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Step by step integration of the Hull-White SDE

I'm struggling to understand the integration process of the Hull-White equation: \begin{equation} dr(t)=[\nu(t)-ar(t)]dt+\sigma dW(t) \end{equation} In the majority of the references that I have ...
vsa's user avatar
  • 61
1 vote
0 answers
35 views

Inverse differencing in continuous time

I want to fit a continuous time ARMA (CARMA) model to traffic data $T_t$. After removing trend and seasonality I need first order differencing to obtain stationarity. Then I fit a CARMA model (yuima ...
Valentin's user avatar
  • 135
0 votes
2 answers
77 views

Obtain B-S-M from a binomial tree as n goes to infinty using Lebesgue integral

My question is simple, consider a European call with payoff max(S_T-K, 0), Let's suppose that the underlying stock follows a binomial tree with up and down factors I know as we take n goes to infinity ...
nic's user avatar
  • 1
2 votes
0 answers
100 views

Path integral approach to price call option on zero coupon bonds

I am given the following identities: $$ Z[J,t_1,t_2]=\int D W e^{\int_{t_1}^{t_2}dtJ(t)W(t)}e^{S}=e^{\frac{1}{2}\int_{t_1}^{t_2}dtJ(t)^2} $$ $$ \int_t^Tdx\alpha(t,x)=\frac{1}{2}\left[\int_t^Tdx\sigma(...
TheHunter's user avatar
  • 133
2 votes
2 answers
290 views

Calculate value of Integral of Wiener process $\int_{0}^t e^{\lambda u } dZ_u$

I am not quite sure how to solve this integral to be able to do numerical calculations with it. $\lambda$ is a constant, $u$ is time, and $Z_u$ is a wiener process. Can anyone provide some direction ...
NC520's user avatar
  • 274
1 vote
0 answers
94 views

Choice of grid for numerical integration

I have to compute an integral involving the characteristic function for pricing options in a model and it so happens that accurate approximation seems to be mostly about putting lots of points in ...
Stéphane's user avatar
  • 2,486
0 votes
1 answer
114 views

How to evaluate the following integral?

I stumbled across an expression and I wonder how to evaluate this: $-\int_ {0} ^ {+\infty} {v(x)} dw^{+} (1-p(x))$ where $v(x)$ is some utility function and $w(p(x)) $ is a decision weighting function,...
T123's user avatar
  • 545
2 votes
0 answers
34 views

How to derive Parameter Derivative within an FFT integral

I have the following function (Carr-Madan) of which I am trying to take the derivative wrt $\theta$: $c(k)=\int_0^\infty \frac{e^{-iuk}}{\alpha^2 + \alpha - u^2 + i(2\alpha+1)u} e^{\phi_T(u-(\alpha+1)...
Schmied's user avatar
  • 21
6 votes
1 answer
697 views

Is there a closed-form solution for the following integral?

The integral under consideration is as follows: $$ F=\int_{a}^{1} \exp\Big\{c\Phi^{-1}(x+b) + d\Big\}\; \mathrm dx, $$ where $0<a, b<1$, and $c>0, d\in\mathbb{R}$ are constants, and the ...
user53249's user avatar
  • 419
5 votes
2 answers
419 views

Expectation of integral where one of limits of integration is a random variable

Is it correct to write \begin{equation} E_t \int_0^{X_T} f(z) dz = \int_0^\infty \left(\int_0^x f(z) dz \right) p(x)dx \,\,? \end{equation} Here $X_T$ is a positive random variable with density $p(x)...
user avatar
1 vote
1 answer
108 views

Regression of stochastic integral on Wiener process

This question is a follow-up from the following: conditional expectation of stochastic integral so I won't repeat myself regarding assumptions and notation. Using Brownian bridge approach, we know ...
Gabriele Pompa's user avatar
1 vote
2 answers
250 views

Show that $\int_0^T(T-t)dW_t=\int_0^TW_t \ dt$

I want to show that $$\int_0^T(T-t)dW_t=\int_0^TW_t \ dt \tag1$$ By Ito's lemma we can write $$d((T-t)W_t)=(T-t)dW_t-W_t \ dt.\tag{2}$$ Now If I integrate this expression and use that $W_0=0$ I should ...
Parseval's user avatar
  • 221
6 votes
1 answer
115 views

Esscher Premium: Integral Transform Proof

I have some difficulty understanding the following proof and I hope someone can help me with that. Claim: I want to show that $E_\alpha(S)=\frac{d}{dr} \log M_S(r)|_{r=\alpha} $, where $M_S(r)=E(\exp(...
Wombat's user avatar
  • 181
4 votes
1 answer
518 views

Ornstein–Uhlenbeck process – integration by parts

While deriving the solution for the stochastic differential equation that models the Ornstein–Uhlenbeck process, Paul Wilmott (Paul Wilmott on Quantitative Finance, chapter 4, page 87) performs the ...
Vinícius Lopes Simões's user avatar
10 votes
2 answers
1k views

conditional expectation of stochastic integral

let $M_t$ be the following stochastic integral $$ M_t = \int_0^t \sigma_s dW_s $$ where $\sigma_t$ is a sufficiently regular deterministic function and $W_t$ is a standard Wiener process (that is $...
Gabriele Pompa's user avatar