时间连续金融理论 class2

dt dZ dt 0 0 dZ 0 dt
This gives the following:
dF (X) =
®(X)
@F @X
+
1 2
¾(X)2
@2F @X2
¸
dt
+
¾(X
)dZ
1.2 Multivariate Case
Here we consider two stochastic processes that evolve over time:
=
!W P1
dW
=
!W P1
dP1
=
!W
dP1 P1
Things are more interesting if we have many assets: Ã
!
dW = W
X
i
!i
dPi Pi
4 Overview of Models for Pricing Securities
1. Equilibrium models - most stringent - prices are endogenous - agents optimize - market clearing imposed
a certain date in the future (maturity), at a certain price (exercise or strike price.) European call options therefore payoff max(0; ST ¡ K).
Time: today will be denoted ”t” and maturity ”T”. The time to maturity is given by ¿ = T ¡ t.
dX = ®(X; Y )dt + ¾(X; Y )dZX dY = ¹(X; Y )dt + ±(X; Y )dZY
½ = Corr[dZX; dZY ]
Consider another stochastic process that is a function of the both processes:
5.1 Assumptions
We will assume that the underlying asset (a stock) follows GBM and that there is a riskless asset with a constant rate of return. We also will assume continuous trading and no-arbitrage (law of one price.)
1 2
¾2)t
+
¾(Zt
¡
Z0)
ln(Xt) ¡ ln(X0)
=
(®
¡
1 2
¾2)t
+
¾(Zt
¡
Z0)
Xt
=
X0
+
e(®¡
1 2
¾2)t+¾(Zt
¡Z0
)
Note that (Zt ¡ Z0) v N(0; t) as we talked about when discussing standard Wiener processes. We can also see that zero is an absorbing barrier. If X0 > 0, then Xt ¸ 0 8 t.
1 Review of Ito’s Lemma
1.1 Univariate Case
Consider a stochastic process that evolves over time (here we supress all the time subscripts):
dX = ®(X)dt + ¾(X)dZ
dX = ®Xdt + ¾XdZ
dX X
= ®dt + ¾dZ
where ® and ¾ are constants. The starting price X(t = 0) = X0. Now make the following transformation:
Y = ln(X)
dY
=
@Y @X
dX
3
Review of Portfolio Dynamics
Suppose we invest our wealth in an asset:
W = wealth or value of nancial portfolio N1 = # of shares of asset 1 P1 = price of per unit of asset 1
dF (X; Y )
=
@F @X
dX
+
@F @Y
dY
+
1 2
@2F @X2
dX2
+
@2F @X@Y
dXdY
+
1 2
@2F @Y 2
dY
2
The multiplication rules are:
dt dZX dZY
dt 0
0
0
dZX 0 dt ½dt
dZY 0 ½dt dt
This gives:
dF (X; Y ) =
+
1 2
@2Y @X2
dX 2
µ
¶
=
1 X
(®X dt
+
¾XdZ)
+
1 2
¡1 X
¾2X2dt
=
(®
¡
1 2
பைடு நூலகம்
¾2)dt
+
¾dZ
This looks like the Arithmetic Brownian Motian (ABM) we studied last class. We want to know the distribution of
No solve for the portfolio dynamics:
dP P
=
!
(®dt
+
¾dZ)
+
(1
¡
!)
1 F
µ ®S
@F @S
+
1 2
@2F @S2
¾2S2
¡
@F @¿
¶
dt
+
(1
¡
!)
¾S F
@F @S
dZ
Combine the "dt" and "dZ" terms and simplify
dP P
½
=
!®
+
(1
¡
!)
1 F
µ
®S
@F @S
+
1 2
@2F @S2
¾2S2
¡
@F @¿
¶¾
dt
+
½ !¾
+
(1
¡
!)
¾S F
@F @S
¾
dZ
The portfolio will be riskless when the "dZ" term is zero:
0
=
!¾
+
(1
¡
!)
¾S F
@F @S
®S µ
@F @S
dt
®S µ
@F @S
+
1 F
®S
@F @S
+@@21+2S@¾@F2122SSFd@@2@@S2S¾FSF222d+S¾Z22@¶@S¡F¿2+@¡d@F¿¿12@@+¶@@F¿2Sd12F¶2t@@¾+d2¿2tFS¾2+S2dd¿¾@@Ft2SFS¡@@d@ZF@SF¿dZdt
®(X;
Y
)
@F @X
+
¹(X;
Y
)
@F @Y
+
1 2
¾(X;
Y
)2
@2F @X2
+
½¾(X;
Y
)±(X;
Y
)
@2F @X@Y
+
1 2
±(X;
Y
)2
@2F @Y 2
¸
dt
+¾(X; Y )dZX + ±(X; Y )dZY
2
Review of Geometric Brownian Motion (GBM)
2. Optimality or partial equilibrium models - all prices except one assumed exogenously - market clearing assumed exogenously - agents optimize.
3. No arbitrage models - least stringent - all prices except one assumed exogenously - market clearing assumed exogenously - optimizing behavior assumed.
!
=
S
@F @S
S
@F @S
¡F
1¡!
=
F
S
@F @S
¡F
Given these portfolio weights, the riskless portfolio dynamics are now:
where the value additions = 0: In a strictly self- nancing portfolio: