Collocation Boundary Element Method for the pricing of Geometric Asian Options

Aimi, A.; Guardasoni, C.

doi:10.1016/j.enganabound.2017.10.007

Cited by 9 publications

(17 citation statements)

References 18 publications

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“…For geometric Asian option, a formal and deep argumentation was described in the work of Aimi and Guardasoni, and it proceeds as follows. The integral representation formula in the domain of the differential problem The value V ( S , A , t ) of an up‐and‐out barrier call option satisfying the differential problem is given by the integral representation formula proven in the work of Aimi and Guardasoni

V false(S, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), S, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

at every point ( S , A , t ) in the existence domain Ω × [0, T ) with

normalΩ : = false(0, B false) \times double-struckR

. The boundary integral equation (BIE) In the integral formula ,

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

is unknown so we cannot apply directly to get the solution over the whole domain Ω × [0, T ). However, if we consider the limit for S → B in and apply the boundary condition , we obtain the BIE

0 = V false(B, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), B, A)

…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…However, if we consider the limit for S → B in and apply the boundary condition , we obtain the BIE

0 = V false(B, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), B, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), B, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

in the sole unknown

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

.The next goal is to implement a strategy for numerically solving . The approximation of

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

will be then inserted in the representation formula to get the solution V at every desired point of the domain Ω × [0, T ). Collocation method for the numerical resolution of We are going to approximate the BIE unknown

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

by collocation method as done in the work of Aimi and Guardasoni

normalΔ t : = \frac{T}{N_{t}}, N_{t} \in N^{+}, t_{k} : ...

…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…The value V ( S , A , t ) of an up‐and‐out barrier call option satisfying the differential problem is given by the integral representation formula proven in the work of Aimi and Guardasoni

V false(S, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), S, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

at every point ( S , A , t ) in the existence domain Ω × [0, T ) with

normalΩ : = false(0, B false) \times double-struckR

.…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…We are going to approximate the BIE unknown

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

by collocation method as done in the work of Aimi and Guardasoni …”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…The existence and uniqueness for are stated in the work of Polidoro, and the exact option value V ( S , A , t ) is known to be the payoff expected value

V false(S, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{+ \infty} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true},

provided that the transition probability density function

G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, \overset{t}{true})

is available. From the works of Barucci et al and Aimi and Guardasoni, we have

\begin{matrix} alignleft \\ align-1 G (S, A, t; \tilde{S}, \tilde{A}, \tilde{t}) = & align-2 \frac{\sqrt{3} H [\tilde{t} - t]}{π σ^{2} {(\tilde{t} - t)}^{2}} \exp \{- \frac{2}{σ^{2} (\tilde{t} - t)} \log^{2} (\frac{S}{\tilde{S}}) + \frac{6}{σ^{2} {(\tilde{t} - t)}^{2}} \log (\frac{S}{\tilde{S}}) (A - \tilde{A} + (\tilde{t} - t) \log (S)) \end{matrix}

…”

Section: The Differential Model Problemmentioning

confidence: 99%

See 4 more Smart Citations

Numerical pricing of geometric asian options with barriers

Aimi

Diazzi

Guardasoni

2018

Math Methods in App Sciences

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V false(S, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), S, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

at every point ( S , A , t ) in the existence domain Ω × [0, T ) with

normalΩ : = false(0, B false) \times double-struckR

. The boundary integral equation (BIE) In the integral formula ,

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

is unknown so we cannot apply directly to get the solution over the whole domain Ω × [0, T ). However, if we consider the limit for S → B in and apply the boundary condition , we obtain the BIE

0 = V false(B, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), B, A)

…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…However, if we consider the limit for S → B in and apply the boundary condition , we obtain the BIE

0 = V false(B, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), B, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), B, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

in the sole unknown

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

.The next goal is to implement a strategy for numerically solving . The approximation of

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

by collocation method as done in the work of Aimi and Guardasoni

normalΔ t : = \frac{T}{N_{t}}, N_{t} \in N^{+}, t_{k} : ...

…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…The value V ( S , A , t ) of an up‐and‐out barrier call option satisfying the differential problem is given by the integral representation formula proven in the work of Aimi and Guardasoni

V false(S, A, t false) = true \int_{- \infty}^{+ \infty} true \int_{0}^{B} V ((), \overset{S}{true}, \overset{A}{true}, T) G ((), S, A, t; \overset{S}{true}, \overset{A}{true}, T) normald \overset{S}{true} normald \overset{A}{true} + true \int_{t}^{T} true \int_{- \infty}^{+ \infty} \frac{σ^{2}}{2} B^{2} \frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true}) G ((), S, A, t; B, \overset{A}{true}, \overset{t}{true}) normald \overset{A}{true} normald \overset{t}{true},

at every point ( S , A , t ) in the existence domain Ω × [0, T ) with

normalΩ : = false(0, B false) \times double-struckR

.…”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…We are going to approximate the BIE unknown

\frac{\partial V}{\partial \overset{S}{true}} ((), B, \overset{A}{true}, \overset{t}{true})

by collocation method as done in the work of Aimi and Guardasoni …”

Section: Semianalytical Methods For Barrier Options Pricingmentioning

confidence: 99%

“…The existence and uniqueness for are stated in the work of Polidoro, and the exact option value V ( S , A , t ) is known to be the payoff expected value