Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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2
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1answer
137 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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0answers
40 views

SDE of a Geometric Levy process with compound Poisson process

Suppose that a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is given. A geometric Levy process is defined in the form of $S_t=S_0 exp(X_t)$ where $S_0$, let's say, is the initial price and $...
2
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0answers
97 views

Largest class of real world probability models admitting explicit risk-neutral change of measure

Assume we have two assets, a random asset $A_t$ and deterministic risk-free bond $B_t = e^{rt}$. Let $P$ be a model of the real-world probabilities of $S$ and $Q$ the unique associated risk-neutral ...
4
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1answer
222 views

Pricing a contract

I'm currently trying to price some different kinds of contracts. I'm stuck on this following exercise, which I can't seems to find a good solution for. The following is assumed: We are in a standard ...
2
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3answers
241 views

Integral of brownian increments

I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following $$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$ I've been ...
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2answers
201 views

Why would exchange rates follow a geometric brownian motion?

I'm reading Shreve's Stochastic Calculus for Finance. On page 382, he begins talking about exchange rates: Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
1
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0answers
32 views

From the perspective of a company, when is the right time to start paying dividends?

I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments. I am supposing that a company has a revenue stream $f(t)$. This is just $...
1
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1answer
195 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 $r = 0.04$....
0
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0answers
65 views

what is $\int t dW$ and $\int W dt$? [duplicate]

More explicitly, if $W(t)$ is Brownian motion, what would be $$f(t) := \int_0^t u dW(u)$$ and $$g(t) := \int_0^t W(u) du$$?
2
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0answers
109 views

Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?

I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
0
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1answer
129 views

Calibration/estimation of the CEV model

The CEV model for a stock price $S(t)$, interest rate $r$ and variance $\delta$ $dS(t)=rS(t)dt+\delta S(t)^{\gamma}dW(t)$ where the volatility for the stock is given by $\sigma(t)=\delta S(t)^{\gamma -...
3
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1answer
186 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
121 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 ...
3
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2answers
144 views

American Option Valuation - Induction algorithm

The price of an American put option is given by $$V_k = \sup_{\tau\in\mathcal{T}, \tau\ge t_K} E\{e^{-\int_{t_k}^\tau r_sds} (K-S_{\tau})^+|\mathcal{F}_{t_k}\}$$ I found in one book the following: $$\...
4
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0answers
87 views

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 ...
0
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0answers
88 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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0answers
173 views

Seeking criticism of model assumptions

I have been trying to publish a new calculus and options model for seven years. I have been consistently desk rejected, so what I am trying to do is get criticism of my assumptions because they ...
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0answers
71 views

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 ...
4
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1answer
218 views

Difficulty with stochastic calculus problem

I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is: Consider a stock whose price differential is $$ dS(t) = r(t) S(...
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0answers
35 views

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(...
0
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1answer
166 views

Game theory and stochastic calculus

Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. ...
2
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1answer
89 views

Ansatz and HJB equation

Suppose we have an HJB equation of the form $$ \frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
3
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0answers
145 views

Deriving Bachelier Greeks

I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link: Bachelier model call option pricing formula This is fairly easy to code ...
4
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0answers
97 views

How to integrate Itô integral w.r.t time?

Let $W_t$ be a Brownian motion. How to calculate the following integral $$ I:=\int_0^t\left( \int_0^u(u-s)dW_s\right) du? $$ My attempt so far is: First note that $$ \int_0 ^u (u-s)dW_s = \int_0^u ...
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1answer
136 views

Clarification on Paul Wilmott's derivation of Ito's Lemma

I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
-1
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1answer
143 views

Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? [closed]

Re close votes: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's pedagogical. Context: I'm actually ...
1
vote
1answer
89 views

Compare errors in estimating a probability

Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion. Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ ...
3
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2answers
157 views

Derivation of static replication formula

I know that a way of computing the price of a derivative paying $S^2$ at time $T$ is by making use of the following strategy: $V=\int_{0}^{\infty} s^2 \frac{\partial^2 C}{\partial K^2}(K=s)ds$ Where $\...
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0answers
90 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 ...
6
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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 ...
2
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1answer
75 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 ...
0
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0answers
59 views

Vasicek Model Expected Short Rate

Please note I am new to quantitative finance and more so to stochastic calculus. I have what should be a relatively simple problem using the Vasicek model for estimating future parameters of the short ...
11
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5answers
512 views

How to compute $E[W(T)\exp(W(T)]$

I have got this interview question twice. Does anyone know from which interview question book or another source this question comes from? It may be some well known source as two different interviewers ...
10
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2answers
780 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}...
7
votes
2answers
350 views

Heston: Variance of Integrated Variance

Consider the standard Heston model\begin{align*} dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\ dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\ dBdW&=\rho dt. \end{align*} Computing $\mathbb{E}\...
2
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0answers
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^...
0
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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 ...
2
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1answer
70 views

Equivalent local martingale measure vs. equvalent martingale measure in a Brownian setup

Assume you have the standard financial market built up of a Brownian motion. I have seen some books say that an equivalent local martingale measure imples no arbitrage, and some say that an equivalent ...
2
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1answer
115 views

Finding Option Probability Density Using Local Volatility from Dupire Model

This question is different than pricing using dupire local volatility model and Is Dupire's local volatility model path independent to recover historical option price? I also asked this on Math ...
0
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1answer
97 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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0answers
103 views

Quantitative interview questions - fixed income [duplicate]

I will pass a quant interview (fixed income). The interviewer said the questions do not have a right answer, they are not math exercises. He said the questions are like their job,the objective is to ...
2
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1answer
51 views

Showing that the shortfall-to-quantile ratio of a normal distribution goes to one

I dont get why $$\lim_{x \to \infty} \frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} } = \lim_{x \to \infty} \frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \...
1
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0answers
47 views

Help in Bernoulli's differential equation

I want to solve the following Bernoulli differential equation: $$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$ where $\sigma$ and $a$ are real numbers. Until now I have divided both sides of the equation with $A^...
2
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1answer
97 views

FX Asian Option Moment-matching in Harmonic case

I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is: $$\left(\frac{A_T - K}{A_T}\right)^{+} = \left(1 - \frac{K}{A_T}\right)^{+} = K \left(\frac{...
2
votes
2answers
129 views

Sampling change in the driving brownian motion of a CIR process

I have volatility driven by a CIR process: $$\mathrm{d}v_t = \kappa (\bar{v}-v_t)\mathrm{d}t + \omega \sqrt{v_t}\mathrm{d}W_v\text{.}\tag{1}$$ I am working with several (complicated) approximations of ...
3
votes
1answer
305 views

Ito Lemma for Poisson Process

I'm new to stochastic calculus on jump processes and encountered a difficulty. I would appreciate some clarification from the community on the following question. Let $g_t$ be a $\mathcal{F_t}$-...
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0answers
65 views

How to Discretize this SDE found in finance? (cross-posted)

Continuous-Time In continuous-time form, the "Heston model" is written as $$ dS_t = \mu S_t dt + \sqrt{\nu_t} S_t dW_t^S \\ d\nu_t = \kappa (\theta - \nu_t) dt + \xi \sqrt{v_t} dW_t^{\nu} $$ ...
1
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0answers
102 views

Equivalence of Call Option on $S_T$ and Put Option on $\frac{1}{S_T}$ in FX Markets

Part 1: I am trying to price an option in the FX world. It naturally pays in the domestic currency, but in this case the payout currency must be the foreign currency. For example, consider the payoff: ...
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0answers
78 views

Price difference digital option : constant vol vs local vol

I got the following interview question: Consider a digital option, it will be priced by using two approaches: 1)constant volatility; 2)local volatility. At the strike, both volatilities are equal. (...
8
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0answers
147 views

Why were Laguerre polynomials a good choice of basis functions for American Monte Carlo?

I am implementing LSMC to price American options based on a custom model. I now need to make a choice of basis functions, so I am looking for the theoretical justification for using Laguerre ...

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