Questions tagged [stochastic-calculus]
A branch of mathematics that operates on stochastic processes.
715
questions
2
votes
0
answers
44
views
Local volatility implied spot vol correlation
I have a question about local volatility models.
In a lot of articles it is stated that the implied spot vol correlation of this model is -1 and we usually compare this with stochastic volatility ...
4
votes
2
answers
358
views
Transformation of local volatility model
Assume we have an SDE
$$dX_t=\mu(X_t)dt + \sigma(X_t)dW_t$$
where $\sigma>0$ and $W_t$ is a Wiener process. Is there a transformation $y(X_t)$ that will make the dynamics of the transformed process ...
0
votes
0
answers
31
views
Maximum entropy probability distribution for $S_T$ implied from discrete market quotes
Consider a maturity $T$, for this maturity I have some implied volatility from market denoted $\sigma^{0}_{i}$. I want to interpolate these volatility using Entropy approach, by using $\sigma^{0}_{i}$...
0
votes
0
answers
57
views
How to calculate mean and variance in Vasicek Model
In the Vasicek model, the short rate of interest under the risk-neutral probability measure is
given by:
where k, θ, σ > 0 and W is a standard Brownian motion. Consider the related process
where ...
3
votes
1
answer
181
views
Numerically stable method for estimating $\partial_t \mathbb{E}[f(X_t)]$ where $X_t$ is an n-dim Ito process and $f:\mathbb{R}^n\rightarrow\mathbb{R}$
Assume $(X_t)_{t\geq 0}$ follows an SDE of the form:
$$dX_t = a(t, X_t) dt + b(t, X_t) dW_t$$
where $W$ is a standard $n$-dimensional Brownian motion, $a$ and $b$ are mappings from $\mathbb{R}_+\times\...
0
votes
1
answer
70
views
I want to know stochastic derivation of zero coupon bond formula
I'm elementary level of stochastic calculus.
In the above picture, from equation (11) to (12) I don't know what is the clue of $μ(t)$ is the expectation of $r(t)$ and how from this identity we can get ...
2
votes
0
answers
44
views
Relation between SABR parameters and Taylor expansion parameters
Suppose a SABR model framework (with $\beta=1$)
$$dF_t=\sigma_t S_t dW^{S}_{t}$$
$$d\sigma_t=\alpha \sigma_t dW^{\sigma}_{t}$$
$$dW^{S}_{t}dW^{\sigma}_{t}=\rho dt$$
I know that the Implied Volatility ...
0
votes
1
answer
64
views
Integration of exponential raised with Brownian Motion wrt the Brownian Motion
I have to derive several things for my thesis, however, I have the following expression:
$$
\int^{t}_{0} \exp\{\sigma W_{t}\}.dW_{t}
$$
Does anyone know what the solution for this is?
Kind regards.
3
votes
0
answers
72
views
Implied Volatility is the harmonic average of Local Volatility
I am trying to demonstrate the famous result that states that when $T \rightarrow 0$, the Implied Volatility is the harmonic average of Local Volatility.
I am st the final stage, and I have the ...
5
votes
0
answers
80
views
Closed formula for computing Implied Volatility from Local Volatility function
The main result of this paper (Asymptotics and Calibration in Local Volatility Models, Berestycki, Busca, and Florent. Quantitative Finance, 2002) is equation (16) on page 63, that states that:
In the ...
0
votes
2
answers
138
views
The meaning of filteration ( coin toss example )
Reference book is 'Steven Shreve: Stochastic Calculus and Finance'
What I don't understand is $F_3$ below picture
I understand that 'filteration' have accumulative information.
So when we tossed the ...
0
votes
1
answer
113
views
Black-Scholes differential equation rewritten [closed]
I have seen that the Black-Scholes equation
$$\frac{\partial V}{\partial t}+\frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+
rS\frac{\partial V}{\partial S}-rV=0$$
can also be written in the ...
1
vote
1
answer
115
views
Calculating Expectation of Stochastic Volatility
I have a question while reading THE NELSON–SIEGEL MODEL OF THE TERM
STRUCTURE OF OPTION IMPLIED VOLATILITY
AND VOLATILITY COMPONENTS by Guo, Han, and Zhao.
I don't understand why the above equations ...
1
vote
0
answers
57
views
Simultaneous Stochastic Differential Equations
I was thinking about cointegrated time series and came up with the following simultaneous equations model:
$dY_t = \alpha (Y_t - \gamma X_t)dt + \sigma dB_t$
$dX_t = \beta (Y_t - \delta X_t)dt + \tau ...
1
vote
0
answers
73
views
Differential vs. derivative in the Vasicek model [closed]
Can anyone help me in understanding how we get the line I have marked with a red arrow?
I guess I have trouble in understanding the difference between differentials and derivatives, i.e. what is the ...
3
votes
1
answer
145
views
Where does the term $\gamma$ come from when moving from measure $\mathbb Q^{N}$ to $\mathbb Q^{M}$?
Consider two measures $\mathbb Q^{M}$ and $\mathbb Q^{N}$, as well as the two numéraires $M$ and $N$, furthermore assume that $X\frac{N}{M}$ is a $\mathbb Q^{M}$-martingale. Furthermore, the ...
4
votes
1
answer
238
views
How am I supposed to understand the following statement on the convexity adjusted rate
Given, a numéraire $(N(t))_{0\leq t \leq T}$ and an index $(X(t))_{0\leq t\leq T}$ that is a $\mathbb Q^{N}$-martingale, we consider the natural payoff $V_{N}(T)$, where it pays
$$V_{N}(T):=X(T)N(T) \...
2
votes
0
answers
26
views
Finding the distribution of $I(T_{1},T_{n})$ under an appropriate measure if the forwards are lognormal? [duplicate]
My question follows beneath the "lengthy" setting I describe:
Given a tenor discretization $0 = T_{0}< ... < T_{n} =T$,
and under the assumption that under $\mathbb P$, for all $i = 1,....
6
votes
3
answers
585
views
Why does the diffusion term remain the same when we change pricing measure?
Consider some Itô process $dS(t)=\mu(t)dt+\sigma(t)dW^{\mathbb P}_{t}$ under the measure $\mathbb P$, where $W^{\mathbb P}$ is a $\mathbb P$-Brownian motion
In plenty of interest rate examples, I have ...
5
votes
1
answer
584
views
Where does 1/2 in Fourier Transform method of pricing options come from?
I am reading Jianwe Zhu's Applications of Fourier Transform to Smile Modeling. On page 26, the author is describing how to use the Fourier tranform to price vanilla European call options. If $f_j$ is ...
2
votes
1
answer
177
views
Obtaining the dynamics of the Vasicek model using Itô
Consider the following expression for the short-term interest rate
$$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},$$
which is ...
0
votes
0
answers
62
views
SDE of a Geometric Levy process with compound Poisson process
Suppose that a probability space $(\Omega, \mathcal{F}, \mathbb{P})$ is given. A geometric Levy process is defined in the form of $S_t=S_0 exp(X_t)$ where $S_0$, let's say, is the initial price and $...
2
votes
0
answers
100
views
Largest class of real world probability models admitting explicit risk-neutral change of measure
Assume we have two assets, a random asset $A_t$ and deterministic risk-free bond $B_t = e^{rt}$. Let $P$ be a model of the real-world probabilities of $S$ and $Q$ the unique associated risk-neutral ...
4
votes
1
answer
233
views
Pricing a contract
I'm currently trying to price some different kinds of contracts. I'm stuck on this following exercise, which I can't seems to find a good solution for. The following is assumed:
We are in a standard ...
2
votes
3
answers
262
views
Integral of brownian increments
I'm stuck at a problem and I'm not sure on how to proceed. My question is how would one go about and integrate the following
$$\sigma\int_{t}^{T}\mathrm{e}^{a\cdot u}\cdot (W_{u}-W_{t})du.$$
I've been ...
1
vote
2
answers
308
views
Why would exchange rates follow a geometric brownian motion?
I'm reading Shreve's Stochastic Calculus for Finance.
On page 382, he begins talking about exchange rates:
Finally, there is an exchange rate $Q(t)$, which gives units of domestic currency per unit ...
1
vote
0
answers
33
views
From the perspective of a company, when is the right time to start paying dividends?
I am trying to understand geometric Brownian motion as it relates to the present discounted value of future dividend payments.
I am supposing that a company has a revenue stream $f(t)$. This is just $...
1
vote
1
answer
219
views
Jump Diffusion Process question
I have a European call option with time maturity $T=3$ years,$K=50$, and given that $S(t)$ refers to the derivative is being described by the geometric Brownian motion with $S_{0}=100$ and $r = 0.04$....
0
votes
0
answers
161
views
what is $\int t dW$ and $\int W dt$? [duplicate]
More explicitly, if $W(t)$ is Brownian motion, what would be
$$f(t) := \int_0^t u dW(u)$$
and $$g(t) := \int_0^t W(u) du$$?
2
votes
0
answers
121
views
Are Stochastic Differential Equation diffusion terms always invariant under a change of measure?
I'm struggling with learning change of numeraire, and stochastic differential equations. I'm reading the beginning of Brigo and Mercurio's Interest Rate Models- Theory and Practice, and I'm on the ...
0
votes
1
answer
173
views
Calibration/estimation of the CEV model
The CEV model for a stock price $S(t)$, interest rate $r$ and variance $\delta$
$dS(t)=rS(t)dt+\delta S(t)^{\gamma}dW(t)$
where the volatility for the stock is given by
$\sigma(t)=\delta S(t)^{\gamma -...
3
votes
1
answer
199
views
Pricing of European options on two underlying assets
Is anybody able to give the solution to the following problem?
Suppose we have two assets, each of which follows a GBM process, and where $dW_S$ and $dW_X$ are correlated $(dW_SdW_X=\rho)$.
$dS=\mu_s ...
0
votes
1
answer
239
views
Solving SDE using integration factor and Ito's lemma
I don't understand how to define such integration factor in order to solve SDE, for example, as was shown in Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$ and Solving Stochastic Differential ...
3
votes
2
answers
151
views
American Option Valuation - Induction algorithm
The price of an American put option is given by
$$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
0
answers
93
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
0
answers
106
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 ...
7
votes
0
answers
210
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
0
answers
102
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
1
answer
238
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
$$
dS(t) = r(t) S(...
0
votes
0
answers
38
views
Instantaneous correlations in multi-currency G2++ model
Hey in "Interest Rate Models - Theory and Practice With Smile, Inflation and Credit" by Damiano Brigo, Fabio Mercurio we have dynamics for two interest rates and FX rate between them:
$$r_1(...
0
votes
1
answer
205
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
1
answer
116
views
Ansatz and HJB equation
Suppose we have an HJB equation of the form
$$
\frac{\partial v}{\partial t}+\frac{1}{2}\sigma^{2}\frac{\partial^{2}v}{\partial s^{2}}+max_{\delta^{a}}\left\{ \lambda^{a}(\delta^{a})\left[v(t,s,x+s+\...
3
votes
0
answers
250
views
Deriving Bachelier Greeks
I am working on the Bachelier Model with r not equal to 0 as described in the first and most upvoted answer in following link:
Bachelier model call option pricing formula
This is fairly easy to code ...
4
votes
0
answers
105
views
How to integrate Itô integral w.r.t time?
Let $W_t$ be a Brownian motion.
How to calculate the following integral
$$
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
1
answer
155
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 ...
0
votes
1
answer
156
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
1
answer
95
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
2
answers
248
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
0
answers
168
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
1
answer
230
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 ...