background economics and the weiner process

Question 1

time value of money:

Answer 1

interest rate

Question 2

simple interest rate:

Answer 2

value of an investment V(t) at time t is V(t)=P(1+rt), where V(0)=P is the initial investment and r is the interest rate

Question 3

compound interest rate:

Answer 3

V(t) at t after mt payments have been made is V(t)=P(1+(r/m))^mt where m is the number of payments per year and t is in years

Question 4

continuously compounded interest rate:

Answer 4

V(t)=Pe^rt, cause m->∞ and lim(z->∞) (1+(1/z))^z=e

Question 5

return on investment:

Answer 5

(value at expiry - initial investment)/initial investment, aka relative increase in price

Question 6

return on a share:

Answer 6

dS/S=μdt+σdW - μ is the deterministic part, generally =μ(S,t), σdW is the stochastic part where dW=ΔW=W(t+Δt)-W(t) as Δt->0, W(t) is the wiener process, σ is called the volatility

Question 7

volatility:

Answer 7

standard deviation of returns for a share price, generally a percentage per year^(1/2)

Question 8

notation:

Answer 8

Δt is a small but finite change in time
ΔS, ΔV, etc. are changes in quantities that depend on t over small but finite change in time
dt is an infinitesimally small change in time
dS, dV, etc. are the corresponding changes in quantities

Question 9

normal distribution:

Answer 9

continuous probability distribution with density f(x; μ,σ)=(1/σroot(2π))exp(-(x-μ)^(2)/2σ^2) where μ is the mean and σ^2 is the variance
if a random variable Y is drawn from a normal distribution with mean μ and variance σ^2, we write Y ~ N(μ,σ^2)

Question 10

properties of the normal distribution:

Answer 10

X+Y ~ N(μx+μy, σx^2+σy^2)
suppose Z=a+bX where X~N(0,1), Z~N(a,b^2)

Question 11

random walk:

Answer 11

basically taking successive draws and adding them
random distribution is ΔW, Wk is the value of the random walk at the k-th step, mean is 0 variance is Δt
W(k+1)=Wk+ΔW where W0=0 and ΔW~N(0,Δt)

Question 12

weiner process:

Answer 12

W(t), the continuous limit of a random walk process, so take the limit Δt->0

Question 13

properties of the wiener process:

Answer 13

W(0)=0
W(t) has independent increments - if u<v<s<t, W(t)-W(s) and W(v)-W(u) are independent
W(s+t)-W(s)~N(0,t)
W(t) has continuous paths, so is continuous in t
W(t)~N(0,t) but we might have to prove that

Question 14

expected value:

Answer 14

the weird E[W], also the mean in a normal distribution

Question 15

variance:

Answer 15

var[W]=E[W^2]-(E[W])^2, also the variance in a normal distribution

Question 16

probability density for a wiener function:

Answer 16

f(y; 0,t^(1/2))=(1/root(2πt))exp(-y^2/2t)

Question 17

probability that W(t) is contained within some interval [a,b] at a future time t:

Answer 17

P(a<=W(t)<=b)=(b)∫(a)f(y; 0,t^(1/2)) dy

Question 18

approximation to the model of the share price:

Answer 18

ΔS~N(μSΔt, σ^(2)S^(2)Δt)
S(t+Δt)=S+ΔS~N(S+μSΔt, σ^(2)S^(2)Δt)