In this section, we are interested in the particular case of the operator Γ derived from the progressive regularization of Lipschitz continuous and pseudomonotone variational inequalities. As mentionned in Lemma 5.3.2, the constants X, Y , Z, T and t appearing in Assumptions 5.3.2 evaluate as
X = γ, Y = γ, Z = γ, T = γ + A, t = γ − A .
Of course, the results of Theorem 5.3.1 apply to the case considered here. In the following, we give a convergence theorem for Algorithm 5.1.3. We failed to prove such a theorem for the sequential version of Algorithm 5.1.4.
Theorem 5.4.1. Assume that problem (1.1) has solutions and that Ψ is Lipschitz continuous with con-stantA and pseudomonotone. In addition, suppose that γ > A and that the auxiliary functions K and L are differentiable, strongly convex with constant c overX and constant d over Xad, respectively and with the G-derivatives Lipschitz continuous with constant C overX and constant D over Xad, respectively. Then, starting from any(x0, y0) ∈ X × X, Algorithm 5.1.3 generates a well-defined sequence{(xk, yk)}. Moreover, there exists a function h depending on a positive parameter α1and onρ, such that:
(i) for allα > 0, if
0 < ρ < min d
γ, 2α(1 + α)(γ − A)d (1 + α)2(A + γ)2− αγ(γ − A)
, (5.18)
thenh(α, ρ) > 0 ; (ii) if
0 < ε < h(α, ρ) , (5.19)
thenkxk− ykk tends toward zero, both sequences {xk} and {yk} are bounded and converge toward a solution of (1.1).
5.5 Conclusions
This paper is an extension of the Moreau-Yosida regularization and the progressive regularization methods to continuous, weakly monotone and pseudomonotone variational inequalities.
Chapter 6
Presentation of our Option Pricing Theory
In this chapter, we give a presentation of our option pricing theory related to [5, 6, 8, 9].
We are concerned with a new theory of European option princing based upon a different, proba-bility free, market model called the interval model. An option is a contract that always involves two parties, one who writes the contract, the “seller”, usually a banking company, and another one who buys the contract: the “buyer” or “holder”. The (european) option contract gives the holder the right to buy or sell an asset, called the underlying asset, at a certain date T , called as the exercise time, for an agreed priceK, called as the exercise price, or the strike. If the holder has the right to buy the asset it is called a call option, if the holder has the right to sell the asset it is called a put option.
The holder of an option is not obliged to exercise this right. If the market price at exercise time is more advantageous, i.e. lower for a call, higher for a put, he will not exercise his right and turn to the market instead. Hence, an option contract provides an insurance against an unfavourable evolution of the market price, without preventing the holder from reeping the benefits of a favourable evolution of the market. We like to give as example of a call option contract, the one between an airline compagny and the bank, where the asset is kerosene; and as example of a put option contract, the one concerning a farmer who cultivates and sells his products.
The buyer of an option contract has to pay a price or “premium” for it up-front. The purpouse of the theory of option pricing, ever since Black and Scholes [66], is to find an objective basis to determine this premium. The fundamental idea, due to Black and Scholes, is for the seller, when selling a contract, to create an initial portfolio made of some of the underlying asset and some riskless bonds (each possibly in a negative quantity through short selling or borrowing) which, when cleverly managed thereafter in a self-financed manner, may be worth at exercise time at least as much as the amount of money necessary to abide by the contract. Such a portfolio is called a hedge, or a hedging portfolio, and the way to manage it a hedging strategy. The optimal premium is then the cost of creating the cheapest possible of such hedging portfolio.
The representation that we use for the option pricing and hedging strategy is based upon a robust control approach with no use of any probability, where the future prices of the underlying asset are not known by trader, but where the set of possible trajectories of the price is limited, the relative rate of change of the asset’s price being in a fixed interval. While it is now well known that, with the classical, stochastic, market model of Black and Scholes, “There is no non-trivial hedging portfolio for option pricing with transaction costs” ([67]) our approach allows us to construct a consistent theory of option pricing and hedging portfolios, including arbitrary proportional transaction costs, and either
continuous or discrete time trading, with the convergence of the latter to the former for vanishing time steps. It also provides us with a fast algorithm to compute a hedging portfolio and a realistic (discrete time) hedging strategy.
These interval models were introduced independently, and almost simultaneously, by P. Bernhard, then at the University of Nice-Sophia Antipolis, Patrick Saint-Pierre, Dominique Pujal and Jean-Pierre Aubin in Paris Dauphine University, and by Jacob Engwerda, Berend Roorda, Hans Schumacher of Tilburg University in the Netherlands. We also cite Vassili Kolokoltsov, in the university of Warwick, who contributed to the interval models theory.
The most classic results of dynamic portfolio management can be found in Merton’s optimal portfolio [68], Black and Scholes’ pricing theory [66], and the Cox, Ross and Rubinstein model [69].
The Merton’s optimal portfolio is based on a stochastic model, in finite or infinite horizon, includ-ing continuous tradinclud-ing, and generalizinclud-ing the Samuelson model [70], which is also called as “geometric diffusion”. The Black and Scholes’ pricing theory is also based on the Samuelson model.
The Cox, Ross and Rubinstein model is a non stochastic approach to the theory of option pricing in a discrete time setting. Their market model is related to our interval market model in that where we allow for an interval of possible future prices of the asset, they allow only the end points of such an interval. Kolokoltsov generalizes this approach of Cox, Ross, and Rubinstein by allowing the same interval model as we use in our theory.
Many authors have proposed a non-stochastic version of the well-known Black and Scholes the-ory. McEneaney [71] may have been the first to replace the stochastic framework with a robust control approach. He derives the so-called stop loss strategy for bounded variation trajectories, and also re-covers the Black and Scholes theory, but this is done at the price of artificially modifying the portfolio model with no other justification than recovering the Itˆo calculus and the Black and Scholes P.D.E.
We will present in the following our theory of European option hedging in the interval market model with proportional transaction costs, hedging with worst case design, and yielding the minimal seller’s price. Our theory includes continuous and discrete trading and the convergence of the latter to the former, as well as it provides us with a fast algorithm.
We will use the following notations, that we also use in chapters 7, 8, 9 and 10:
Constants
• ρ: interest rate of the riskless bonds.
• τ− < 0 and τ+ > 0: fixed known parameters: extreme relative rates of change of the asset’s price.
• h: time step for the discrete trading problem.
• K ∈ N: Kh = T .
• K = {0, 1, . . . , K − 1}.
• τh− = eτ−h− 1, τh+= eτ+h− 1.
• C+ > 0 and C− < 0: fixed known parameters: the rates of the proportional transaction costs for buying resp. selling assets.
• c+ ∈ [0, C+] and c− ∈ [C−, 0]: fixed known parameters: same as C+and C−but for closing costs.
Further, for Z ∈ {τ, C, c} the notation Zεstands for (Zε, ε ∈ {−, +}), and for any a ∈ R, Zεa will mean Zsign(a)a. We also set
τ?= max
ε |τε| , C? = min
ε |Cε| . (6.1)
Variables with time
• t ∈ [0, T ]: time, 0 = date when the option is sold, T = exercise time.
• tk = kh: trading time instants in discrete trading.
• R(t) = eρ(t−T ): the value of the bond in the portfolio and an end-time value coefficient.
• S(t): asset’s market price.
• X(t): number of assets in the portfolio.
• Y (t): number of riskless bonds in the portfolio.
• u(t) = S(t)/R(t): normalized asset’s price.
• v(t) = X(t)S(t)/R(t): normalized worth of the risky part of the portfolio.
• w(t) = v(t) + Y (t): normalized portfolio worth.
• τ (t) = ˙u(t)/u(t) ∈ [τ−, τ+]: relative rate of change of u, i.e. action of the market.
• (uk, vk, wk) = (u(tk), v(tk), w(tk)).
• τk= (uk+1− uk)/uk: one step relative rate of change of uk.
• ξ(t): control of the seller of the option, or “trader”: the rate of buying (selling if negative) of assets.
• ξk: intensity of the Dirac in ξ(.) at time tk, and jump in v at time tk. Sets
• U: set of admissible assets’ price trajectories u(.).
• Uh: set of sequences=sampled elements ofU.
• Ω: set of admissible relative rate of change histories, mesurable functions τ (.) from [0, T ] to [τ−, τ+].
• Ωh: set of admissible sequences {τk}k∈K=sampled elements of Ω.
• Ξ: set of admissible trader’s control, i.e. the set of time distributions defined over [0, T ] which are the sum of a measurable function ξ(·) and a finite number of weighted translated Dirac impulses ξkδ(t − tk).
• Φ: set of admissible trader’s strategies (non-anticipative mappings) Ω → Ξ.
• Φh: set of admissible discrete trading strategies (non-anticipative mappings) Ωh → R.
Functions
• M (S(T )) = M (u(T )): terminal payment, i.e. the cost to the seller of the option.
• N (u, v): terminal payment augmented by closing costs.
• ϕ ∈ Φ: strategy of the trader: ϕ : Ω → Ξ : τ (·) 7→ ξ(t) = ϕ(τ (·))(t) nonanticipative strategy including impulses that cause “jumps” ξk.
• W (t, u, v): Value function of the minimax problem to be solved, giving the premium sought as P (u(0)) = W (0, u(0), 0).
• Wkh(u, v): Value of the discrete time minimax game at time tk= kh.
• Wh(t, u, v): interpolated function of the sequence {Wkh}.
The model we present is Bernhard’s model, its description in details can be found in [6, 72]. In the following, we present our theory of obtaining the optimal price of the option, with a robust control approach to hedging whatever the asset price scenario, this is in the interest of the trader, and moreover by proposing to the buyer of the option the minimal price of the option. This theory leads us to solve a minimax impulse game problem.
6.1 Continuous Trading
All the monetary values are expressed in end-time value computed at a fixed riskless rate ρ, that represents the evolution of the value of the money in our economy. This ρ is the interest rate of the riskless bonds. At time T , we assume that R(T ) = 1, meaning that we normalize the value of the bond R at the exercise time. The bond R(t) follows this differential equation
dR(t)
dt = ρR(t) , R(T ) = 1 .
Then R(t) = exp (ρ(t − T )), that is our end-time value coefficient. Consequently, we will deal, instead of the asset’s market price S(t), with its normalized price, i.e its end-time value at rate ρ, u(t) = S(t)/R(t).
Our interval market model is defined by a setU of possible price trajectories, i.e, the set of all absolutely continuous functions u(·) and two real numbers τ−< 0 and τ+> 0, such that for any two time instants t1and t2, we have
eτ−(t2−t1)≤ u(t2)
u(t1) ≤ eτ+(t2−t1). (6.2) The equivalent description in the continuous trading theory is the following characterization
˙
u = τ u , τ ∈ [τ−, τ+] . (6.3)
where τ (·) is a measurable function from [0, T ] into [τ−, τ+], that plays the role of the control of the market. We denote by Ω the set of such functions.
We denote by X(t) the number of underlying asset in the portfolio, and hence the exposed part of the portfolio worth is v(t) = X(t)u(t). We denote by Y (t) the number of riskless bonds in the
portfolio. We also denote by ξ(t) the buying rate (selling rate, in this case ξ(t) is negative), so we have in continuous time
˙v = τ v + ξ . (6.4)
In order to avoid mathematical ill-posedness, we expressly allow “infinite” buying or selling rate in the form of instantaneous block buy or sale of a finite amount of asset, at time instants chosen by the trader together with the amount. Thus the control of the trader also involves the choice of finitely many time instants tkand trading amounts ξk. The model is then augmented with
v(t+k) = v(t−k) + ξk, (6.5)
meaning that v(·) has a jump discontinuity of size ξk at time tk. Equations (6.4) and (6.5) can be equivalently written as the generalized differential equation:
˙v = τ v + ξ +X
tk≤t
ξkδ(t − tk) ,
where δ is the standard Dirac function.
Let then Ξ be the set of real time functions (or rather distributions) defined over [0, T ] which are the sum of a measurable function ξ(·) and a finite number of weighted translated Dirac impulses ξkδ(t − tk).
At initial time t = 0, the trader needs to hold a hedging portfolio which contains X0 shares of the risky asset, at the market price S(0), for a normalized end-time value of v0 = X0u(0) = X0S(0)/R(0), and Y0(typically negative for a call) normalized riskless bonds. The total value of his initial portfolio is thus, in normalized form, or end-time value, w0 = v0 + Y0. At time t = 0, the trader has no portfolio, hence v(0) = 0. The hedging strategy begins with a jump from v(0) = 0 to v(0+) = v0, for a transaction cost C(v0) to be added to w0 to determine the premium to be charged to the buyer. Afterward, the quantities v(t) and w(t) will both vary over time, since, on the one hand, u(t) varies with the market, and, on the other hand, the trader will manage this portfolio in a best way that we will present later, so he can decide to buy or sell parts of the risky asset. We assume that these transactions by the trader have costs, that we denote by C(ξ) in the presentation below. In [5, 6, 8, 9], these transactions costs are assumed to be proportional to the transaction amount.
The trader’s portfolio will always be assumed self financed, that is, the buy, respectively the sale of the riskless bonds must exactly pay for the sale, respectively the buy of the risky assets, plus the transaction costs. So,
Y = −ξ − C(ξ) ,˙ and at the jump instants,
Y (t+k) = Y (t−k) − ξk− C(ξk) . The worth w(t) = v(t) + Y (t) of the portfolio satisfies then
˙
w = τ v − C(ξ) , (6.6)
and at jump instants,
w(t+k) = w(t−k) − C(ξk) (6.7)
This is equivalent to
w(t) = w(0) + Z t
0
(τ (s)v(s) − C(ξ(s))ds − X
k|tk≤t
C(ξk) . (6.8)