4.7 Digression on Expectation Values
45
function e
2πir/λ
/r plays the role of the Green’s function. Here λ is the wavelength
of the light and r is the distance between the source point and the observation point
on the screen.
After pointing out the analogies in physics, let’s get straight back to one of the
core topics of finance, the Black-Scholes equation.
Exercises
1. Use a binomial tree with two layers to calculate the value of a call option that,
after one year, has a strike price K . The annual growth rate is ρ = 0.05/year, and
the annual volatility is σ = 0.3/
√ year.
2. Verify that, in three dimensions, the Green’s function G(r ) = 1/4πr satifies
G(r ) = δ(r ).
3. Determine the Green’s function for a damped harmonic oscillator, described by
¨
x + 2α ˙
x + ω
2 x = v 0 δ(t). Here ω is the oscillation frequency, α is the damping
constant, and v 0 is an initial change in the velocity, instantaneously applied to
the oscillator at t = 0. The solution to this equation is the Green’s function.
It is equivalent to the impulse response of an instantaneous change in velocity
˙
x = v 0 .
4. Equation 4.15 can be used to describe the dependence of the temperature T → ψ,
where D → σ
2
/2 is the heat diffusion constant of the material. At t = 0 heat,
which causes an instantaneous temperature rise T 0 , is injected at a distance d
from the insulated end of a semi-infinite slab with heat diffusion constant D.
Calculate the temperature at the insulated end as a function of time. Hint: think
of “image heat loads” and that ∂ T /∂ x = 0 at the insulated end.
5. Show that ψ(z, τ ) from (4.31) (a) is normalized and (b) satisfies (4.30).
6. What is the probability that a stock with an annual return ρ = 0.1/year and an
annual volatility of σ = 0.3/
√ year at least doubles its value in 2 years?
7. What is the probability that the stock from Exercise 6 has less than half its value
after two years?
8. You invent a new option O that has a payoff function that grows linearly with the
stock value S in the range from (1 − u)K to K and decreases linearly from K to
(1 + u)K , where u is in the range 0.05 < u < 0.2. (a) Sketch the payoff function.
(b) Calculate the price of the option for a stock with annual return ρ = 0.1/year
and volatility σ = 0.3/
√ year.
9. Huygen’s principle tells us that the image of an aperture can be calculated by
summing all emanating spherical waves with the point-spread function e
2πir/λ
/r ,
where λ is the wavelength of the light. Considering only one dimension, calculate
the image of an aperture of width a on a screen at a distance d a from the
aperture. Hint: you can assume that r in the denominator of the point-spread
function is approximately equal to d.
45
function e
2πir/λ
/r plays the role of the Green’s function. Here λ is the wavelength
of the light and r is the distance between the source point and the observation point
on the screen.
After pointing out the analogies in physics, let’s get straight back to one of the
core topics of finance, the Black-Scholes equation.
Exercises
1. Use a binomial tree with two layers to calculate the value of a call option that,
after one year, has a strike price K . The annual growth rate is ρ = 0.05/year, and
the annual volatility is σ = 0.3/
√ year.
2. Verify that, in three dimensions, the Green’s function G(r ) = 1/4πr satifies
G(r ) = δ(r ).
3. Determine the Green’s function for a damped harmonic oscillator, described by
¨
x + 2α ˙
x + ω
2 x = v 0 δ(t). Here ω is the oscillation frequency, α is the damping
constant, and v 0 is an initial change in the velocity, instantaneously applied to
the oscillator at t = 0. The solution to this equation is the Green’s function.
It is equivalent to the impulse response of an instantaneous change in velocity
˙
x = v 0 .
4. Equation 4.15 can be used to describe the dependence of the temperature T → ψ,
where D → σ
2
/2 is the heat diffusion constant of the material. At t = 0 heat,
which causes an instantaneous temperature rise T 0 , is injected at a distance d
from the insulated end of a semi-infinite slab with heat diffusion constant D.
Calculate the temperature at the insulated end as a function of time. Hint: think
of “image heat loads” and that ∂ T /∂ x = 0 at the insulated end.
5. Show that ψ(z, τ ) from (4.31) (a) is normalized and (b) satisfies (4.30).
6. What is the probability that a stock with an annual return ρ = 0.1/year and an
annual volatility of σ = 0.3/
√ year at least doubles its value in 2 years?
7. What is the probability that the stock from Exercise 6 has less than half its value
after two years?
8. You invent a new option O that has a payoff function that grows linearly with the
stock value S in the range from (1 − u)K to K and decreases linearly from K to
(1 + u)K , where u is in the range 0.05 < u < 0.2. (a) Sketch the payoff function.
(b) Calculate the price of the option for a stock with annual return ρ = 0.1/year
and volatility σ = 0.3/
√ year.
9. Huygen’s principle tells us that the image of an aperture can be calculated by
summing all emanating spherical waves with the point-spread function e
2πir/λ
/r ,
where λ is the wavelength of the light. Considering only one dimension, calculate
the image of an aperture of width a on a screen at a distance d a from the
aperture. Hint: you can assume that r in the denominator of the point-spread
function is approximately equal to d.