Some Financial Mathematics
The one-period rate of return of an asset at time t.
pt  pt 1
Rt 
pt 1
where pt = the asset price at time t.
Note:
pt  pt 1 1  Rt 
Also if there was continuous compounding during the
period t – 1 to t at a rate rt then the value of the asset
would be:
pt  pt 1e
rt
, thus rt  ln 1  Rt 
rt = ln (1 + Rt) is called the log return.
Note: If we have an initial capital of A0, an nominal
interest rate of r with continuous compounding then the
value of the capital at time T is:
AT  A0e
rT
The k-period rate of return of an asset at time t.
pt  pt k
Rt  k  
pt k
again
pt  pt k 1  Rt (k ) 
Also
pt  pt 1 1  Rt   pt 2 1  Rt 1 1  Rt 
 pt k 1 Rt k 1 1 Rt k 2 
1 Rt 
Thus the k-period rate of return of an asset at time t.
1  Rt (k )  1  Rt k 1 1  Rt k 2 
1 Rt 
If there was continuous compounding during the period
t – k to t at a rate rt (k) then the value of the asset would
be at time t:
pt  pt k e t 
r k k
 pt k 1  Rt (k ) 
 pt k 1 Rt k 1 1 Rt k 2 
 pt k e
rt k 1 rt k 2
e
e  pt k e
rt
1 Rt 
rt k 1  rt k 2  rt
Thus the k-period continuous compounding rate of
return of an asset at time t is:
1
rt  k    rt  k 1  rt k  2 
k
rt 
Taking t = k, let p0 = the value that an asset is purchased
at time t = 0. Then the value of the asset of time t is:
pt  p0er1 r2 
rt
ln pt  ln p0  r1  r2 
rt
If r1 , r2 , rt are independent identically distributed
mean 0, then ln pt is a random walk.
Some distributional properties of
Rt and rt
Ref: Analysis of Financial Time Series
Ruey S. Tsay
Conditional Heteroscedastic
Models
Models for asset price volatility
• Volatility is an important factor in the trading of
options (calls & puts)
• A European call option is an option to buy an
asset at a fixed price (strike price) on a given
date (expiration date)
• A European put option is an option to sell an
asset at a fixed price (strike price) on a given
date (expiration date)
• If you can exercise the option prior to the
expiration date it is an American option.
Black-Scholes pricing formula
ct = the cost of the call option,
Pt = the current price,
K = the strike price,
l = time to expiration
r = the risk-free interest rate
st = the conditional standard
deviation of the log return
of the specified asset
F(x) = the cumulative
distribution function for
the standard normal
distribution
Conditional Heteroscedastic Models
for log returns {rt}
Let Pt-1 denote the information available at time
t–1
(i.e. all linear functions of { …, rt-3 , rt-2 , rt-1 })
Let m t = E [rt | Pt-1] and ut = rt – mt.
s  var  rt Pt-1] = E[(rt – mt)2| Pt-1]
2
t
Assume an “ARMA(p,q)” model, i.e.
p
p
i 1
i 1
rt  mt  ut with mt     i rt i  i ut i
The conditional heteroscedastic models are
concerned with the evolution of
s t2  var  rt Pt-1] = E[(rt – mt)2| Pt-1]
= var[ ut | Pt-1]
The ARCH(m) model
Auto-regressive conditional heteroscedastic model
ut  s t zt , s  0   u 
2
t
2
1 t 1
 u
2
m t m
where {zt} are independent identically
distributed (iid) variables with mean zero
variance 1.
with 0  0 and i  0 if i  0
The GARCH(m,s) model
Generalized ARCH model
m
s
i 1
j 1
ut  s t zt , s t2  0  iut2i   js t2 j
where { zt} are independent identically
distributed (iid) variables with mean zero
variance 1.
with 0  0,i  0 , j  0
max  m, s 
and
      1
i 1
i
i
Excel files illustrating ARCH and
GARCH models
CH models