Questions tagged [stochastic-calculus]

A branch of mathematics that operates on stochastic processes.

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8
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2answers
4k views

How to compute the variance of this stochastic integral?

I'm new to stochastic calculus and I did an exercise but I don't know if it is correct, so I need somebody with more experience to check if it is true. I am trying to compute the variance of the ...
8
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1answer
932 views

Girsanov's Theorem - Change of Measure

I have trouble understanding Girsanov's theorem. The Radon Nikodym process $Z$ is defined by: $$Z(t)=\exp\left(-\int_0^t\phi(u) \, \mathrm dW(u) - \int_0^t\frac{\phi^2(u)}{2} \, \mathrm du\right)$$ ...
8
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1answer
334 views

Non-arbitrage theory and existence of a risk premium

Consider a probability filtred space $(\Omega, \mathcal F, \mathbb F, \mathbb P)$, where $\mathbb F = (\mathcal F_t)_{0\leq t\leq T}$ satisfing the habitual conditions and isgenerated by $1 d $- ...
8
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1answer
509 views

How to measure a non-normal stochastic process?

If I understand right, Itô's lemma tells us that for any process $X$ that can be adapted to an underlying standard normal Wiener measure $\mathrm dB_t$, and any twice continuously differentiable ...
8
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1answer
934 views

What is the forward rate for a Black-Karasinski interest rate model?

I was wondering if anyone could help me with the instantaneous forward rate equation for a Black-Karasinski interest rate model? I was also after the Black-Karasinski Bond Option Pricing Formula.
8
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1answer
181 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 ...
8
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3answers
490 views

Stochastic Calculus Rescale Exercise

I have the following system of SDE's $ dA_t = \kappa_A(\bar{A}-A_t)dt + \sigma_A \sqrt{B_t}dW^A_t \\ dB_t = \kappa_B(\bar{B} - B_t)dt + \sigma_B \sqrt{B_t}dW^B_t $ If $\sigma_B > \sigma_A$ I ...
8
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1answer
273 views

Stochastic control (HJB) for wealth process involving stopping times

Given a wealth process that evolves as $$d w_t = r w_t dt + \theta_t ( \sigma dW_t + (\mu-r) dt) - c_t dt.$$ where $\theta_t$ is the worth of holding at time $t$ and $c_t$ is the consumption stream. ...
8
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1answer
528 views

Integral-differential equation for forward rates

I am struggling in this question: Let $P(t,T)$ denote the price of a zero-coupon bond (with marturity at time $T$) at time $t \in [0,T]$. As usual, at time $t$ for maturity $T$, the forward rate is ...
8
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1answer
391 views

What is a good Computer Algebra System for financial engineering?

I would like to know if there exists some computer algebra systems adapted to calculate pricing based on particular models, i.e. pricing YoY Inflation Swap under Jarrow Yildirim Model. I know that ...
8
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2answers
282 views

Why won't Bjork ever show that the integrability condition is satisfied?

A major technique employed throughout Bjork's "Arbitrage theory in Continuous Time" is that when taking the expectation of a stochastic integral, the result is 0. This is based on a result presented ...
7
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2answers
2k views

What is the average stock price under the Bachelier model?

Let's say stock price follows following process: $$dS(t) = \sigma dW(t)$$ where $W(t)$ is Standard Brownian motion. The initial level for the stock is $S(0)$. Define the average of stock price $Z(t)...
7
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4answers
2k views

Can strike prices of options be negative?

I am trying to understand the stochastic model of a financial market in one period by [Föllmer, Schied]. They introduce call and put options for the primary assets, which are non-negative. They do not ...
7
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240 views

A question on Ito

If we know the dynamics of $S$, then we can estimate the value of $S$ at a time point, $t$. Here, I have a question concerning how to solve for $S_t$ by Itô because I obtained different results by ...
7
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1answer
4k 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 ...
7
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1answer
882 views

How to show that this process is “normally distributed”?

Say we have following SDE (Vasicek): $$dr(t) =(b-ar_t) dt + \sigma dW_t$$ I am able to reach an integral form of this SDE : $$r(t) = r(0) e^{-at} + \frac{b}{a}[1 - e^{-at}] + \sigma e^{-at}\int_0^t e^...
7
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1answer
259 views

Show that $(W_t, \int_0^t W_s ds)$ has a normal joint distribution

I have to show that, if $W_t$ is a 1-d Brownian motion then $\biggl(W_t, \int_0^t W_s ds\biggr)$ has normal distribution. Hint: apply Ito formula to this bivariate process. Any idea or suggestion on ...
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1answer
607 views

Baxter & Rennie HJM: differentiating Ito integral

From Baxter and Rennie, page 138: $$f(t,T)=\sigma W_t+f(0,T)+\int_0^t\alpha(s,T)ds$$ $$Z_t=\exp-\bigg(\sigma(T-t)W_t+\sigma\int_0^tW_sds+\int_0^Tf(0,u)du+\int_0^t\int_s^T\alpha(s,u)ds\bigg)$$ $$dZ_t=...
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347 views

Stochastic Integral Graph

As we can represent the integration of $f(x)$ on $[a,b]$ with the graph below, I was wondering how to represent the following integral with $X(t)$ a Brownian motion, $f(t)$ any function and $t_j = ...
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1answer
1k views

How to use the Girsanov theorem to prove $\hat{W_t}$ is a $\hat{\mathbb P}$-Brownian motion?

Let $T > 0$, and let $(\Omega, \mathscr F, \{\mathscr F_t\}_{t \in [0,T]}, \mathbb P)$ be a filtered probability space where $\mathbb P = \tilde{\mathbb P}$ (risk-neutral measure) and $\mathscr F_t ...
7
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1answer
913 views

generalized black scholes

I understand how to derive the black scholes solution if $dS_t$ = $\mu S_tdt$ + $\sigma S_tdW_t$ and r is constant. The solution is c(t, x) = $xN(d_{+}(T - t), x))$ - K$e^{-r(T - t)}N(d\_(T - t), x))$ ...
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1answer
3k views

Implications of shifting the lognormal model for forward rates from a probability perspective

I have a question regarding the application of a shift to the Black-Scholes formula for negative forward rates. I am reading in the Brigo book that "increasing the shift $\alpha$ shifts the ...
7
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1answer
191 views

Proof that the stopping time for a Brownian Motion is finite for given target levels

Given a standard brownian motion $W_t$ and defining $\tau$ as: $\tau :=inf\{t\geq0:W_t=1$ or $W_t=-2\}$ The proof below shows that the stopping time is finite: $P(\tau < t) \geq (|W_t|>2)\\$ ...
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1answer
216 views

Option pricing with Brownian Bridge

Say I have an asset following arithmetic Brownian motion $$ dX(t) = \sigma dW^\bot (t) $$ with $\sigma$ constant, and I have prices of vanilla options on $X$. I introduce a Brownian bridge $$ dY(t) = ...
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0answers
142 views

Random variable minus Integral of Ito Generator is a Martingale under what conditions?

I am reading about american option pricing and the variational inequality, and the book I am reading states, in the derivation of the variational inequality, the following is a martingale: $$M_s = U(s,...
6
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3answers
593 views

Expectation of exponential of 3 correlated Brownian Motion

Consider, are correlated Brownian motions with a given I want to calculate the, , I can't think of a way to solve this although I have solved an expectation question with only a single exponential ...
6
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1answer
4k views

Can I always use quadratic variation to calculate variance?

Suppose we have a Brownian Motion $BM(\mu,\sigma)$ defined as $X_t=X_0 + \mu ds + \sigma dW_t$ The quadratic variation of $X_t$ can be calculated as $dX_t dX_t = \sigma^2 dW_tdW_t = \sigma^2 dt$ ...
6
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2answers
420 views

Clarification on Deriving Ito's Lemma

The classical approach to deriving Ito's Lemma is to assume we have some smooth function $f(x,t)$ which is at least twice differentiable in the first argument and continuously differentiable in the ...
6
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2answers
596 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 ...
6
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1answer
4k views

Girsanov Theorem for Quanto/Compo adjustment

Assume that I have a foreign asset $$Y_t = Y_0 \exp \left((r_f-\frac{1}{2}\sigma^2_Y)t+\sigma_Y W_t^1\right)$$ and an exchange rate $$X_t = X_0 \exp\left((r_d-r_f-\frac{1}{2}\sigma^2_X)t+\sigma_X ...
6
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2answers
540 views

Using Black-Scholes to price a geometric average price call

Sorry if this is the wrong exchange for this question. It seems to be the most relevant, anyway. I'm trying to learn and understand the Black-Scholes framework, with a focus on the stochastic ...
6
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2answers
193 views

Radon Nikodym derivative when changing numeraires

I note from Wikipedia that if $Q$ and $Q^N$ are two measures corresponding to numeraires $M$ and $N$, then the Radon Nikodym derivative is given by: $$\frac{dQ^N}{dQ} = \frac{M(0)}{M(T)}\frac{N(T)}{N(...
6
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1answer
4k views

Extended Hull White Interest Rate Model for Zero Coupon Bond

Let's take the following three SDEs: $$dr=u(r,t)dt + w(r,t)dX$$ $$u(r,t)=a(t)-br$$ $$w(r,t)=c$$ where $b$ and $c$ are constants and $a(t)$ an arbitrary function of time $t$. If Zero Coupon Bond $Z(...
6
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1answer
187 views

Why is GARCH more often applied in risk analysis than stochastics?

I am trying to look out for something I can engage in for my final year project (M.Sc) but my interests lie more in risk analysis (specifically credit risk). I have tried searching the web but really ...
6
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2answers
103 views

Solution for a SDE for a Bond found in Bugard & Kjaer

I'm going over the paper -Partial Differential Equation Representation of Derivatives with Bilateral Counterparty Risk and Funding Costs- from Burgard and Kjaer. There the following SDE is given for ...
6
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1answer
245 views

Derivation of the Stochastic Vol PDE

I'm trying to follow the derivation of the stochastic vol pde for an option price - as given in Gatheral (The vol surface), Wilmott on Quant Finance and many other places. As usual one starts off with ...
6
votes
1answer
214 views

Solving $dX_{t} = \mu X_{t} dt + \sigma dW_{t}$

I want to solve the following SDE: $$ dX_{t} = \mu X_{t} dt + \sigma dW_{t} \quad X_{0} = x_{0}$$ Integrating, I get: $$ X_{t} - x_{0}= \mu \int_{0}^{t} X_{s} ds + \sigma \int_{0}^{T} dW_{t} $$ $$ ...
6
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1answer
154 views

Why is the numeraire in the LGM model tradeable?

I'm trying to understand the LGM model, which Hagan defines as follows. The state variable $X$ evolves according to $$dX(t) = \alpha(t) dW^N(t)$$ wrt the numeraire $$N(t) = \frac{1}{P(0,t)} e^{H(t)X(...
6
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1answer
171 views

Application of Vibrato Montecarlo methods

Ciao, I was studying Vibrato Montecarlo methods and I came up with a very simple question: what is an real application of this method? Let me explain. In short the main idea of the method is the ...
6
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1answer
176 views

Computation of Expectation

This question has so long preoccupied my mind.Please help me to solve it. Question: Assume $X_t$ described by the following stochastic differential equation $$dX_t^{\,\alpha}=\alpha X_t^{\,\alpha} ...
6
votes
2answers
346 views

Itô diffusion processes in finance with unknown distribution at a terminal value

In several papers it is argued that for many Itô diffusion processes, $$dX_t = a(t,X_t)dt+b(t,X_t)dB_t,$$ in mathematical finance the distribution of $X_T$ for fixed $T>0$ is unknown, which makes ...
6
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1answer
304 views

Upper bound concerning Snell envelope

Consider a non-negative continuous process $X = \left (X_t \right)_ {t\geq 0}$ satisfying $ \mathbb E \left \{ \bar X \right\}< \infty $ (where $ \bar X =\sup _{0\leq t \leq T} X_t $) and its ...
6
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0answers
164 views

Expectation over Markov Process and discrete Ito integral (discrete stochastic calculus)

I am doing a research on communication protocol design. A file of $n$ blocks is transferred in several rounds and $R_i$ denotes the number of blocks received in the $i$-th round. The sender sends $n-...
5
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1answer
188 views

Periodic functions when determining No Arbitrage price

Is it possible to value a T-claim which has a periodic component? For example a claim such as $X = cos(S(T))$. We assume here that $S(T)$ is the stock price derived from the dynamics $dS(t)=rS(t)dt+\...
5
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3answers
455 views

Determine $E[W_p W_q W_r]$

Given prob space $(\Omega, \mathscr{F}, P)$ and a Wiener process $(W_t)_{t \geq 0}$, define filtration $\mathscr{F}_t = \sigma(W_u : u \leq t)$ Let 0 < p < q < r. Determine $E[W_p W_q W_r]$. ...
5
votes
2answers
200 views

Stochastic growth model

In this problem we consider a model of stochastic growth. In particular, consider the following system of SDEs: \begin{align} dX_t &= Y_t dt + \sigma_XdZ_{1t}\\ dY_t &= -\lambda Y_t dt + \...
5
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1answer
4k 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. ...
5
votes
1answer
1k 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(-\...
5
votes
1answer
688 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 ...
5
votes
1answer
172 views

Stochastic Differential

Let $W_t$ be a Wiener process. It is clear to me that $dW_t$ is of size $\sqrt{dt}$. This can be seen because $$ \mathrm{Var}(W_{t+\Delta} - W_{t})=\Delta. $$ But am I allowed to actually write $(...

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