3.2 The Private Price and its Black & Scholes
counterpart
During the previous subsection we have derived an explicit
form to the Executive Private Price. We have built an incomplete market
framework in order to understand how could be the behaviour of an executive
with an ESO. We know that in Black&Scholes (B & S) framework failed to
fair-valued a such option since assumptions such that unconstrained portfolio
and riskless agent are unrealistic. Intuitively, we could say that B & S
valuation overstate the fair value of an ESO: the B & S price is an upper
bound of the fair price. The idea here is to define an approximate expression
of the Private Price derived previously and compare it by the B & S value.
First of all we are going to deal with some key statistical concepts and
subsequently use an analytical tool (perturbation expansion) in order to
approximate the Executive Private Price. This will allow us to derive the
Executive Private Price as the B & S price plus a negative pertubation. And
finally we are going to conclude that B & S price overstate the fair-value
of an ESO.
3.2.1 Skewness and Kurtosis
A random variable could be defined with its moment. Mean and
variance which are the most wellknown moment of a random variable are
respectively the first moment and second central moment. But some higher moment
are interesting such that skewness and kurtosis which are respectively the
third and fourth central moment.
This moment are interesting since it measure respectively the
lopsidedness and the degree to which a statistical frequency curve is peaked.
But in our problem, this moments will serve us to give an polynomial expression
to the Private price.
Considere first the expression of the skewness and secondly the
one of kurtosis.
By the definition of skew(X) we get:
~u3skew(X) = u3 -- 3u1u2 + 2u2 1 Where uk = E
[(X)k] , ?k E N (34)
Definition 3.2.2. Kurtosis
The kurtosis is the relative peakness or flatness of a
distribution compared with the Gaussian distribution. Let X the same random
variable as previously. Thus the fourth standardized central moment of X is
written by kurt(X) and is defined by:
E[(X -- u1)4]
kurt(X) :=
3 (35)
u4
By definition of kurt(X) we get:
u4kurt(X) = u4 -- 3u22 + 12u21u2
-- 4u1u3 -- 60. (36)
In the next part we will derive a polynomial form for the
Executive Private Price.
3.2.2 The perturbative expansion
By Taylor argument we can approximate each n-differentiable
function by its n orders differentials. The idea in this part is to derive a
tractable polynomial expression of the Executive Private Price in order to
reveal the B & S valuation and thus to be able to compare this two
valuation. By equation (3.1) we have an explicit form of the private price.
?ñ2)(S,--K)+ | St = S, Xt = x] )
e--r(T--t)
p(t, s) =-- log(EP0 [e?ã(1
ã(1 -- ñ2)
Let E := ã(1-- ñ2), z := (St --
K)+, f (Ez) = e-€z - 1 and y := EP0 [f
(Ez) | St = S, Xt = x] = Ellt°,08,x [Ez]. Thus the polynomial expansion of
the function e-€z and log(1 + y) at 0 to the order 4 for the
first one and to order 1 for the other one as the following expression:
8
<>> >
>>>:
(Ez)2
e-€z = 1 -- Ez
2!
(Ez)3 + (Ez)4 O(E4)
3! 4!
log(1 + y) = y + O(y),?|y|= 1
Suppose that uk := EP0 ((ST --K)+)k | St =
S, Xt = x = Ellt',08,x
e
p(t, s) --
log (1 -- Eu1 + E2 u2 --Eu3 +E4u41
2! 3! E4!
-r(T-t)
~e-r(T-t)2 E2 2 E3 3
Eu1 + 2! (u2 -- + 2! u1 -- 3!
(u3 -- 3u1u2 + 2u21) +
3!3 -- 3! 1 2E3 u2 + E4 4! u4 )
·
·
·
~ ~
e-r(T-t) E2 E
EEPt ,x 2! ,8 [z] + V tP0,x [z] -- E3
skewP0 (z) + 4! kurtP 0 (z) +
3! é
E
e-r(T-t) (4,08,x [z] vir0
L 2! .,8,x !
[z] + E 32 skewP0 (z) -- 4E! kurtP0 (z) --
é)
E
(37)
Where é = €22u21 +
€33! (3u1u2 -- 2u21) + €43! (3u2 2 -- 12u21u2 +
4u1u3 +
6u41)
3.2.3 Comments
We have found in the previous part a nice form of the Executive
Private Price. In fact we can write the Indifference Price as a linear function
of the n-moments of the option payoff.
Without complex calculus in order to find the explicit Private
Price expression with all variables defined in the model we can reduce the
previous expression and consider only terms with epsilon-power strictly lower
than two. In fact we assume that ~n 0, {?n E N n n = 2}
Thus the simplified form of the Executive Private Price is:
~ ~
[(ST - K)+] - ã(1 - ñ2)
p(t, s) e-r(T -t) E 0 V 0
[(ST - K)+] + E 0 [((ST -
K)+)2]))
t,s,x t,s,x t,s,x
2
p(t, s) pBS(t, s) - Ø(f, S)
Where Ø(f, S) > 0 and
pBS (t, s) is the price of an european call option in B & S
framework.
(38) Finally we have shown by a polynomial approximation that
the fair-price of an ESO is lower than the price derived in B & S. This
inequality come from the risk-aversion of the Executive which cannot perfectly
hedge her risk with the set of constraints which are imposed to her.
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