Upload
leo-love
View
20
Download
2
Embed Size (px)
DESCRIPTION
MGT 821/ECON 873 Martingales and Measures. Derivatives Dependent on a Single Underlying Variable. Forming a Riskless Portfolio. Market Price of Risk. This shows that ( m – r )/ s is the same for all derivatives dependent on the same underlying variable , q - PowerPoint PPT Presentation
Citation preview
1
MGT 821/ECON 873Martingales and Martingales and
MeasuresMeasures
Derivatives Dependent on a Single Underlying Variable
2
process thefollows that security) tradeda of
price y thenecessaril(not variableaConsider
dzsdtmd
Suppose . and prices
with on dependent sderivative twoImagine
21 ƒƒ
dzσ dtμƒ
df dzσ dtμ
f
df22
2
211
1
1 ,
3
Forming a Riskless Portfolio
tƒƒσ샃σμ=
ƒƒσƒƒσ
ƒσ
ƒσ
)(
)()(
derivative 2nd theof
and derivative1st theof +
of consisting
portfolio riskless a upset can We
21122121
211122
11
22
4
or
:gives This
:riskless is portfolio the Since
2
2
1
1
121221
σ
rμ
σ
rμ
r σr σσμσμ
t=r
Market Price of RiskMarket Price of Risk
This shows that ( – r )/ is the same for all derivatives dependent on the same underlying variable,
We refer to ( – r )/ as the market price of risk for and denote it by
5
Extension of the AnalysisExtension of the Analysisto Several Underlying Variablesto Several Underlying Variables
then
with
variablesunderlying severalon depends If
1
12
1
σλrμ
dzσμ dtf
df
f
n
iii
n
iii
6
MartingalesMartingales
A martingale is a stochastic process with zero drift
A variable following a martingale has the property that its expected future value equals its value today
7
Alternative Worlds
dzfdtfrdf
dzσfdtrfdf
)(
is risk of price market the where worlda In
worldneutral-risk ltraditiona the In
8
The Equivalent Martingale The Equivalent Martingale Measure ResultMeasure Result
fgf
g
pricessecurity derivative allfor martingale a is that shows
lemma sIto' then ,security a of volatility the to equal set weIf
9
Forward Risk Neutrality
We will refer to a world where the market price of risk is the volatility of g as a world that is forward risk neutral with respect to g.
If Eg denotes a world that is FRN wrt g
f
gE
f
ggT
T
0
0
10
Alternative Choices for the Numeraire Security g
Money Market Account Zero-coupon bond price Annuity factor
11
Money Market Accountas the Numeraire
The money market account is an account that starts at $1 and is always invested at the short-term risk-free interest rate
The process for the value of the account is
dg=rg dt This has zero volatility. Using the money
market account as the numeraire leads to the traditional risk-neutral world where =0
12
Money Market Accountcontinued
worldneutral-risk ltraditiona the in nsexpectatio denotes where
becomes
equation the ,= and 1= Since
E
feEf
g
fE
g
f
egg
T
rdt
T
Tg
rdt
T
T
T
ˆ
ˆ 0
0
0
0
0
0
13
Zero-Coupon Bond Maturing at time T as Numeraire
price bond the wrtFRN is that worlda in nsexpectatio denotes and price bond coupon-zero the is ),( where
becomes
equation The
T
TT
T
Tg
ETP
fETPf
g
fE
g
f
0
][),0(0
0
0
14
Forward Prices
In a world that is FRN wrt P(0,T), the expected value of a security at time T is its forward price
15
Interest Rates
In a world that is FRN wrt P(0,T2) the expected value of an interest rate lasting between times T1 and T2 is the forward interest rate
16
Annuity Factor as the Numeraire
)()0(0
0
0
TA
fEAf
g
fE
g
f
TA
T
Tg
becomes
equation The
17
Annuity Factors and Swap Rates
Suppose that s(t) is the swap rate corresponding to the annuity factor A.
Then:
s(t)=EA[s(T)]
18
Extension to Several Extension to Several Independent FactorsIndependent Factors
m
iiig
m
iigi
m
iiif
m
iifi
m
iiig
m
iiif
dztgtdttgttrtdg
dztftdttfttrtdf
dztgtdttgtrtdg
dztftdttftrtdf
1,
1,
1,
1,
1,
1,
)()()()()()(
)()()()()()(
)()()()()(
)()()()()(
consistent internally are that worldsother For
worldneutral-risk ltraditiona the In
19
Extension to Several Independent Factors
hold. results the of rest the and martingalea is case, factor-one the in As
where worldas wrtFRN is that worlda define We
gf
σλg
igi ,
20
Applications
Extension of Black’s model to case where inbterest rates are stochastic
Valuation of an option to exchange one asset for another
Black’s ModelBlack’s Model
By working in a world that is forward risk neutral with respect to a P(0,T) it can be seen that Black’s model is true when interest rates are stochastic providing the forward price of the underlying asset is has a constant volatility
c = P(0,T)[F0N(d1)−KN(d2)]
p = P(0,T)[KN(−d2) − F0N(−d1)]
21
T
TKFd
T
TKFd
F
F
F
F
20
2
20
1
)/ln()/ln(
Option to exchange an asset Option to exchange an asset worth U for one worth Vworth U for one worth V This can be valued by working in a world that
is forward risk neutral with respect to U
22
23
Change of NumeraireChange of Numeraire
wvwghwv
vhg
w
vwv
and between ncorrelatio the is and of volatility the is of
volatility the is whereby increases variable a of drift the , to from
security numeraire the change weWhen
,,,