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with a martingale measure Q that is not unique because of the market frictions. Every other mar-

e tingale measure Q is related to Q by the density formula

dQt

e dQt

= e X t

2

[X]

t

= Z t ,

(8)

where [X]t is the quadratic variation process. For geometric Brownian motion,

X t =

Z t 0

r

dZ =

r

σσ

Z t ,

(9)

where the parameter

can be positive or negative. In the incomplete market, the parameter

is

not specified in an arbitrage free model. Hence, let the market choose the parameter way. Define the theoretical price

in the following

t,T ( ) =

t

[ S T ] = S t e

r+ )τ

.

(10)

e In other words, for any martingale measure Q there exists a parameter

˜

where

t

[ S T ] = S t

e

r+˜)τ

.

We follow Bjoerk (1998) to determine

by solving the least squares minimization problem

= argmin

" n X i = 1

(

i( )

) 2 i

#

,

(11)

where

i

are n historically observed prices. The minimization problem ensures that the theoretical

and historic prices are as close as possible.

The parameter approach.

is assumed to be constant. Allowing for a time dependent

would generalize the

The upper bound of the property spread p then follows

pτ =

ln[(1 + k1b + k2)e

  • +

    (k1s

τ

  • +

    k2)e

r+

)τ ]

r.

(12)

Next we consider the lower bound of the property spread, p. The actual property hedger (APH), t h e s e l l e r , h a s a n i n i t i a l w e a l t h o f W t t i e d u p i n t h e p r o p e r t i e s S t t h a t h e s e l l s a t t . A f t e r e x p i r y o f t hedge horizon T , he buys the properties back and gets h e

WAPH,T

= (1

= (1

k1s

k2

k1s

k2

k3)S e

t

(1 + k1b + k2)ST + ST

k3)S e

t

(k1b + k2)ST ,

(13)

8

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