170
10 Quantum Finance and Path Integrals
Exercises
1. Is the operator x
∂
∂ x
hermitian or anti-hermitian? Prove your answer!
2. (a) Verify that the expectation value of the Hamilton operator H = p
2
/2m +
mω
2 x
2
/2 of the harmonic oscillator, where p = i∂/∂ x and for the wave function ψ(x) =
β
2
/π
1/4 e
−β
2 x
2 /2 with β
2
= mω/ is ψ|H |ψ = ω/2. Calculate the expectation values of (b) the position ψ|x|ψ, (c) the momentum
ψ| p|ψ, (c) the kinetic energy ψ|( p
2
/2m)|ψ, and (d) the potential energy
ψ|(mω
2 x
2
/2)|ψ.
3. Show that the pricing kernel from (10.22) and the Green’s function from (5.11)
describe the same quantity.
4. Calculate the integral I using Monte-Carlo methods with
I =
2
1
ln(x)e
−x
3/4 dx .
(10.71)
How many random numbers do you need until the result stabilizes within I /I ≈
10
−3 ? Check your result with another numerical integration tool of your choice.
Document, how well your Monte-Carlo compares to it.
5. Use the Metropolis-Hastings algorithm to build a random number generator that
generates random numbers according to the Cauchy distribution. Make a histogram of the numbers and verify that the numbers are distributed according the
Cauchy distribution. Explore different values of β.
References
1. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, vol. 2 (Wiley, New York, 1977)
2. B. Baaquie, Quantum Finance (Cambridge University Press, Cambridge, 2004)
3. R. Feynman, Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20,
367 (1948)
4. R. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals, emended edn. (Dover, New
York, 2005)
5. J. Dash, Path Integrals and Options, Part I, CNRS Preprint CPT-88, PE.2206, see also J Dash,
Quantitative Finance and Risk Management (World Scientific, Singapore, 1988), p. 2004
6. H. Goldstein, J. Safko, C. Poole, Classical Mechanics (Pearson, Harlow, 2014)
10 Quantum Finance and Path Integrals
Exercises
1. Is the operator x
∂
∂ x
hermitian or anti-hermitian? Prove your answer!
2. (a) Verify that the expectation value of the Hamilton operator H = p
2
/2m +
mω
2 x
2
/2 of the harmonic oscillator, where p = i∂/∂ x and for the wave function ψ(x) =
β
2
/π
1/4 e
−β
2 x
2 /2 with β
2
= mω/ is ψ|H |ψ = ω/2. Calculate the expectation values of (b) the position ψ|x|ψ, (c) the momentum
ψ| p|ψ, (c) the kinetic energy ψ|( p
2
/2m)|ψ, and (d) the potential energy
ψ|(mω
2 x
2
/2)|ψ.
3. Show that the pricing kernel from (10.22) and the Green’s function from (5.11)
describe the same quantity.
4. Calculate the integral I using Monte-Carlo methods with
I =
2
1
ln(x)e
−x
3/4 dx .
(10.71)
How many random numbers do you need until the result stabilizes within I /I ≈
10
−3 ? Check your result with another numerical integration tool of your choice.
Document, how well your Monte-Carlo compares to it.
5. Use the Metropolis-Hastings algorithm to build a random number generator that
generates random numbers according to the Cauchy distribution. Make a histogram of the numbers and verify that the numbers are distributed according the
Cauchy distribution. Explore different values of β.
References
1. C. Cohen-Tannoudji, B. Diu, F. Laloe, Quantum Mechanics, vol. 2 (Wiley, New York, 1977)
2. B. Baaquie, Quantum Finance (Cambridge University Press, Cambridge, 2004)
3. R. Feynman, Space-time approach to non-relativistic quantum mechanics. Rev. Mod. Phys. 20,
367 (1948)
4. R. Feynman, A. Hibbs, Quantum Mechanics and Path Integrals, emended edn. (Dover, New
York, 2005)
5. J. Dash, Path Integrals and Options, Part I, CNRS Preprint CPT-88, PE.2206, see also J Dash,
Quantitative Finance and Risk Management (World Scientific, Singapore, 1988), p. 2004
6. H. Goldstein, J. Safko, C. Poole, Classical Mechanics (Pearson, Harlow, 2014)