Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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246 views

Applying Ito's formula to complex functions

Within my lecture notes, the following definition is given: We say that the stochastic process $X_t$ has stochastic differential $$ dX_t = b_t dt + \sigma_t dW_t $$ if and only if $$ X_t = ...
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1answer
103 views

Girsanov's Theorem for Multiple Risky Assets

Girsanov's theorem provides the measure transformation from probability measure P to Q such that- $dW_t^Q=dW_t^P+\lambda dt\implies \xi_tW_t^Q$ is a martingale under the P measure where $\xi_t=e^{-\...
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1answer
200 views

Black Scholes in the case of dividends

Let's take the case where the underlying stock has the continuous dividend yield $\delta$. Then, in the risk-neutral world, $\frac{dS}{S}=(r-\delta)dt+\sigma dW^Q$. Suppose we want to price a ...
2
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1answer
281 views

Show that the Ito integral is Gaussian

Let $f(t), 0 \leq t \leq T$ be a deterministic function with $f(t) = \sum_{i=1}^na_{i-1}1_[t_{i=1}, t_i)(t)$ with $0 \leq t_0<t_1<...<t_{n-1} = T$. Show that the stochastic integral $I_t(f) ...
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1answer
103 views

squaring stochastic calculus and other solutions [closed]

It is well-known that the solution to the stochastic SDE $$ dS = S_0(\mu dt + \sigma dWt) $$ is $$ S_t=S_0 e^{(\mu-\frac{\sigma^2}{2})t+W_t} $$ Were $\sigma=0$, this is simply the formula for ...
2
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1answer
91 views

Distribution in Heston

$$dV_t=-k(V_t-1)dt+ \epsilon\sqrt{V_t}dW_t$$ $W_t$ is wiener process and the rest is just some parameters. For $T_{i+1}>T_{i}$ how do I find the expectation and variance of $V_{T_{i+1}}$ ...
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1answer
108 views

Finding the process of $X/Y$

This comes from Mark Joshi's concepts of mathematical finance exercise 4 chapter 11. If $$dX_t = \alpha X_t dt + \beta X_t dW_t$$ $$dY_t = \alpha Y_t dt + \gamma Y_t d\tilde{W}_t$$ with $W$ ...
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2answers
166 views

Isn't GBM the equivalent of adding infinitessimally small normally distributed returns?

The classic treatment of GBM for asset pricing leads to a point where eventually one gets a solution that is the same as assuming an underlying arithmetic Brownian motion, $X_t$, which has (over unit ...
2
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1answer
85 views

How to derive the dynamic of the log forward price?

I have the following Schwartz model: $$dS_t=a(\mu-\ln S_t)S_tdt+\sigma S_tdW_t$$ $$X_t=\ln S_t$$ $$dX_t=a(\hat{\mu}-X_t)dt+\sigma dW_t$$ with $\hat{\mu}=\mu-\frac{\sigma^2}{2a}\sigma$ $$F_t(T)= \exp\...
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157 views

Properties of Stochastic Exponential

Let $\{X_t\}_{t \ge 0},\{Y_t\}_{t \ge 0}$ be a continuous semi-martingale with $X_0 = Y_0 = 0$, let ${\cal E}(X)$ to be the unique solution of: $dZ_t = Z_t dX_t$ with $Z_0=1$. We can show that ${\cal ...
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1answer
2k views

Integral of Wiener process w.r.t. time

I have a doubt with regards to the calculation of the below integral- $\int_0^t W_sds$ where $W_s$ is the Wiener Process. This has been solved very ably in the following page. It turns out to be a ...
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64 views

Computing Malliavin Derivative for European Call Payoff

Let $X_t$ be a continuous local-martingale modeling the stock price given by $$ X_t = \int_0^t \sigma_t(T,K)dW_t , $$ and $\sigma_t(T,K)$ is an $L^2$-measurable process not adapted to $W_t$'s ...
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2answers
138 views

Show that the two solutions of the SDE are equivalent

I have a process: $$dr_t = (W_t^1 - ar_t)dt +\sigma dW_t^2$$ where $W_t^1$ and $W_t^2$ are brownian motions with instantaneous correlation coefficient $\rho$. I want to show that the solution of this ...
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1answer
151 views

Have I used correct state space formulation of Bivariate Trending OU process for Kalman Filter estimation?

Introduction I'm trying to estimate the parameters of an Ornstein Uhlenbeck process for a risky asset using the Kalman Filter but have doubts about the state space formulation that I am using. Also, ...
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1answer
144 views

Integral of the OU (Ornstein Uhlenbeck) process conditioned on hitting a threshold value for the first time

Let say I have a zero-mean OU process as follows: $dX_t = -\alpha X_t + dW_t$ The process starts at $x_0 = 0%$ and I'm interested in the event in which the process hits the value $x_{\tau} = a$ for ...
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1answer
428 views

How to take the differential of a stochastic integral?

Denote $$X_t = \int^t_0\sigma e^{-k(t-s)}dW_s$$ here $W_s$ is the Brownian motion, $k,\sigma$ are constants. I want to calculate $d X_t$ and the variance $Var[X_t].$ I know how to take the ...
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2answers
234 views

Show a process is Martingale

$$Z(t)=(\frac{S(t)}{H})^p$$where $S$ has a standard Black-scholes Dynamics for a stock, $H$ is a postive constant and $p =1 - \frac{2r}{\sigma^2}$. How can I show that $Z(t)/Z(0)$ is a postive Q-...
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1answer
482 views

Equivalent Martingale Measure(EMM) of Inverse of Stock Price

I met this question says how to price a vanilla call option $C(St,t,T,K) = \frac{1}{S_T}$which pays the inverse of a stock $V_{t} = \frac{1}{S_{t}}$ at maturity if the stock price follows a geometric ...
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164 views

Pricing a structured note instrument

I am trying to work out the following fixed income problem, where I am asked to price a structured note in Excel, which seems to me to be a reverse collar. My purpose was replicating this structured ...
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98 views

Complete Financial Market: Integrability condition for Contingent Claims

Consider an arbitrage-free and complete financial market with underlying filtered probability space $(\Omega,\mathcal{F},\{\mathcal{F}_{t}\}_{t\,\in\,[0,T]},\mathbb{Q})$, where $T\in(0,\infty)$ is ...
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69 views

Stochastic Calculus: How to test for dependency of random variables

If I let $g(x)$ be a deterministic function of a real variable $x$ and define $X(t)$ as: $$X_T=\int_{0}^{T}f(u)dW_u$$ with $W_t$ being a wiener process. For $s<t$, Will $X_s$ and $X_s-X_t$ then be ...
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89 views

Geometric Brownian Motion: Drawdown as a function of time

Suppose I have a strategy (model it as the usual geometric Brownian motion with a drift). Question is, how does max drawdown grow as a function of duration?
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1answer
261 views

Test if a process (with no drift) is a martingale

Consider the process $$Z(t)=\int_{0}^{t} \frac{u^a}{t^a}dW_u$$ for some real constant $a$ and $W_t$ is a wiener process. I want to check whether this process is a $F_t^W$-martingale. I noticed Lemma 4....
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1answer
2k views

What is an adapted process

I am reading Björk, Arbitrage theory in Continous Time and I have noticed that he uses the term adapted proces a lot. I can't seem to understand what an 'adapted proces' is by the wikipedia article. ...
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1answer
128 views

How to find correct change of measure

I'm trying to figure out how to find the correct equivalent martingale measure to change into. First of, since I am on mobile and find it hard to write LaTeX here, I will refer to Wikipedia's version ...
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2answers
93 views

Is a wiener proces measurable? (exercise from Bjork)

I will claim $$E[W(T) \vert F_t] = 0$$ for $t<T$. Anyway, in an exercise in Bjork the results requires that $$E[W(t) \vert F_t] = 0$$ But why? Isn't $W(t)$ measurable at time $t$ and hence not ...
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885 views

Girsanov Theorem, Radon-Nikodym Derivative backward

Given a filtered probablity space $(\Omega,\mathcal{F},{\mathcal{F}}_t,\mathbb{P})$ and a standard Brownian motion $W_t$. Normally, in Girsanov Theorem, we use the exponential martingale $Z_t=\exp(-\...
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208 views

Binomial model's Radon-Nikodym derivative

Related: Dumb question: is risk-neutral pricing taking conditional expectation? In the one-step binomial model... For $\frac{d \mathbb Q}{d \mathbb P}$, I think it's $\frac{d \mathbb Q}{d \mathbb P}...
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2answers
368 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 ...
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1answer
79 views

Spot Interest Rate at time $t$

I know that the general model for the dynamics of the spot interest rate is $$dr(t)=\mu(r,t)dt+\sigma(r,t)dB(t)$$ My question is, if $P(t,T)$ is the bond value at time $t$, how would I derive $dP$?
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1answer
82 views

Deriving $dR(t)$ For Reverse Exchange Rate

Say $Q(t)$ is the exchange rate at time $t$. It's the price in domestic currency of one unit of foreign currency and converts foreign currency into domestic currency. The model for the dynamics of ...
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2answers
373 views

Variance of $\int_{t=o}^{T}\sqrt{|B(t)|}$ $dB(t)%$

I'm new to stochastic calculus. Could someone please explain how I would calculate the variance of $\int_{t=o}^{T}\sqrt{|B(t)|}$ $dB(t)%$ I'm aware that I would first have to calculate the ...
3
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1answer
586 views

Ito's Lemma: Multiplication Rule

I have a conceptual question about Ito's lemma, in particular, the multiplication. Ito's multiplication rule states, that multiplying dt by itself or by dx (the stochastic differential) equals zero. ...
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1answer
229 views

Feynman Kac Terminal value problem two variables

So, I need some help to move forward with this problem. $$ \begin{cases} \frac{\partial F(t,x,y)}{\partial t}+\frac{1}{2}\frac{\partial^2 F(t,x,y)}{\partial x^2}+\frac{9}{2}\frac{\partial^2 F(t,x,y)}...
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2answers
396 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^{-...
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1answer
167 views

Ito representation unique up to indistinguishability? Proof?

Given an Ito-process $X(t)$, $t\in[0,T]$ $$X(t)=X_{0}+\int_{0}^{t}F(s)ds + \int_{0}^{t}G(s)dW(s)$$ with $F\in \mathbb{L}^{1}(0,T)$ and $G\in\mathbb{L}^{2}(0,T)$. It is now often claimed that this ...
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1answer
405 views

Code examples of solving Stochastic Optimal Control problems

I'm currently reading a book demonstrating how Stochastic Optimal Control can solve common optimization problems encountered within quantitative finance. I haven't covered much continuous mathematics ...
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1answer
323 views

Change of numeraire from bank account to Zcb [closed]

Why is there no drift adjustment when numeraire is changed from bank account (risk neutral measure) to zero coupon bond who matures at time of payoff (fwd risk neutral measure) ?
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1answer
437 views

Merton's jump diffusion

Can someone help me finding the expected value of the solution to Merton's jump diffusion model: \begin{align} S_t &= S_0 \exp \left( \left(r - \frac{\sigma^2}{2} - \lambda k \right) t + \sigma ...
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1answer
327 views

Stochastic integrals wrt to independent Wiener processes are uncorrelated, but potentially dependent?

In Proof of Proposition 1.2.20 in the following lectures notes http://math.uni-heidelberg.de/studinfo/reiss/sode-lecture.pdf I found following quote " stochastic integrals with respect to ...
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0answers
464 views

stochastic modeling and machine learning [closed]

For a little bit of background, I've been studying stochastic calc and a few of it's applications (currently I'm still at the early stages of learning applications) and have been curious as to whether ...
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0answers
161 views

Normalized Gains Process is a Q-Martingale - Proof and Intuition

I'm trying to work the proof that the normalized gains process, $G^z_t = \frac{S_t}{B_t}+\int^t_0\frac{1}{B_s}dD_s$ is a Q-martingale under Q (the risk-neutral measure). I'll show what I've worked ...
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1answer
334 views

Vector of differences of Brownian motion integrals is multivariate normal

Given a 2-dimensional Wiener process $(W_{1},W_{2})$ with correlation $\rho$. Let \begin{equation*} X(t):= F(t) + \int_{0}^{t} f(s) dW_{1}(s) + \int_{0}^{t} g(s) dW_{2}(s)\end{equation*} for some ...
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1answer
264 views

Change-of-measure: Dynamics of $\log(S_t)$ with $S_t$ as numeraire [duplicate]

Let $S$ be a GBM with dynamics $dS_t/S_t=rdt+\sigma dW_t$. We want to compute the following expected value: \begin{align*} \mathbb{E}(S_T\log(S_T)). \end{align*} Using a change of measure we can write ...
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1answer
131 views

Zero value of cash flow for future in Shreve's book

Here is the statements of future price in Shreve's book Stochastic Calculus for Finance II page 244 to proof the ...
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45 views

Is the 'constant weight in the risky asset' portfolio-strategy self-financing?

My question concerns a topic in quantitative finance that I feel is often brushed under the table: is a given strategy self-financing. We have two assets, one risky and one riskless, defined by the ...
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1answer
286 views

Trouble understanding jump part in Kou double exponential jump diffusion model

I am trying to work with Kou's double exponential Jump-diffusion model and simulate a price path in a programming language. So the dynamics of the asset price in Kou's model follow: \begin{equation} ...
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63 views

Laplace Exponent of a Jump-Diffusion Process

I'm currently reading a paper (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2543702) which uses the following process to describe the dynamics of a firm's asset value: \begin{equation} V_t = ...
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1answer
50 views

Payoff of an odd indicator of one stock being greater than another

Suppose $S_t^1$ and $S_t^2$ are two stocks following GBMs and have current value $s_1$ and $s_2$ respectively. How can I explicitly compute the payoff $$ V(t,s_1,s_2)\triangleq \mathbb{E}\left[ 1_{\{...
3
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1answer
270 views

Pricing the Passport option

Suppose underlying asset $S$ $$dS = \mu Sdt + \sigma Sd W$$ our portfolio $\pi$ consist with $q(t)$ stock $S$ and cash $\pi - qS$...