-1) + v_1(u_1-P(t-1))+\sqrt{\sigma(t)}\cdot \epsilon_1(t) \\ \sigma(t)...
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1answer
169 views

How to incorporate momentum in Ornstein Uhlenbeck to capture overshooting in financial markets?

In modelling asset prices, it is a good idea to model it using a fair value or target price concept. Recently Carr & Prado explored this idea to find optimal stop loss/take profit levels when the ...
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1answer
93 views

what's the difference between instantaneous short rate and instantaneous forward rate?

In the short rate models, sometimes it models the instantaneous short rate and sometimes it models the instantaneous forward rate. Does instantaneous short rate = F(0, t + tau) and instantaneous ...
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93 views

Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$

Suppose a stock follows the stochastic differential equation $$dS=\frac{P-S}{\omega}dt+SdW_t,$$ such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
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1answer
220 views

Parametric Stochastic Integral

I need help. Defining the parametric stochastic integral $$ F_t = \int_t^T\xi(t,s)g(s)ds $$ $\\\\$ with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
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1answer
77 views

HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$

I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
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1answer
187 views

Understanding out-of-sample performance metrics for Realized Volatility

I fitted several models on a realized volatility process and then proceeded to obtain out-of-sample results. I'm struggling to interpret these results apart from to tell model A seems better than ...
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38 views

Equivalence of expectation condition for contingent claims attainable and contingent claims super replicable

We have the following definitions for set of contingent claims attainable and contingent claims super replicable I want to prove the following result How do I show iii $\implies $ ii.I understand ...
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1answer
195 views

Proving the discounted stock price is martingale

Let $\mathcal{K}_s$ be $$ \mathcal{K}_s=\{\tilde{V}_t(\theta):0\leq t<\infty,\,\theta\text{ a simple strategy}\},$$ where $\tilde{V}_t(\theta)$ is the discounted value process of the self financing ...
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72 views

Ruin theory with infinite-mean Pareto-distributed claims: how to characterize the ruin time and the reserve prior to ruin

Consider the Cramér–Lundberg model $$\hspace{8em}R(t)=u+c\,t-\sum_{j=1}^{N(t)}V_{j}\,,\hspace{8em}(1)$$ where $c$ and $u$ are positive constants, $N(t)$ is a Poisson process with a rate $\lambda$ (in ...
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61 views

Speculation with quanto option - how to see the realized correlation

From this question, on vanilla option vol speculation, we can gain intuition on the impact of realized vol on the gamma, and consequently on the efficiency of the speculation trade. Asuming long ...
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1answer
58 views

What do Future price and Forward price represent

In Shreve's Finance and Stochastic calculus, definitions are: Forward Price: The $T$-forward price $For_S(t,T)$ of this asset at time $t$, where $0\leq t\leq T$, is the value of $K$ that makes the ...
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810 views

Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?

Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete. (a) $\beta_{t}=e^{t}, S_{t}...
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77 views

Wealth process in the Black-Scholes model with discrete dividends

Good evening, The following problem is the sequel of a previous post I made here a few days ago. Consider the Black-Scholes model with discrete dividends in the interval $[0,T]$. This means that ...
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What interpolation methods are standard to use for interpolating on equity volatility surfaces?

The answer to this question (Volatility surface interpolation for Black-Scholes delta hedging) names Cubic Spline Interpolation and Guassian Process interpolation (is this exactly the same thing as ...
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146 views

Derivation of Bergomi model

In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) : $$ \frac{dP}{dt}+(r-q)S\frac{dP}{dS}+\frac{\xi^t}{2}S^2\frac{d^2P}{dS^2}+\frac{1}{2}\int_t^Tdu\int_t^...
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1answer
128 views

Question on Ito's lemma involving $\mathrm{d}W(t)$

I am new to Ito-calculus, so please forgive me if the question is stupid. Let $W(t)$ be a Brownian-Motion and $f(W(t))=W(t)^2$. If I want to calculate the differential $\mathrm{d}f(W(t))$, Ito's lemma ...
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1answer
87 views

Can you shift a standard libor market model with regard to only at-the-money options?

Suppose I have an LMM defined using the spot measure as in Brigo and Mercurio: $dF_k(t) = \sigma_k(t)F_k(t)\sum^k_{j=\beta(t)}\frac{\tau_j\rho_{j,k}\sigma_j(t)F_j{t}}{1+\tau_jF_k(t)}dt + \sigma_k(t)...
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55 views

Bergomi's model normalisation

On his book https://www.amazon.fr/dp/B019FNKQS8/ref=dp_kinw_strp_1 Bergomi derives a multifactor mean reversible volatility of the volatility such that : \begin{equation*} d \xi_{t}^{T}=\omega(\tau) \...
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1answer
261 views

Understanding Monte Carlo to solve option price with local volatility

I have read this question pricing using dupire local volatility model which seems to have an answer from here https://www.csie.ntu.edu.tw/~d00922011/python/cases/LocalVol/DUPIRE_FORMULA.PDF Both of ...
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1answer
73 views

Why does the LMM in Hull seem so different from the LMM in Brigo and Mercurio?

When I look at Hull's "Options Futures and Other Derivatives" the process for $F_k(t)$ in the rolling forward risk neutral world is specified as $\frac{dF_k(t)}{F_k(t)} = \sum^k_{i=m(t)}\...
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1answer
99 views

In what cases characteristic function of (log-)price process is known?

Hey I know that we can use characteristic function of log-price process to price different options. But when we know the characteristic function? I know that we can take Levy processes and constant ...
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41 views

Programming the Milstein method and computing the increments

In the wikipedia article on the Milstein method, the following python code to simulate a geometric Brownian motion is presented: ...
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1answer
599 views

How to simulate Levy processes

Hey how to simulate Levy processes? I have no problem with Wiener process and compound Poisson process, I also know how to simulate Variance Gamma process but I have no idea how to simulate for ...

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