The Randomized American Option as a Classical Solution to the Penalized Problem

TheBlack-Scholesmodel is the lingua franca [1], the vehicular language, of option pricing. Yet pricing American options in the Black-Scholes remains the topic of a significant literature because every known method requires significant numerical calculations. For surveys onAmerican option pricingwe refer to [2–4]. Among all methods, Howison et al. [5] argue that, with a quadratic convergence [6], the so-called penalty method for the value of the American option in the Black-Scholes model is “the most efficient numerical approximation methods presently available for American option valuation.” It was used among others in [5–14].The penalty method transforms the free boundary problem associated with the price of the American option into a partial differential equation (PDE) of the form


Introduction
The Black-Scholes model is the lingua franca [1], the vehicular language, of option pricing.Yet pricing American options in the Black-Scholes remains the topic of a significant literature because every known method requires significant numerical calculations.For surveys on American option pricing we refer to [2][3][4].Among all methods, Howison et al. [5] argue that, with a quadratic convergence [6], the so-called penalty method for the value of the American option in the Black-Scholes model is "the most efficient numerical approximation methods presently available for American option valuation."It was used among others in [5][6][7][8][9][10][11][12][13][14].The penalty method transforms the free boundary problem associated with the price of the American option into a partial differential equation (PDE) of the form    (, ) = A (, ) −  (, ) +  (ℎ () ,  (, )) ,  (, 0) = ℎ () , where is the infinitesimal generator of a risk neutral geometric Brownian motion  with volatility  and  is the risk-free rate and where the penalty term, (ℎ(), (, )), is a term which typically is zero when ℎ() ≤ (, ) allowing the option to behave like a European option and which drastically pushes the value of the option higher when ℎ() > (, ).Indeed, the "canonical" [5] penalty term is for some large value of  > 0.
In studies [5,6,[9][10][11][12]14] where such canonical penalty method was used for approximating the American option value in the Black-Scholes model, the solution to the associated PDE is seen as a viscosity solution or as a weak solution that is the solution to a variational problem.It is well known that, in general, viscosity and weak solutions do not possess the regularity properties of classical solutions which can actually be differentiated in the classical sense to solve the PDE.
In this paper, we connect the randomized American option [15] to the penalty method, showing that not only does its value  solve the canonical penalty problem (1), but also it is a classical solution to this Cauchy problem and, for a given maturity, A is bounded.
Many of the above cited papers using the canonical penalty method as well as other papers using penalized problems such as [16,17] were actually concerned not by estimating the value of American options but rather by determining the exact speed of convergence of option values under tree schemes approximations of the Black-Scholes model, a difficult [18] and long lasting problem still unsolved when the maturity is not allowed to float.Indeed, randomized American options can be used as a tool to help determine this exact speed of convergence.It is well known that payoff smoothness drastically affects this rate of convergence.We believe that our result may contribute to solving this problem.Yet the submitted paper answers the very natural question of whether or not the canonical penalty problem has a classical solution.
A randomly exercisable American option is an option which, prior to maturity, can be exercised only at some exercisable times following each other independently after an exponentially distributed waiting time of average 1/.Under the label "option with random intervention time," randomly exercisable American options were first introduced in Dupuis and Wang [19] for American perpetuities, then in Guo and Liu [20] for American lookback perpetuities, and then in Leduc [15] for American options.Note that the exercisability randomized American option considered in this paper differs from Carr's maturity randomized option [21] which can be exercised any time up to some random maturity.In contrast, the exercisability randomized option can be exercised only at random times up to a fixed maturity.
We denote by V R   ℎ() the value of a randomly exercisable American option with maturity  and payoff function ℎ, when the spot price  0 of the underlying at time 0 is .The value of this randomized American option V where T  [0, ] is the set of exercisable stopping times in [0, ] and where   is the expectation of  given that  0 = .As shown in [15], V R   ℎ() is the only solution to the following evolution equation: where, for functions  : R → R, the expression E  () denotes the discounted expectation It is also shown in [15] that V where and where U is the semigroup associated with ; that is, for functions  : R → R, Recall that a Lipschitz function ℎ : R + → R + is absolutely continuous and almost surely differentiable.In a slight abuse of notion, we replace the Lipschitz constant  of ℎ by ‖ℎ  ‖ ∞ so that for every The proof of our main result is divided into several steps.In Section 2, we show that V R   ℎ() is continuous.In Section 3, we prove that V R   ℎ() is Lipschitz with respect to .In Section 4, we show that V R   ℎ() is twice continuously differentiable with respect to , and the bounds for (/)V (11).

Lipschitz Property
We show here that V Proof.Note first that because ℎ is Lipschitz, for every 0 ≤ ,  there exists a quantity , with || ≤ ‖ℎ  ‖ ∞ such that Let   and  + be, respectively, the optimal (exercisable) stopping time for the randomized American option with maturity  when  0 =  and when  0 =  + .
Step 1.Note first that when  0 =  + , the stopping time   is suboptimal and therefore for some random variable  satisfying || ≤ ‖ℎ  ‖ ∞ .Hence, since  is risk neutral, Step 2. By definition, for some random variable  satisfying || ≤ ‖ℎ  ‖ ∞ .By suboptimality of  + when  0 = , we conclude that 4.  (2) Solution and for functions  : R → R and integers ℓ ≥ 0 and  > 0, let where A consequence of Lemma 2 is that since ℎ is bounded and Lipschitz, is also a bounded Lipschitz function.It follows from Theorem A.1 in Appendix that   (m  (  )) is infinitely differentiable and that there exist constants  0 such that Because ‖m ⋅ ℎ‖ ∞ ≤ ‖ℎ‖ ∞ , it follows from (5) that The fact that ‖m This in turn yields As (m  (  )) is infinitely differentiable with respect to  > 0, function  2 exp(−)(

Classical Solution
Recall G   ℎ from (8).Equation ( 7) can be rewritten as from which it follows that, for 0 < ,  +  ≤ , Letting  go to zero, we get AV where A is the infinitesimal generator of the GBM.In other words, (43)

Conclusion
In this paper, we showed that the exercisability randomized American option is a classical solution (and therefore the only classical solution) to the canonical penalized problem.This relationship can be extended to a broader class of models than the Black-Scholes model.Indeed, the key property is to obtain uniform bounds for   (  /  )E  ℎ() in terms of powers of 1/√ as in Theorem A.1 in Appendix.
Proof.See [22,Th. 4.1] where the assumption that ℎ is polynomially bounded, continuous, and piecewise continuously differentiable with polynomially bounded derivative can be replaced by our assumptions on ℎ without any change in the argument, as a Lipschitz function is Lebesgue almost everywhere differentiable and absolutely continuous allowing the required integration by parts.Proof.As ℎ is Lipschitz it is Lebesgue almost everywhere differentiable and the proof of [22, Lemma 6.3] can be followed without any change.