Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework

Transcription

1 Valuation and Optimal Decision for Perpetual American Employee Stock Options under a Constrained Viscosity Solution Framework Quan Yuan Joint work with Shuntai Hu, Baojun Bian candy5191@163.com Tongji University, China 02 Jul 2012 Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

2 Outline 1 Introduction Background and motivation Overview of Our Model 2 Problem Formulation HJB equation 3 Value Function Value Function as the Constrained Viscosity Solution Comparison Principle 4 Numerical Simulation Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

3 Introduction Employee Stock Options (ESOs) Call options issued by a company Based on the company s common stock Granting the holder the right to buy a certain number of stocks at a predetermined price (Strike Price) during a specified period Extensive Use Granted as compensation or reward to employees by companies Employees profit from exercising ESOs Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

4 Features of ESOs American-style: can be exercised at any time before expiration Long maturity: from 5-10 years Job termination risk: get fired or leave the company voluntarily A vesting period: during which exercising is forbidden Transfer and hedging restrictions: transfer of ESOs and short selling the underlying stock are forbidden Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

5 Valuation and Optimal Decision for ESOs Companies (Valuation) estimate the cost of ESOs report to the Financial Accounting Standards Board(FASB) Employees (Optimal decision) how to exercise a block of ESOs over time maximize the returns Difficulties ESOs operate in an incomplete market. Standard valuation methods fail. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

6 Related Literature Exercise at a single date: treat every option equally J.Carpenter,The exercise and valuation of executive stock options, Journal of Financial Economics(1998)48, B.J.Hall, K.J.Murphy, Stock option for undiversified executives, Journal of Accounting and Economics(2002)33, Multi-period Model: Utility Maximization Ashish Jain, Ajay Subramanian, The intertemporal exercise and valuation of employee options, The Accounting Review(2004)Vol.79 No.3, L.C.G. Rogers, JosWScheinkman, Optimal exercise of executive stock options, Finance Stoch(2007)11, Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

7 Our Model Assumptions An employee holding a block of perpetual American ESOs Exercise process: Continuous fluid model with a bounded exercise rate Method Maximize the overall discounted exercise returns Stochastic control approach: works well in an incomplete market Results Value-based maximum defines the cost of ESOs Determine the optimal exercise strategy Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

8 Problem Formulation Time horizon: t [0, ) (t = 0: Grant Date) Stock price of the company: X t dx t = µx t dt + σx t dw t, X 0 = x µ: expected stock return rate σ: constant volatility W t : standard Browian motion (BM) Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

9 Problem Formulation The employee is granted N shares of perpetual American ESOs with strike price K at grant date 0 Exercise process: the accumulated number of exercised options up to t: Y t a continuous fluid model dy t = u t dt, Y 0 = y control set Γ = [0, λ] {Y t } t 0 : a non-negative non-decreasing right-continuous process not exceeding N Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

10 Problem Formulation State variable (X t, Y t ) S: S = (0, ) (0, N) Admissible Control u( ) with initial value (x, y) S Definition (Admissible) 1 u( ) is F t = σ{x s : s t} adapted 2 u(t) Γ for all t 0 3 the corresponding state process (X t, Y t ) S for all t 0 Denote by A = A(x, y) the set of all admissible controls. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

11 Optimization Objective function [ J(x, y; u.) = E 0 ] e ρt (X t K) + u t dt X 0 = x, Y 0 = y discount rate ρ: ρ > µ > 0 payoff function: (X t K) + = Value function v(x, y) = { Xt K, X t > K 0, X t K sup J(x, y; u( )) u( ) A(x,y) Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

12 HJB Equation HJB equation (Stochastic analysis theory) where Boundary Condition Lv + max(ubv) = 0, (x, y) [0, ) [0, N] (1) u Γ Lv = µxv x + σ2 2 x2 v xx ρv, Bv = v y + (x K) +. v(0, y) = 0, 0 y N. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

13 Properties of Value Function 1 For each x [0, ), v(x, y) is non-increasing in y 2 For each y [0, N], v(x, y) is non-decreasing in x 3 v(x, y) is Lipschitz continuous in (x, y), that is { } λ v(x 1, y 1 ) v(x 2, y 2 ) min ρ µ, 2(N y 1) x 1 x 2 +(2x 2 + 1) y 1 y 2 for any (x i, y i ) [0, ) [0, N] (i = 1, 2). Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

14 Definition for Constrained Viscosity Solution Semicontinuous functions in S USC( S) = {v : S R { } v is upper semicontinuous}, LSC( S) = {v : S R {+ } v is lower semicontinuous}. Viscosity Supersolution w(x, y) LSC( S) is a supersolution of (1) in S if and only if Lϕ + max u Γ (ubϕ) (x0,y 0 ) 0 (2) whenever ϕ(x, y) C 2,1 and w(x, y) ϕ(x, y) has a local minimum at (x 0, y 0 ) S with w(x 0, y 0 ) = ϕ(x 0, y 0 ). Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

15 Definition for Constrained Viscosity Solution Viscosity Subsolution w(x, y) USC(S) is a subsolution of (1) in S if and only if Lϕ + max u Γ (ubϕ) (x0,y 0 ) 0 (3) whenever ϕ(x, y) C 2,1 and w(x, y) ϕ(x, y) has a local maximum at (x 0, y 0 ) S with w(x 0, y 0 ) = ϕ(x 0, y 0 ). Definition (Constrained Viscosity Solution) A continuous function w is a constrained viscosity solution of (1) if it is both a viscosity supersolution of (1) in S and a viscosity subsolution of (1) in S. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

16 Comparison Principle Theorem Let v USC(S) is a subsolution of (1) in S and v LSC( S) is a supersolution of (1) in S. Furthermore, v, v grow linearly in X, i.e. there exists a constant C > 0 such that, v(x), v(x) C X for X S, and v(0) = v(0) = 0. Then if ρ > 2µ + σ 2, we have v v in S. Proof by contradiction. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

17 Value Function as a Constrained Viscosity Solution Theorem The continuous value function v(x, y) growing linearly in (x, y) is the unique constrained viscosity solution of (1) in S. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

18 Optimal Exercise Decision HJB equation: Lv + max (ubv) = 0, (x, y) [0, ) [0, N] 0 u λ No-exercise Region(NR) and Exercise Region(ER) NR := {(x, y) : Bv(x, y) 0}; ER := {(x, y) : Bv(x, y) > 0} Optimal Exercise Rate u (x, y) = (the feedback control depending on v) { 0, if (x, y) NR, λ, if (x, y) ER. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

19 Numerical Simulation Model Parameters: µ = 0.1, σ = 0.3, ρ = 0.15, λ = 1, K = 2, M = 5, N = 30. Value Function v(x, y): Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

20 NR,ER and the Threshold Boundary NR (u = 0): hold and wait ER (u = λ): exercise immediately Threshold-style Strategy: easy to implement in practice Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

21 Impact of Discount Rate ρ ρ = 0.15, 0.18, 0.21 ρ: time scale Larger ρ encourages earlier exercise action. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

22 Impact of Expected Stock Return Rate µ µ = 0.15, 0.1, 0.12, 0.14 µ: increasing capacity of the stock price Larger µ encourages more patient waiting. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

23 Impact of Exercise Rate Restriction λ λ = 1, 2, 3 λ: upper bound of exercise rate Larger λ encourages more patient waiting. Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24

24 Thank You! Q. Yuan (Tongji University) Valuation and Optimal Decision for ESOs 02 Jul / 24