Valuation Methods of Executive Stock Options

3 The Executive's Optimal Exercise Policy: the general approach

3.1 utility-based pricing

3.1.1 Introduction

By using the complete market argument, the standard financial theory can valuate a contingent claim by replicating its future payoff via the use of risky and riskfree stocks in the market. The derivative security price found is unique.

However in the incomplete market case, the future payoff cannot be replicated since the agent's environ- ment is constrained and thus there are not enough assets in the market that allow the fully replication of the terminal payoff.

This issue can be solved by considering the utility-based Pricing method.

Suppose that the agent has a utility function dependant on her risk aversion parameter and her initial wealth. Then by founding an optimal trading rule which involves the investment of her wealth between risky and riskfree assets we can find a price p which makes the agent indifferent between having a stock option and paying p or paying nothing and not having the derivatives. In the economic literature, p is called indifference price or private price.

Given the general formulation of the Excutive Investment Problem (17) we can formulate the indifference price idea via the definition (1.5.1).

The main problem in this approach come from the technical difficulty to find an explicit solution. Thus a set of assumptions is imposed to the utility function form. Because the optimization program , technically speaking is hard, it is supposed an exponential form of the utility function in order to allow an easy variable separation.

By this way we can formulate the Executive's Optimal Policy in its general form through a general method.

3.1.2 The general form of the EIP with 1 unit of ESO

By assumption, the ESO's holder is not allowed to sell her option or to trade her company stock. Therefore, it is central to consider her risk aversion. The general result found on the Merton Problem allow us to generalize it in the case where the Executive is endowed by 1 unit of ESO.

Assume all constraints imposed in the previous section hold (recall: the Executive is allowed to trade only in the risk-less asset and the Market Index) and Mt = M (The Market Index price at t is M).

The aim of the Executive is to maximize her expected utility among all trading strategy before the Terminal time T.

Then at time u E [0, T], the EIP associated to the value function G(u,X,M) is defined by:

G(u,X,S) = sup è?È0,T = sup è?È0,T

EP [U(T,XT + (ST - K)^+)| Xt = X,St = s] EP [ ] -e-ã(XT +(ST -K)+)er(T _ u)e - (T _u) 2 ( u_r ó ) 2 | Xt = X, St = s

(18)

Here we have reformulate the EIP general form by introducing only 1 unit of ESO.

Remark : It can be easily formulated this EIP with n identical ESO by putting n as factor before the derivative's payoff.

3.1.3 Private Price of 1 unit of ESO

The second step of this methodology consists of finding the Private Price p. To achieve this objective we are going to use the Private Price definition.

( G(u, X, S) = sup E^P ~U(T, XT + (ST - K)⁺) | Xt = X, St = s

è?È0,T (19)

G(u, X - p, S) = G(u, X, 0)

Using the Bellman dynamic programming principle, G(u,X,S) have to satisfy the following Partial Differential Equation:

(

sup LG = 0
è?È0,T G(u, X - p, S) = G(u, X, 0)

Where L define the inifinisetimal generator of (X,S) under the historical probability measure P: D

L = + [9(sa - r) + rX] D DX + (í - q)S D DS + ¹ ₂(9 ó)2 D2

DX2 + ¹ 2(çS)2 D2

DS2 + (ñçó9S) D2

DSDX (20)

Du

Remark :

1. the differential operator L is not linear in 9. Then if we focus us on the optimization problem we are face on a non linear Hamilton-Jacobi-Bellman equation. By 1.11 argument we are allowed to linearize it by introducing a power transformation;

2. given that G(u,X,S) could be written as G(u, X, S) = e_ã^r(T_^u)XG(u, 0, S) we can reduce the dimension of the original problem (18)

Remark : By using the Girsanov's change of measure argument we know that G(u,X,S) is a martingale under the optimal strategy 9^* define by equation (17). And moreover this martingale is MEMM by 1.10 argument.

According to the last argument we can define the MEMM P0 relative to the historical probability P such that the Utility process is a P0-martingale.

[{ ó )2T )} ]

e(_ u-r

ó WT _ 1 2 ( u-r

P0(A) = E IA, A?FT (21)

The last argument point out that all other strategy are not optimal and define a supermartingale.

1

Given G(u, 0,S) = p 1-ñ2 and using the Bellman dynamic programming principle and the 1.11 argument,

p have to satisfy the programming system:

sup è?È0,T

Lp=0 (22)

With the following boundary conditions:

f

p(T,X,S) = e_ã(1_p2)(ST_K)+ p(T,X,0) = 0 The simplest form of the PDE is:

Lp(t, s) = 0 (23)

Where L is the infinitesimal generator of the process (St) under P0:

L = D + (í - q - sa - r D 2(çs)2 D2

1

DS +

çñ)s (24)

Dt ó DS²

(25)

Proof. We can rewrite the stock price diffusion process (St) under P0 as:

dS_u = (í - q)S_udu + çSudW u 2

= (í - q)S_udu + çñS_udW u By Girsanov argument (cf.1.9)

= (í - q)S_udu + çñSu(dW P0

u - u_r

ó du)

= (í - q - u_r

ó çñ)S_udu + çñSudW 1,P0

u

Now we can write an explicit form for the intermediate function p(t, S).

By Feynman & Kac argument the PDE (23) has the following form under the measure P0:

1

1-ñ2 (27)

]p(t, S) = EP0 [ _e_ã(1_p²)(ST _K)+ | Xt = X, St = s(26) And with this expression we deduce the form of the value function G(t,X,S):

]

G(t, X, S) = -e_ãXer(T -t)_ (T -t)

2 ( u-r

ó )2EP0 [ _e_ã(1_p²)(ST _K)+ | Xt = X, St = s

Now by using the system (19) we can deduce the Private Price of 1 unit of ESO:

Proposition 3.1. Executive Indifference Price

The Executive's indifference price for her ESO according to 1.5.1 as the following form:

2)(S-K)+ | St =S,Xt = x])

_e-r(T-t)

(28)

p(t, s) = P° [-ã(1

ã(1 - ñ2) log(Ee

Or equivalently

G(t, x, s) = V (t, x)e-ãp(t,s)e-r(T-t) (29)

Proof. By definition the Executive's Indifference Price is such that:

G(t, x - p, s) = G(t, x, 0) = V (t, x)

Then by separation variables argument we get:

1

G (t , x - p, 0)p 1- P2 = G(t, x, 0) = V (t, x)

t] -e-ã(x-p)e^r(T-t) e- 2 (T-t)(ur) 2 } EP° [e-ã(1-ñ2)(ST-K)+ |Xt = X, St = s

1 1-P^2

= V(t, x)

1

teãp^er(T-t)V(t, x)} EP° [e-ã(1-ñ2)(ST-K)+| Xt = X, St = s] 1-P² = V(t, x)

1

teãper(T-t)} _EP° [e-ã(1-ñ2)(ST-K)+ | Xt = X, St = s] 1-P² = 1 (Since V(t, x) =6 0)

1

eãper(T-t) = _EP ° [e -ã(1-ñ2)(ST-K)+ | Xt = X, St = s] 1-P²

ãper(T-t) _=-1-1ñ2 log(EP° [e-ã(1-ñ2)(ST-K)+ Xt = X St = s])|

log(EP° [e-ã(1-ñ2)(ST-K)+|Xt = X, St = s])

p =-ã(1-- ñ²)

Now we can deduce the PDE of the private price.

3.1.4 The Partial Differential Equation of the Private Price

From the Private Price's expression we know that p(t,s) have to satisfied the following PDE (which has been defined in the equation (23)):

_e-r(T -t)

p(t, s) ^=-ã(1 - ñ2) log(p) (30)

By Feynman-Kac argument p solves:

Lp= 0

With the boundary conditions:

f /3(T, s) = eã(1-ñ2)(ST-K)+

1 p(T, 0) = 0

And where L defined by (24) is the inifinitesimal generator of the company's stock price process under the MEMM P0.

Thus the Private Price p(t,s) satisfies the following Partial Differential Equation:

Lp(t, s) - rp(t, s) -

1 T --t 819(4 s) )2

(31)

ã(çs)²(1 - ñ2)er( ) = 0

?s

With boundary condition:

p(t, s) = (ST - K)⁺

Proof. By equation(23) we have:

-- r1

pt(t, s) + (í--q ñ)sp_s(t, s) + ₂ (çs)2

.73ss(t, s) = 0

ó

With:

· pt =

ap
at

· p_s = a p

a_s

_a2

· fIC^.1

s s = p

a_s2

But:

_e-r(T-t)

p(t, s) = --

y(1 -- ñ2) log(p(t, s)) ? p(t, s) = e-ã(1-ñ2)p(t,s)er(T-t)

Then:

·

pt(t, s) = --y(1 -- ñ2)e-r(T-t)e-ã(1-ñ2)p(t,s)e-r(T-t) (pt(t, s) -- rp(t, s))

· p_s(t, s) = --y(1 -- ñ2)e-r(T-t)e-ã(1-ñ2)p(t,s)e-r(T-t)

_p2)_e--r(T--t)_e--^y(1--P^2-r(T-t) e_y(i _p 2)_e--r(T--t)_ps /
·,

· /3_ss(t, s) = y(1-- --p_ss(t, s))
p _s (t,s)

Then we get the following PDE in term of p after divided each part of the equation by:

--y(1 -- _ñ2)_e-r(T-t)_e-ã(1-ñ²)p(t,s)e^-r(T-t)

for ñ =6 1 and y =6 0

pt(t, s) -- rp(t, s) + (í -- q -- u-r _óçñ)sp_s(t, s) + 12 (çs)² _(pss (t, s) + y(1-- ñ2)e-r(T-t)p2s(t,s)) = 0

By grouping we get: (32)

Lp(t, s) -- rp(t, s) -- ã2 (ç s)² (1 -- ñ²)e^-r(T-t)p²_s(t, s) = 0