STAT0018: Stochastic Methods in Finance II
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 1 / 28
Module outline (Part 1)
Introduction (Chs 1-2)
Mathematical Background (Chs 3-4)
Investment Timing (Ch 5)
Operational Timing (Ch 6)
Entry, Exit, Lay-Up, and Scrapping (Ch 7)
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 2 / 28
Lecture outline
Basic model
Solution via dynamic programming
Solution via contingent claims analysis
Characteristics of optimal investment
Alternative stochastic processes
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 3 / 28
Basic model: Optimal timing
Suppose project value, V , evolves according to a GBM, i.e.,
dV = αVdt + σVdz , which may be obtained at a sunk cost of I
When is the optimal time to invest?
I A perpetual option, i.e., calendar time is not important
I Ignore temporary suspension or other embedded options
I Use both dynamic programming and contingent claims methods
Problem formulation: maxT E[(VT ? I)e?ρT ]
I Assume δ ≡ ρ? α > 0, otherwise it is always better to wait indefinitely
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 4 / 28
Basic model: Deterministic case
Suppose that σ = 0, i.e., V (t) = V0eαt for V0 ≡ V (0)
I F (V ) ≡ maxT e?ρT (VeαT ? I)
I If α < 0, then F (V ) = max{V ? I, 0}
I Otherwise, for 0 < α < ρ, waiting may be better because either (i)
V < I or (ii) V ≥ I, but discounting of future sunk cost is greater that
that in the future project value
I Thus, the FONC is
dF (V )
dT 0? (ρ?α)Ve?(ρ?α)T = ρIe?ρT ? T ? = max
{ 1
α ln{ ρI(ρ?α)V }, 0
}
I Reason for delaying is that the MC is depreciating over time by more
than the MB
Substitute T ? to determine V ? = ρIρ?α > I
And, F (V ) = αIρ?α [
(ρ?α)V
ρI ]
ρ
α if V ≤ V ? (F (V ) = V ? I otherwise)
Figure 5.1 indicates that greater α increases V ?
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 5 / 28
Basic model: Figure 5.1
Figure 5.1. Value of Investment Opportunity, F (V ), for σ = 0 and ρ = 0.1 (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 6 / 28
Dynamic programming solution
Bellman equation for continuation is ρFdt = E[dF ]
Expand the RHS via It?’s lemma: dF = F ′(V )dV + 12F ′′(V )(dV )2 ?
E[dF ] = F ′(V )αVdt + 12F ′′(V )σ2V 2dt
Substitution into the Bellman equation yields the ODE
1
2F ′′(V )σ2V 2 + F ′(V )αV ? ρF (V ) = 0
I Equivalently, 12F ′′(V )σ2V 2 + F ′(V )(ρ? δ)V ? ρF (V ) = 0I Three boundary conditions: (i) F (0) = 0, (ii) F (V ?) = V ? ? I, and (iii)
F ′(V ?) = 1
I General solution to the ODE is F (V ) = A1V β1 + A2V β2
I Taking derivatives, we have F ′(V ) = A1β1V β1?1 + A2β2V β2?1 and
F ′′(V ) = A1β1(β1 ? 1)V β1?2 + A2β2(β2 ? 1)V β2?2
I Substitution into the ODE yields A1V β1
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 7 / 28
Dynamic programming solution
The characteristic quadratic, Q(β) = 12σ2β(β ? 1) + (ρ? δ)β ? ρ, has
two roots such that β1 > 1 and β2 < 0
I Q(β) has a positive coefficient for β2, i.e., it is an upward-pointing
parabola
I Note that Q(1) = ?δ < 0, which means that β1 > 1
I Q(0) = ?ρ, which means that β2 < 0 (Figure 5.2)
Consequently, the first boundary condition implies that A2 = 0, i.e.,
F (V ) = A1V β1
I Using the VM and SP conditions, we obtain V ? = β1β1?1 I and
A1 = V
??I
(V ?)β1 =
(β1?1)β1?1
β
β1
1 Iβ1?1I Since β1 > 1, we also have V ? > I
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 8 / 28
Dynamic programming solution: Figure 5.2
Figure 5.2. The Fundamental Quadratic (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 9 / 28
Dynamic programming solution: Comparative statics
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 10 / 28
Dyn. progr. solution: Comparison to neoclassical theory
Marshallian analysis is to compare
V0 ≡ Epi0
Tobin’s q is the ratio of the value of the existing capital goods to the
their current reproduction cost
I Rule is to invest when q ≥ 1
I If we interpret q as being VI , then the real options threshold is
q = β1β11 > 1I Hence, the real options definition of q adds option value to the PV of
assets in place
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 11 / 28
Contingent claims solutions: Background
Instead of using an arbitrary discount rate, ρ, we now try to ground it
more firmly using market principles
I Assume that x is the price of an asset that is perfectly correlated with
V , i.e., ρxm = ρVM
I If x pays no dividends, then dx = μxdt + σxdz
I From CAPM, μ = r + φρxmσ > α, where α is the expected percentage
rate of change of V
I Let δ = μ? α be the dividend rate, and if it were equal to zero, then it
would imply that the option would always be held to maturity
I In other words, there would be no opportunity cost to delaying exercise
of the option since the entire return comes from the price movement,
i.e., one would never invest
I Thus, we assume δ > 0, and if δ →∞, then invest either now or never,
i.e. opportunity cost of waiting is high and option value goes to zero
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 12 / 28
Contingent claims solutions:
Find F (V ) by constructing a risk-free portfolio, Φ, which consists of
one unit of F (V ) and n = F ′(V ) units short of the underlying project
(or correlated assets)
I Recall that n = bFxBX in order for the synthetic portfolio to be risk freeI Φ = F ? F ′(V )V , which means that n must change over time even if it
is kept constant for the next dt time units
I Short position requires dividend payment of δVF ′(V )
I Thus, the total portfolio return is
dΦ? δF ′(V )Vdt = dF ? F ′(V )dV ? δF ′(V )Vdt
I From It?’s lemma, we have dF = F ′(V )dV + 12F ′′(V )(dV )2I Substitution yields the total portfolio return is
1
2F ′′(V )(dV )2 ? δF ′(V )Vdt = 12F ′′(V )V 2σ2dt ? δF ′(V )VdtI The no-arbitrage condition implies 12F ′′(V )V 2σ2dt ? δF ′(V )Vdt =
r [F ? F ′(V )V ]dt ? 12F ′′(V )V 2σ2 + (r ? δ)F ′(V )V ? rF = 0
I Hence, F (V ) = A1V β1 , where β1 = 12 ? r?δσ2 +
√[ r?δ
σ2 ? 12
]2 + 2rσ2
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 13 / 28
Characteristics of the optimal investment rule
Use numerical examples to illustrate how investment values and
thresholds change using I = 1, r = 0.04, δ = 0.04, and σ = 0.20
I This implies that β1 = 2, V ? = 2I = 2, and A1 = 14 , i.e., real options
says to invest when project value is twice as high as the investment cost
I Furthermore, F (V ) = 14V 2 for V ≤ 2 and F (V ) = V ? 1 otherwise
(Figure 5.3)
I Note that F (V ) and V ? increase with σ: greater uncertainty increases
value of waiting and, thus, the opportunity cost of investing (Figure 5.4)
I Greater δ increases the opportunity cost of delaying the investment and,
thus, reduces the option value and the investment threshold (Figures 5.5
and 5.6)
I Caveat: σ and δ are related via δ = μ? α = r + φσρxm ? α, but we
treat them as being independent for sake of exposition
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 14 / 28
Characteristics of the optimal investment rule: Figure 5.3
Figure 5.3. Value of Investment Opportunity, F (V ), for σ = 0, 0.2, and 0.3 (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 15 / 28
Characteristics of the optimal investment rule: Figure 5.4
Figure 5.4. Critical Value V? as a Function of σ (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 16 / 28
Characteristics of the optimal investment rule: Figure 5.5
Figure 5.5. Value of Investment Opportunity, F (V ), for δ = 0.04 and 0.08 (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 17 / 28
Characteristics of the optimal investment rule: Figure 5.6
Figure 5.6. Critical Value V? as a Function of δ (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 18 / 28
Characteristics of the optimal investment rule
Use numerical examples to illustrate how investment values and
thresholds change using I = 1, r = 0.04, δ = 0.04, and σ = 0.20
I Increasing r increases F (V ) and V ? because the PV of expenditure at
future time T , Ie?rT , is reduced while the PV of revenue, Ve?δT , is
unaffected (Figure 5.7)
I Thus, it is worthwhile to wait more even if the value of the option
increases
I Cast results in terms of Tobin’s q = V ?I =
β1
β1?1 , i.e., use definition
without option value
I Plot contours of constant q? for combinations of 2rσ2 and
2δ
σ2 (Figure 5.8)I Find that q? is large when either δ is small or r is large: intuitively,
higher dividend rate reduces value of waiting, while higher interest rate
does the opposite
I Finally, note that all estimated parameters, such as α and σ, may be
changing over time
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 19 / 28
Characteristics of the optimal investment rule: Figure 5.7
Figure 5.7. Critical Value V? as a Function of r (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 20 / 28
Characteristics of the optimal investment rule: Figure 5.8
Figure 5.8. Curves of Constant q? = β1
β1?1 (Source: Dixit and Pindyck (1994))
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 21 / 28
Alternative stochastic processes: GMR process
Suppose V follows a GMR process: dV = η(Vˉ ? V )Vdt + σVdz
I Expected percentage change of V is 1dtE[
dV
V ] = η(Vˉ ? V )
I Thus, expected absolute rate of change is 1dtE[dV ] = ηV Vˉ ? ηV 2,
which is a parabola that is zero at V = 0 and V = Vˉ with a maximum
at Vˉ2I Let μ be the risk-adjusted rate of return for the project and define the
dividend rate to be δ(V ) = μ? 1dtE[ dVV ] = μ? η(Vˉ ? V )I End up with same ODE as before using contigent claims, but adjust for
δ(V ) : 12σ2V 2F ′′(V ) + [r ? μ+ η(Vˉ ? V )]VF ′(V )? rF = 0I Must satisfy the same three boundary conditions as before
I Typically, a closed-form solution is difficult to find
I Express the solution as F (V ) = AV θh(V ) and substitute it back into
the ODE
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 22 / 28
Alternative stochastic processes: GMR process
I Use substitution x = 2ηVσ2 to transform into Kummer’s equation,
xg ′′(x) + (b ? x)g ′(x)? θg(x), which has the solution
H(x ; θ, b) = 1 + θb x +
θ(θ+1)x2
b(b+1)2! +
θ(θ+1)(θ+2)x3
b(b+1)(b+2)3! + . . .
I Hence, F (V ) = AV θH( 2ησ2V ; θ, b)
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 23 / 28
Alternative stochastic processes: Investment characteristics
Use numerical example with same parameters as before pluss μ = 0.08
and varying η and V
I As Vˉ increases, so does the value of waiting and, thus, both F (V ) and
V increase (Figure 5.11)
I Variation with η: if Vˉ > I, then F (V ) increases in η (but decreases
otherwise) as V is likely to rise above I and remain there (Figures 5.12
and 5.13)
I Shape of F (V ) becomes concave for small V because the absolute rate
of mean reversion rises rapidly
I V increases with η as long as Vˉ is large (Figure 3.14)
Dr Sebastian Maier (Dept of Stat Science) Stochastic Methods in Finance II Week 3 24 / 28
Characteristics of the optimal investment rule: Figure 5.11
Figure 5.11. Mean Reversion – F (V ) for η = 0.05 and Vˉ = 0.5, 1.0, and 1.5 (Source: Dixit and Pindyck (1994))