# Questions tagged [stochastic]

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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 ...
57 views

### 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 ...
188 views

### 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: \begin{equation} dS_t = \mu S_t dt + \sigma S_tdW_t \\ S_t(0) =S_0 \end{...
80 views

### 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 ...
60 views

### 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 ...
47 views

### 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.
31 views

### 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. ...
203 views

### 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 + \...
43 views

### Calculation of upper stochastic dominance bound of an option

I’d like to calculate, for a call option on a stock, the upper stochastic dominance bound as proposed by Constantinides et al. in their 2002 paper 'Stochastic dominance bounds on derivatives prices in ...
Can someone help me solve this following Itô Calculus problem? Let $Z(t):= [B(t)*X(t)]/S(t)$ We have the following dynamics of B(t), X(t) and S(t): $dS(t)=\alpha S(t)dt+\sigma S(t)dW(t)$ $dB(t)=rB(... 1answer 75 views ### Independence of increments of the stochastic process$\frac{1}{t}\int_0^t u dW_u $Let$X_t$be a stochastic process such that $$X_{t} =\frac{1}{t}\int_0^t u dW_u$$ I know that for $$Y_{t} =\int_0^t u dW_u$$$Y_t-Y_s$is independent of$Y_s$where$t>s$. But is this also true ... 1answer 123 views ### Heston model with jumps in both variance and underlying dynamic How can I build on Matlab a Heston model using characteristic function adding jumps in both variance and underlying dynamic ? Suppose that the number of jumps is Poisson-distributed but the jump size ... 1answer 132 views ### Problem of stochastic differential equation (SDE) Please help to answer this stochastic differential equation (SDE). Thank you very much. 1answer 192 views ### Evaluating the SDE$dX_t = t\,dS_t$The process$S$is a geometric Brownian motion with an SDE:$dS_t = S_t(\sigma\, dB_t + \mu\, dt)$. I'm stuck evaluating$E(X_t)$and$V(X_t)$, where$dX_t = t\,dS_t$. 1answer 80 views ### SDE Parameter Estimation Have a question about "How to estimate parameters for SDE with multiple Brownian Motions ?" Let's say$X_t$follows the process:$dX_t=\mu dt+\sigma_1 dW_t^1 + \sigma_2 dW_t^2 $I think I've checked ... 1answer 145 views ### Invariance Scaling of Brownian Motion Prove$\frac{1}{\sqrt{t}}\log\left(\int_0^t \exp(B_s)\mathrm{d}s\right)$converges to$\sup\limits_{t\in [0,1]}B_t$in distribution as$t\to\infty$. I have a sense to use scaling invariance, but no ... 1answer 260 views ### integration of squared brownian motion w.r.t time How to prove$\int_0^1 B_s^2ds$is a random variable and compute its first two moments? From excercise 1.15 on the book martingales and brownian motion. 0answers 87 views ### The Ho-Lee Model (1986) (My question) I solved the following questions. However, if you know the other solutions, please let me know those along with computation processes. Besides,$W_t$is a S.B.M. (Thank you for your ... 1answer 249 views ### Expected value of exponential of hitting time of GBM We have a stopping time $$\tau=\inf\{t\geq 0: S_0e^{\sigma B_t+(r-\sigma^2/2)t}=S^* \}$$ where$S_0,\sigma,r,S^*$are constants and$S^*<S_0$, and$B_t$is a brownian motion. I wish to compute ... 0answers 69 views ### stochastic volatility and smile Can we say that the volatility smile contain for sure stochastic volatility information ? If yes why ? Saying that BlackScholes does not explain the smile does not necessary mean there is an ... 2answers 333 views ### Detect trend of an index My question is about determining the trend and it can break down to 3 parts. To clarify, a trend in my point of view, and in simple form, is the last close at time t relative to its time reference, i.... 1answer 124 views ### change of measure expectation How to find expectation of this stochastic process? Also, to show that the expectation of a stochastic process expression [Xt - St] in one measure is equal to expectation of another expression (of the ... 2answers 363 views ### Application of Itô's lemma - Forward process How would be applied the itô's lemma in the following equation: And we know that: 0answers 51 views ### Negative theta in Log-linear stochastic volatility model I was asked to simulate the following geometric Brownian motion to get paths for the SPX stock price. the process follows a Log-Linear stochastic volatility.$dS_t = \mu S_tdt+e^VS_tdW_1 $where ... 0answers 40 views ### A Soft Problem: Application of Stochastic Differential Equations in Hilbert Space Beyond HJM Interest Rate Model I am reading books on stochastic differential equations (SDE) in Hilbert spaces. It seems that every book just discusses HJM interest rate model as an application when discussing financial ... 1answer 226 views ### Good references on Heston Model? I am looking for good bibliographic references on Heston Model and Stochastic volatility models in general. Does anyone know any good introductory/intermediate references on this topic? 1answer 258 views ### CIR calibration I'm using a CIR short rate model to forecast interest rate paths. I've been thinking and also searching online about different ways of estimating its parameters (a, b and sigma). While there are a ... 2answers 2k views ### Integral of Brownian Motion w.r.t Time: what is wrong with this solution? [duplicate] My question is about a stochastic integral of brownian motion w.r.t time. Let$W(t)$the Wiener process (or brownian motion). I want to calculate this: \begin{eqnarray} X(t)=\int_{0}^t dt' W(t'). \... 0answers 283 views ### Forward implied vol vs Instantaneous vol In the Discrete Stochastic Implied Volatility Model which is from the standard Heston Model, the model shows the evolution of forward implied volatilities with time. I thought forward implied ... 0answers 43 views ### Total Variance of an asset in case of stochastic rates Let's suppose the underlying S follows a BS dynamic with the drift being the short rate that follows a short dynamic model. the "local volatility" of the equity should be the implied volatility from ... 3answers 300 views ### Heging against stochastic interest rate I am working on an Index and I am trying to price Call options on it. I work with the 3 Months LIBOR as Cash. I use the following Black-Scholes formula $$C_{t} = S_{t}e^{-q_{t}(T-t)}\mbox{N}[d_{1}(t)]... 0answers 64 views ### \int_{0}^1W_x(t)dW_y(t)/(\int_{0}^1W_x^2(t)dt)^{1/2} normally-distributed? I have came across the following stochastic integrals:$$\frac{\int_{0}^1W_x(t)dW_y(t)}{(\int_{0}^1W_x^2(t)dt)^{1/2}}$$which was claimed to be standard normally distributed (W_x and W_y are ... 2answers 645 views ### What is meant by innovations in volatility? I am currently reading about stocks with "high sensitivity to innovations in aggregate volatility". I am not a native speaker so this might be a trivial question, but I truly cannot find an answer ... 1answer 2k views ### Expected Value of Stochastic Process Given the following stochastic process:$$ dX = a(X,t)dt + b(X,t)dz $$where:$$ dz = A \sqrt{dt}$$and A is a random variable with mean zero and variance 1. Is there a way to calculate the ... 1answer 199 views ### Calibration of stochastic volatility models Which are good references to know about different calibration methods for stochastic volatility models such as Heston? I know that there are a lot of way of carrying this task out and I was just ... 2answers 533 views ### Hawkes process intensity solution Hail to all, I am struggling to solve the following SDE for intensity: d\lambda_t = \kappa(\rho(t) - \lambda_t)dt + \delta dN_t I know to expect the solution in the form of \lambda_t = c(0)e^{-... 1answer 297 views ### FX options pricing exchange rate regimes how can we estimate the impact of a exchange rate regime switch ( from fixed to float) on the options prices i'm talking about the moroccan case (EUR/MAD USD/MAD) options OTC , is there any stochastic ... 2answers 279 views ### Why won't Bjork ever show that the integrability condition is satisfied? A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ... 1answer 216 views ### Question on implied vol (surface) and strikes there have been loads of papers on skews ATM / OTM, volatility premium and such. Lots of explanations for why iv is different on same stock with different strikes focused on preference of informed ... 1answer 172 views ### Piecewise Ito formula Usually Ito's lemma is stated for C^{1,2}(\mathbb{R}^{d+1},\mathbb{R}) functions. My question is does Ito still hold if the domain is restricted. That is if the semi-martingale Z_t is only ... 0answers 213 views ### Is there a way I could find a matlab or R code to estimate a regime switching stochastic volatility model (discrete)? Sorry to bother you with this request but, does anyone know where I could find a matlab or R code to estimate a regime switching stochastic volatility model (discrete)? Thank you very much. 1answer 132 views ### Merton portfolio allocation problem proportions/weights >1 or <0? In the classical Merton portfolio problem, lets assume:$$ dX_t \, = \, \frac{\pi_t X_t}{S_t} S_t(\mu dt +\sigma dW_t) = \pi_t X_t (\mu dt +\sigma dW_t) $$ie: zero interest rates for simplicity. ... 1answer 148 views ### Problem with derivating integral I have a doubt : I know that if x_{t}=\int_{0}^{t}\gamma(s)dW_{s} (with W_{s} a brownian motion), we have : dx_{t}=\gamma(t)dW_{t} What about if x_{t}=\int_{0}^{t}\gamma(s,t)dW_{s}. Do I have ... 3answers 228 views ### existence of implied volatility I read a book where it was written : 1/ "implied volatility is the market's consensus on the volatility of the asset between now and the maturity of the option". -> Could someone explain me this ... 4answers 6k views ### Is it really possible to create a robust algorithmic trading strategy for intraday trading? I'm an engineer doing academic research for my master thesis in the area of quantitative finance, basically the purpose is to study the possibility to create an intraday-trading algorithm. I've tried ... 2answers 3k views ### Intergral of Brownian motion w.r.t. Brownian motion I don't understand why S (highlight on picture), I learned$$\int_0^t W(s) dW(s) = \left. \frac{1}{2} (W^2(s)-s) \right \vert_0^t$$everyone please explain for me. Thank you 1answer 124 views ### Why$W_{t}^3$is not a martigale?(by Definition) If$W_t$be a wiener process then,how can i show that$W_{t}^{3}$is not a martingale by definition? 1answer 167 views ### stochastic calculus - brownian motion I don't know how to prove this : let be$X_t = \int_{0}^{t}\sigma_{u}dW_{u}$where$\sigma_{t}$is a predictable process. If$|\sigma_{t}| = c$a.s. how can I prove that$X_{t}=c*\beta_{t}$(... 1answer 244 views ### stochastic calculus - Itô formula? I encounter a problem in the proof below: I don't know how to proove the first line in yellow (cf below): it makes me think about the Itô formula a lot I don't undertand the deduction (ok$\gamma^{\...
I encounter the following problem : I have the equality in distribution: for all $\lambda >0, ((1/\lambda)*\int_{0}^{\lambda t}\sigma_{u}^{2}du,t\geq0)=(\int_{0}^{t}\sigma_{u}^{2}du,t\geq0)$ ...