Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
697
questions
2
votes
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 ...
0
votes
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
votes
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
votes
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
votes
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 ...
1
vote
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
vote
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
vote
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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
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:
$$\...
4
votes
0answers
87 views
Where is the Quadratic Variation Coming from in this One-Factor Cheyette Model?
I am having difficulty switching from a general interest rate model (the quasi-gaussian or cheyette model) and a specific version of this model. In particular, I assume the following instantaneous ...
0
votes
0answers
88 views
Black Scholes derivation: Why treat Delta as a constant?
In the derivation of the Black-Scholes equation, it is argued (e.g. in the original paper and in Hull) that
$$dV(S_t, t)=(…)dt + \frac{\partial V}{\partial S} dS_t,$$
where $V(S_t, t)$ is the value at ...
6
votes
0answers
173 views
Seeking criticism of model assumptions
I have been trying to publish a new calculus and options model for seven years. I have been consistently desk rejected, so what I am trying to do is get criticism of my assumptions because they ...
1
vote
0answers
71 views
Change of Numeraire technique (Cross-currency models)
Hey I have problem with understanding change of numeraire technique. For example we have
$dr^d(t)=\kappa_1(\theta_1(t)-r^d(t))dt+\sigma_1 dW_1$ (under measure $Q^1$ associated with domestic bank ...
4
votes
1answer
218 views
Difficulty with stochastic calculus problem
I'm currently working through Shreve's Volume II, and I'm having some difficulty on Exercise 5.4 of Chapter 5. The problem statement is:
Consider a stock whose price differential is
$$
dS(t) = r(t) S(...
0
votes
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:
$$r_1(...
0
votes
1answer
166 views
Game theory and stochastic calculus
Does anybody know any details of game theory literature combined with stochastic calculus in finance? If yes, please recommend some papers of any authors who are doing exceptional work on the filed. ...
2
votes
1answer
89 views
Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\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
votes
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
votes
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
$$
\int_0 ^u (u-s)dW_s = \int_0^u ...
0
votes
1answer
136 views
Clarification on Paul Wilmott's derivation of Ito's Lemma
I'm currently self-studying to be quant and have been thoroughly enjoying PW's book. I have some questions regarding his derivation of Ito's lemma. Specifically, I can see that the first line in his ...
-1
votes
1answer
143 views
Is it fair in an introductory stochastic calculus/derivatives pricing class to ask for the price when absence of arbitrage is violated? [closed]
Re close votes: I believe this is a fair kind of opinion-based question because it's like those ethics questions in academia se or workplace se or because it's pedagogical.
Context: I'm actually ...
1
vote
1answer
89 views
Compare errors in estimating a probability
Let $X_t$ be a geometric Brownian motion: $dX_t = \mu(X_t,t)dt + \sigma(X_t,t)dW_t$ with $W_t$ a standard Brownian motion.
Given the intervals $[t_{j-1}, t_{j}]$ for $j\in {1,...,U,...,N}$, let $M_j$ ...
3
votes
2answers
157 views
Derivation of static replication formula
I know that a way of computing the price of a derivative paying $S^2$ at time $T$ is by making use of the following strategy:
$V=\int_{0}^{\infty} s^2 \frac{\partial^2 C}{\partial K^2}(K=s)ds$
Where $\...
0
votes
0answers
90 views
Ito's Lemma in option pricing for a stock satisfying $dS=\frac{P-S}{\omega}dt+SdW_t$
Suppose a stock follows the stochastic differential equation
$$dS=\frac{P-S}{\omega}dt+SdW_t,$$
such that $W_t$ is a wiener process, $\omega\in\mathbb{R}^+$, and $P_t,S_t\in\mathbb{R}$. If the value ...
6
votes
1answer
220 views
Parametric Stochastic Integral
I need help.
Defining the parametric stochastic integral
$$
F_t = \int_t^T\xi(t,s)g(s)ds
$$
$\\\\$
with $\xi$ a generic stochastic process such that $d\xi(t,s) = \mu(t,s)dt + \sigma(t,s)dW_t$, I'm ...
2
votes
1answer
75 views
HJM drift condition problem: Show that the HJM drift condition implies $b(t) \equiv b, \rho^{2}(t) \equiv a$
I need your help with understanding and solving the HJM framework. I am hoping I can get some help as I feel so lost with HJM and learning online because of the pandemic is adding more stress. Anyway ...
0
votes
0answers
59 views
Vasicek Model Expected Short Rate
Please note I am new to quantitative finance and more so to stochastic calculus.
I have what should be a relatively simple problem using the Vasicek model for estimating future parameters of the short ...
11
votes
5answers
512 views
How to compute $E[W(T)\exp(W(T)]$
I have got this interview question twice. Does anyone know from which interview question book or another source this question comes from? It may be some well known source as two different interviewers ...
10
votes
2answers
780 views
Change of measure and Girsanov's Theorem: Do the following models admit arbitrage and are they complete?
Let $S_{t}$ denote the price of stock, $\beta_{t}$ denote the savings account. For each model below state with reason whether it admits arbitrage and whether it is complete.
(a) $\beta_{t}=e^{t}, S_{t}...
7
votes
2answers
350 views
Heston: Variance of Integrated Variance
Consider the standard Heston model\begin{align*}
dX&=\left(r-\frac{1}{2}v\right)dt+\sqrt{v}dB,\\
dv&=\kappa(\theta-v)dt+\xi\sqrt{v}dW, \\
dBdW&=\rho dt.
\end{align*}
Computing $\mathbb{E}\...
2
votes
0answers
146 views
Derivation of Bergomi model
In Stochastic Volatility Modeling, L. Bergomi introduces in Chapter 7 the pricing equation (7.4) :
$$
\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
votes
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
votes
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
votes
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
votes
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 ...
0
votes
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
votes
1answer
51 views
Showing that the shortfall-to-quantile ratio of a normal distribution goes to one
I dont get why $$\lim_{x \to \infty}
\frac{\mu \{1 - \Phi(x)\} + \sigma \phi(x)}{(\mu + \sigma x) \{1 - \Phi(x)\} }
= \lim_{x \to \infty}
\frac{1}{1 - \sigma \frac{1 - \Phi(x)}{(\mu + \...
1
vote
0answers
47 views
Help in Bernoulli's differential equation
I want to solve the following Bernoulli differential equation:
$$A'(t)=A^2(t)[-2\sigma +1]-2aA(t)$$
where $\sigma$ and $a$ are real numbers.
Until now I have divided both sides of the equation with $A^...
2
votes
1answer
97 views
FX Asian Option Moment-matching in Harmonic case
I need to price a "foreign-paying" fixed-strike Asian (i.e., average) option. Thus, the payoff is:
$$\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 $\mathcal{F_t}$-...
0
votes
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
vote
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:
...
1
vote
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
votes
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 ...