Viable Capture Basin for Studying Differential and Hybrid Games ...
International Game Theory Review c World Scientific Publishing Company
Viable Capture Basin for Studying Differential and Hybrid Games : Application to Finance
Patrick Saint-Pierre Centre de Recherche Viabilit´ e, Jeux, Contrˆ ole Universit´ e Paris-Dauphine F-75016 Paris, France
Abstract Viability theory can be applied for determining viable capture basin for control problem in presence of uncertainty. We first recall the concepts of viability theory which allow to develop numerical methods for computing viable capture basin for control problems and guaranteed control problems. Recent developments of option pricing in the framework of dynamical games with constraints lead to the formulation of guaranteed valuation in terms of guaranteed viable-capture basin of a dynamical game. As an application we show how the viability/capturability algorithm evaluates and manages portfolios. Regarding viability/capturability issues, stochastic control is a particular use of tychastic control. We replace the standard translation of uncertainty by stochastic control problem by tychastic ones and the concept of stochastic viability by the one of guaranteed viability kernel. Considering the Cox-Rubinstein model, we extend algorithms for hedging portfolios in the presence of transaction costs and dividends using recent developments on hybrid calculus. Keywords: Viability, Capturability, Tychastic control, Dynamical Games, Hedging portfolio.
1. Introduction In this paper we present some applications of Viability Theory and Set Valued Numerical Analysis to the problem of hedging portfolio with transaction costs. We first recall and illustrate the concept of Viable Capture Basin in the framework of control problems and the concept of Guaranteed Viable Capture Basin in the framework of differential Games. Then we show how these notions of “capturability” of a target and of viability of a system, under constraints can be applied for evaluating portfolios in the general case and how the Capture Basin Algorithm can be fruitfully used to determine numerically the rules for managing a portfolio (“Pujal (2000)”, “Pujal & Saint-Pierre (2001)”). Our scope is to emphasize the articulation between Viability, Games theory and Mathematical Finance, following the ideas developed in “Bernhard (2000), (2002)” and “Pujal (2000)” that appeared simultaneously and independently at the end of the year 2000. They consider the evolution of the prices
Viable Capture Basin for Studying Differential and Hybrid Games . . .
governed by ∀ i = 1, . . . , n, x0i (t) = xi (t)ρi (x(t), v(t)) where v(t) ∈ Q(x(t)) where v(t) is regarded as a tyche, a perturbation, a disturbance. We consider the problem of evaluation of portfolio in the presence of transaction cost. This has been studied in “Aubin, Pujal & Saint-Pierre (2001)” and “Bernhard (2002)”. We present here new algorithms for evaluating portfolio and finding hedging strategies and we provide some numerical results. We will not give any proof of existence of solutions or convergence of algorithm that can be found in referenced papers. 2. Viability Kernels & Viable Capture Basins Let us consider the differential control system x0 = f (x, u), where u ∈ U (x)
(2.1)
The viable capture basin of a closed target C viable in a closed set K is the set of elements x ∈ K such that there exists a continuous feedback u e(x) ∈ U (x) and t∗ ∈ R+ such that the solution x(·) to x0 = f (x, u e(x)) exists and satisfies i) x(t) ∈ K, ∀ t ∈ [0, t∗ ] ii) x(t∗ ) ∈ C This set is denoted CaptF (K, C). Existence and properties of this set can be found in “Aubin (1991)”. In general there is no way to describe it analytically. However it can be numerically approximated thanks to the Viability Kernel Algorithm. Let FC be the set-valued map which coincides with F outside C, equals to 0 on Interior(C) and equals to Conv({0} ∪ F (x)) on ∂C. If K is a repeller under F , then CaptF (K, C) = V iabFC (K). Otherwise, one can characterize the viable capture basin as the domain of the Minimal Time function defined by ϑK C (x) :=
inf
{τ | x(τ ) ∈ C, x(t) ∈ K, ∀t ≤ τ }
x(·)∈SF (x)
+ which satisfies the relation Epigraph(ϑK C ) = V iabFC ×{−1} (K × R ) so that
CaptF (K, C) = P rojX V iabFC ×{−1} (K × R+ ) To illustrate this we consider the following examples: Example 1. [Minimal time for the Labyrinth Problem] Let us consider the dynamical system
(x0 (t), y 0 (t), z 0 (t)) = f (x(t), y(t), z(t), u, v) := (u, v, −1), with u2 + v 2 ≤ 1 with the target C = {(x, y) ∈ IR2 | x2 + y 2 ≥ 1} and the constraints K = {[−1.2, 1.2] × [−1.2, 1.2]}\M where M are obstacles. The Capture Basin of C under K for f is the domain of the minimal time function - which coincides with K - and the epigraph of the minimal time function
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 1. Minimal time function with constraints : graph and optimal solution.
is the Viability Kernel of K × R+ for f . This function is the smallest lower semi-continuous super solution of the Hamilton Jacobi-Belmann equation: min (u
(u,v)∈B
∂ϑ ∂ϑ + v ) ≤ −1 ∂x ∂y
on K\C 2.1. Viable Guaranteed and Conditional Capture Basins Let us now consider the two-player differential game characterized by the differential system x0 = f (x, u, v), where u ∈ U (x) and v ∈ V (x) and the two-players discrete game described by the recursive system xn+1 = g(xn , un , v n ) with a constraint set K. Two kinds of games with constraints involve targets, the first one is a capture problem with an open target and the second one is a minimal time problem with a close target (see “Cardaliaguet, Quincampoix & Saint-Pierre (1995), (2001)”). We extend the concept of Viable Capture Basin up to differential games and define - the guaranteed viable-capture basin which is the set of x ∈ K such that there exists a continuous selection u e(x) ∈ U (x) such that ∀v(·) ∈ V (x(·)), ∃t∗ ∈ R+ such that the solution x(·) to x0 (t) = f (x(t), u e(x(t)), v(t))
exists and satisfies x(t) ∈ K, ∀ t ∈ [0, t∗ ] and x(t∗ ) ∈ C. - the conditional viable-capture basin which is the set of x ∈ K such that for any continuous selection ve(x) ∈ V (x), ∃u(·) ∈ U (x(·)) such that ∃t∗ ∈ R+ such that the
Viable Capture Basin for Studying Differential and Hybrid Games . . .
solution x(·) to x0 (t) = f (x(t), u(t), ve(x(t)))
exists and satisfies x(t) ∈ K, ∀ t ∈ [0, t∗ ] and x(t∗ ) ∈ C. These domains can be characterized by geometric condition: • The Viability Kernel: V iabF (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x) : f (x, u) ∈ TD (x) −−→ • The Discrete Viability Kernel: V iabG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x) : g(x, u) ∈ D −−−→ • The Discrete Conditional Viability Kernel: CondG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∀v ∈ V (x), ∃u ∈ U (x) : g(x, u, v) ∈ D −−−→ • The Discrete Guaranteed Viability Kernel: GuarG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x), ∀v ∈ V (x) : g(x, u, v) ∈ D 2.2.
Discrete Constrained Games Algorithms
Thanks to these properties we can design algorithms for approximate viable guaranteed or conditional capture basin. For this task let us consider a discrete game given by xn+1 = gρ (xn , un , v n ) = xn + ρfρ (xn , un , v n ) where ρ denotes the time step and fρ = f + ϕ(ρ)B. We can approach Discrete Guaranteed Viability Kernel and Discrete Conditional Viability Kernel by extending the Viability Kernel Algorithm and constructing decreasing sequences of closed sets defined recursively as follows (“Cardaliaguet, Quincampoix & Saint-Pierre (1999)”) • Discrete Guaranteed Viability Kernel: 0 n+1 n n Kρ,g = K, Kρ,g = {x ∈ Kρ,g | ∃u ∈ U (x), ∀v ∈ V (x), gρ (x, u, v) ∈ Kρ,g } ∞ and GuarFρ (K) = Kρ,g
• Discrete Conditional Viability Kernel 0 n+1 n n Kρ,c = K, Kρ,c = {x ∈ Kρ,c | ∀v ∈ V (x), ∃u ∈ U (x), gρ (x, u, v) ∈ Kρ,c } ∞ and CondFρ (K) = Kρ,c
3.
Application: Hedging Portfolio without or with Transaction Costs
Let us now consider the problem of evaluating an hedging portfolio in the framework of dynamical games with constraints where the epigraph of the claim function will play the role of the target.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
We discuss here only the discrete model of evaluation but we will comment some questions arising when considering continuous models with both uncertainty and transaction costs. Let be T the time left until maturity. Let S0 , S1 and W denote the bond, the values of the risky asset and of the hedging portfolio. Let be S = (S0 , S1 ). The variable x corresponds to (T, S, W ) and p is the control. Parameters γ0 (S0 ) and γ1 (S1 , v) measure the rates of price evolution, v describes tychastic uncertainty. T is the exercise time and K is the striking price∗ . Let us consider a given time-independent function u : R2 7→ R ∪ {+∞}, called the contingent claim† . The general model can be interpreted, in the discrete formulation, as a problem of finding initial conditions (T, S1 , W ) such that there exists a feedback S 7→ pe(S) ∈ P such that, whatever the perturbation v n ∈ [vm , vM ] is, the hedging portfolio satisfies a viability condition ∀n ≤ N, W n ≥ b(T − tn , S n )
(T − tn , S n , W n ) ∈ Epi(b)
⇔
(3.2)
where b is a given constraint function defined on R × R+ , with values in R ∪ +∞, and a capturability condition : 1. European Option: N N N N W N = pN 0 S0 + p1 S1 ≥ u(S )
⇔
(S N , W N ) ∈ Epi(u)
(3.3)
2. American Option: W n = pn0 S0n + pn1 S1n ≥ u(S n ), ∀n ≤ N,
⇔
(S n , W n ) ∈ Epi(u)
(3.4)
3. First Time Options: the option is exercised at the first time step n∗ ≤ N when ∗
∗
∗
∗
∗
∗
W n = pn0 S0n + pn1 S n ≥ u(S n )
⇔
∗
∗
(S n , W n ) ∈ Epi(u)
(3.5)
In order to treat the three rules of the three games as particular cases of a more general framework, we introduce a nonnegative extended functions c (objective function) satisfying ∀ (t, S) ∈ R × R+ , 0 ≤ b(t, S) ≤ c(t, S) ≤ +∞ and ∀ t < 0, b(t, S) = c(t, S) = +∞ ∗K
is the usual notation in Finance, without confusion with the previous notation for the constraint state. † For exemple, one can choose u(S) = (S − K)+ but any lower semicontinuous map can be 1 considered.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
By associating with the initial function u adequate pairs (b, c) of extended functions, we shall replace the requirements (3.3,3.4,3.5) by the requirement i) ∀ n ≤ N, (T − tn , S n , W n ) ∈ Epi(b) (dynamical constraint) ∗ ∗ ∗ (3.6) ii) (T − tn , S n , W n ) ∈ Epi(c) (final objective)
Allowing the functions to take infinite values (i.e., to be extended), allows us to acclimate many examples. So the three rules associated with a same function u : R2 × R 7→ R ∪ {+∞} can be written in the form (3.6) by adequate choices of pairs (b, c) of functions associated with u. Indeed, denoting by 0 the function defined by 0 if t ≥ 0, 0(t, S) = +∞ if not
and by u∞ the function defined by u∞ (t, S) :=
u(S) if +∞ if not
t=0
(3.7)
we can recover the three rules of the game 1. We take b(t, S) := 0 and c(t, S) = u∞ (t, S), we obtain the rule for the European option, 2. We take b(t, S) := u(S) and b(t, S) := u∞ (t, S), we obtain the rule for the American option, 3. We take b(t, S) := 0 and c(t, S) = u(S), we obtain the rule for the first time option. 3.1. Portfolio Evaluation without transaction cost P Without transaction cost, since W n = 1i=0 pni Sin , we introduce the self-financing P1 − pni )Sin+1 = 0 and the discrete evolution of the finance assumption i=0 (pn+1 i system becomes n+1 T = Tn − ρ n+1 = Sin (1 + γρi (Sin , v n )), i = 0, 1 Si n+1 W = W n (1 + γρ0 (S0n )) + pn1 S1n (γρ1 (S1n , v n ) − γρ0 (S0n )) where pn ∈ P (S n ) and v n ∈ Qρ (S n ) where quantities p0 and p1 play the role of controls, p0 ≤ 0, p1 ∈ [0, 1] and N denotes the number of intervals of time at which end the hedging portfolio is reevaluated which length is ρ. The uncertainty over the period of time [tn , tn + ρ] is represented by a tyche v ∈ Qρ (S) (which may depends on n).
Viable Capture Basin for Studying Differential and Hybrid Games . . .
The choice of Qρ (S) determines the degree of uncertainty distributed on each T small time intervals ρ = N . λ λ If γρ1 (S1 , v) = ργ1 (S1 ) + v with v ∈ Qρ (S) = [vm , vM ] = [e−σρ − 1, eσρ − 1], • the “tychastic”” or “contingent” uncertainty corresponds to λ = 1. In this case we get the guaranteed or conditional contingent evaluation corresponding to situations where the uncertainty is not stochastic. • the Cox & Rubinstein model corresponds to λ = 12 that is the up and down volatility terms in the time refined binomial model‡ . The target is C = (T, Epi(u)) and the constraint set is K = Epi(b). The Discrete Guaranteed and Conditional Capture Basin Algorithms that we briefly recalled in the previous section amounts to define sequences of functions n+1 n+1 n n Vρ,g and Vρ,c which epigraphs are precisely sets Kρ,g and Kρ,c . This leads to the following construction. Let us start the sequences with if t < 0 +∞, 0 0 u(S1 ), if 0 ≤ t < ρ Vρ,g (t, S1 ) = Vρ,c (t, S1 ) = b(t, S1 ), if ρ ≤ t ≤ T
Fig. 2. Guaranteed Evaluation of Standard European Call with Cox-Rubinstein type of uncertainty.
ALGORITHM 0 (without transaction cost) The approximation of the guaranteed valuation function is given by n+1 n Vρ,g (t, S1 ) := max Vρ,g (t, S1 ), inf inf sup β∈[−ϕ(ρ),ϕ(ρ)] p1 ∈[0,1] v∈[vm ,vM ]
n Vρ,g (t − ρ, S1 (1 + γρ1 (S1 , v) + β)) − p1 S1 (γρ1 (S1 , v) − γρ0 ) ) 1 + γρ0 ‡ Let
us recall that when hN goes to 0 the evaluation function of European option converges to the Black & Scholes value that is to say to stochastic uncertainty (“Bernhard (2002)”). In this case we can approximate the value given by the Black & Scholes formula which in some sense calibrates the Algorithm.
Viable Capture Basin for Studying Differential and Hybrid Games . . . n We set Vρ,g (t, s) = Vρ,g (t, S1 ), ∀t ∈](n − 1)ρ, nρ].
The approximation of the conditional valuation function is given by n+1 n Vρ,c (t, S1 ) := max Vρ,c (t, S1 ),
sup
inf
inf
v∈[vm ,vM ] β∈[−ϕ(ρ),ϕ(ρ)] p1 ∈[0,1]
n Vρ,c (t − ρ, S1 (1 + γρ1 (S1 , v) + β)) − p1 S1 (γρ1 (S1 , v) − γρ0 ) ) 1 + γρ0 n We set Vρ,c (t, s) = Vρ,c (t, S1 ), ∀t ∈](n − 1)ρ, nρ]. Example 2. [Guaranteed Evaluation of European Call with Cox-Rubinstein type of uncertainty] The following figures provide numerical results obtained with Algorithm O. Figure 2 left shows the target Epigraph(u) at t = 0 (in the plane t = 0), the graph of the evaluation function Vρ,c (t, s) and the value function of the call at maturity (in the plane t = T = 1). Figure 2 right shows the same elements projected on the plane (S, W ). Figure 3 left shows the optimal guaranteed policy for hedging portfolio by a color scaling superposed on the graph of the evaluation function. Table 1 gives values of a call using the Capture Basin Method when the riskless asset growth rate is γ0 = 5%, and the maturity price is K = 100.
Number of periods N = 512 N = 512 N = 512 N = 512 N = 512
value of S1 80 90 100 110 120
Value of the Call 4.53 8.61 14.24 21.04 28.84
p0 -22.10 -34.94 -48.17 -59.94 -69.63
p1 0.3332 0.4844 0.6240 0.7363 0.8207
Table 1.
Fig. 3. On the left: optimal part π0 (T, x) of non risky asset. On the right, contingent (tychastic) conditional evaluation of a non standard European call with varying rate and uncertainty.
Example 3. [Guaranteed Evaluation of European Call with varying rate and tychastic uncertainty]
Viable Capture Basin for Studying Differential and Hybrid Games . . .
In this example function is given by the +claim ((S−K) )2 sup . u(S) = min K sup −K , K √
S 1 with ρ1 (S) = 1000 and σρ (t) = 0.3ρ 0.01+t 2. Figure 3 right shows the graphical result obtained with Algorithm 0 for computing the evaluation function of such a European call with varying rate and varying uncertainty.
3.2. Portfolio Evaluation with transaction cost We can extend the Capture Basin Algorithm (CBA) designed for evaluating options for self-financed portfolio without transaction costs (“Pujal & Saint-Pierre (2001)”) to the case when in the presence of proportional and/or fix transaction costs. Functions u, b and c are now depending on variables (t, S, P ) and variable x corresponds to (T, S, P, W ). Parameter p1 is considered as a variable which derivative denoted u now play the role of the control. Let α1 (p1 ) represent the rate of the transaction cost. Let β1 represent the cost that one have to pay as soon as the composition of portfolio is altered: β1 (u) is equal to β1 if u 6= 0 and 0 if u = 0. Even if the theoretical study of this problem is out of the scope of this paper, we can easily extend the algorithm up to the case of fix transaction costs as long as we keep N bounded. The self-financing assumption becomes 1 X
p0i (t)xi (t) + α1 |p01 (t)|S1 (t) + β1 (p1 (t)) = 0
i=0
and the discrete system describing the evolution of the finance items reads: n+1 T = Tn − ρ n+1 = Sin (1 + γρi (Sin , v n )), i = 0, 1 Si n+1 p = pn1 + ρun 1 n+1 W = W n (1 + γρ0 (S0n )) + pn1 S1n (γρ1 (S1n , v n ) −γρ0 (S0n )) − ρ|un1 |α1 S1n − β1 (pn1 (t)) where un ∈ U (pn ) = [− ρ1 pn1 , 1ρ (1 − pn1 )] and v n ∈ Qρ (S n )
(3.8)
We look for the subset of initial conditions (S10 , p01 , W0 ) for which there exists a feedback S 7→ u e(S) ∈ U such that, whatever the perturbation v n ∈ Q(S n , ρ) is, the successive values of the portfolio satisfy the viability-capturability conditions. The Discrete Guaranteed Capture Basin Algorithm with Transition Costs leads to the construction of a sequence of sets Kρn recursively defined by Kρ0
and, for n ≥ 0,
= {(τ, S1 , p1 , y) ∈ [0, T ] × R+ × [0, 1] × R+ such that (T − τ, S1 , p1 , y) ∈ Epi(c), if τ ≤ ρ (T − τ, S1 , p1 , y) ∈ Epi(b), if τ ∈]ρ, T ]}
(3.9)
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Kρn+1
= {(τ, S1 , p1 , y) ∈ Kρn such that ∃φ ∈ ϕ(ρ)B, ∃u ∈ U (p), ∀v ∈ Qρ (S1 ), g(τ, S1 , p1 , y, u, v) + ρφ ∈ Kρn }
(3.10)
where g(τ, S1 , p1 , y, u, v) denotes the right hand side of relation (3.8) and ϕ(ρ)B denotes the ball contained in R3 of radius ϕ(ρ) which decreases to 0 when ρ → 0. The function ϕ(ρ) depends on the regularity parameters of the map g; in the Lipschitz case, ϕ(ρ) = 12 M `ρ (“Saint-Pierre (1994)”). This corrective function ϕ(ρ) needs to be introduced when we are looking for limit solutions when N → +∞. If we only consider discrete evolution of portfolio with fixed N steps, we choose ϕ(ρ) ≡ 0 as we have done in the next numerical examples. ALGORITHM I (with proportional transaction costs) We suppose here that there is no fix transaction cost: β1 (p1 ) = 0. With the sequence Kρn we associate the sequence of maps (τ, S1 , p1 ) → Wρn (τ, S1 , p1 ) defined by Wρn (τ, S1 , p1 ) := inf{y such that (τ, S1 , p1 , y) ∈ Kρn } and from relation (3.10) it comes Wρn+1 (τ, S1 , p1 ) := max Wρn (τ, S1 , p1 ),
inf
inf
sup
φ∈ϕ(ρ)B u∈U (τ,p1 ) v∈Q(S1 ,ρ)
Wρn (τ − ρ, S1 (1 + ργ1 (S1 , v)) + ρφ, p1 + ρu) − ρ(p1 + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρα1 |u|S1 1 + ργ0
with
+∞ c(S1 , p1 ) Wρ0 (t, S1 , p1 ) = b(S1 , p1 )
(3.11)
if t < 0 if t ≤ ρ if t > ρ
(3.12)
and the discrete guaranteed evaluation function is given by Wρ (t, S1 , p1 ) = Wρn (t, S1 , p1 ), ∀t ∈](n − 1)ρ, nρ] which epigraph is precisely the viable capture basin associated with (3.8) and Wρ (T, S1 , p1 ) is the minimal value of the hedging portfolio for the exercise time T is the value of the risky asset is S1 and if the portfolio contains p1 . Example 4. [European call with transaction costs under tychastic uncertainty] In the next example we choose T = 1, K = 100, σ := 0.3, γ0 = 0. Uncertainty is tychastic with γ1 (v) = 0.1+v, v ∈ [−σ, +σ] and the transaction cost rate α1 = 0.01. On figure 5, evaluation functions (x, p) 7→ Vρ (T, S1 , p) are computed for different values of T with α1 = 1%, and on figure 6, they are superposed. 3.3. The underlying Hamilton-Jacobi-Isaacs Equation Without fix transaction cost we can prove that, when N → ∞, the approximated valuation function recovers the solution of the Hamilton-Jacobi-Isaacs inequality we have obtained above. From the definition of Wρ given by (3.11), we deduce
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 4. ALGORITHM I : Guaranteed evaluation function with Cox-Rubinstein uncertainty of European call with transaction costs.
Wρ (τ + ρ, S1 , p1 ) := max
Wρ (τ, S1 , p1 ),
inf
inf
sup
u∈U (p en 1 ) φ∈[−ϕ(ρ),ϕ(ρ)] v∈Q(S1 ,N )
Wρ (τ, S1 (1 + ρ(γ1 (S1 , v) + φ)), p1 + ρu) − ρ(p1 + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0
(3.13)
So that Wρ (τ + ρ, S1 , p1 ) − Wρ (τ, S1 , p1 ) = max (0, inf n inf sup u∈U (p e1 ) φ∈[−ϕ(ρN ),ϕ(ρ)]] v∈Q(S1 ,N ) ρ Wρ (τ, S1 (1 + ρ(γ1 (S1 , v) + φ), p1 + ρu) − Wρ (τ, S1 , p1 ) ρ(1 + ργ0 ) −γ0 Wρ (τ, S1 , p1 ) − (p1 + ρu)S1 (γ1 (S1 , v) − ρ0 ) + |u|α1 S1 + ) 1 + ργ0
(3.14)
Assume that γ1 (S1 , v) = γ1 + v, it comes ∆t Wρ (τ, S1 , p1 )
−
inf
inf
sup
u∈U (pn 1 ) φ∈[−ϕ(ρ),ϕ(ρ)] v∈Q(S1 ,N )
(∆S1 Wρ (τ, S1 , p1 ).((γ1 + v)S1 + φ) + ∆p Wρ (τ, S1 , p1 ).u +(−γ0 Wρ (τ, S1 ) − (p1 + ρu)S1 (γ1 + v − γ0 ) + |u|α1 S1 )) ≥ 0 + o(ρ) (3.15) The infimum with respect to φ expresses the property that we approximate the Epi-
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 5. Evaluation functions (x, p) 7→ Vρ (T, S1 , p) are computed for different values of maturity T with α1 = 1% under tychastic uncertainty.
Fig. 6. Superposed evaluation functions computed for different values of maturity.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
derivative D ↑ W (τ, S1 , p1 )(−1, vx , vp ) = Wρ (τ − ρ, S1 + ρ(vx + φ), p1 + ρ(vp + φ)) − Wρ (τ, S1 , p1 ) lim inf ρ→0 φ∈B(0,ϕ(ρ)) ρ Wρ (τ − ρ, S1 + ρv’x , p1 + ρv’p ) − Wρ (τ, S1 , p1 ) lim inf . ρ ρ→0,v’x →vx ,v’p →vp where vx and vp are the derivatives of S1 and p1 . Then, when ρ → 0, Wρ (τ, S1 , p1 ) converges to the lowest solution to the HamiltonJacobi-Isaacs variational inequality (“Frankowska (1993)”), as formulated in the previous sections ∂ ∂τ
W (τ, S1 ) −
inf
sup
u∈U (p1 ) v∈Q(x)
D ↑ W (τ, S1 , p1 ).f (τ, S1 , p1 , u, v) + l(S1 , p1 , u, v) + m(S1 , p1 , v)W (τ, S1 , p1 ) ≥ 0 (3.16) where f (τ, S1 , p1 , u, v) describes the evolution of t, S1 and p1 , m(S1 , p1 , v) = −γ0 and l(S1 , p1 , u, v) = −(p1 + ρu)S1 (γ1 + v − γ0 ) + |u|α1 S1 . 3.4. The One Period Case or the First Step Evaluation in the N Periods
case. If T = 1 and N = 1 for the one period case or N fixed for the N period case, then the following formula gives the evaluation function for the hedging portfolio for an one step T procedure with ρ = N : W1 (T, S1 , p1 )
:= max (b(S1 ), (S1 (1 +
inf
sup
p p u∈[− ρ1 ,1− ρ1 ] v∈[−σ,+σ] + ργ1 (S1 , v)) − K) − ρ(p1 + ρu)S1 (γ1 (S1 , v)
− γ0 ) + ρα1 |u|S1 + β1 (u)
1 + ργ0
(3.17)
Choosing p1 = 0, assuming b = 0; γ0 = 0, setting γ1+ = γ1 (S1 , σ) and γ1− = γ1 (S1 , −σ) with σ = 0.3 the previous expression reads: W1 (T, S1 ) := min { (S1 (1 + ργ1+ ) − K)+ ,
max [ (S1 (1 + ργ1+ ) − K)+ − ρS1 (γ1+ + α1 ), S1 (1 + ργ1− ) − K)+ − ρS1 (γ1− + α1 ) ] + β1 } (3.18)
When the riskless asset rate is set to γ0 = 5%, the exercise price K = 100, T = 1 and N = 1, γ1 (v) = 0.1 + v, v ∈ [−0.3, 0.3] and the fix cost β1 = 0.00 and β1 = 0.01, we get the following numerical values: α1 S1 \β 70 90 100 120
0.00 0.00-0.01 0.49-0.50 9.97-9.98 14.71-14.72 24.76-24.77
0.01 0.02 0.03 0.04 0.00-0.01 0.00-0.01 0.00-0.01 0.00-0.01 0.51-0.52 0.53-0.54 0.56-0.57 0.58-0.59 10.45-10.46 10.93-10.94 11.40-11.41 11.88-11.89 15.42-15.43 16.12-16.13 16.83-16.84 17.53-17.54 25.90-25.91 27.05-27.06 28.19-28.20 29.33-29.34 Fig. 8. Option Value in the one period case
3.5. The N Periods Case
0.05 0.00-0.01 0.60-0.61 12.36-12.37 18.23-18.24 30.48-30.49
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 7. Value function after one step: comparison results for N = 1 or N = 1000 and tychastic q ±σ
T
T N − 1) uncertainty. ) or Cox & Rubinstein (γ1 = e (γ1 = ±σ N Curves 1: (S1 (1 + γ1+ ) − K)+ − S1 (γ1+ + α1 ) + β1 ; 2: (S1 (1 + γ1− ) − K)+ − S1 (γ1− + α1 ) + β1 ; 3: (S1 (1 + γ1+ ) − K)+ , 4: W1 (T, S1 )= min(Curve 3 , max(Curve 1,Curve 2)).
Viable Capture Basin for Studying Differential and Hybrid Games . . .
We are now looking for the behavior of the numerical approximation of the value function for a small number of steps N ≤ 20 first when there is a proportional cost and second when there is both a proportional and a fix cost.
Fig. 9. Evaluation Function with transaction costs: CBATC algorithm with α 1 = 1% and β1 = 0, Cox & Rubinstein uncertainty
Let us first remark that if N is small, tychastic and Cox & Rubinstein uncertainty lead to similar evaluation but when N goes to infinity, the tychastic evaluation converges whereas the Cox & Rubinstein model diverges (see figure 17 when N = 1000). Indeed in presence of transaction costs, in the tychastic case the portfolio is not often revalued and in the Cox & Rubinstein case it must be revalued at each step as we can see on figure 12 which represents the evolution of the control u(t, S 1 ) in both situations. In the lack of fix transaction cost, when β = 0, for larger values of N the following examples show a strong regularity of the value function with respect to time tn : 1, ..., N , for N = 200 fixed. (figure 9). 3.5.1. Proportional and fix transaction costs In this situation, the method we have presented above cannot be applied directly since in the presence of fix transaction costs the problem falls within the study of impulse dynamical games. Indeed, from a mathematical point of view, the right hand side of the dynamical system which describes the evolution of the financial items is lower semicontinuous. However, as we will see in the last section, this problem can be studied in the frame of impulse systems since whenever p(t) is non null, that is to say that a transaction occurs at time t, the trajectory (x( t), p1 (t), y(t)) is reset on a new position (x( t), p1 (t), y(t) + β1 ). ALGORITHM II (with transaction cost and terminal adjustment) The guaranteed evaluation function W depends on S1 and p1 .
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Let us consider the minimal value of Wρ (T, S1 , p1 ) with respect to p1 ∈ [0, 1] and the Capital Minimal function S1 7→ Wρ (T, S1 ) = min Wρ (T, S1 , p1 ) = Wρ (T, S1 , pe1 (T )) p1 ∈[0,1]
Fig. 10. The graph of Wρ (T, S1 , p) and the Capital Minimal function Wρ (T, x) .
The Argmin value of the function (T, S1 , p1 ) → Wρ (T, S1 , p1 ) provides the optimal rule for the constitution of the replicating portfolio if the value of the share is S1 and the maturity is T . Figure 10 shows the graph of the map (S1 , p) 7→ Wρ (T, S1 , p). Figure 11 represent the projection on the plane (S1 , W ) of the graph of the Capital Minimal function Wρ .
Fig. 11. The Capital Minimal function x 7→ Wρ (T, x).
Figure 12 shows the optimal buy & sell strategy u∗ = γρ (nρ, S1 , p) which minimizes at each step n the right hand side of the equation defining Wρ (n + 1, S1 , p). Figure 13 shows variations of function Wρ (T, S1 , p) with respect to α1 for different values of N ∈ {10, 20, 30, 40, 50, 60, 70, 80, 90},
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 12. Evolution of the optimal buy & sell strategy (ρ.u) for the Cox-Rubinstein model.
Fig. 13. Variations of the evaluation function with respect to α1 for tychastic uncertainty σ, when N varies from 10 to 90.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Figure 14 shows variations of function Wρ (T, S1 , p) with respect to the rate of transaction cost α1 ∈ 0.00,0.01,0.02,0.03,0.04,0.05,0.06,0.07, 0.08,0.09,0.10 when N varies from 1 to 125.
Fig. 14. Variations of the evaluation function with transaction cost with respect to N for different values of α1 ∈ {0, 0.01, ..., 0.1} when N varies from 1 to 125.
ALGORITHM III (Uncoupling Algorithm) From a numerical point of view, the major difference between the Capture Basin Algorithm designed for approximating the Value Function in the lack of transaction cost (presented in (“Pujal & Saint-Pierre (2001)”)), and the Capture Basin Algorithm for Transaction Cost (presented in “Aubin, Pujal & Saint-Pierre” (2001)) lies in the larger number of variables§ involved. The valuation function W is defined on the set {0, ρ, ..., N ρ} × R + × [0, 1]. When n = 0, the state (0, x0 , p0 ) must have reached the target Epigraph(c) Let us assume that at maturity the replicating portfolio must not contain any quantity of securities. This amounts to impose p0 = 0. If not there will be a final transaction cost to be added to the value of the call since, is the case, the seller of a call for instance will have to sell the risky part, if positive, of the replicating portfolio which will entail transaction cost. Moreover we assume that functions b and c does not depend on p. Thank to this assumption, we can define parallel sequences of functions Wρn and Pnρ defined on R × R+ with values in R+ satisfying Wρ0 (S1 ) = c(0, S1 ). Since p + ρu = p0 = 0 and p ∈ [0, 1], at the first step we determine for any S1 the minimal § Computing
the Value function W (t, S, p) needs roughly 4.Nt .NS .Np . octets where Nt = N , Nx and Np the grids size for representing S1 and p1 . For instance, the memory space required for implementing the second algorithm is 500M o if Nt = NS = Np = 500 and 4Go if Nt = NS = Np = 1000. For implementing the first algorithm the memory space required is 4M o if N t = NS = 1000.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
f0 (S1 ) ∈ [− 1 , 0] of the function argument u ρ
Wρ0 (S1 (1 + ργ1 (S1 , v))) − ρ(−ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0 v∈Q sup
and we set Wρ1 (S1 ) = max [b(ρ, S1 ), # Wρ0 (S1 (1 + ργ1 (S1 , v))) + ρ2 uS1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 inf sup 1 + ργ0 u∈[− ρ1 ,0] v∈Q and f0 (S1 ). P1ρ (S1 ) = −ρu
In other words, at step one, if the value of the risky asset is S1 then the replicating portfolio contains P1ρ (S1 ) quantities of this asset and the its value is equal to Wρ1 (S1 ). At the following steps we define in the same way from Wρn (S1 ) and from Pnρ (S1 ): " n+1 W (S ) := max b((n + 1)ρ, S1 ), inf sup 1 ρ u | Pn ρ (nρ,S1 )+ρu∈[0,1] v∈Q Wρn (S1 (1 + ργ1 (S1 , v))) − ρ(Pρn (S1 ) + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0 n+1 n (S )) f Pρ (S1 ) := Pn 1 ρ (S1 ) + ρu
n (S ) is the minimal argument which defines W n+1 (t, S ) f where u 1 1 ρ Remark: in general equality between Wρn (S1 ) and Wρ (nρ, S1 , Pn ρ (S1 )) defined in Algorithm II does not hold.
Application : Sensibility with respect to the type of uncertainty
3.6.
Between tychastic and stochastic process there exists a wide spectrum of different type of uncertainty which can be represented by the choice of λ in the definition of the λ T uncertainty range on small time intervals of length ρ = N : v ∈ [vm , vM ] = [e−σρ − λ
1, eσρ − 1]. Let us point out that tychastic uncertainty corresponds to λ = 1 and Cox & Rubinstein uncertainty corresponds to λ = 12 .
Numerical observations When N increases, the value of WρN (S1 ) jumps from a lower constant value Wm = 5 to an upper constant value WM = 48 if λ = 1 but, if λ = it remains to a lower value Wm = 14 and explodes when N becomes sufficiently high.
1 2
Analyzing values given in table 15 looking at figures 16, 17, 18, 19, 20 and 21 we can notice the central roles played by the two types of uncertainty: the tychastic one when λ = 1 and the Cox & Rubinstein one when λ = 12 . • 1) if λ = 1 the value of the call first decreases then jumps from a minimal value to W = 46.7. Then it remains constant. • 2) if λ > 1, uncertainty is super-tychastic, the value of the call first decreases and then jumps sooner to a lower value than W . Then it continually decreases.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
N \λ
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
1 11 21 31 33 34 35 36 41 51 81 91 101 150 201 301 401 501 601 701 801 871 881 891 901 911 921 1001 2000
15.41 57.16 47.21 50.70 51.22 51.33 51.68 51.77 52.83 54.22 56.25 56.74 56.97 57.67 57.97 58.13 58.07 58.13 58.11 58.20 58.10 58.12 58.14 58.07 58.10 58.13 58.09 58.11 58.
15.41 24.24 27.64 29.85 30.21 30.79 30.56 31.12 31.45 32.63 35.42 36.17 37.19 39.17 41.03 43.74 44.69 44.92 47.23 46.17 47.17 49.52 49.60 49.12 48.70 48.29 48.06 47.72 51.00
15.41 15.34 15.12 14.98 15.27 15.03 15.29 15.05 15.09 15.11 15.15 15.16 15.18 15.20 15.21 15.23 15.28 15.35 15.42 15.48 15.56 15.65 16.08 22.65 23.95 24.44 9 106 13 106 70 106
15.41 10.11 9.05 8.70 8.68 8.61 8.58 8.56 8.47 8.27 7.97 10.87 176.7 205.6 229.9 271.41 307.17 339.98 368.18 394.98 422.06 440.71 445.03 446.25 447.08 451.43 452.06 477.97 678.77
15.41 7.31 6.71 7.73 7.90 8.02 8.22 46.64 46.66 46.68 46.72 46.73 46.73 46.75 46.75 46.76 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.78 46.78 46.78 46.77
15.41 6.23 25.71 24.18 23.95 23.85 23.74 23.64 23.18 22.45 21.03 20.70 20.42 19.38 18.71 17.85 17.29 16.88 16.56 16.31 16.10 15.96 15.95 15.94 15.93 15.90 15.88 15.76 14.81
15.41 5.87 17.05 15.78 15.60 15.54 15.44 15.36 15.02 14.49 13.55 13.35 13.18 12.59 12.24 11.83 11.59 11.42 11.29 11.20 11.12 11.07 11.07 11.06 11.05 11.05 11.05 11.01
15.41 15.26 13.22 12.40 12.29 12.24 12.19 12.15 11.95 11.65 11.44 11.07 10.99 10.73 10.59 10.43 10.35 10.30 10.26 10.23 10.20 10.19 10.20 10.19 10.19 10.19 10.18 10.18
15.41 12.83 11.48 11.00 10.94 10.91 10.89 10.86 10.76 10.61 10.38 10.34 10.31 10.20 10.15 10.10 10.08 10.06 10.05 10.05 10.03 10.04 10.04 10.03 10.03 10.04 10.04 10.04 10.02
Fig. 15. Evolution “at the value ” (S1 = K) of the value of a call when N varies from 1 to 1000. Comparative results with respect to λ.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
• 3) if 12 < λ < 1, uncertainty is sub-tychastic and super-CR-stochastic, the value function is first greater than the tychastic reference and it jumps later to a value greater than W . Then it continually increases. • 4) if λ =
1 2,
uncertainty is CR-stochastic, the value of the call is constant
W = 15.2 on interval n ∈ [1, W ]. This value W is lower than W it suddenly explodes as soon as n becomes greater than N (W = 964602). • 5) if λ
0 is a virtual laps of time which indicates that a reset occured. We define the hybrid system (t0 , S10 , p1 , W 0 ) ∈ {−1} × F (S1 , p1 , W ), a.e. t n+1 (tn+1 , S1n+1 , pn+1 , W ) = Φ(tn , S n , W n ), t = T0 1 Since the Impulse Kernel Hyb{1}×F,Φ (K) is empty, the capture domain of the epigraph of u coincides with Hyb{1}×FC ,Φ (K) where FC = F oc C c and FC = 0 on C. −−→ CaptFC ,Φ (Epi(u)) = V iabGρ (K)Hyb{1}×FC ,Φ (K) We consider the discrete model define above and the Algorithm II. Exemple: Let us consider the basic European call for Cox & Rubinstein type of uncertainty and assume that the payment of dividend is done at time T0 = 0.1, the strike is K = 100 and the maturity time is T = 1. We get the following numerical values which correspond to values given by standard methods.
Viable Capture Basin for Studying Differential and Hybrid Games . . . Fig. 22. Evolution of the valuation with respect to the dividend
dividend 0 2 5 10
value of S1 100 100 100 100
Value of the Call 22.07 17.96 16.42 12.98
4.3. Application 2: Hedging Portfolio with Transaction Costs But we can also interpret the evaluation problem of a call in term of impulse control system under uncertainty. Since transaction cost may now appear at any time, the reset set is the whole space C = {(t, S1 , p1 , W ) ∈ [0, T ] × R+ × [0, 1] × R+} and the reset set-valued map becomes (t, S1 , p1 , W ) → Φ(t, S1 , p1 , W ) := {(t, S1 , q, W + |p1 − q|α1 S1 ), q ∈ [0, 1]} We define formally the (continuous) hybrid system (t0 , S10 , p01 , W 0 ) (tn+1 , S1n+1 , pn+1 , W n+1 ) 1
∈ {−1} × F (S1 , p1 , W ), a.e. t ∈ [0, T ] ∈ Φ(tn , S1n , pn1 , W n )
(4.21)
Then considering the discrete dynamical system associated with (4.21) we have tk+1 Sik+1 pk+1 1 W k+1
= = = =
tk − µρ Sik (1 + γρi (Sik , v)), i = 0, 1 pk1 + (1 − µ)ρu W k (1 + γρ0 (S0k )) + pk1 S1k (γρ1 (S1k , v) − γρ0 (S0k )) − (1 − µ)ρ|u|α1 S1k − (1 − µ)β1 pk
pk
where u ∈ [− ρ1 , 1 − ρ1 ] and µ belongs to the discrete set {0, 1}. If µk = 0 a transaction is effective at time n = tk . If µk = 1, there is no transaction at time n = tk . References AUBIN J.-P. (1991) Viability Theory Birkh¨ auser, Boston, Basel, Berlin AUBIN J.-P. (2000) “Viability Kernels and Capture Basins of Sets under Differential Inclusions”, SIAM J. Control AUBIN J.-P. (1999) “Impulse Differential Inclusions and Hybrid Systems: A Viability Approach”, Lecture Notes, University of California at Berkeley AUBIN J.-P., PUJAL D. & SAINT-PIERRE P. (2001) “Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty”, Preprints di Matematica n - 15, Scuola Normale Superiore, Pisa, Italy. BERNHARD P. (2000) “Une approche d´eterministe de l’´evaluation des options, in Optimal Control et Partial Differential Equations”, IOS Press BERNHARD P. (2000) A robust control approach to option pricing, Cambridge University Press BERNHARD P. (2002) Robust control approach to option pricing, including coˆ ut de transaction, Annals of Dynamic Games
Viable Capture Basin for Studying Differential and Hybrid Games . . .
CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1995) “Contribution a ` l’´etude des jeux diff´erentiels quantitatifs et qualitatifs avec contrainte sur l’´etat”, Comptes-Rendus de l’Acad´emie des Sciences, 321, 1543-1548 CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1999) “Set-valued numerical methods for optimal control and differential games”, In Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, 177-247 Birkh¨ auser CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (2001) “Pursuit Differential Games with States Constraints”, SIAM J. on Control and Optimization, Vol 39,No5, pp1615-1632 FRANKOWSKA H. (1993) “Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation”, SIAM J. on Control and Optimization, PUJAL D. & SAINT-PIERRE P. (2001) “L’algorithme du bassin de capture appliqu´e pour ´evaluer des options europ´eennes, am´ericaines ou exotiques”, preprint PUJAL D. (2000) “Valuation et gestion dynamiques de portefeuilles”, Th`ese de l’Universit´e de Paris-Dauphine SAINT-PIERRE P. (1994) “Approximation of the viability kernel”, Applied Mathematics & Optimisation, 29, 187-209 SAINT-PIERRE P. (2001) “Approximation of Viability Kernels and Capture Basins for Hybrid Systems”. Proceedings of the European Control Conference 2001, Porto
Viable Capture Basin for Studying Differential and Hybrid Games : Application to Finance
Patrick Saint-Pierre Centre de Recherche Viabilit´ e, Jeux, Contrˆ ole Universit´ e Paris-Dauphine F-75016 Paris, France
Abstract Viability theory can be applied for determining viable capture basin for control problem in presence of uncertainty. We first recall the concepts of viability theory which allow to develop numerical methods for computing viable capture basin for control problems and guaranteed control problems. Recent developments of option pricing in the framework of dynamical games with constraints lead to the formulation of guaranteed valuation in terms of guaranteed viable-capture basin of a dynamical game. As an application we show how the viability/capturability algorithm evaluates and manages portfolios. Regarding viability/capturability issues, stochastic control is a particular use of tychastic control. We replace the standard translation of uncertainty by stochastic control problem by tychastic ones and the concept of stochastic viability by the one of guaranteed viability kernel. Considering the Cox-Rubinstein model, we extend algorithms for hedging portfolios in the presence of transaction costs and dividends using recent developments on hybrid calculus. Keywords: Viability, Capturability, Tychastic control, Dynamical Games, Hedging portfolio.
1. Introduction In this paper we present some applications of Viability Theory and Set Valued Numerical Analysis to the problem of hedging portfolio with transaction costs. We first recall and illustrate the concept of Viable Capture Basin in the framework of control problems and the concept of Guaranteed Viable Capture Basin in the framework of differential Games. Then we show how these notions of “capturability” of a target and of viability of a system, under constraints can be applied for evaluating portfolios in the general case and how the Capture Basin Algorithm can be fruitfully used to determine numerically the rules for managing a portfolio (“Pujal (2000)”, “Pujal & Saint-Pierre (2001)”). Our scope is to emphasize the articulation between Viability, Games theory and Mathematical Finance, following the ideas developed in “Bernhard (2000), (2002)” and “Pujal (2000)” that appeared simultaneously and independently at the end of the year 2000. They consider the evolution of the prices
Viable Capture Basin for Studying Differential and Hybrid Games . . .
governed by ∀ i = 1, . . . , n, x0i (t) = xi (t)ρi (x(t), v(t)) where v(t) ∈ Q(x(t)) where v(t) is regarded as a tyche, a perturbation, a disturbance. We consider the problem of evaluation of portfolio in the presence of transaction cost. This has been studied in “Aubin, Pujal & Saint-Pierre (2001)” and “Bernhard (2002)”. We present here new algorithms for evaluating portfolio and finding hedging strategies and we provide some numerical results. We will not give any proof of existence of solutions or convergence of algorithm that can be found in referenced papers. 2. Viability Kernels & Viable Capture Basins Let us consider the differential control system x0 = f (x, u), where u ∈ U (x)
(2.1)
The viable capture basin of a closed target C viable in a closed set K is the set of elements x ∈ K such that there exists a continuous feedback u e(x) ∈ U (x) and t∗ ∈ R+ such that the solution x(·) to x0 = f (x, u e(x)) exists and satisfies i) x(t) ∈ K, ∀ t ∈ [0, t∗ ] ii) x(t∗ ) ∈ C This set is denoted CaptF (K, C). Existence and properties of this set can be found in “Aubin (1991)”. In general there is no way to describe it analytically. However it can be numerically approximated thanks to the Viability Kernel Algorithm. Let FC be the set-valued map which coincides with F outside C, equals to 0 on Interior(C) and equals to Conv({0} ∪ F (x)) on ∂C. If K is a repeller under F , then CaptF (K, C) = V iabFC (K). Otherwise, one can characterize the viable capture basin as the domain of the Minimal Time function defined by ϑK C (x) :=
inf
{τ | x(τ ) ∈ C, x(t) ∈ K, ∀t ≤ τ }
x(·)∈SF (x)
+ which satisfies the relation Epigraph(ϑK C ) = V iabFC ×{−1} (K × R ) so that
CaptF (K, C) = P rojX V iabFC ×{−1} (K × R+ ) To illustrate this we consider the following examples: Example 1. [Minimal time for the Labyrinth Problem] Let us consider the dynamical system
(x0 (t), y 0 (t), z 0 (t)) = f (x(t), y(t), z(t), u, v) := (u, v, −1), with u2 + v 2 ≤ 1 with the target C = {(x, y) ∈ IR2 | x2 + y 2 ≥ 1} and the constraints K = {[−1.2, 1.2] × [−1.2, 1.2]}\M where M are obstacles. The Capture Basin of C under K for f is the domain of the minimal time function - which coincides with K - and the epigraph of the minimal time function
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 1. Minimal time function with constraints : graph and optimal solution.
is the Viability Kernel of K × R+ for f . This function is the smallest lower semi-continuous super solution of the Hamilton Jacobi-Belmann equation: min (u
(u,v)∈B
∂ϑ ∂ϑ + v ) ≤ −1 ∂x ∂y
on K\C 2.1. Viable Guaranteed and Conditional Capture Basins Let us now consider the two-player differential game characterized by the differential system x0 = f (x, u, v), where u ∈ U (x) and v ∈ V (x) and the two-players discrete game described by the recursive system xn+1 = g(xn , un , v n ) with a constraint set K. Two kinds of games with constraints involve targets, the first one is a capture problem with an open target and the second one is a minimal time problem with a close target (see “Cardaliaguet, Quincampoix & Saint-Pierre (1995), (2001)”). We extend the concept of Viable Capture Basin up to differential games and define - the guaranteed viable-capture basin which is the set of x ∈ K such that there exists a continuous selection u e(x) ∈ U (x) such that ∀v(·) ∈ V (x(·)), ∃t∗ ∈ R+ such that the solution x(·) to x0 (t) = f (x(t), u e(x(t)), v(t))
exists and satisfies x(t) ∈ K, ∀ t ∈ [0, t∗ ] and x(t∗ ) ∈ C. - the conditional viable-capture basin which is the set of x ∈ K such that for any continuous selection ve(x) ∈ V (x), ∃u(·) ∈ U (x(·)) such that ∃t∗ ∈ R+ such that the
Viable Capture Basin for Studying Differential and Hybrid Games . . .
solution x(·) to x0 (t) = f (x(t), u(t), ve(x(t)))
exists and satisfies x(t) ∈ K, ∀ t ∈ [0, t∗ ] and x(t∗ ) ∈ C. These domains can be characterized by geometric condition: • The Viability Kernel: V iabF (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x) : f (x, u) ∈ TD (x) −−→ • The Discrete Viability Kernel: V iabG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x) : g(x, u) ∈ D −−−→ • The Discrete Conditional Viability Kernel: CondG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∀v ∈ V (x), ∃u ∈ U (x) : g(x, u, v) ∈ D −−−→ • The Discrete Guaranteed Viability Kernel: GuarG (K) is the largest closed subset D ⊂ K such that ∀x ∈ D, ∃u ∈ U (x), ∀v ∈ V (x) : g(x, u, v) ∈ D 2.2.
Discrete Constrained Games Algorithms
Thanks to these properties we can design algorithms for approximate viable guaranteed or conditional capture basin. For this task let us consider a discrete game given by xn+1 = gρ (xn , un , v n ) = xn + ρfρ (xn , un , v n ) where ρ denotes the time step and fρ = f + ϕ(ρ)B. We can approach Discrete Guaranteed Viability Kernel and Discrete Conditional Viability Kernel by extending the Viability Kernel Algorithm and constructing decreasing sequences of closed sets defined recursively as follows (“Cardaliaguet, Quincampoix & Saint-Pierre (1999)”) • Discrete Guaranteed Viability Kernel: 0 n+1 n n Kρ,g = K, Kρ,g = {x ∈ Kρ,g | ∃u ∈ U (x), ∀v ∈ V (x), gρ (x, u, v) ∈ Kρ,g } ∞ and GuarFρ (K) = Kρ,g
• Discrete Conditional Viability Kernel 0 n+1 n n Kρ,c = K, Kρ,c = {x ∈ Kρ,c | ∀v ∈ V (x), ∃u ∈ U (x), gρ (x, u, v) ∈ Kρ,c } ∞ and CondFρ (K) = Kρ,c
3.
Application: Hedging Portfolio without or with Transaction Costs
Let us now consider the problem of evaluating an hedging portfolio in the framework of dynamical games with constraints where the epigraph of the claim function will play the role of the target.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
We discuss here only the discrete model of evaluation but we will comment some questions arising when considering continuous models with both uncertainty and transaction costs. Let be T the time left until maturity. Let S0 , S1 and W denote the bond, the values of the risky asset and of the hedging portfolio. Let be S = (S0 , S1 ). The variable x corresponds to (T, S, W ) and p is the control. Parameters γ0 (S0 ) and γ1 (S1 , v) measure the rates of price evolution, v describes tychastic uncertainty. T is the exercise time and K is the striking price∗ . Let us consider a given time-independent function u : R2 7→ R ∪ {+∞}, called the contingent claim† . The general model can be interpreted, in the discrete formulation, as a problem of finding initial conditions (T, S1 , W ) such that there exists a feedback S 7→ pe(S) ∈ P such that, whatever the perturbation v n ∈ [vm , vM ] is, the hedging portfolio satisfies a viability condition ∀n ≤ N, W n ≥ b(T − tn , S n )
(T − tn , S n , W n ) ∈ Epi(b)
⇔
(3.2)
where b is a given constraint function defined on R × R+ , with values in R ∪ +∞, and a capturability condition : 1. European Option: N N N N W N = pN 0 S0 + p1 S1 ≥ u(S )
⇔
(S N , W N ) ∈ Epi(u)
(3.3)
2. American Option: W n = pn0 S0n + pn1 S1n ≥ u(S n ), ∀n ≤ N,
⇔
(S n , W n ) ∈ Epi(u)
(3.4)
3. First Time Options: the option is exercised at the first time step n∗ ≤ N when ∗
∗
∗
∗
∗
∗
W n = pn0 S0n + pn1 S n ≥ u(S n )
⇔
∗
∗
(S n , W n ) ∈ Epi(u)
(3.5)
In order to treat the three rules of the three games as particular cases of a more general framework, we introduce a nonnegative extended functions c (objective function) satisfying ∀ (t, S) ∈ R × R+ , 0 ≤ b(t, S) ≤ c(t, S) ≤ +∞ and ∀ t < 0, b(t, S) = c(t, S) = +∞ ∗K
is the usual notation in Finance, without confusion with the previous notation for the constraint state. † For exemple, one can choose u(S) = (S − K)+ but any lower semicontinuous map can be 1 considered.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
By associating with the initial function u adequate pairs (b, c) of extended functions, we shall replace the requirements (3.3,3.4,3.5) by the requirement i) ∀ n ≤ N, (T − tn , S n , W n ) ∈ Epi(b) (dynamical constraint) ∗ ∗ ∗ (3.6) ii) (T − tn , S n , W n ) ∈ Epi(c) (final objective)
Allowing the functions to take infinite values (i.e., to be extended), allows us to acclimate many examples. So the three rules associated with a same function u : R2 × R 7→ R ∪ {+∞} can be written in the form (3.6) by adequate choices of pairs (b, c) of functions associated with u. Indeed, denoting by 0 the function defined by 0 if t ≥ 0, 0(t, S) = +∞ if not
and by u∞ the function defined by u∞ (t, S) :=
u(S) if +∞ if not
t=0
(3.7)
we can recover the three rules of the game 1. We take b(t, S) := 0 and c(t, S) = u∞ (t, S), we obtain the rule for the European option, 2. We take b(t, S) := u(S) and b(t, S) := u∞ (t, S), we obtain the rule for the American option, 3. We take b(t, S) := 0 and c(t, S) = u(S), we obtain the rule for the first time option. 3.1. Portfolio Evaluation without transaction cost P Without transaction cost, since W n = 1i=0 pni Sin , we introduce the self-financing P1 − pni )Sin+1 = 0 and the discrete evolution of the finance assumption i=0 (pn+1 i system becomes n+1 T = Tn − ρ n+1 = Sin (1 + γρi (Sin , v n )), i = 0, 1 Si n+1 W = W n (1 + γρ0 (S0n )) + pn1 S1n (γρ1 (S1n , v n ) − γρ0 (S0n )) where pn ∈ P (S n ) and v n ∈ Qρ (S n ) where quantities p0 and p1 play the role of controls, p0 ≤ 0, p1 ∈ [0, 1] and N denotes the number of intervals of time at which end the hedging portfolio is reevaluated which length is ρ. The uncertainty over the period of time [tn , tn + ρ] is represented by a tyche v ∈ Qρ (S) (which may depends on n).
Viable Capture Basin for Studying Differential and Hybrid Games . . .
The choice of Qρ (S) determines the degree of uncertainty distributed on each T small time intervals ρ = N . λ λ If γρ1 (S1 , v) = ργ1 (S1 ) + v with v ∈ Qρ (S) = [vm , vM ] = [e−σρ − 1, eσρ − 1], • the “tychastic”” or “contingent” uncertainty corresponds to λ = 1. In this case we get the guaranteed or conditional contingent evaluation corresponding to situations where the uncertainty is not stochastic. • the Cox & Rubinstein model corresponds to λ = 12 that is the up and down volatility terms in the time refined binomial model‡ . The target is C = (T, Epi(u)) and the constraint set is K = Epi(b). The Discrete Guaranteed and Conditional Capture Basin Algorithms that we briefly recalled in the previous section amounts to define sequences of functions n+1 n+1 n n Vρ,g and Vρ,c which epigraphs are precisely sets Kρ,g and Kρ,c . This leads to the following construction. Let us start the sequences with if t < 0 +∞, 0 0 u(S1 ), if 0 ≤ t < ρ Vρ,g (t, S1 ) = Vρ,c (t, S1 ) = b(t, S1 ), if ρ ≤ t ≤ T
Fig. 2. Guaranteed Evaluation of Standard European Call with Cox-Rubinstein type of uncertainty.
ALGORITHM 0 (without transaction cost) The approximation of the guaranteed valuation function is given by n+1 n Vρ,g (t, S1 ) := max Vρ,g (t, S1 ), inf inf sup β∈[−ϕ(ρ),ϕ(ρ)] p1 ∈[0,1] v∈[vm ,vM ]
n Vρ,g (t − ρ, S1 (1 + γρ1 (S1 , v) + β)) − p1 S1 (γρ1 (S1 , v) − γρ0 ) ) 1 + γρ0 ‡ Let
us recall that when hN goes to 0 the evaluation function of European option converges to the Black & Scholes value that is to say to stochastic uncertainty (“Bernhard (2002)”). In this case we can approximate the value given by the Black & Scholes formula which in some sense calibrates the Algorithm.
Viable Capture Basin for Studying Differential and Hybrid Games . . . n We set Vρ,g (t, s) = Vρ,g (t, S1 ), ∀t ∈](n − 1)ρ, nρ].
The approximation of the conditional valuation function is given by n+1 n Vρ,c (t, S1 ) := max Vρ,c (t, S1 ),
sup
inf
inf
v∈[vm ,vM ] β∈[−ϕ(ρ),ϕ(ρ)] p1 ∈[0,1]
n Vρ,c (t − ρ, S1 (1 + γρ1 (S1 , v) + β)) − p1 S1 (γρ1 (S1 , v) − γρ0 ) ) 1 + γρ0 n We set Vρ,c (t, s) = Vρ,c (t, S1 ), ∀t ∈](n − 1)ρ, nρ]. Example 2. [Guaranteed Evaluation of European Call with Cox-Rubinstein type of uncertainty] The following figures provide numerical results obtained with Algorithm O. Figure 2 left shows the target Epigraph(u) at t = 0 (in the plane t = 0), the graph of the evaluation function Vρ,c (t, s) and the value function of the call at maturity (in the plane t = T = 1). Figure 2 right shows the same elements projected on the plane (S, W ). Figure 3 left shows the optimal guaranteed policy for hedging portfolio by a color scaling superposed on the graph of the evaluation function. Table 1 gives values of a call using the Capture Basin Method when the riskless asset growth rate is γ0 = 5%, and the maturity price is K = 100.
Number of periods N = 512 N = 512 N = 512 N = 512 N = 512
value of S1 80 90 100 110 120
Value of the Call 4.53 8.61 14.24 21.04 28.84
p0 -22.10 -34.94 -48.17 -59.94 -69.63
p1 0.3332 0.4844 0.6240 0.7363 0.8207
Table 1.
Fig. 3. On the left: optimal part π0 (T, x) of non risky asset. On the right, contingent (tychastic) conditional evaluation of a non standard European call with varying rate and uncertainty.
Example 3. [Guaranteed Evaluation of European Call with varying rate and tychastic uncertainty]
Viable Capture Basin for Studying Differential and Hybrid Games . . .
In this example function is given by the +claim ((S−K) )2 sup . u(S) = min K sup −K , K √
S 1 with ρ1 (S) = 1000 and σρ (t) = 0.3ρ 0.01+t 2. Figure 3 right shows the graphical result obtained with Algorithm 0 for computing the evaluation function of such a European call with varying rate and varying uncertainty.
3.2. Portfolio Evaluation with transaction cost We can extend the Capture Basin Algorithm (CBA) designed for evaluating options for self-financed portfolio without transaction costs (“Pujal & Saint-Pierre (2001)”) to the case when in the presence of proportional and/or fix transaction costs. Functions u, b and c are now depending on variables (t, S, P ) and variable x corresponds to (T, S, P, W ). Parameter p1 is considered as a variable which derivative denoted u now play the role of the control. Let α1 (p1 ) represent the rate of the transaction cost. Let β1 represent the cost that one have to pay as soon as the composition of portfolio is altered: β1 (u) is equal to β1 if u 6= 0 and 0 if u = 0. Even if the theoretical study of this problem is out of the scope of this paper, we can easily extend the algorithm up to the case of fix transaction costs as long as we keep N bounded. The self-financing assumption becomes 1 X
p0i (t)xi (t) + α1 |p01 (t)|S1 (t) + β1 (p1 (t)) = 0
i=0
and the discrete system describing the evolution of the finance items reads: n+1 T = Tn − ρ n+1 = Sin (1 + γρi (Sin , v n )), i = 0, 1 Si n+1 p = pn1 + ρun 1 n+1 W = W n (1 + γρ0 (S0n )) + pn1 S1n (γρ1 (S1n , v n ) −γρ0 (S0n )) − ρ|un1 |α1 S1n − β1 (pn1 (t)) where un ∈ U (pn ) = [− ρ1 pn1 , 1ρ (1 − pn1 )] and v n ∈ Qρ (S n )
(3.8)
We look for the subset of initial conditions (S10 , p01 , W0 ) for which there exists a feedback S 7→ u e(S) ∈ U such that, whatever the perturbation v n ∈ Q(S n , ρ) is, the successive values of the portfolio satisfy the viability-capturability conditions. The Discrete Guaranteed Capture Basin Algorithm with Transition Costs leads to the construction of a sequence of sets Kρn recursively defined by Kρ0
and, for n ≥ 0,
= {(τ, S1 , p1 , y) ∈ [0, T ] × R+ × [0, 1] × R+ such that (T − τ, S1 , p1 , y) ∈ Epi(c), if τ ≤ ρ (T − τ, S1 , p1 , y) ∈ Epi(b), if τ ∈]ρ, T ]}
(3.9)
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Kρn+1
= {(τ, S1 , p1 , y) ∈ Kρn such that ∃φ ∈ ϕ(ρ)B, ∃u ∈ U (p), ∀v ∈ Qρ (S1 ), g(τ, S1 , p1 , y, u, v) + ρφ ∈ Kρn }
(3.10)
where g(τ, S1 , p1 , y, u, v) denotes the right hand side of relation (3.8) and ϕ(ρ)B denotes the ball contained in R3 of radius ϕ(ρ) which decreases to 0 when ρ → 0. The function ϕ(ρ) depends on the regularity parameters of the map g; in the Lipschitz case, ϕ(ρ) = 12 M `ρ (“Saint-Pierre (1994)”). This corrective function ϕ(ρ) needs to be introduced when we are looking for limit solutions when N → +∞. If we only consider discrete evolution of portfolio with fixed N steps, we choose ϕ(ρ) ≡ 0 as we have done in the next numerical examples. ALGORITHM I (with proportional transaction costs) We suppose here that there is no fix transaction cost: β1 (p1 ) = 0. With the sequence Kρn we associate the sequence of maps (τ, S1 , p1 ) → Wρn (τ, S1 , p1 ) defined by Wρn (τ, S1 , p1 ) := inf{y such that (τ, S1 , p1 , y) ∈ Kρn } and from relation (3.10) it comes Wρn+1 (τ, S1 , p1 ) := max Wρn (τ, S1 , p1 ),
inf
inf
sup
φ∈ϕ(ρ)B u∈U (τ,p1 ) v∈Q(S1 ,ρ)
Wρn (τ − ρ, S1 (1 + ργ1 (S1 , v)) + ρφ, p1 + ρu) − ρ(p1 + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρα1 |u|S1 1 + ργ0
with
+∞ c(S1 , p1 ) Wρ0 (t, S1 , p1 ) = b(S1 , p1 )
(3.11)
if t < 0 if t ≤ ρ if t > ρ
(3.12)
and the discrete guaranteed evaluation function is given by Wρ (t, S1 , p1 ) = Wρn (t, S1 , p1 ), ∀t ∈](n − 1)ρ, nρ] which epigraph is precisely the viable capture basin associated with (3.8) and Wρ (T, S1 , p1 ) is the minimal value of the hedging portfolio for the exercise time T is the value of the risky asset is S1 and if the portfolio contains p1 . Example 4. [European call with transaction costs under tychastic uncertainty] In the next example we choose T = 1, K = 100, σ := 0.3, γ0 = 0. Uncertainty is tychastic with γ1 (v) = 0.1+v, v ∈ [−σ, +σ] and the transaction cost rate α1 = 0.01. On figure 5, evaluation functions (x, p) 7→ Vρ (T, S1 , p) are computed for different values of T with α1 = 1%, and on figure 6, they are superposed. 3.3. The underlying Hamilton-Jacobi-Isaacs Equation Without fix transaction cost we can prove that, when N → ∞, the approximated valuation function recovers the solution of the Hamilton-Jacobi-Isaacs inequality we have obtained above. From the definition of Wρ given by (3.11), we deduce
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 4. ALGORITHM I : Guaranteed evaluation function with Cox-Rubinstein uncertainty of European call with transaction costs.
Wρ (τ + ρ, S1 , p1 ) := max
Wρ (τ, S1 , p1 ),
inf
inf
sup
u∈U (p en 1 ) φ∈[−ϕ(ρ),ϕ(ρ)] v∈Q(S1 ,N )
Wρ (τ, S1 (1 + ρ(γ1 (S1 , v) + φ)), p1 + ρu) − ρ(p1 + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0
(3.13)
So that Wρ (τ + ρ, S1 , p1 ) − Wρ (τ, S1 , p1 ) = max (0, inf n inf sup u∈U (p e1 ) φ∈[−ϕ(ρN ),ϕ(ρ)]] v∈Q(S1 ,N ) ρ Wρ (τ, S1 (1 + ρ(γ1 (S1 , v) + φ), p1 + ρu) − Wρ (τ, S1 , p1 ) ρ(1 + ργ0 ) −γ0 Wρ (τ, S1 , p1 ) − (p1 + ρu)S1 (γ1 (S1 , v) − ρ0 ) + |u|α1 S1 + ) 1 + ργ0
(3.14)
Assume that γ1 (S1 , v) = γ1 + v, it comes ∆t Wρ (τ, S1 , p1 )
−
inf
inf
sup
u∈U (pn 1 ) φ∈[−ϕ(ρ),ϕ(ρ)] v∈Q(S1 ,N )
(∆S1 Wρ (τ, S1 , p1 ).((γ1 + v)S1 + φ) + ∆p Wρ (τ, S1 , p1 ).u +(−γ0 Wρ (τ, S1 ) − (p1 + ρu)S1 (γ1 + v − γ0 ) + |u|α1 S1 )) ≥ 0 + o(ρ) (3.15) The infimum with respect to φ expresses the property that we approximate the Epi-
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 5. Evaluation functions (x, p) 7→ Vρ (T, S1 , p) are computed for different values of maturity T with α1 = 1% under tychastic uncertainty.
Fig. 6. Superposed evaluation functions computed for different values of maturity.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
derivative D ↑ W (τ, S1 , p1 )(−1, vx , vp ) = Wρ (τ − ρ, S1 + ρ(vx + φ), p1 + ρ(vp + φ)) − Wρ (τ, S1 , p1 ) lim inf ρ→0 φ∈B(0,ϕ(ρ)) ρ Wρ (τ − ρ, S1 + ρv’x , p1 + ρv’p ) − Wρ (τ, S1 , p1 ) lim inf . ρ ρ→0,v’x →vx ,v’p →vp where vx and vp are the derivatives of S1 and p1 . Then, when ρ → 0, Wρ (τ, S1 , p1 ) converges to the lowest solution to the HamiltonJacobi-Isaacs variational inequality (“Frankowska (1993)”), as formulated in the previous sections ∂ ∂τ
W (τ, S1 ) −
inf
sup
u∈U (p1 ) v∈Q(x)
D ↑ W (τ, S1 , p1 ).f (τ, S1 , p1 , u, v) + l(S1 , p1 , u, v) + m(S1 , p1 , v)W (τ, S1 , p1 ) ≥ 0 (3.16) where f (τ, S1 , p1 , u, v) describes the evolution of t, S1 and p1 , m(S1 , p1 , v) = −γ0 and l(S1 , p1 , u, v) = −(p1 + ρu)S1 (γ1 + v − γ0 ) + |u|α1 S1 . 3.4. The One Period Case or the First Step Evaluation in the N Periods
case. If T = 1 and N = 1 for the one period case or N fixed for the N period case, then the following formula gives the evaluation function for the hedging portfolio for an one step T procedure with ρ = N : W1 (T, S1 , p1 )
:= max (b(S1 ), (S1 (1 +
inf
sup
p p u∈[− ρ1 ,1− ρ1 ] v∈[−σ,+σ] + ργ1 (S1 , v)) − K) − ρ(p1 + ρu)S1 (γ1 (S1 , v)
− γ0 ) + ρα1 |u|S1 + β1 (u)
1 + ργ0
(3.17)
Choosing p1 = 0, assuming b = 0; γ0 = 0, setting γ1+ = γ1 (S1 , σ) and γ1− = γ1 (S1 , −σ) with σ = 0.3 the previous expression reads: W1 (T, S1 ) := min { (S1 (1 + ργ1+ ) − K)+ ,
max [ (S1 (1 + ργ1+ ) − K)+ − ρS1 (γ1+ + α1 ), S1 (1 + ργ1− ) − K)+ − ρS1 (γ1− + α1 ) ] + β1 } (3.18)
When the riskless asset rate is set to γ0 = 5%, the exercise price K = 100, T = 1 and N = 1, γ1 (v) = 0.1 + v, v ∈ [−0.3, 0.3] and the fix cost β1 = 0.00 and β1 = 0.01, we get the following numerical values: α1 S1 \β 70 90 100 120
0.00 0.00-0.01 0.49-0.50 9.97-9.98 14.71-14.72 24.76-24.77
0.01 0.02 0.03 0.04 0.00-0.01 0.00-0.01 0.00-0.01 0.00-0.01 0.51-0.52 0.53-0.54 0.56-0.57 0.58-0.59 10.45-10.46 10.93-10.94 11.40-11.41 11.88-11.89 15.42-15.43 16.12-16.13 16.83-16.84 17.53-17.54 25.90-25.91 27.05-27.06 28.19-28.20 29.33-29.34 Fig. 8. Option Value in the one period case
3.5. The N Periods Case
0.05 0.00-0.01 0.60-0.61 12.36-12.37 18.23-18.24 30.48-30.49
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 7. Value function after one step: comparison results for N = 1 or N = 1000 and tychastic q ±σ
T
T N − 1) uncertainty. ) or Cox & Rubinstein (γ1 = e (γ1 = ±σ N Curves 1: (S1 (1 + γ1+ ) − K)+ − S1 (γ1+ + α1 ) + β1 ; 2: (S1 (1 + γ1− ) − K)+ − S1 (γ1− + α1 ) + β1 ; 3: (S1 (1 + γ1+ ) − K)+ , 4: W1 (T, S1 )= min(Curve 3 , max(Curve 1,Curve 2)).
Viable Capture Basin for Studying Differential and Hybrid Games . . .
We are now looking for the behavior of the numerical approximation of the value function for a small number of steps N ≤ 20 first when there is a proportional cost and second when there is both a proportional and a fix cost.
Fig. 9. Evaluation Function with transaction costs: CBATC algorithm with α 1 = 1% and β1 = 0, Cox & Rubinstein uncertainty
Let us first remark that if N is small, tychastic and Cox & Rubinstein uncertainty lead to similar evaluation but when N goes to infinity, the tychastic evaluation converges whereas the Cox & Rubinstein model diverges (see figure 17 when N = 1000). Indeed in presence of transaction costs, in the tychastic case the portfolio is not often revalued and in the Cox & Rubinstein case it must be revalued at each step as we can see on figure 12 which represents the evolution of the control u(t, S 1 ) in both situations. In the lack of fix transaction cost, when β = 0, for larger values of N the following examples show a strong regularity of the value function with respect to time tn : 1, ..., N , for N = 200 fixed. (figure 9). 3.5.1. Proportional and fix transaction costs In this situation, the method we have presented above cannot be applied directly since in the presence of fix transaction costs the problem falls within the study of impulse dynamical games. Indeed, from a mathematical point of view, the right hand side of the dynamical system which describes the evolution of the financial items is lower semicontinuous. However, as we will see in the last section, this problem can be studied in the frame of impulse systems since whenever p(t) is non null, that is to say that a transaction occurs at time t, the trajectory (x( t), p1 (t), y(t)) is reset on a new position (x( t), p1 (t), y(t) + β1 ). ALGORITHM II (with transaction cost and terminal adjustment) The guaranteed evaluation function W depends on S1 and p1 .
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Let us consider the minimal value of Wρ (T, S1 , p1 ) with respect to p1 ∈ [0, 1] and the Capital Minimal function S1 7→ Wρ (T, S1 ) = min Wρ (T, S1 , p1 ) = Wρ (T, S1 , pe1 (T )) p1 ∈[0,1]
Fig. 10. The graph of Wρ (T, S1 , p) and the Capital Minimal function Wρ (T, x) .
The Argmin value of the function (T, S1 , p1 ) → Wρ (T, S1 , p1 ) provides the optimal rule for the constitution of the replicating portfolio if the value of the share is S1 and the maturity is T . Figure 10 shows the graph of the map (S1 , p) 7→ Wρ (T, S1 , p). Figure 11 represent the projection on the plane (S1 , W ) of the graph of the Capital Minimal function Wρ .
Fig. 11. The Capital Minimal function x 7→ Wρ (T, x).
Figure 12 shows the optimal buy & sell strategy u∗ = γρ (nρ, S1 , p) which minimizes at each step n the right hand side of the equation defining Wρ (n + 1, S1 , p). Figure 13 shows variations of function Wρ (T, S1 , p) with respect to α1 for different values of N ∈ {10, 20, 30, 40, 50, 60, 70, 80, 90},
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Fig. 12. Evolution of the optimal buy & sell strategy (ρ.u) for the Cox-Rubinstein model.
Fig. 13. Variations of the evaluation function with respect to α1 for tychastic uncertainty σ, when N varies from 10 to 90.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
Figure 14 shows variations of function Wρ (T, S1 , p) with respect to the rate of transaction cost α1 ∈ 0.00,0.01,0.02,0.03,0.04,0.05,0.06,0.07, 0.08,0.09,0.10 when N varies from 1 to 125.
Fig. 14. Variations of the evaluation function with transaction cost with respect to N for different values of α1 ∈ {0, 0.01, ..., 0.1} when N varies from 1 to 125.
ALGORITHM III (Uncoupling Algorithm) From a numerical point of view, the major difference between the Capture Basin Algorithm designed for approximating the Value Function in the lack of transaction cost (presented in (“Pujal & Saint-Pierre (2001)”)), and the Capture Basin Algorithm for Transaction Cost (presented in “Aubin, Pujal & Saint-Pierre” (2001)) lies in the larger number of variables§ involved. The valuation function W is defined on the set {0, ρ, ..., N ρ} × R + × [0, 1]. When n = 0, the state (0, x0 , p0 ) must have reached the target Epigraph(c) Let us assume that at maturity the replicating portfolio must not contain any quantity of securities. This amounts to impose p0 = 0. If not there will be a final transaction cost to be added to the value of the call since, is the case, the seller of a call for instance will have to sell the risky part, if positive, of the replicating portfolio which will entail transaction cost. Moreover we assume that functions b and c does not depend on p. Thank to this assumption, we can define parallel sequences of functions Wρn and Pnρ defined on R × R+ with values in R+ satisfying Wρ0 (S1 ) = c(0, S1 ). Since p + ρu = p0 = 0 and p ∈ [0, 1], at the first step we determine for any S1 the minimal § Computing
the Value function W (t, S, p) needs roughly 4.Nt .NS .Np . octets where Nt = N , Nx and Np the grids size for representing S1 and p1 . For instance, the memory space required for implementing the second algorithm is 500M o if Nt = NS = Np = 500 and 4Go if Nt = NS = Np = 1000. For implementing the first algorithm the memory space required is 4M o if N t = NS = 1000.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
f0 (S1 ) ∈ [− 1 , 0] of the function argument u ρ
Wρ0 (S1 (1 + ργ1 (S1 , v))) − ρ(−ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0 v∈Q sup
and we set Wρ1 (S1 ) = max [b(ρ, S1 ), # Wρ0 (S1 (1 + ργ1 (S1 , v))) + ρ2 uS1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 inf sup 1 + ργ0 u∈[− ρ1 ,0] v∈Q and f0 (S1 ). P1ρ (S1 ) = −ρu
In other words, at step one, if the value of the risky asset is S1 then the replicating portfolio contains P1ρ (S1 ) quantities of this asset and the its value is equal to Wρ1 (S1 ). At the following steps we define in the same way from Wρn (S1 ) and from Pnρ (S1 ): " n+1 W (S ) := max b((n + 1)ρ, S1 ), inf sup 1 ρ u | Pn ρ (nρ,S1 )+ρu∈[0,1] v∈Q Wρn (S1 (1 + ργ1 (S1 , v))) − ρ(Pρn (S1 ) + ρu)S1 (γ1 (S1 , v) − γ0 ) + ρ|u|α1 S1 1 + ργ0 n+1 n (S )) f Pρ (S1 ) := Pn 1 ρ (S1 ) + ρu
n (S ) is the minimal argument which defines W n+1 (t, S ) f where u 1 1 ρ Remark: in general equality between Wρn (S1 ) and Wρ (nρ, S1 , Pn ρ (S1 )) defined in Algorithm II does not hold.
Application : Sensibility with respect to the type of uncertainty
3.6.
Between tychastic and stochastic process there exists a wide spectrum of different type of uncertainty which can be represented by the choice of λ in the definition of the λ T uncertainty range on small time intervals of length ρ = N : v ∈ [vm , vM ] = [e−σρ − λ
1, eσρ − 1]. Let us point out that tychastic uncertainty corresponds to λ = 1 and Cox & Rubinstein uncertainty corresponds to λ = 12 .
Numerical observations When N increases, the value of WρN (S1 ) jumps from a lower constant value Wm = 5 to an upper constant value WM = 48 if λ = 1 but, if λ = it remains to a lower value Wm = 14 and explodes when N becomes sufficiently high.
1 2
Analyzing values given in table 15 looking at figures 16, 17, 18, 19, 20 and 21 we can notice the central roles played by the two types of uncertainty: the tychastic one when λ = 1 and the Cox & Rubinstein one when λ = 12 . • 1) if λ = 1 the value of the call first decreases then jumps from a minimal value to W = 46.7. Then it remains constant. • 2) if λ > 1, uncertainty is super-tychastic, the value of the call first decreases and then jumps sooner to a lower value than W . Then it continually decreases.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
N \λ
0.00
0.25
0.50
0.75
1.00
1.25
1.50
1.75
2.00
1 11 21 31 33 34 35 36 41 51 81 91 101 150 201 301 401 501 601 701 801 871 881 891 901 911 921 1001 2000
15.41 57.16 47.21 50.70 51.22 51.33 51.68 51.77 52.83 54.22 56.25 56.74 56.97 57.67 57.97 58.13 58.07 58.13 58.11 58.20 58.10 58.12 58.14 58.07 58.10 58.13 58.09 58.11 58.
15.41 24.24 27.64 29.85 30.21 30.79 30.56 31.12 31.45 32.63 35.42 36.17 37.19 39.17 41.03 43.74 44.69 44.92 47.23 46.17 47.17 49.52 49.60 49.12 48.70 48.29 48.06 47.72 51.00
15.41 15.34 15.12 14.98 15.27 15.03 15.29 15.05 15.09 15.11 15.15 15.16 15.18 15.20 15.21 15.23 15.28 15.35 15.42 15.48 15.56 15.65 16.08 22.65 23.95 24.44 9 106 13 106 70 106
15.41 10.11 9.05 8.70 8.68 8.61 8.58 8.56 8.47 8.27 7.97 10.87 176.7 205.6 229.9 271.41 307.17 339.98 368.18 394.98 422.06 440.71 445.03 446.25 447.08 451.43 452.06 477.97 678.77
15.41 7.31 6.71 7.73 7.90 8.02 8.22 46.64 46.66 46.68 46.72 46.73 46.73 46.75 46.75 46.76 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.77 46.78 46.78 46.78 46.77
15.41 6.23 25.71 24.18 23.95 23.85 23.74 23.64 23.18 22.45 21.03 20.70 20.42 19.38 18.71 17.85 17.29 16.88 16.56 16.31 16.10 15.96 15.95 15.94 15.93 15.90 15.88 15.76 14.81
15.41 5.87 17.05 15.78 15.60 15.54 15.44 15.36 15.02 14.49 13.55 13.35 13.18 12.59 12.24 11.83 11.59 11.42 11.29 11.20 11.12 11.07 11.07 11.06 11.05 11.05 11.05 11.01
15.41 15.26 13.22 12.40 12.29 12.24 12.19 12.15 11.95 11.65 11.44 11.07 10.99 10.73 10.59 10.43 10.35 10.30 10.26 10.23 10.20 10.19 10.20 10.19 10.19 10.19 10.18 10.18
15.41 12.83 11.48 11.00 10.94 10.91 10.89 10.86 10.76 10.61 10.38 10.34 10.31 10.20 10.15 10.10 10.08 10.06 10.05 10.05 10.03 10.04 10.04 10.03 10.03 10.04 10.04 10.04 10.02
Fig. 15. Evolution “at the value ” (S1 = K) of the value of a call when N varies from 1 to 1000. Comparative results with respect to λ.
Viable Capture Basin for Studying Differential and Hybrid Games . . .
• 3) if 12 < λ < 1, uncertainty is sub-tychastic and super-CR-stochastic, the value function is first greater than the tychastic reference and it jumps later to a value greater than W . Then it continually increases. • 4) if λ =
1 2,
uncertainty is CR-stochastic, the value of the call is constant
W = 15.2 on interval n ∈ [1, W ]. This value W is lower than W it suddenly explodes as soon as n becomes greater than N (W = 964602). • 5) if λ
0 is a virtual laps of time which indicates that a reset occured. We define the hybrid system (t0 , S10 , p1 , W 0 ) ∈ {−1} × F (S1 , p1 , W ), a.e. t n+1 (tn+1 , S1n+1 , pn+1 , W ) = Φ(tn , S n , W n ), t = T0 1 Since the Impulse Kernel Hyb{1}×F,Φ (K) is empty, the capture domain of the epigraph of u coincides with Hyb{1}×FC ,Φ (K) where FC = F oc C c and FC = 0 on C. −−→ CaptFC ,Φ (Epi(u)) = V iabGρ (K)Hyb{1}×FC ,Φ (K) We consider the discrete model define above and the Algorithm II. Exemple: Let us consider the basic European call for Cox & Rubinstein type of uncertainty and assume that the payment of dividend is done at time T0 = 0.1, the strike is K = 100 and the maturity time is T = 1. We get the following numerical values which correspond to values given by standard methods.
Viable Capture Basin for Studying Differential and Hybrid Games . . . Fig. 22. Evolution of the valuation with respect to the dividend
dividend 0 2 5 10
value of S1 100 100 100 100
Value of the Call 22.07 17.96 16.42 12.98
4.3. Application 2: Hedging Portfolio with Transaction Costs But we can also interpret the evaluation problem of a call in term of impulse control system under uncertainty. Since transaction cost may now appear at any time, the reset set is the whole space C = {(t, S1 , p1 , W ) ∈ [0, T ] × R+ × [0, 1] × R+} and the reset set-valued map becomes (t, S1 , p1 , W ) → Φ(t, S1 , p1 , W ) := {(t, S1 , q, W + |p1 − q|α1 S1 ), q ∈ [0, 1]} We define formally the (continuous) hybrid system (t0 , S10 , p01 , W 0 ) (tn+1 , S1n+1 , pn+1 , W n+1 ) 1
∈ {−1} × F (S1 , p1 , W ), a.e. t ∈ [0, T ] ∈ Φ(tn , S1n , pn1 , W n )
(4.21)
Then considering the discrete dynamical system associated with (4.21) we have tk+1 Sik+1 pk+1 1 W k+1
= = = =
tk − µρ Sik (1 + γρi (Sik , v)), i = 0, 1 pk1 + (1 − µ)ρu W k (1 + γρ0 (S0k )) + pk1 S1k (γρ1 (S1k , v) − γρ0 (S0k )) − (1 − µ)ρ|u|α1 S1k − (1 − µ)β1 pk
pk
where u ∈ [− ρ1 , 1 − ρ1 ] and µ belongs to the discrete set {0, 1}. If µk = 0 a transaction is effective at time n = tk . If µk = 1, there is no transaction at time n = tk . References AUBIN J.-P. (1991) Viability Theory Birkh¨ auser, Boston, Basel, Berlin AUBIN J.-P. (2000) “Viability Kernels and Capture Basins of Sets under Differential Inclusions”, SIAM J. Control AUBIN J.-P. (1999) “Impulse Differential Inclusions and Hybrid Systems: A Viability Approach”, Lecture Notes, University of California at Berkeley AUBIN J.-P., PUJAL D. & SAINT-PIERRE P. (2001) “Dynamic Management of Portfolios with Transaction Costs under Tychastic Uncertainty”, Preprints di Matematica n - 15, Scuola Normale Superiore, Pisa, Italy. BERNHARD P. (2000) “Une approche d´eterministe de l’´evaluation des options, in Optimal Control et Partial Differential Equations”, IOS Press BERNHARD P. (2000) A robust control approach to option pricing, Cambridge University Press BERNHARD P. (2002) Robust control approach to option pricing, including coˆ ut de transaction, Annals of Dynamic Games
Viable Capture Basin for Studying Differential and Hybrid Games . . .
CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1995) “Contribution a ` l’´etude des jeux diff´erentiels quantitatifs et qualitatifs avec contrainte sur l’´etat”, Comptes-Rendus de l’Acad´emie des Sciences, 321, 1543-1548 CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (1999) “Set-valued numerical methods for optimal control and differential games”, In Stochastic and differential games. Theory and numerical methods, Annals of the International Society of Dynamical Games, 177-247 Birkh¨ auser CARDALIAGUET P., QUINCAMPOIX M. & SAINT-PIERRE P. (2001) “Pursuit Differential Games with States Constraints”, SIAM J. on Control and Optimization, Vol 39,No5, pp1615-1632 FRANKOWSKA H. (1993) “Lower semicontinuous solutions of Hamilton-Jacobi-Bellman equation”, SIAM J. on Control and Optimization, PUJAL D. & SAINT-PIERRE P. (2001) “L’algorithme du bassin de capture appliqu´e pour ´evaluer des options europ´eennes, am´ericaines ou exotiques”, preprint PUJAL D. (2000) “Valuation et gestion dynamiques de portefeuilles”, Th`ese de l’Universit´e de Paris-Dauphine SAINT-PIERRE P. (1994) “Approximation of the viability kernel”, Applied Mathematics & Optimisation, 29, 187-209 SAINT-PIERRE P. (2001) “Approximation of Viability Kernels and Capture Basins for Hybrid Systems”. Proceedings of the European Control Conference 2001, Porto
