Questions tagged [integral]

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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

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
  • 131
1 vote
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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
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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
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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

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
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5 votes
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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)...
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1 vote
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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

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
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6 votes
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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

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

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