1. 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 [2–4].

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(1)∂∂tux,t=Aux,t-rux,t+βhx,ux,t,ux,0=hx,where(2)Afx=rxf′x+12x2σ2f′′xis the infinitesimal generator of a risk neutral geometric Brownian motion ξ with volatility σ and r is the risk-free rate and where the penalty term, βh(x),ux,t, is a term which typically is zero when h(x)≤ux,t allowing the option to behave like a European option and which drastically pushes the value of the option higher when h(x)>ux,t. Indeed, the “canonical” [5] penalty term is(3)βhx,ux,t=maxhx-ux,t,0nfor some large value of n>0.

In studies [5, 6, 9–12, 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 u solve the canonical penalty problem (1), but also it is a classical solution to this Cauchy problem and, for a given maturity, Au 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/n. 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 vtRnhx the value of a randomly exercisable American option with maturity t and payoff function h, when the spot price ξ0 of the underlying at time 0 is x. The value of this randomized American option vtRnhx is given by(4)vtRnhx≝supτ∈Tn0,tExe-rτhξτ,where Tn[0,t] is the set of exercisable stopping times in [0,t] and where Ex is the expectation of ξ given that ξ0=x. As shown in [15], vtRnhx is the only solution to the following evolution equation:(5)vtRnhx=e-ntEthx+∫0tEsmaxh,vt-sRnhxne-nsds,where, for functions ψ:R→R, the expression Etψx denotes the discounted expectation(6)Etψx≝e-rtExψξt.It is also shown in [15] that vtRnhx solves(7)vtRnhx=Uthx+∫0tUsGt-snhxds,where(8)Gt-snhy≝maxhy-vt-sRnhy,0n-vt-sRnhyrand where U is the semigroup associated with ξ; that is, for functions ψ:R→R,(9)Utψx≝Exψξt.

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 C of h by h′∞ so that for every x,y∈R+(10)hx-hy≤h′∞x-y.Finally, we denote by I the identity function: Iz=z for every z.

Theorem 1.
If h is a Lipschitz function and Ih′∞<∞, then vtRnhx is the unique classical solution to the Cauchy problem:(11)∂∂tux,t=Aux,t-rux,t+maxhx-ux,t,0n,ux,0=hx.Furthermore,(12)∂∂xvtRnhx≤h′∞,and for every T, there exists a constant Q depending only on r, σ, T, h∞, and Ih′∞ such that, for every 0<t≤T and 0≤x,(13)x∂∂xvtRnhx≤Q,x2∂2∂x2vtRnhx≤Qt.

The proof of our main result is divided into several steps. In Section 2, we show that vtRnhx is continuous. In Section 3, we prove that vtRnhx is Lipschitz with respect to x. In Section 4, we show that vtRnhx is twice continuously differentiable with respect to x, and the bounds for I∂/dxvtRn and I2∂2/dx2vtRn are established. In Section 5, we show that vtRnhx is a classical solution to (11).

4.
<inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M111"><mml:mrow><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mrow><mml:mfenced separators="|"><mml:mrow><mml:mn>2</mml:mn></mml:mrow></mml:mfenced></mml:mrow></mml:msup></mml:mrow></mml:math></inline-formula> Solution
Define(24)ζsz≝expsσz+r-12σ2s,and for functions f:R→R and integers l≥0 and s>0, let(25)Eslfx≝e-rs∫-∞∞fxζszϕlzdz,where ϕ is the probability density function of a standard normal random variable. Note that if f is bounded then(26)Es0f∞, Es1f∞≤f∞.For any family of functions ft:R→R, 0≤t≤T, set(27)f·∞≝sup0≤t≤Tft∞.

A consequence of Lemma 2 is that since h is bounded and Lipschitz,(28)mthx≝maxhx,vtRnhxis also a bounded Lipschitz function. It follows from Theorem A.1 in Appendix that Exmuξu is infinitely differentiable and that there exist constants α0 such that (29)xe-ru∂∂xExmuξu=α0u -1Eu1muhx≤α0u-1muhx∞.Because m·h∞≤h∞, it follows from (5) that (30)x∂∂xvtRnhx=e-ntx∂∂xEthx+∫0te-nuxe-ru∂∂xExmaxmuξudu=e-ntx∂∂xEthx+∫0te-nsα0u-1Eu1mt-uhxn du.Using Lemma A.2 in Appendix we get(31)I∂∂xEth∞=EtIh′x∞≤Ih′∞.From this we obtain(32)I∂∂xv·Rnh∞≤Ih′∞+h∞Tα0.

For a fixed n, set(33)mu′hz≝∂∂zmaxhz,vuRnhz,which is Lebesgue almost everywhere since muh is Lipschitz. Note that(34)Im·′h∞≤Ih′∞,I∂∂xv·Rnh∞<∞.Again, from Theorem A.1 in Appendix, there exist constants β0 and β1 such that(35)x2e-ru∂2∂x2Exmaxmuξu=β0Eu0Imu′hx+β1u-1Eu1Imu′hx.The fact that Im·′h∞<∞ implies that(36)x2e-ru∂2∂x2Exmaxmuξu≤β0+β1u -1Im·′h∞.This in turn yields (37)x2∂2∂x2vtRnhx=e-ntx2∂2∂x2Ethx+∫0te-nux2e-ru∂2∂x2Exmaxmuξudu=e-ntx2∂2∂x2Ethx+∫0te-nu∑l=01βlu-lEulImt-u′xdu.As Emuxξu is infinitely differentiable with respect to x>0, function x2exp-ru∂2/∂x2Exmaxmuξu is continuous in x>0 and, with (36), dominated convergence gives that x2∂2/∂x2vtRnhx is continuous. From Theorem A.1 in Appendix we obtain that, for some constant K,(38)sup0≤t≤T sup0<xtx2∂2∂x2Ethx≤Ih′∞K,and hence (39)sup0≤t≤T sup0<xtx2∂2∂x2vtRnhx≤Ih′∞K+β0TIm·′h∞+β1TIm·′h∞.