Chapter 18 Option Pricing
Without Perfect Replication
Framework

On the Edges of Arbitrage

One-period Good-deal Bounds

Multiple Periods and Continuous Time

Extensions, Other Approaches, and Bibliography
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Asset Pricing
2
On the Edges of Arbitrage

No Arbitrage Principle is Obvious, But…





Not Trade Continuously, and Transactions Costs
Stochastic Interest Rate or Stochastic Stock Volatility
Underlying Asset Not Traded, Real Option for Example, or
Short Selling Forbidden
Internal Inconsistency of Arbitrage Portfolio
So Unavoidable Basis Risk Arises. Many Authors simply
add market price of risk assumptions. But this leaves the
questions, How Sensitive are the Results to MPR
Assumption? What are the reasonable Values for MPR
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Asset Pricing
3
On the Edges of Arbitrage



We are still willing to take as given the prices of lots of
assets in determining the price of an option, and in
particular assets that will be used to hedge the option.
We Form an “Approximate” hedge or portfolio of basis
asset closest to the focus payoff, hedge most of the
option’s risk. The uncertainty about the option value is
reduced only to figuring out the price of the residual.
Good-Deal Option Price Bounds: Systematically
Searching over all Possible Assignments of the MPR to
a reasonable Value, and Impose No Arbitrage
Opportunities and Find Upper and Lower Bounds on the
Option Price
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Asset Pricing
4
One-period Good-deal Bounds

We want to price the payoff xc, where xc = Max(ST - K ,0)
for a call option. Given an N-dimensional vector of basis
payoffs x, whose prices p we can observe. The goodc
deal bound finds the minimum and maximum value of x
by search over all positive discount factors that price the
basis assets and have limited volatility
C  min E (mx c )
{m}
s.t. p  E (mx);
m  0;
 ( m )  h / R f  E ( m 2 )  A2 ,
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Asset Pricing
1 h2
A  f2
R
2
5
One-period Good-deal Bounds

In order of calculation, all the combinations of binding
and nonbinding constrains:
-
-
Volatility Constraint Binds, Positivity Constrain Slack
Positivity Constraint Binds, Volatility Constraint Slack
Volatility and Positivity Constraints both Bind

Volatility Constraint Binds, Positivity Constrain Slack
-
C  min E (mx c ) s.t. p  E (mx), E (m2 )  A2
{m}
-
-
Lagrange Multipliers
Orthogonal Decomposition
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Asset Pricing
6
Volatility Constraint Binds, Positivity Constrain Slack
1, Given X , find x*
c
c
2, $x = proj ( x c | X ) = E ( x c x ') E ( xx ')- 1 x, w = x c - $x
3, p = E (mx ) = E (( x* + m* ) x ) = E ( x* x ) + E (m* x )
Þ E ( m* x ) = 0 Þ E ( m* x * ) = 0
4, m > 0, so triangular region
5, E (m 2 ) £ A2 , so circle
m
c
c
6, E (mx c ) = E[( x* + m* )($x + w)] = E ( x* $x ) + E (m* w)
*
m
m
Þ m = x* + vw, m = x* - vw = x* -
m
Þ v=
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A2 - E ( x*2 )
w
E ( w2 )
A2 - E ( x*2 )
E ( w2 )
Asset Pricing
7
Volatility Constraint Binds, Positivity Constrain Slack

So the call price bound:
C = E(mxc ) = E( x* xc ) - vE(w2 )

Interpretation:
-
The first term: approximate hedge portfolio
E( x* xc )  E( x* xˆ c )  E(mxˆ c )
-
The second term: Residual Consistent with Volatility Bound
vE(w2 )  E(vww)  E[( x*  vw)w]  E(mw)

Explicit Option Pricing Formula:
C / C = p ' E ( xx ')- 1 E ( xx c ) m A2 - p ' E ( xx ')- 1 p E ( x c 2 ) - E ( x c x ') E ( xx ')- 1 E ( xx c )
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8
Volatility Constraint Binds, Positivity Constrain Slack
Algebraic Expressions

Any discount factor m that satisfies p =
decomposed as:
E (mx) can
be
m = x* + vw + e, x* = p ' E( xx ')- 1 x


where E( x*w) = E( x*e) = E(we)
So minimization problem is then:
Min E (mx c ) s.t. E (m 2 ) £ A2 ,
{v ,e}
c
Min E[( x* + vw + e )($x + w) s.t. E ( x*2 ) + v 2 E ( w2 ) + E (e 2 ) £ A2 ,
{v ,e}
c
Min E ( x* $x ) + vE ( w2 ) s.t. E ( x*2 ) + v 2 E ( w2 ) + E (e 2 ) £ A2
{v ,e}
Þ e = 0, v = 2015/4/8
( A2 - E ( x*2 )) / E ( w2 )
Asset Pricing
9
Volatility and Positivity Constraints both Bind
Direct Approach

Introducing Lagrange multipliers, the problem is:

C  min max E (mx )   '[ E (mx)  p]  [ E (m 2 )  A2 ]
{m  0} { ,  0}
2
c

Introducing Kuhn-Tucker multiplier p (s)v(s) on m( s) > 0, and
taking partial derivatives wrt. m in each state
C = Min å
{m}
s
dé
+ êå
2 êë s
é
ù
ê
p ( s ) m ( s ) x ( s ) + l ' å p ( s ) m( s ) x ( s ) - p ú
êë s
ú
û
ù
2
2
p ( s )m( s ) - A ú+ å p ( s )v( s )m( s )
ú
û s
c
1 ¶
: x c ( s) + l ' x( s) + dm( s) + v( s) = 0
p ( s) ¶ s
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10
Volatility and Positivity Constraints both Bind
Direct Approach

Kuhn Multiplier Discussion
-
If the positivity constraint is slack, v(s) = :0
-
x c ( s ) + l ' x( s )
m( s ) = d
If the positivity constraint binds:
m( s ) = 0

So the solution:
+
æ xc + l ' x ÷
ö é xc + l ' x ù
ú
m = Max çç, 0÷
= ê÷
çè
ú
÷
d
d
ø êë
û

And Maximize:
2
  x  'x
 2
C  Max E{( 
}


'
p

A

{ , 0}
2
 
2
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11
Volatility and Positivity Constraints both Bind
Dual Approach


Hansen, Heaton and Luttmer (1995)
Interchanging Min and Max,

C  max min E (mx c )   '[ E (mx)  p]  [ E (m 2 )  A2 ]
{ ,  0} {m  0}
2

We may obtain:
2
  x  'x
 2
C  Max E{( 
}


'
p

A

{ , 0}
2
 
2

Search numerically over (l , d) to find the solution to this
problem.
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12
Positivity Constraint Binds, Volatility Constraint Slack

The problem:
C  Min E (mxc ), s.t. p  E (mx), m  0
m

These are the Arbitrage Bounds, Denote the lower
arbitrage bound by Cl . The minimum-variance discount
factor that generates the arbitrage bound Cl solves:
2
E (m )min
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 p
x
 Min E ( x ), s.t.    E (m  c ), m  0
{m}
x 
Cl 
2
Asset Pricing
13
Positivity Constraint Binds, Volatility Constraint Slack

Introducing the Lagrange Multiplier:
é
ù
ê
L = å p ( s )m( s ) + 2l å p ( s )m( s ) x ( s ) - p ú
êë s
ú
s
û
é
ù
+ 2d êå p ( s) m( s ) x c ( s ) - Cl ú+ å p ( s )v( s )m( s )
êë s
ú
û s
2


Using the same conjugate method, the optimal m :
+
c
é
ù
m = êë- (dx + l ' x)úû
This problem is equivalent to:
E (m2 )min  max{ E{[( x c   ' x)]2  2 ' p  2 Cl }
{ , }

We search numerically for (l , d) to solve this problem.
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14
Good-Deal Comparisons with BS and Arbitrage Bounds
We can obtain closer bounds on prices with more
information about the discount factor. In particular, if we
know the correlation of discount factor with xc, we could
price the option better.
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15
Multiple Periods and Continuous Time


The central fact that makes good-deal bounds tractable
in dynamic environments that bounds are recursive.
Consider two periods version:
C 0  Min E0 (m1m2 x2c ) s.t.
{m1 , m2 }
pt  Et (mt 1 pt 1 ), Et (mt21 )  At2 , mt 1  0, t  0,1

Be equivalent to a series of one-period problems:
C1  Min E1 (m2 x2c )
{m2 }

C 0  Min E0 (m1 C 1 )
{ m1 }
E0 (m1m2 x2c ) must optimize
So the two-period problem {Min
m ,m }
c
Min E1 (m2 x2 ) in each state of nature at 1
{m }
The recursive property only holds if we impose m > 0
1
2
2

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16
Basis Risk and Real Options (One Dimension)



Consider pricing a European call option on asset V that
is not a traded asset, but correlated with a traded asset S
that can be used as an approximate hedge
Terminal Payoff: xTc  Max(VT  K ,0)
dS
Basis Asset:
    dz
S



S
S
dV
 V dt   Vz dz   Vw dw
V
Underling Asset:
The dw risk cannot be hedged by the basis asset, so the
market price of dw risk will matter to the option price
Goal: Discount factor to price S and r f , has instantaneous
volatility A
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17
Basis Risk and Real Options (One Dimension)

SDF solution form:
*
dL d L
= * ±
L
L



dL *
L*
*
m- r
dL
A - h dw, * = - rdt - hS dz, hS = S
sS
L
2
2
S
éd L 2 ù
Et ê 2 ú= A2
êL ú
ë
û
Interpretation:
,
So the Good-deal bound is given by:
And the Result:
1
éL T
ù
ê
C t = Et
Max(VT - K , 0)ú
êL t
ú
ë
û
1
sV T )
2
2
ln(V0 / K ) + (h + r )T
dV 2
2
2
s V = Et 2 = s Vz2 + s Vw
,d =
,
V
sV T
C or C = V0 ehT F (d +
s V T ) - K - rT F (d -
é
æ
öù
A2
çç
2÷
ê
ús , h = mS - r , h = mV - r
÷
h = êhV - hS çr - a
1
1
r
÷
V
úV S
÷
hS2
sS
sV
÷
çè
êë
øú
û
ìïï + 1 Upper Bound
ædV dS ö
s Vz
r = corr çç
, ÷
=
,
a
=
÷
í
÷ s
çè V S ø
ïïî - 1 Lower Bound
V
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18
Basis Risk and Real Options (One Dimension)

Market Prices of Risk: if an asset has a price process P
that loads on a shock s dw , then its expected return:
Et

With Sharpe ratio:
l =

æd L ÷
ö
dP
- r f dt = - s Et çç
dw÷
÷
çè L
ø
P
Et
dP
- r f dt
æd L ÷
ö
dP
P
= - Et çç
dw÷
Þ
E
- r f dt = l s
t
÷
ç
èL
ø
s
P
In the above Real Option Example:
lz=
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mS - r
,l w =
sS
A2 - hS2
Asset Pricing
19
Continuous time (High Dimensions)

Basis Assets: nS -dimension, Dividends rate
D( S ,V , t )dt
dS
  S ( S ,V , t )dt   S ( S ,V , t )dz , E (dzdz ')  I
S

Underlying Assets: nV -dimension
dV
 V ( S , V , t )dt   Vz ( S , V , t )dz   Vw dw,
V
E (dwdw ')  I , E (dwdz ')  0

Goal: Value an asset that pays continuous dividends at
rate xc (S ,V , t )dt , and terminal payment xTc (S ,V , T )
s c

xs ds  Et ( T xTc )
s t 
t
t
C t  min Et 
{ s ,t  s T }
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T
Asset Pricing
20
Continuous time (High Dimensions)

For small time intervals, discretize:
C t  t  Min Et {x c  t t  (C t  C )( t   )}
{ }
xtc
E[d (C )]
t  0  0  dt  Min
{d }
C
C

It can also be expressed:
 d  dC 
dC xtc
Et
 dt  r f dt   Min Et 

{ d }
C C

C



Discount Factor Form:
*
*
dL d L
dL
° ' S - 1s dz ,
= * - vdw, * = - rdt - m
S
S
S
L
L
L
° = m + D - r, S = s s '
m
S
S
S S
S
S
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21
Continuous time (High Dimensions)

Volatility Constraint:
1
d 2
d  d *
2
Et 2  A , and
 *  vdw
dt



1
d *2
2
vv '  A  Et *2  A2  s' s 1s
dt


Market price of risk:v is the vector of market prices of
risks of the dw shocks
v

1  d 
Et 
dw 
dt  

Solutions: Guess the lower bound C follows:
dC
= mC ( S ,V , t )dt + s Cz ( S ,V , t )dz + s Cw ( S ,V , t )dw
C
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22
Continuous time (High Dimensions)


The Optimal SDF
Theorem: The lower bound discount factor L t follows:
d  d *
 *  vdw



And C , Cz , Cw satisfy the restriction:
x
1
d *
'
C   r   Et ( *  Cz dz )  v Cw
C
dt


Where:
 Cw
1
d *2
v  A  ( Et *2 )
'
dt

 Cw Cw
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23
Continuous time (High Dimensions)

Proof:
xtc
E[d (C )]
0  dt  Min
{ d }
C
C
d d 
 *  vdw


*

 d ( * C ) 
xtc

dC 
0  dt  Et  *

Min
vE
dw
t 


C
C 

  C  {v }
1  d *2 
2
s.t. vv '  A  Et  *2 
dt   
And Guess: dC = mC ( S ,V , t )dt + s Cz ( S ,V , t )dz + s Cw ( S ,V , t )dw
C

So Problem:
2015/4/8
 d ( * C ) 
xtc 1
'
0
 Et 
 Min v Cw

*
C dt   C  {v}
 d *2 
1
2
s.t. vv '  A  Et  *2 
dt   
Asset Pricing
24
Continuous time (High Dimensions)

This is a linear objective in v with a quadratic constraint.
Therefore, as long as s Cw ¹ 0 , the constraint binds and
the optimal v :
 Cw
1
d *2
v  A  ( Et *2 )
'
dt

 Cw Cw

So plugging the optimal value:
xtc 1  d (* C ) 
'
0   Et  *

v

Cw
C dt   C 

For clarity, and exploiting the fact that d*does not load
on dw , write the term as
x
1 d *
'
C   r   Et ( *  Cz dz)  v Cw
C
dt

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25
Continuous time (High Dimensions)


The Optimal SDF:
Plug in the definition of * , we may obtain:
d
 rdt   s'  s 1 s dz  A2   s'  s 1s

 Cw
 Cw
'
Cw
dw
xc
'
C   r  s' s 1 s Cz  A2  s' s 1s  Cw Cw
C


The Good-Deal Option Price Bounds
Theorem: The lower bound C(S ,V , t ) is the solution to the
partial differential equation
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26
Continuous time (High Dimensions)
¶C 1
¶ 2C
x - rC +
+ å
Si S j s Si s S' j
¶ t 2 i , j ¶ Si ¶ S j
c
1
¶ 2C
+ å
(s Vz s V' z + s Vw s V' w ) +
i
j
i
j
2 i , j ¶ Vi¶ V j
æD
= çç çè S

å
i, j
¶ 2C
Sis Si s V' z
j
¶ Si ¶ V j
'
ö÷
' -1
' -1
'
'
2
°
°
° C' s s ' C
r ÷÷ ( SC S ) + (mS S S s S s Vz - mV )CV + A - mS S S m
V Vw Vw V
S
ø
Subject to the boundary conditions provided by the focus
c
x
asset payoff T . Replacing + with – before the square
root gives the partial differential equation satisfied by the
upper bound
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27
Continuous time (High Dimensions)


In general, the  process depends on the parameters s Cw
, so without solving the PDE we don’t not know how to
spread the loading of d L across the multiple sources of
risk dw whose risk prices we do not observe.
Thus, we cannot use an integration approach to find the
bound; we cannot characterize  enough to calculate:
s c

xs ds  E ( t xTc )
s t 
t
t
Ct  E 
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T
Asset Pricing
28
Continuous time (High Dimensions)


However, if there is only shock dw, then we don’t have to
worry about how loading of d  spreads across multiple
sources of risk. V can be determined simply by the
volatility constraint. in this special casedw and  Cw are
scalars.
Theorem: In the special case that there is only one extra
noise dw driving the V process, we can find the lower
bound discount factor L directly from
' -1
dL
°
= - rdt - mS S S s S dz L
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Asset Pricing
' -1
°
° dw
A - mS S S m
S
2
29
Extensions, Other Approaches, and Bibliography



The one-period good-deal bound is the dual to the HJ
bound with positivity --- Hansen and Jagannathan (1991)
study the minimum variance of positive discount factors
that correctly price a given set of assets.
The Good-deal bound interchanges the position of the
option pricing equation and the variance of the discount
factor. The techniques for solving the bound, are exactly
those of the HJ bound in this one-period setup.
This kind of problem needs weak but credible discount
factor restrictions that lead to tractable and usefully tight
bounds
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30
Extensions, Other Approaches, and Bibliography




Several other similar restrictions:
Monotonicity Restrictions: Levy (1985), Constantinide
(1998), the discount factor declines monotonically with a
state variable; marginal utility should decline with wealth.
The good-deal bounds allow the worst case that
marginal utility growth is perfectly correlated with a
portfolio of basis and focus assets. So more credible
correlation setup obtains tighter bounds
Gain-Loss Restriction: Bernardo and Ledoit (2000) use
the restriction a ³ m ³ b to sharpen the no-arbitrage
restriction ¥ ³ m > 0. They show that this restriction has a
beautiful portfolio inter-pretation -- a < m < b corresponds to
limited gain-loss
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31
Extensions, Other Approaches, and Bibliography

Define[Re ]+ = Max(Re ,0) and [Re ]- = - Min(- Re ,0) as the gains and
losses of an excess return R e , then:
[ Re ]+
sup(m)
Max
=
Min
{m:0= E ( mRe )} inf(m)
{Re Î Re } [ Re ]-



Bernardo and Ledoit also suggest a ³ m / y ³ b , where y is
an explicit discount factor, such as the consumption
based model or CAPM
There alternatives are really not competitors
The continuous-time treatment has not considered jumps
and if with jumps, both positivity and volatility constraints
will bind
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32
Thanks
Your suggestion is welcome!
2015/4/8
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