Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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9 votes
2 answers
2k views

Ho and lee derivation for short rates model

A silly question that is bugging me. I am working my way through Baxter and Rennie (again) and I am getting my wires crossed on the short rate models in particular the straight forward Ho and Lee ...
0 votes
0 answers
44 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 $\sigma^{0}_{i}$...
3 votes
1 answer
204 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
148 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
152 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 ...
-1 votes
1 answer
247 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.
2 votes
0 answers
176 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 ...
6 votes
0 answers
239 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
366 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
166 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
155 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
86 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
264 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 ...
4 votes
1 answer
262 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) \...
3 votes
1 answer
171 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 ...
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,....
1 vote
1 answer
375 views

Transition density of geometric Brownian motion with time-dependent drift and volatility

Can you provide a reference to the transition density of the scalar geometric Brownian Motion with time-dependent drift and volatility, i.e. the scalar process $X = (X_t)_{t\geq 0}$ defined by the SDE ...
5 votes
1 answer
659 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
431 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 ...
2 votes
0 answers
104 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 ...
2 votes
3 answers
305 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 ...
17 votes
9 answers
10k views

Why the expected return rate of a stock has nothing to do with its option price?

OK, I admit that this is a frequently asked question. But I couldn't find a satisfying answer after I read the explanations of books, went through the derivations of B-S formula, and searched answers ...
4 votes
1 answer
249 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 ...
1 vote
1 answer
306 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 $r = 0.04$....
2 votes
2 answers
830 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
36 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
758 views

If $W_t$ is standard Brownian motion, what is $\int_0^T W_t \ln(W_t) dW_t$?

If $W_t$ is standard Brownian motion, what is meant by $\int_0^T W_t dW_t$ in finance? Furthermore, what then is the meaning of $\int_0^T W_t \ln(W_t) dW_t$?
0 votes
0 answers
437 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
189 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 ...
1 vote
0 answers
325 views

First Hitting Time and Monte Carlo simulation

I am interested in implementing a Monte Carlo simulation in Python of a first hitting time (first passage time) of an Ornstein-Uhlenbeck process (or similar). Specifically interested in fatter tails ...
1 vote
1 answer
476 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
348 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
558 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. ...
3 votes
2 answers
168 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
177 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 ...
1 vote
0 answers
231 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 ...
8 votes
0 answers
282 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 ...
4 votes
1 answer
341 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(...
1 vote
0 answers
322 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 ...
3 votes
0 answers
1k 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
151 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
342 views

Stochastic Interest Rates in Option pricing

My lecturer has written the slide below. The function B^T(t) is a zero coupon bond. I don't understand how V(t) can be a negative integral from 0 to ...
0 votes
1 answer
276 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 ...
14 votes
3 answers
2k views

Deterministic interpretation of stochastic differential equation

In Paul Wilmott on Quantitative Finance Sec. Ed. in vol. 3 on p. 784 and p. 809 the following stochastic differential equation: $$dS=\mu\ S\ dt\ +\sigma \ S\ dX$$ is approximated in discrete time by $$...
3 votes
2 answers
892 views

Dumb question: is risk-neutral pricing taking conditional expectation?

Dumb question: is risk-neutral pricing taking conditional expectation? $\tag{1}$ In trying to recall intuition for risk-neutral pricing, I think I read that we should price derivatives risk-neutrally ...
3 votes
0 answers
435 views

What exactly is/How exactly do we interpret the binomial model's Radon-Nikodym derivative?

Related: Dumb question: is risk-neutral pricing taking conditional expectation? Maybe there's not quite an interpretation given Lewis' triviality result if $E^Q[X]$ is a real world conditional ...
1 vote
1 answer
106 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
627 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 $\...
6 votes
1 answer
256 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 ...
0 votes
0 answers
392 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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