Perpetual American options with asset-dependent discounting
PPERPETUAL AMERICAN OPTIONS WITH ASSET-DEPENDENT DISCOUNTING
JONAS AL-HADAD AND ZBIGNIEW PALMOWSKI
Abstract.
In this paper we consider the following optimal stopping problem V ω A ( s ) = sup τ ∈T E s [ e − (cid:82) τ ω ( S w ) dw g ( S τ )] , where the process S t is a jump-diffusion process, T is a family of stopping times and g and ω are fixedpayoff function and discounting function, respectively. In a financial market context, if g ( s ) = ( K − s ) + or g ( s ) = ( s − K ) + and E is the expectation taken with respect to a martingale measure, V ω A ( s ) describes theprice of a perpetual American option with a discount rate depending on the value of the asset process S t .If ω is a constant, the above problem produces the standard case of pricing perpetual American options.In the first part of this paper we find sufficient conditions for the convexity of the value function V ω A ( s ) .This allows us to determine the stopping region as a certain interval and hence we are able to identify theform of V ω A ( s ) . We also prove a put-call symmetry for American options with asset-dependent discounting.In the case when S t is a geometric Lévy process we give exact expressions using the so-called omega scalefunctions introduced in [49]. We prove that the analysed value function satisfies HJB and we give sufficientconditions for the smooth fit property as well. Finally, we analyse few cases in detail performing extensivenumerical analysis. Keywords.
American option (cid:63)
Lévy process (cid:63) diffusion (cid:63)
Black-Scholes market (cid:63) optimal stopping problem (cid:63) convexity
Contents
1. Introduction 32. Main results 62.1. Jump-diffusion process 62.2. Assumptions 72.3. Convexity of the value function 72.4. American put option and the optimal exercise time 82.5. Spectrally negative geometric Lévy process 82.6. HJB, smooth and continuous fit properties 102.7. Put-call symmetry 112.8. Black-Scholes model 122.9. Exponential crashes market 123. Numerical analysis 14
Date : July 21, 2020.2010
Mathematics Subject Classification.
Primary: 60G40; Secondary: 60J60; 91B28.Jonas Al-Hadad and Zbigniew Palmowski have been partially supported by the National Science Centre under the grant2016/23/B/HS4/00566. a r X i v : . [ q -f i n . M F ] J u l J. Al-Hadad — Z. Palmowski erpetual American options with asset-dependent discounting Introduction
In this paper the uncertainty associated with the stock price S t is described by a jump-diffusion processdefined on a complete filtered risk-neutral probability space (Ω , F , {F t } t ≥ , P ) , where F t is a natural filtra-tion of S t satisfying the usual conditions and P is a risk-neutral measure under which the discounted (withrespect to the risk-free interest rate) asset price process S t is a local martingale. We point out that, asnoted in [24], introducing jumps into the model, implies lost of completeness of the market which resultsin the lack of a uniqueness of equivalent martingale measure. The main our goal is the analysis of thefollowing optimal stopping problem(1) V ω A ( s ) := sup τ ∈T E s (cid:104) e − (cid:82) τ ω ( S w ) dw g ( S τ ) (cid:105) , where T is a family of F t -stopping times and g and ω are fixed payoff function and discounting function,respectively. Above E s denotes the expectation with respect to P when S = s . We assume that the function g is convex. We allow in this paper for ω to have negative values. In the case when g ( s ) = ( K − s ) + or g ( s ) = ( s − K ) + and P is a martingale measure, this function can be interpreted as the value function ofthe perpetual American option with functional discounting ω and payoff function g . In the case of generaltheory of stochastic processes multiplying by discounting factor e − (cid:82) τ ω ( S w ) dw corresponds to killing of agenerator of S t by potential ω .This problem extends the classical theory of option pricing, where the deterministic discount rate is con-sidered, that is if ω ( s ) = r , then we obtain the standard form V A ( s ) := sup τ ∈T E s (cid:2) e − rτ g ( S τ ) (cid:3) of the perpetual American option’s value function with constant discount rate r .The main objective of this paper is to find a closed expression of (1) and identify the optimal stoppingrule τ ∗ for which supremum is attained. To do this we start from proving in Theorem 2 an inheritanceof convexity property from the payoff function to the value function. This corresponds to preserving theconvexity by the solution of a certain obstacle problem.Using this observation and classical optimal stopping theory presented e.g. in [56] one can identify theoptimal stopping region as an interval [ l ∗ , u ∗ ] , that is, τ ∗ = inf { t ≥ S t ∈ [ l ∗ , u ∗ ] } . Hence, in general, onecan obtain in this case a double continuation region .Later we focus on the case when S t is a geometric spectrally negative Lévy process. In this case, using thefluctuation theory of Lévy processes, we identify value function (1) in terms of the omega scale functionsintroduced in [49].For optimal stopping problem (1) we formalise the classical approach here as well. In particular, in Theorem6 we prove that the value function V ω A ( s ) is the unique solution of a certain Hamiltonian-Jacobi-Bellman(HJB) system. Moreover, in the case of geometric Lévy process of the asset price S t , we prove that theregularity of for the half-lines ( −∞ , and (1 , + ∞ ) gives the smooth fit property at the ends of thestopping region. In Theorem 8 we show the put-call symmetry as well.These theoretical results allow us to find the price of the perpetual American option with asset-dependentdiscounting for some particular cases. We take for example a put option, that is g ( s ) = ( K − s ) + , and ageometric Brownian motion for the asset price S t . We model S t also by the geometric Lévy process withexponentially distributed downward jumps. We analyse various discounting functions ω . In Section 3 weprovide extensive numerical analysis.The discount rate changing in time or a random discount rate are widely used in pricing derivatives infinancial markets. They have proved to be valuable and flexible tools to identify the value of various options. J. Al-Hadad — Z. Palmowski
Usually, either the interest rate is independent from the asset price or this dependence is introduced viataking correlation between gaussian components of these two processes. Our aim is slightly different. Wewant to understand an extreme case when we have strong, functional dependence between the interest rateand the asset price. For example, for the American put option, if the asset price is in ’higher’ region onecan expect that the interest rate will be higher as well. The opposite effect one expect for ’smaller’ rangeof asset’s prices.One can look at optimisation problem (1) from a wider perspective though. The killing by potential ω has been known widely in physics and other applied sciences. Then (1) can be seen as a certain functionaldescribing gain or energy and the goal is to optimise it by choosing some random time horizon. We focushere on financial applications only.Our research methodology is based on combining the theory of partial differential equations with thefluctuation theory of Lévy processes.To prove the convexity we start from proving in Theorem 15 the convexity of(2) V ω E ( s, t ) := E s,t (cid:104) e − (cid:82) Tt ω ( S w ) dw g ( S T ) (cid:105) for fixed time horizon T , where E s,t is the expectation E with respect to P when S t = s . In the proof wefollow the idea given by Ekström and Tysk in [34]. Namely, the value function V ω E ( s, t ) given in (2) canbe presented as the unique viscosity solution of a certain Cauchy problem for some second-order operatorrelated to the generator of the process S t . In fact, applying similar arguments like in [58, Proposition 5.3]and [34, Lemma 3.1], one can show that, under some additional assumptions, this solution can be treatedas the classical one. Then we can formulate the sufficient locally convexity preserving conditions for theinfinitesimal preservation of convexity at some point. This characterisation is given in terms of a differentialinequality on the coefficients of the considered operator. It also allows to prove the convexity of V ω E ( s, t ) .Then, in Theorem 2 and Lemma 19 we apply the dynamic programming principle (see [33]) in order togeneralise the convexity property of V ω E ( s, t ) to the value function V ω A ( s ) .Later we focus on the American put option, hence when g ( s ) = ( K − s ) + for some strike price K > .Using the convexity property mentioned above we can conclude that the optimal stopping rule is definedas the first entrance of the process S t to the interval [ l, u ] , that is,(3) τ l,u := inf { t ≥ S t ∈ [ l, u ] } . In the next step, one has to identify(4) v ω A Put ( s, l, u ) := E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ) + (cid:105) and take maximum over levels l and u to identify the optimal stopping rule τ ∗ and to find the price V ω A Put ( s ) of the American put option when S = s . This is done for the geometric spectrally negative Lévy process S t = e X t where X t is a spectrally negative Lévy process starting at X = log S = log s . We recall thatspectrally negative Lévy processes do not have positive jumps. Hence, in particular, our analysis could beused for Black-Scholes market where X t is a Brownian motion with a drift. To execute this plan we express v ω A Put ( s, l, u ) in terms of the laws of the first passage times and then we use the fluctuation theory developedin [49]. In the whole analysis the use of the change of measure technique developed in [54] is crucial as well.Optimal levels l ∗ and u ∗ of the stopping region [ l ∗ , u ∗ ] and the price V ω A Put ( s ) of the American put optioncould be found by application of the appropriate HJB equation. We prove this HJB equation and thesmooth fit condition relying on the classical approach of [47] and [56].Finally, to find the price of the American call option we prove the put-call symmetry in our set-up. Theproof is based on the exponential change of measure introduced in [54]. erpetual American options with asset-dependent discounting We analyse in detail the Black-Scholes model and the case when a logarithm of the asset price is a linear driftminus compound Poisson process with exponentially distributed jumps and various discounting functions ω . The first example shows behaviour of the American prices in a gaussian and continuous market while thelatter is to model the market including downward shocks in the assets’ behaviour. In this paper we presentsome numerical analysis for these two cases. In particular, we show how two different approaches, namelysolving HJB equation, finding function v ω A Put ( s, l, u ) and maximising it over l and u , can be executed indaily practice.Our paper seems to be the first one analysing the optimal problem of the form (1) in this generality forjump-diffusion processes. For the classical diffusion process Lamberton in [48] proved that the value functionin (1) is continuous and can be characterised as the unique solution of a variational inequality in the senseof distributions. Another crucial paper for our considerations is [11] which introduced discounting via apositive continuous additive functional of the process S t and used the technique of Bensoussan and Lions[12] to characterise the value function. Note that t → (cid:82) t ω ( S w ) dw is indeed additive functional. Anotherinteresting paper of Rodosthenous and Zhang [59] who studied the optimal stopping of an American calloption in a random time-horizon under a geometric spectrally negative Lévy model. The random time-horizon is modeled by Omega default clock which is in their case the first time when the occupation timeof the asset price below a fixed level y exceeds an independent exponential random variable with mean /(cid:37) . This corresponds to the special case of our discounting with ω ( s ) = r + (cid:37) { s ≤ y } , where r is a risk-freeinterest rate.The convexity of the value function and convexity preserving property, which is a key ingredient of ouranalysis, have been studied quite extensively, see e.g. [13, 14, 20, 32, 36, 37, 40, 41] for diffusion models,and [35, 39] for one-dimensional jump-diffusion models.We model dynamics of the asset price in a financial market by the jump-diffusion process. The reasonto take into account more general class of stochastic processes of asset prices than in the seminal Black-Scholes market is the empirical observation that the log-prices of stocks have a heavier left tail than thenormal distribution, on which the seminal Black-Scholes model is founded. The introduction of jumps inthe financial market dates back to [51], who added a compound Poisson process to the standard Brownianmotion to better describe dynamics of the logarithm of stocks. Since then, there have been many papersand books working in this set-up, see e.g. [24, 61] and references therein. In particular, [24, Table 1.1,p. 29] gives many other reasons to consider this type of market. Apart from the classical Black-Scholesmarket one can consider the normal inverse Gaussian model of [53], the hyperbolic model of [30], thevariance gamma model of [50], the CGMY model of [18], and the tempered stable process analysed in[15, 43]. American options in the jump-diffusion markets have been studied in many papers as well; seee.g. [1, 2, 4, 9, 15, 19, 21, 42, 52].Identifying the solution of the optimal stopping problem by solving the corresponding HJB equation (asit is done in this paper as well) has been widely used; see [44, 56] for details. In the context of Americanoptions with constant discounting both methods of variational inequalities and viscosity solutions of theboundary value problems in the spirit of Bensoussan and Lions [12] are also well-known; see e.g. [47, 57, 58].To determine the unknown boundary of stopping region usually the smooth fit condition is used; see e.g.[46, 47] for the geometric Lévy process of asset prices. As Lamberton and Mikou [47] and Kyprianou andSurya [46] showed the continuous fit is always satisfied but not necessary the smooth fit property. Thereforewe focus on identifying the sufficient conditions for the smooth fit in our model which are generalisationsof the classical ones. What we want to underline here is that using our approach (proving convexity andmaximising over ends l and u of the stopping interval [ l, u ] ) one can avoid these calculations. Apart of this,the interval form of the stopping region (hence producing double-sided continuation region) is much more J. Al-Hadad — Z. Palmowski rare. It comes from the fact that when at time t = 0 the discount rate is negative then it is worth to waitsince discounting might increase the value of payoff. This phenomenon has been already observed for fixednegative discounting (see [6, 7, 8, 26, 63]) or in the case of American capped options with positive interestrate (see [16, 27]).In this paper we also prove that in this general setting of asset-dependent discounting, one can expressthe price of the call option in terms of the price of the put option. It is called the put-call symmetry (orput-call parity). Our finding supplements [31, 38] who extend to the Lévy market the findings by [17]. Ananalogous result for the negative discount rate case was obtained in [6, 7, 8, 26]. A comprehensive reviewof the put-call duality for American options is given in [28]. We also refer to [29, Section 7] and otherreferences therein for a general survey on the American options in the jump-diffusion model.The paper is organised as follows. In Section 2 we introduce basic notations and assumptions that we usethroughout the paper and we give the main results of this paper. In Section 3, we perform the numericalanalysis for the case of put option and Black-Scholes market and the market with prices being modeled bygeometric Lévy process with downward exponential jumps. Section 4 contains proofs of all main theorems.We put into Appendix proofs of auxiliary lemmas. The last section includes our concluding remarks.2. Main results
Jump-diffusion process.
In this paper we assume a jump-diffusion financial market defined formallyas follows. On the basic probability space we define a couple ( B t , v ) adapted to the filtration F t , where B t is a standard Brownian motion and v = v ( dt, dz ) is an independent of B t homogeneous Poisson randommeasure on R +0 × R for R +0 = [0 , + ∞ ) . Then the stock price process S t solves the following stochasticdifferential equation(5) dS t = µ ( S t − , t ) dt + σ ( S t − , t ) dB t + (cid:90) R γ ( S t − , t, z )˜ v ( dt, dz ) , where • ˜ v ( dt, dz ) = ( v − q )( dt, dz ) is a compensated jump martingale random measure of v , • v is a homogenous Poisson random measure defined on R +0 × R with intensity measure q ( dt, dz ) = dt m ( dz ) . If additionally, the jump-diffusion process has finite activity of jumps, i.e. when λ := (cid:90) R m ( dz ) < ∞ , then N t = v ([0 , t ] × R ) is a Poisson process and m can be represented as m ( dz ) = λ P (cid:0) e Y i − ∈ dz (cid:1) , where { Y i } i ∈ N are i.i.d. random variables independent of N t with distribution µ Y . Note that B t and N t are independent of each other as well. When additionally µ ( s, t ) = µs , σ ( s, t ) = σs and γ ( s, t, z ) = sz , thenthe asset price process S t is the geometric Lévy process, that is,(6) S t = e X t , where X t is a Lévy process starting at x = log s with a triple ( ζ, σ, Π) for(7) ζ := µ − σ , Π( dx ) := λµ Y ( dx ) . This observation follows straightforward from Itô’s rule. erpetual American options with asset-dependent discounting Assumptions.
Before we present the main results of this paper, we state now the assumptions onthe model parameters used later. We denote R + := (0 , + ∞ ) . Assumptions (A) (A1) The drift parameter µ : R + × R +0 → R and the diffusion parameter σ : R + × R +0 → R are continuousfunctions, while the jump size γ : R + × R +0 × R → R is measurable and for each fixed z ∈ R , thefunction ( s, t ) → γ ( s, t, z ) is continuous.(A2) There exists a constant C > such that µ ( s, t ) + σ ( s, t ) + γ ( s, t, z ) ≤ Cs for all ( s, t, z ) ∈ R + × R +0 × R .(A3) There exists a constant C > such that | µ ( s , t ) − µ ( s , t ) | + | σ ( s , t ) − σ ( s , t ) | + | γ ( s , t, z ) − γ ( s , t, z ) | ≤ C | s − s | for all ( s, t, z ) ∈ R + × R +0 × R .(A4) There exists a constant C > − such that γ ( s, t, z ) > Cs for all ( s, t, z ) ∈ R + × R +0 × R .(A5) g ( s ) ∈ C pol ( R + ) , where C pol ( R + ) denotes the set of functions of at most polynomial growth.(A6) ω ( s ) is bounded from below. Assumptions (A1), (A2), (A3) guarantee that there exists a unique solution to (5). Moreover, (A2) and(A4) imply that P ( S t ≤ for some t ∈ R +0 ) = 0 which is a natural assumption since the process S t describes the stock price dynamics and its value has tobe positive. Additionally, assumptions (A5) and (A6) give that V ω A ( s ) is finite. Remark 1.
Note that assumptions (A1)–(A4) are all satisfied for the geometric Lévy process.2.3.
Convexity of the value function.
Our first main result concerns the convexity of the value function V ω A ( s ) . Theorem 2.
Let Assumptions (A) hold. Assume that the payoff function g is convex, ω is concave, thestock price process S t follows (5) , and the following inequalities are satisfied (8) ∂ γ ( s, t, z ) ∂s ≥ , (9) (cid:18) ∂ µ ( s, t ) ∂s − dω ( s ) ds (cid:19) ∂V ω E ( s, t ) ∂s ≥ , where V ω E ( s, t ) is defined in (2) . Then the value function V ω A ( s ) is convex as a function of s . The proof of the above theorem is given in Section 4.
Remark 3.
We give now sufficient conditions in terms of model parameters for (9) to be satisfied. If S t isthe geometric Lévy process (hence µ ( s, t ) = µs , σ ( s, t ) = σs and γ ( s, t, z ) = sz ) then (8) is satisfied. Letadditionally g ( s ) = ( K − s ) + . Then our optimal stopping problem is equivalent to pricing American putoption with functional discounting. If ω is increasing function then the function s → V ω E ( s, t ) is decreasing. J. Al-Hadad — Z. Palmowski
Hence in this case condition (9) is satisfied as well. Concluding, if ω is concave and increasing, then thevalue function of American put option in geometric Lévy market is convex as a function of the initial assetprice.2.4. American put option and the optimal exercise time.
Assume now the particular case of (1)with the payoff function g ( s ) = ( K − s ) + , that is, the value function V ω A ( s ) gives the price of American put option. The value function for this specialchoice of payoff function is denoted by(10) V ω A Put ( s ) := sup τ ∈T E s (cid:104) e − (cid:82) τ ω ( S w ) dw ( K − S τ ) (cid:105) . Note that above we used the fact that the option will not be realised when it equals to zero, hence the plusin the payoff function could be skipped.From [56, Thm. 2.7, p. 40] it follows that the optimal stopping rule is of the form τ ∗ = inf { t ≥ V ω A Put ( S t ) = ( K − S t ) } . From Theorem 2 we know that V ω A Put ( s ) is convex. Moreover, from the definition of the value functionit follows that V ω A Put ( s ) ≥ ( K − s ) . Having both these facts in mind together with linearity of the payofffunction, it follows that V ω A Put ( s ) and g can cross each other in at most two points. This observationproduces straightforward the following main result. We recall that in (3) and (4) we introduced theentrance time τ l,u = inf { t ≥ S t ∈ [ l, u ] } into interval [ l, u ] and the corresponding value function v ω A Put ( s, l, u ) = E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ) (cid:105) , respectively. Theorem 4.
The value function defined in (10) equals to V ω A Put ( s ) = v ω A Put ( s, l ∗ , u ∗ ) , where v ω A Put ( s, l ∗ , u ∗ ) := sup ≤ l ≤ u ≤ K v ω A Put ( s, l, u ) . The optimal stopping rule is τ l ∗ ,u ∗ , where l ∗ , u ∗ realise the supremum above. Theorem 4 indicates that the optimal stopping rule in our problem is the first time when the process S t enters the interval [ l ∗ , u ∗ ] for some l ∗ ≤ u ∗ . In the case when l ∗ = u ∗ the interval becomes a point which ispossible as well. In some cases the above observation allows to identify the value function in a much moretransparent way. Finally, note that if the discounting function ω is positive, then it is never optimal to waitto exercise the option for small asset prices, that is, always l ∗ = 0 in this case and the stopping region isone-sided.2.5. Spectrally negative geometric Lévy process.
We can express the value function V ω A Put ( s ) explic-itly for the spectrally negative geometric Lévy process defined in (6), that is when S t = e X t , where X t is a spectrally negative Lévy process with X = x = log s and hence S = s . This means that X t does not have positive jumps which corresponds to the inclusion of the support of Lévy measure m on the negative half-line. This is very common assumption which is justified by some financial crashes;see e.g. [2, 5, 19]. One can easily observe that the dual case of spectrally positive Lévy process X t canbe also handled in a similar way. We decided to skip this analysis and focus only on a more natural erpetual American options with asset-dependent discounting spectrally negative case. We express the value function in terms of some special functions, called omegascale functions; see [49] for details.To introduce these functions let us define first the Laplace exponent via ψ ( θ ) := 1 t log E [ e θX t | X = 0] , which is finite al least for θ ≥ due to downward jumps. This function is strictly convex, differentiable,equals to zero at zero and tends to infinity at infinity. Hence there exists its right inverse Φ( q ) for q ≥ .The key functions for the fluctuation theory are the scale functions; see [23]. The first scale function W ( q ) ( x ) is the unique right continuous function disappearing on the negative half-line whose Laplace transform is(11) (cid:90) ∞ e − θx W ( q ) ( x ) dx = 1 ψ ( θ ) − q for θ > Φ( q ) .For any measurable function ξ we define the ξ -scale functions {W ( ξ ) ( x ) , x ∈ R } , {Z ( ξ ) ( x ) , x ∈ R } and {H ( ξ ) ( x ) , x ∈ R } as the unique solutions to the following renewal-type equations W ( ξ ) ( x ) = W ( x ) + (cid:90) x W ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy, (12) Z ( ξ ) ( x ) = 1 + (cid:90) x W ( x − y ) ξ ( y ) Z ( ξ ) ( y ) dy, (13) H ( ξ ) ( x ) = e Φ( c ) x + (cid:90) x W ( c ) ( x − z )( ξ ( z ) − c ) H ( ξ ) ( z ) dz, (14)where W ( x ) = W (0) ( x ) is a classical zero scale function and in equation (14) it is additionally assumed that ξ ( x ) = c for all x ≤ and some constant c ∈ R . We also need function {W ( ξ ) ( x, z ) , ( x, z ) ∈ R } solving thefollowing equation W ( ξ ) ( x, z ) = W ( x − z ) + (cid:90) xz W ( x − y ) ξ ( y ) W ( ξ ) ( y, z ) dy. (15)We introduce the following S t counterparts of the scale functions (12), (13), (14) and (15) W ( ξ ) ( s ) := W ( ξ ◦ exp) (log s ) , (16) Z ( ξ ) ( s ) := Z ( ξ ◦ exp) (log s ) , (17) H ( ξ ) ( s ) := H ( ξ ◦ exp) (log s ) , (18) W ( ξ ) ( s, z ) := W ( ξ ◦ exp) (log s, z ) , (19)where ξ ◦ exp( x ) := ξ ( e x ) .For α for which the Laplace exponent is well-defined we can define a new probability measure P ( α ) via(20) d P ( α ) d P (cid:12)(cid:12)(cid:12)(cid:12) F t = e αX t − ψ ( α ) t . By [54], under P ( α ) , the process X t is again spectrally negative Lévy process with the new Laplace exponent(21) ψ ( α ) ( θ ) := ψ ( θ + α ) − ψ ( α ) . For this new probability measure P ( α ) we can define ξ -scale functions which are denoted by adding subscript α to the regular counterparts, hence we have W ( ξ ) α ( s ) , Z ( ξ ) α ( s ) , H ( ξ ) α ( s ) and W ( ξ ) α ( s, z ) .Let ω u ( s ) := ω ( su ) and ω αu ( s ) := ω u ( s ) − ψ ( α ) . J. Al-Hadad — Z. Palmowski
The main result is given in terms of the resolvent density at z of X t starting at log s − log u killed by thepotential ω u and on exiting from positive half-line given by(22) r ( s, u, z ) := W ( ω u ) (log s − log u ) c W ( ωu ) / W ( ωu ) ( z ) − W ( ω u ) (log s − log u, z ) , where c W ( ωu ) / W ( ωu ) ( z ) := lim y →∞ W ( ω u ) (log y, z ) W ( ω u ) (log y ) . Theorem 5.
Assume that the stock price process S t is described by (6) with X t being the spectrally negativeLévy process. Let ω be a measurable function such that (23) ω ( s ) = c for all s ∈ (0 , and some constant c ∈ R .Then v ω A Put ( s, l, u ) = H ( ω ) ( s ) H ( ω ) ( l ) ( K − l ) { s
HJB, smooth and continuous fit properties.
The classical approach via HJB system is possiblein our set-up as well. More precisely, as before in (6) we have S t = e X t for the Lévy process X t with the triple ( ζ, σ, Π) . We start from the observation that using [60, Thm. 31.5,Chap. 6] and Itô’s formula one can conclude that the process S t is a Markov process with an infinitesimalgenerator A f ( s ) = A C f ( s ) + A J f ( s ) , where A C is the linear second-order differential operator of the form A C f ( s ) = σ s f (cid:48)(cid:48) ( s ) + (cid:18) ζ + σ (cid:19) sf (cid:48) ( s ) and A J is the integral operator given by A J f ( s ) = (cid:90) ( −∞ , (cid:0) f ( se z ) − f ( s ) − s | z | f (cid:48) ( s ) {| z |≤ } (cid:1) Π( dz ) . The domain D ( A ) of this generator consists of the functions belonging to C ( R + ) if σ > and C ( R + ) if σ = 0 . In this paper we prove that V ω A ( s ) satisfies the following HJB equation with appropriate smooth fitconditions. erpetual American options with asset-dependent discounting Theorem 6.
Assume that V ω A ( s ) ∈ D ( A ) and that g ( s ) ∈ C ( R + ) . Then V ω A ( s ) solves uniquely thefollowing equations (24) A V ω A ( s ) − ω ( s ) V ω A ( s ) = 0 , s / ∈ [ l, u ] ,V ω A ( s ) = g ( s ) , s ∈ [ l, u ] . Moreover, if is regular for ( −∞ , and for the process S t then there is a smooth fit at the right end ofthe stopping region ( V ω A ) (cid:48) ( u ) = g (cid:48) ( u ) Similarly, if is regular for (1 , + ∞ ) and for the process S t then there is a smooth fit at the left end of thestopping region ( V ω A ) (cid:48) ( l ) = g (cid:48) ( l ) . Remark 7.
Let us consider the American put option. Then from Theorems 4 and 5, we can concludethat smoothness of the value function V ω A Put ( s ) corresponds to the smoothness of the scale functions for ω , ω u and ω αu . From the definition of these scale functions given in (12), (13) and (14) it follows that thesmoothness of the latter functions is equivalent to the smoothness of the first scale function observed undermeasures P and P ( α ) . By [45, Lem. 8.4] the smoothness of the first scale function does not change underthe exponential change of measure (20). Thus from [23, Lem. 2.4, Thms 3.10 and 3.11] if follows that • if σ > then V ω A Put ( s ) ∈ C ( R + ) ; • if σ = 0 and the jump measure Π is absolutely continuous or (cid:82) − | x | Π( dx ) = + ∞ , then V ω A Put ( s ) ∈ C ( R + ) .Moreover, by [2, Prop. 7], is regular for both ( −∞ , and (1 , + ∞ ) if σ > . Hence HJB system (24)with the smooth fit property could be used without any additional assumptions as long as σ > . If onehas single continuation region [ u ∗ , + ∞ ) and σ = 0 then by [2, Prop. 7] to get the smooth fit condition at u ∗ it is sufficient to assume that the drift ζ of the process X t is strictly negative.2.7. Put-call symmetry.
The put-call parity allows to calculate the American call option price havingthe put one. We formulate this relation again for S t being a general geometric Lévy process defined in (6),that is, S t = e X t for X t being a general Lévy process having triple ( ζ, σ, Π) for ζ and Π defined in (7) and starting position X = log S = s . Apart from function v ω A Put ( s, K, ζ, σ, Π , l, u ) := E s [ e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ) + ] defined in (4) we denote v ω A Call ( s, K, ζ, σ, Π , l, u ) := E s [ e − (cid:82) τl,u ω ( S w ) dw ( S τ l,u − K ) + ] . Theorem 8.
Assume that ψ (1) = log E e X = log E S is finite. Let l ≤ u ≤ K . Then we have v ω A Call ( s, K, ζ, σ, Π , l, u ) = v ϑ (1) A Put (cid:16)
K, s, − ζ, σ, ˆΠ , su K, sl K (cid:17) , where ˆΠ( dx ) := e − x Π( − dx ) , (25) ϑ (1) ( · ) := ω (cid:18) · sK (cid:19) − ψ (1) . J. Al-Hadad — Z. Palmowski
Remark 9.
Note that price of the American call option is expressed in terms of the American put optioncalculated for the Lévy process ˆ X t being dual to X t process observed under the measure P (1) . In particular,the jumps of the process ˆ X t have opposite direction to the jumps of the process X t for which the call optionis priced.2.8. Black-Scholes model.
We can give more detailed analysis in the case of classical Black–Scholesmodel in which the stock price process S t satisfies the following stochastic differential equation dS t = µS t dt + σS t dB t with a constant µ ∈ R and σ > . That is,(26) S t = se ( µ − σ ) t + σB t is a geometric Brownian motion. Theorem 10.
Let (A6) hold and assume that the stock price process S t follows (26) . Then the function v ω A Put ( s, l, u ) defined in (4) is given by v ω A Put ( s, l, u ) = h ( s ) h ( l ) ( K − l ) { s
The optimal boundaries l ∗ and u ∗ can be found from the smooth fit property given inTheorem 6.2.9. Exponential crashes market.
We can construct more explicit equation for the value function forthe case of classical Black–Scholes model with additional downward exponential jumps, that is, as in (6), S t = e X t for(29) X t = log s + (cid:18) µ − σ (cid:19) t + σB t − N t (cid:88) i =1 Y i , where N t is the Poisson process with intensity λ > independent of Brownian motion B t and { Y i } i ∈ N arei.i.d. random variables independent of B t and N t having exponential distribution with mean /ϕ > .For this model the price of American put option is easier to determine. Theorem 12.
Assume that ω is nonnegative, concave and increasing. For geometric Lévy model (6) with X t given in (29) we have l ∗ = 0 . Furthermore,(i) V ω A Put ( s ) = sup u> (cid:26) (cid:18) K − uϕϕ + 1 (cid:19) (cid:16) Z ( ω u ) (cid:16) su (cid:17) − c Z ( ωu ) / W ( ωu ) W ( ω u ) (cid:16) su (cid:17)(cid:17) (cid:27) (30) erpetual American options with asset-dependent discounting if σ = 0 , where W ( ω u ) and Z ( ω u ) are given in (16) and (17) , respectively and (31) c Z ( ωu ) / W ( ωu ) := lim z →∞ Z ( ω u ) ( z ) W ( ω u ) ( z ) . (ii) (32) V ω A Put ( s ) = sup u> (cid:26) (cid:18) K − uϕϕ + 1 (cid:19) (cid:16) Z ( ω u ) (cid:16) su (cid:17) − c Z ( ωu ) / W ( ωu ) W ( ω u ) (cid:16) su (cid:17)(cid:17) + ( K − u ) (cid:16) lim α →∞ (cid:16) Z ( ω αu ) α (cid:16) su (cid:17) − c Z ( ωαu ) α / W ( ωαu ) α W ( ω αu ) α (cid:16) su (cid:17)(cid:17)(cid:17) (cid:27) if σ > . In this case the optimal boundary u ∗ is determined by the smooth fit condition ( V ω A Put ) (cid:48) ( u ∗ ) = − . Thus to identify the price V ω A Put ( s ) of American put option we have to identify the scale functions W ( ξ ) ( s ) and Z ( ξ ) ( s ) for ξ equals to ω u or ω αu under measure P and measure P ( α ) defined in (20). Note that from(21) (see also [54, Prop. 5.6]) with the Laplace exponent(33) ψ ( θ ) = (cid:18) µ − σ (cid:19) θ + σ θ − λθϕ + θ of the process X t , under P ( α ) , the Lévy process X t given in (29) is of the same form with µ and σ unchangedand with new intensity of Poisson process λ ( α ) := λϕ/ ( ϕ − α ) and new parameter of exponential distributionof Y i given by ϕ ( α ) := ϕ − α . To find the scale functions W ( ξ ) ( s ) and Z ( ξ ) ( s ) it enough then to identifythem under original measure P . To do so, we recall that in (16) and (17) we introduced them via regularomega scale functions, that is W ( ξ ) ( s ) = W ( ξ ◦ exp) ( x ) and Z ( ξ ) ( s ) = Z ( ξ ◦ exp) ( x ) for x = log s . It sufficesto find omega scale functions W ( ξ ) ( x ) and Z ( ξ ) ( x ) for given generic function ξ . We recall that both omegascale functions are given as the solutions of renewal equations (12) and (13) formulated in terms of theclassical scale function W ( x ) . From the definition of the first scale function given in (11) with q = 0 and(33) we derive W ( x ) = (cid:88) i =1 Υ i e γ i x , where γ i solves ψ ( γ i ) = 0 and Υ i := 1 ϕ (cid:48) ( γ i ) . If σ = 0 then Υ := 0 , γ := 0 , γ := λ − ϕµµ , Υ := − ϕλ − ϕµ and Υ := λµ ( λ − ϕµ ) . Observe that Υ + Υ = µ .Next theorem gives the ordinary differential equations whose solutions are the omega scale functions. Weuse this result later in the numerical analysis. Theorem 13.
We assume that the function ξ is continuously differentiable. For geometric Lévy model (6) with X t given in (29) we have(i) If σ = 0 then the function W ( ξ ) ( x ) solves the following ordinary differential equations (34) W ( ξ ) (cid:48)(cid:48) ( x ) = ((Υ + Υ ) ξ ( x ) + γ ) W ( ξ ) (cid:48) ( x ) + ((Υ + Υ ) ξ (cid:48) ( x ) − γ Υ ξ ( x )) W ( ξ ) ( x ) J. Al-Hadad — Z. Palmowski with (35) W ( ξ ) (0) = Υ + Υ and (36) W ( ξ ) (cid:48) (0) = Υ γ + (Υ + Υ ) ξ (0) W ( ξ ) (0) . Moreover, the function Z ( ξ ) ( x ) solves the same equation (34) with the following boundary conditions Z ( ξ ) (0) = 1 and (37) Z ( ξ ) (cid:48) (0) = (Υ + Υ ) ξ (0) Z ( ξ ) (0) . (ii) If σ > then the function W ( ξ ) ( x ) solves the following ordinary differential equations W ( ξ ) (cid:48)(cid:48)(cid:48) ( x ) = ((Υ + Υ + Υ ) ξ ( x ) + γ + γ ) W ( ξ ) (cid:48)(cid:48) ( x )+ (2(Υ + Υ + Υ ) ξ (cid:48) ( x ) + Υ ( γ − γ ) ξ ( x ) − (Υ + Υ + Υ ) γ ξ ( x ) − γ γ − γ Υ ξ ( x )) W ( ξ ) (cid:48) ( x )+ ((Υ + Υ + Υ ) ξ (cid:48)(cid:48) ( x ) + Υ ( γ − γ ) ξ (cid:48) ( x ) − γ (Υ + Υ + Υ ) ξ (cid:48) ( x ) + γ γ Υ ξ ( x ) − γ Υ ξ (cid:48) ( x )) W ( ξ ) ( x ) (38) with W ( ξ ) (0) = Υ + Υ + Υ , W ( ξ ) (cid:48) (0) = Υ γ + Υ γ + (Υ + Υ + Υ ) ξ (0) W ( ξ ) (0) and W ( ξ ) (cid:48)(cid:48) (0) = Υ γ ( γ − γ ) + (Υ + Υ + Υ )( ξ (cid:48) (0) W ( ξ ) (0) + ξ (0) W ( ξ ) (cid:48) (0))+ Υ ( γ − γ ) ξ (0) W ( ξ ) (0) + γ W ( ξ ) (cid:48) (0) − γ Υ ξ (0) W ( ξ ) (0) . Moreover, the function Z ( ξ ) ( x ) solves the same equation (38) with the following boundary conditions Z ( ξ ) (0) = 1 , Z ( ξ ) (cid:48) (0) = (Υ + Υ + Υ ) ξ (0) Z ( ξ ) (0) and Z ( ξ ) (cid:48)(cid:48) (0) = (Υ + Υ + Υ )( ξ (cid:48) (0) Z ( ξ ) (0) + ξ (0) Z ( ξ ) (cid:48) (0))+ Υ ( γ − γ ) ξ (0) Z ( ξ ) (0) + γ Z ( ξ ) (cid:48) (0) − γ Υ ξ (0) Z ( ξ ) (0) . Remark 14.
Note that assumption (23) is not required in Theorem 13 because we do not use the function H ( ω u ) ( s ) in the expression for value function (30) and (32).3. Numerical analysis
In this section we present the closed forms of value function (10) for the particular ω and for the Black-Scholes model and Black-Scholes model with downward exponential jumps. In the first scenario, we takeinto account only the case of negative ω which implies a double continuation region, while in the secondexample we focus on the positive ω . erpetual American options with asset-dependent discounting Black-Scholes model revisited.
Let ω ( s ) = − Cs + 1 − D, where C and D are some positive constants. Applying Theorem 10, we obtain v ω A Put ( s, l, u ) = h ( s ) h ( l ) ( K − l ) { s ∈ (0 ,l ) } + ( K − s ) { s ∈ [ l,u ] } + h ( s ) h ( u ) ( K − u ) { s ∈ ( u, + ∞ ) } , where h is a solution to(39) σ s h (cid:48)(cid:48) ( s ) + µsh (cid:48) ( s ) − (cid:18) − Cs + 1 − D (cid:19) h ( s ) = 0 which satisfies(40) h ( s ) = g ( s ) , s ∈ [ l ∗ , u ∗ ] , lim s →∞ h ( s ) = const . We first solve above equation and then we look for boundaries l ∗ and u ∗ such that v ω A Put ( s, l ∗ , u ∗ ) =sup l,u v ω A Put ( s, l, u ) . The general solution to (39) is given by(41) h ( s ) = K s d F ( a , b ; c ; − s ) + K s d F ( a , b ; c ; − s ) , where L := − µσ , M := (cid:113) L − Dσ , G := (cid:113) L − C + D ) σ , while a := − M + G , b := M + G , c := 1+2 G , d := G + L , a := M − G , b := − M − G , c := 1 − G , d := − G + L and K , K are some constants.Using formula (41) and the boundary conditions given in (40) we can identify the form of value function(10). Since we consider the negative ω we obtain a double continuation region. We take one of the summandfrom (41) for s ∈ (0 , l ∗ ) and the second one for s ∈ ( u ∗ , + ∞ ) . This choice is made in a such a way that onthe given interval we impose to have a greater function of these two. Hence we derive V ω A Put ( s ) = K s d F ( a , b ; c ; − s ) , s ∈ (0 , l ∗ ) ,K − s, s ∈ [ l ∗ , u ∗ ] ,K s d F ( a , b ; c ; − s ) , s ∈ ( u ∗ , + ∞ ) . Using the smooth and continuous fit properties we can find K and K and show that l ∗ and u ∗ solve thefollowing equation F ( a i , b i , c i , − s ) K i D i + s d i P i = 0 , (42)where K i := ( K − s ) s − d i F ( a i , b i , c i , − s ) ,D i := d i s d i − ,P i := − a i b i F ( a i + 1 , b i + 1 , c i + 1 , − s ) c i for i = 1 , . We numerically calculate the roots of (42) for i = 1 , . We assign the smaller result to l ∗ , andthe greater one to u ∗ .Let us assume that C = 0 . , D = 1% , K = 20 , µ = 5% and σ = 20% . Above numerical procedureproduces l ∗ ≈ . and u ∗ ≈ . . The figure of the value function is depicted in Figure 1. J. Al-Hadad — Z. Palmowski
Figure 1.
The value and payoff functions for the given set of parameters: C = 0 . , D = 1% , K = 20 , µ = 5% and σ = 20% .3.2. Exponential crashes market revisited.
We consider the stock price process S t given in (6) with X t defined in (29) for σ = 0 , i.e. X t = x + µt − N t (cid:88) i =1 Y i where x = log s , N t is the Poisson process with intensity λ > and { Y i } i ∈ N are i.i.d. exponential randomvariables with parameter ϕ > . Let ω ( s ) = Cs, where C is some positive constant. Note that this discounting function is nonegative. Hence from Theorem12 it follows that l ∗ = 0 . Let η ( x ) := ω ( e x ) = ω ( s ) and η u ( x ) := η ( x + log u ) . (43)From Theorem 12 (see equation (30)) with σ = 0 and using (16) and (17) we can conclude that V ω A Put ( s ) = sup u> v ω A Put ( s, , u ) , where v ω A Put ( s, , u ) = (cid:18) K − uϕϕ + 1 (cid:19) (cid:16) Z ( η u ) ( x − log u ) − c Z ( ηu ) / W ( ηu ) W ( η u ) ( x − log u ) (cid:17) , (44)and from (31) c Z ( ωu ) / W ( ωu ) = c Z ( ηu ) / W ( ηu ) := lim z →∞ Z ( η u ) ( z ) W ( η u ) ( z ) . From Theorem 13 it follows that W ( η ) ( x ) solves the following ordinary differential equation(45) W ( η ) (cid:48)(cid:48) ( x ) = ( Ae x + B ) W ( η ) (cid:48) ( x ) + De x W ( η ) ( x ) , erpetual American options with asset-dependent discounting with A := Cµ , B := λ − ϕµµ and D := C ϕµ . The above equation is also satisfied by Z ( η ) ( x ) .From (35)–(37) we conclude that W ( η ) (0) = µ , W ( η ) (cid:48) (0) = C + λµ and Z ( η ) (0) = 1 , Z ( η ) (cid:48) (0) = Cµ . We solve (45) numerically which allows us to plot its solution. In this way we can produce figures of W ( η ) ( x ) and Z ( η ) ( x ) . By shifting these scale functions by log u we can produce figures of W ( η u ) ( x − log u ) and Z ( η u ) ( x − log u ) . Then c Z ( ηu ) / W ( ηu ) is calculated numerically as the ratio of the scale functions for largeenough arguments (when the ratio stops changing). In this way we can derive value function (44). Finally,by the continuous fit condition we choose the optimal u in such a way that at s = u the value function isequal to the payoff function.Let us assume that C = 1 , K = 20 , µ = 5% , σ = 20% , λ = 6 , ϕ = 2 . Above numerical procedure produces u ∗ ≈ . . The figure of the value function is shown in Figure 2. Figure 2.
The value and payoff functions for the given set of parameters: C = 1 , K = 20 , µ = 5% , σ = 20% , λ = 6 , ϕ = 2 . 4. Proofs
Before we prove main Theorem 2 we show the convexity of European option price V ω E ( s, t ) defined in (2) asa function of s . It is done in Theorems 15 and 18. In the proof we apply idea demonstrated in [34, Prop.4.1]. Later, in the proof of Theorem 2, we use a variant of the maximum principle.Let us introduce a set E ⊂ R × [0 , T ] . We use the following notations • C α ( E ) is the set of locally Hölder( α ) functions with α ∈ (0 , , J. Al-Hadad — Z. Palmowski • C pol ( E ) is the set of functions of at most polynomial growth in s , • C p,q ( E ) is the set of functions for which all the derivatives ∂ k ∂s k (cid:16) ∂ l f ( s,t ) ∂t l (cid:17) with | k | + 2 l ≤ p and ≤ l ≤ q exist in the interior of E and have continuous extensions to E , • C p,qα ( E ) and C p,q pol ( E ) are the sets of functions f ∈ C p,q ( E ) for which all the derivatives ∂ k ∂s k (cid:16) ∂ l f ( s,t ) ∂t l (cid:17) with | k | + 2 l ≤ p and ≤ l ≤ q belong to C pol ( E ) and C α ( E ) , respectively.We need the following conditions in the proofs. Assumptions (B)
There exist constants
C > and α ∈ (0 , such that(B1) µ ( s, t ) ∈ C , α ( R + × [0 , T ]) ;(B2) σ ( s, t ) ≥ Cs for all ( s, t ) ∈ R + × [0 , T ] ;(B3) σ ( s, t ) ∈ C , α ( R + × [0 , T ]) ;(B4) γ ( s, t, z ) ∈ C , α ( R + × [0 , T ]) with the Hölder continuity being uniform in z ;(B5) | ω ( s ) | ≤ C for all s ∈ R + ;(B6) ω ( s ) ∈ C α ( R + ) ;(B7) g ( s ) is Lipschitz continuous;(B8) g ( s ) ∈ C α ( R + ) . Assumptions (C)
There exist a constant
C > such that(C1) | ∂µ ( s,t ) ∂t | ≤ Cs , | ∂ µ ( s,t ) ∂s | ≤ Cs for all ( s, t ) ∈ R + × [0 , T ] ;(C2) | ∂σ ( s,t ) ∂t | ≤ Cs , | ∂ σ ( s,t ) ∂s | ≤ Cs for all ( s, t ) ∈ R + × [0 , T ] ;(C3) | ∂γ ( s,t,z ) ∂t | ≤ Cs , | ∂ γ ( s,t,z ) ∂s | ≤ Cs for all ( s, t, z ) ∈ R + × [0 , T ] × R ;(C4) | dω ( s ) ds | ≤ Cs , | d ω ( s ) ds | ≤ Cs for all s ∈ R + ;(C5) g ( s ) ∈ C pol ( R + ) . Theorem 15.
Let all assumptions of Theorem 2 be satisfied. We assume additionally that conditions (B)and (C) hold true. Then V ω E ( s, t ) is convex with respect to s at all times t ∈ [0 , T ] .Proof. The first part of the proof proceeds in a similar way as the proof of [34, Prop. 4.1].Let L V ω E ( s, t ) = − ∂V ω E ( s, t ) ∂t − A Ct V ω E ( s, t ) − A Jt V ω E ( s, t ) + ω ( s ) V ω E ( s, t ) , where A t is the linear second-order differential operator of the form A Ct V ω E ( s, t ) = β ( s, t ) ∂ V ω E ( s, t ) ∂s + µ ( s, t ) ∂V ω E ( s, t ) ∂s with β ( s, t ) = σ ( s,t )2 and A Jt is the integro-differential operator given by A Jt V ω E ( s, t ) = (cid:90) R (cid:18) V ω E ( s + γ ( s, t, z ) , t ) − V ω E ( s, t ) − γ ( s, t, z ) ∂V ω E ( s, t ) ∂s (cid:19) m ( dz ) . Lemma 16.
Let Assumptions (A) and (B) hold and assume that the stock price process S t follows (5) .Then V ω E ( s, t ) ∈ C , α ( R + × [0 , T ]) ∩ C pol ( R + × [0 , T ]) and it is the solution to the Cauchy problem (46) L V ω E ( s, t ) = 0 , ( s, t ) ∈ R + × [0 , T ) ,V ω E ( s, T ) = g ( s ) , s ∈ R + . erpetual American options with asset-dependent discounting Lemma 17.
Let Assumptions (A), (B) and (C) hold and assume that the stock price process S t follows (5) . Then there exist constants n > and K > such that the value function V ω E ( s, t ) satisfies (cid:12)(cid:12)(cid:12)(cid:12) ∂ V ω E ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ( s − n + s n ) for all ( s, t ) ∈ R + × [0 , T ] . Proofs of both above lemmas are given in Appendix.We introduce the function u ω : R + × [0 , T ] → R + of the form u ω ( s, t ) := V ω E ( s, T − t ) and we prove convexity of u ω ( s, t ) with respect to s . Note that it is equivalent to the convexity of the valuefunction V ω E ( s, t ) in s . Furthermore, based on Lemma 16, the function u ω ( s, t ) solves the Cauchy problemof the form ∂u ω ( s,t ) ∂t = ˆ L u ω ( s, t ) , ( s, t ) ∈ R + × (0 , T ] ,u ω ( s,
0) = g ( s ) , s ∈ R + , where ˆ L u ω ( s, t ) = β ( s, t ) ∂ u ω ( s, t ) ∂s + µ ( s, t ) ∂u ω ( s, t ) ∂s − ω ( s ) u ω ( s, t )+ (cid:90) R (cid:18) u ω ( s + γ ( s, t, z ) , t ) − u ω ( s, t ) − γ ( s, t, z ) ∂u ω ( s, t ) ∂s (cid:19) m ( dz ) with β ( s, t ) = σ ( s,t )2 . Observe that by Lemma 17 there exist constants n > and K > such that(47) (cid:12)(cid:12)(cid:12)(cid:12) ∂ u ω ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ( s − n + s n ) for all ( s, t ) ∈ R + × [0 , T ] .Let us now define a convex function κ : R + → R + of the form κ ( s ) := s n +3 + s − n +1 with d κ ( s ) ds = ( n + 3)( n + 2) s n +1 + n ( n − s − n − and d ( ˆ L κ ( s )) ds = ∂ β ( s, t ) ∂s d κ ( s ) ds + 2 ∂β ( s, t ) ∂s d κ ( s ) ds + β ( s, t ) d κ ( s ) ds + ∂ µ ( s, t ) ∂s dκ ( s ) ds + 2 ∂µ ( s, t ) ∂s d κ ( s ) ds + µ ( s, t ) d κ ( s ) ds − d ω ( s ) ds κ ( s ) − dω ( s ) ds dκ ( s ) ds − ω ( s ) d κ ( s ) ds + (cid:90) R (cid:32) d κ ( s + γ ( s, t, z )) ds (cid:18) ∂γ ( s, t, z ) ∂s (cid:19) + dκ ( s + γ ( s, t, z )) ds ∂ γ ( s, t, z ) ∂s − γ ( s, t, z ) d κ ( s ) ds − (cid:18) ∂γ ( s, t, z ) ∂s (cid:19) d κ ( s ) ds − ∂ γ ( s, t, z ) ∂s dκ ( s ) ds (cid:33) m ( dz ) . J. Al-Hadad — Z. Palmowski
The assumptions that we put on the coefficients µ , σ and γ and function ω and their derivatives imply thateach component of the above expression grows at most like s n +1 for large s and like s − n − for small s . Thesame behaviour characterises d κ ( s ) ds .In addition, we define the function ϑ : R + × [0 , T ] → R given by ϑ ( s, t ) := (cid:18) ∂ µ ( s, t ) ∂s − dω ( s ) ds (cid:19) dκ ( s ) ds which also behaves like d ( ˆ L κ ( s )) ds at + ∞ and −∞ .Hence we claim that there exist a positive constant C such that(48) C d κ ( s ) ds − d ( ˆ L κ ( s )) ds > − ϑ ( s, t ) . In the second part of the proof, we define the auxiliary function(49) u ωε ( s, t ) := u ω ( s, t ) + εe Ct κ ( s ) for some ε > .We carry out a proof by contradiction. Let us then assume that u ωε ( s, t ) is not convex. For this purpose,we denote by Λ the set of points for which u ωε ( s, t ) is not convex, i.e. Λ := { ( s, t ) ∈ R + × [0 , T ] : ∂ u ωε ( s, t ) ∂s < } and we assume that the set Λ is not empty.From Lemma 17 we know that u ω ( s, t ) satisfies (47). Due to this fact and using (49) we claim that thereexist a positive constant R such that Λ ⊆ [ R − , R ] × [0 , T ] . This is a direct consequence of such a choice of u ωε ( s, t ) in (49) so that d κ ( s ) ds grows faster than ∂ u ω ( s,t ) ∂s for both large and small values of s .Consequently, the set Λ is a bounded set. Since the closure of a bounded set is also bounded, we concludethat the closure of Λ , i.e. cl (Λ) , is compact.Due to the fact that a compact set always contains its infimum we can define t := inf { t ≥ s, t ) ∈ cl (Λ) for some s ∈ R + } . From the initial condition, i.e. u ω ( s,
0) = g ( s ) and convexity of g we have d u ωε ( s, ds = d ( g ( s ) + εκ ( s )) ds ≥ ε d κ ( s ) ds > for all s ∈ R + . Hence we can conclude that t > .Moreover, at the point when the infimum is attained, i.e. ( s , t ) for some s ∈ R + ∂ u ωε ( s , t ) ∂s = 0 . This is a consequence of the continuity of the function ∂ u ωε ( s,t ) ∂s in s . In addition, for t ∈ [0 , t ) we have ∂ u ωε ( s ,t ) ∂s > and thus, by applying the symmetry of second derivatives at t = t , we derive(50) ∂ ∂s (cid:18) ∂u ωε ( s , t ) ∂t (cid:19) = ∂∂t (cid:18) ∂ u ωε ( s , t ) ∂s (cid:19) ≤ . erpetual American options with asset-dependent discounting Furthermore, at ( s , t ) we also have ∂ ( ˆ L u ωε ( s , t )) ∂s = ∂ β ( s , t ) ∂s ∂ u ωε ( s , t ) ∂s + 2 ∂β ( s , t ) ∂s ∂ u ωε ( s , t ) ∂s + β ( s , t ) ∂ u ωε ( s , t ) ∂s + ∂ µ ( s , t ) ∂s ∂u ωε ( s , t ) ∂s + 2 ∂µ ( s , t ) ∂s ∂ u ωε ( s , t ) ∂s + µ ( s , t ) ∂ u ωε ( s , t ) ∂s − d ω ( s ) ds u ωε ( s , t ) − dω ( s ) ds ∂u ωε ( s , t ) ∂s − ω ( s ) ∂ u ωε ( s , t ) ∂s + (cid:90) R (cid:32) ∂ u ωε ( s + γ ( s , t , z ) , t ) ∂s (cid:18) ∂γ ( s , t , z ) ∂s (cid:19) + ∂u ωε ( s + γ ( s , t , z ) , t ) ∂s ∂ γ ( s , t , z ) ∂s − γ ( s , t , z ) ∂ u ωε ( s , t ) ∂s − (cid:18) ∂γ ( s , t , z ) ∂s (cid:19) ∂ u ωε ( s , t ) ∂s − ∂ γ ( s , t , z ) ∂s ∂u ωε ( s , t ) ∂s (cid:33) m ( dz ) . Since ∂ u ωε ( s ,t ) ∂s = 0 and ∂ u ωε ( s,t ) ∂s has a local minimum at s = s , we have ∂ u ωε ( s ,t ) ∂s = 0 and ∂ u ωε ( s ,t ) ∂s ≥ . Thus ∂ ( ˆ L u ωε ( s , t )) ∂s ≥ ∂ µ ( s , t ) ∂s ∂u ωε ( s , t ) ∂s − d ω ( s ) ds u ωε ( s , t ) − dω ( s ) ds ∂u ωε ( s , t ) ∂s + (cid:90) R (cid:32) ∂u ωε ( s + γ ( s , t , z ) , t ) ∂s ∂ γ ( s , t , z ) ∂s − ∂u ωε ( s , t ) ∂s ∂ γ ( s , t , z ) ∂s (cid:33) m ( dz ) . Since u ωε ( s, t ) is convex in s and ∂ u ωε ( s ,t ) ∂s = 0 , applying (8) we can conclude that the integral part of theabove expression is nonnegative. Moreover, the concavity of ω and (9) imply that(51) ∂ ( ˆ L u ωε ( s , t )) ∂s ≥ εe Ct (cid:18) ∂ µ ( s , t ) ∂s − d ω ( s ) ds (cid:19) dκ ( s ) ds = εe Ct ϑ ( s , t ) . Combining (48) with (50) and (51) at ( s , t ) we derive that ∂ ∂s (cid:18) ∂u ωε ( s , t ) ∂t − ˆ L u ωε ( s , t ) (cid:19) = εe Ct d ds ( Cκ ( s ) − ˆ L κ ( s )) > − εe Ct ϑ ( s , t ) ≥ ∂ ∂s (cid:18) ∂u ωε ( s , t ) ∂t − ˆ L u ωε ( s , t ) (cid:19) which is a contradiction. This confirms that the set Λ is empty, and thus u ωε ( s, t ) is a convex function.Finally, letting ε → we conclude that u ω ( s, t ) is convex in s for all t ∈ [0 , T ] . (cid:3) Using the same arguments like in the proof of [34, Thm. 4.1], we can resign from Assumptions (B) and (C)in Theorem 15, that is, the following theorem holds true.
Theorem 18.
Let assumptions of Theorem 2 hold true. Then V ω E ( s, t ) is convex with respect to s at alltimes t ∈ [0 , T ] . J. Al-Hadad — Z. Palmowski
We are ready to give the proof of our first main result.
Proof of Theorem 2 . As noted in [34, Sec. 7], under conditions (A1)–(A4), for each p ≥ there exists aconstant C such that the stock price process given in (5) satisfies E s (cid:20) sup ≤ t ≤ T | S t | p (cid:21) ≤ C (1 + s p ) . Together with (A5) and (A6) it implies that the value function given by V ω A T ( s, t ) := sup τ ∈T Tt E s,t [ e − (cid:82) τt ω ( S w ) dw g ( S τ )] is well-defined, where T Tt is the family of F t -stopping times with values in [ t, T ] for fixed maturity T > .Moreover, we denote V ω A T ( s ) := V ω A T ( s, . Let us define now a Bermudan option with the value function of the form V ω B Ξ ( s, t ) := sup τ ∈T Ξ E s,t [ e − (cid:82) τt ω ( S w ) dw g ( S τ )] , where T Ξ is the set of stopping times with values inB Ξ = (cid:110) n Ξ ( T − t ) + t : n = 0 , , ..., Ξ (cid:111) , where Ξ is some positive integer number. To simplify the notation, we denote V ω B Ξ ( s ) := V ω B Ξ ( s, . In contrast to the American options, the Bermudan options are the options that can be exercised at one ofthe finitely many number of times.Now we show that V ω B Ξ ( s, t ) inherits the property of convexity from its European equivalent V ω E ( s, t ) . Next,we generalise this result to the American case V ω A ( s ) . Lemma 19.
Let assumptions of Theorem 2 hold true. Then V ω B Ξ ( s, t ) is convex with respect to s at alltimes t ∈ [0 , T ] . Its proof is given in Appendix.As the possible exercise times of the Bermudan option get denser, the value function V ω B Ξ ( s, t ) converges to V ω A T ( s, t ) . To formalise this result, we proceed as follows. For a given stopping time τ T that takes valuesin [0 , T ] , we define τ Ξ := inf { t ∈ B Ξ : t ≥ τ T } . Then τ Ξ ∈ B Ξ is a stopping time and τ Ξ → τ T almost surely as Ξ → + ∞ . Moreover, by the dominatedconvergence theorem, we obtain (cid:12)(cid:12)(cid:12)(cid:12) E s (cid:104) e − (cid:82) τ Ξ0 ω ( S w ) dw g ( S τ Ξ ) (cid:105) − E s (cid:20) e − (cid:82) τT ω ( S w ) dw g ( S τ T ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ≤ E s (cid:12)(cid:12)(cid:12)(cid:12) e − (cid:82) τ Ξ0 ω ( S w ) dw g ( S τ Ξ ) − e − (cid:82) τT ω ( S w ) dw g ( S τ T ) (cid:12)(cid:12)(cid:12)(cid:12) → as Ξ → ∞ . Therefore, it follows that lim inf Ξ →∞ V ω B Ξ ( s ) ≥ V ω A T ( s ) Since it is obvious that V ω B Ξ ( s ) ≤ V ω A T ( s ) , erpetual American options with asset-dependent discounting we finally derive V ω B Ξ ( s ) → V ω A T ( s ) as Ξ → ∞ . To receive our claim we take the maturity T tending to infinity. (cid:3) Proof of Theorem 5 . Before we proceed to the actual proof let us remind the main exit identities from[49]. Let σ + a := inf { t > X t ≥ a } , σ − a := inf { t > X t ≤ a } for some a ∈ R . Then for the function η defined in (43) we have E (cid:20) e − (cid:82) σ + a η ( X w ) dw ; σ + a < ∞ | X = x (cid:21) = H ( η ) ( x ) H ( η ) ( a ) , (52) E (cid:20) e − (cid:82) σ − η ( X w ) dw ; σ − < ∞ | X = x (cid:21) = Z ( η ) ( x ) − c Z ( η ) / W ( η ) W ( η ) ( x ) , (53)where c Z ( η ) / W ( η ) = lim z →∞ Z ( η ) ( z ) W ( η ) ( z ) and we use condition η ( x ) = c for all x ≤ and some constant c ∈ R inthe first identity. Denoting τ + a := inf { t > S t ≥ a } , τ − a := inf { t > S t ≤ a } and keeping in mind that S t = e X t , from (52) and (53) we can conclude that E s (cid:20) e − (cid:82) τ + a ω ( S w ) dw ; τ + a < ∞ (cid:21) = H ( ω ) ( s ) H ( ω ) ( a ) , (54) E s (cid:20) e − (cid:82) τ − ω ( S w ) dw ; τ − < ∞ (cid:21) = Z ( ω ) ( s ) − c Z ( ω ) / W ( ω ) W ( ω ) ( s ) , where ω ( s ) = ω ( e x ) = η ( x ) and the functions Z ( ω ) ( s ) , W ( ω ) ( s ) , H ( ω ) ( s ) were defined in (16), (17) and(18).We consider three possible cases of a position of the initial state S = s of the process S t .(1) s < l : As the process S t is spectrally negative and starts below the interval [ l, u ] , it can enter thisinterval only in a continuous way and hence τ l,u = τ + l and S τ l,u = l . Thus from (54) v ω A Put ( s, l, u ) = E s (cid:20) e − (cid:82) τ + l ω ( S w ) dw ; S τ + l = l (cid:21) ( K − l )= H ( ω ) ( s ) H ( ω ) ( l ) ( K − l ) . (2) s ∈ [ l, u ] : If the process S t starts inside the interval [ l, u ] which is an optimal stopping region, wedecide to exercise our option immediately, i.e. τ l,u = 0 . Therefore, we have v ω A Put ( s, l, u ) = K − s. (3) s > u : There are three possible cases of entering the interval [ l, u ] by the process S t when it startsabove u : either S t enters [ l, u ] continuously going downward or it jumps from ( u, + ∞ ) to ( l, u ) or S t jumps from the interval ( u, + ∞ ) to the interval (0 , l ) and then, later, enters [ l, u ] continuously.We can distinguish these cases in the following way(55) v ω A Put ( s, l, u ) = E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ); τ − u < τ − l (cid:105) + E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ); τ − u = τ − l (cid:105) . J. Al-Hadad — Z. Palmowski
To analyse the first component in (55), note that E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ); τ − u < τ − l (cid:105) = E s (cid:20) e − (cid:82) τ − u ω ( S w ) dw ( K − S τ − u ); S τ − u ∈ [ l, u ] (cid:21) = (cid:90) ( l,u ) ( K − z ) E s (cid:20) e − (cid:82) τ − u ω ( S w ) dw ; S τ − u ∈ dz (cid:21) + ( K − u ) E s (cid:20) e − (cid:82) τ − u ω ( S w ) dw ; S τ − u = u (cid:21) . We express now above formulas in terms of X t = log S t process. Let x = log s and we recall that in (43)we introduced functions η ( x ) = ω ◦ exp( x ) = ω ( e x ) and η u ( x ) = η ( x + log u ) . Then(56) E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ); τ − u < τ − l (cid:105) = (cid:90) (log l, log u ) ( K − e z ) E (cid:34) e − (cid:82) σ − log u η ( X w ) dw ; X σ − log u ∈ dz | X = x (cid:35) + ( K − u ) E (cid:34) e − (cid:82) σ − log u η ( X w ) dw ; X σ − log u = log u | X = x (cid:35) = (cid:90) (0 , log u − log l ) ( K − e log u − y ) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; − X σ − ∈ dy | X = x − log u (cid:21) + ( K − u ) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; X σ − = 0 | X = x − log u (cid:21) . From the compensation formula for Lévy processes given in [45, Thm. 4.4] we have(57) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; − X σ − ∈ dy | X = x − log u (cid:21) = (cid:90) ∞ r ( η u ) ( x − log u, z )Π( − z − dy ) dz, where r ( η u ) ( x − log u, z ) is the resolvent density of X t killed by potential η u and on exiting from positivehalf-line which is, by [49, Thm. 2.2], given by r ( η u ) ( x − log u, z ) = W ( η u ) ( x − log u ) lim y →∞ W ( η u ) ( y, z ) W ( η u ) ( y ) − W ( η u ) ( x − log u, z ) . Note that r ( η u ) (log s − log u, z ) = r ( s, u, z ) for r ( s, u, z ) given in (22).To find E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; X σ − = 0 | X = x − log u (cid:21) , we consider E (cid:34) e − (cid:82) σ − η u ( X w ) dw + αX σ − ; σ − < ∞ | X = x − log u (cid:35) for some α > . Note that using the change of measure given in (20) it is equal to(58) E ( α ) (cid:20) e − (cid:82) σ − η αu ( X w ) dw ; σ − < ∞ | X = x − log u (cid:21) where E ( α ) is an expectation with respect to P ( α ) and η αu ( x ) := η u ( x ) − ψ ( α ) . From (53) we know that E ( α ) (cid:20) e − (cid:82) σ − η αu ( X w ) dw ; σ − < ∞ | X = x − log u (cid:21) = Z ( η αu ) α ( x − log u ) − c Z ( ηαu ) α / W ( ηαu ) α W ( η αu ) α ( x − log u ) . erpetual American options with asset-dependent discounting Moreover, observe that E (cid:34) e − (cid:82) σ − η u ( X w ) dw + αX σ − ; σ − < ∞ | X = x − log u (cid:35) = E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; X σ − = 0 | X = x − log u (cid:21) + E (cid:34) e − (cid:82) σ − η u ( X w ) dw + αX σ − ; X σ − < | X = x − log u (cid:35) . Taking the limit α → ∞ and using (58) we derive(59) lim α →∞ E ( α ) (cid:20) e − (cid:82) σ − η αu ( X w ) dw ; σ − < ∞ | X = x − log u (cid:21) = E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; X σ − = 0 | X = x − log u (cid:21) and therefore we have(60) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; X σ − = 0 | X = x − log u (cid:21) = lim α →∞ (cid:16) Z ( η αu ) α ( x − log u ) − c Z ( ηαu ) α / W ( ηαu ) α W ( η αu ) α ( x − log u ) (cid:17) . Furthermore, the second component of (55) equals to(61) E s (cid:104) e − (cid:82) τl,u ω ( S w ) dw ( K − S τ l,u ); τ − u = τ − l (cid:105) = E (cid:104) e − (cid:82) τl,u η ( X w ) dw ( K − e X τl,u ); σ − log u = σ − log l | X = x (cid:105) = E (cid:104) e − (cid:82) τl,u η ( X w ) dw ( K − e X τl,u ); X σ − log u < log l | X = x (cid:105) = E (cid:34) e − (cid:82) σ − log u η ( X w ) dw E X σ − log u (cid:104) e − (cid:82) τl,u η ( X w ) dw ( K − e X τl,u ) (cid:105) ; X σ − log u < log l | X = x (cid:35) = (cid:90) ∞ log u − log l E (cid:20) e − (cid:82) σ − η u ( X w ) dw E (cid:104) e − (cid:82) τl,u η u ( X w ) dw ( K − e X τl,u ) | X = log u − y (cid:105) ; − X σ − ∈ dy (cid:21) = (cid:90) ∞ log u − log l H ( η u ) (log u − y ) H ( η u ) (log l ) ( K − l ) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; − X σ − ∈ dy | X = x − log u (cid:21) . Now we have to express all scale functions in terms of the S t scale functions defined in (16)–(19) with x = log s and using (43). Finally, using (55) together with (56), (57), (59) and (61) completes the proof. (cid:3) Proof of Theorem 6 . From the fact that V ω A ( s ) ∈ D ( A ) and using classical arguments it follows that V ω A ( s ) solves uniquely equation (24); see [56, Thm. 2.4, p. 37], [3] and [62, Thm. 1] for details. Moreformally, our function as a convex function is continuous (in whole domain). Since our boundary is suffi-ciently regular we know that the Dirichlet/Poisson problem can be solved uniquely in D ( A ) . This solutioncan then be identified with the value function V ω A ( s ) itself using the stochastic calculus or infinitesimalgenerator techniques in the continuation set; see [56, p. 131] for further details. We are left with the proofof the smoothness at the boundary of stopping set. We prove it at u . The proof at lower end followsexactly in the same way. We choose to follow the idea given in [47] although one can also apply [3] orsimilar arguments as the ones given in [26].Suppose then that is for ( −∞ , . Since V ω A ( s ) ≥ g ( s ) and V ω A ( u ) = g ( u ) , we have V ω A ( u + h ) − V ω A ( u ) h ≥ g ( u + h ) − g ( u ) h . Hence lim inf h ↓ V ω A ( u + h ) − V ω A ( u ) h ≥ g (cid:48) ( u ) . J. Al-Hadad — Z. Palmowski
To get the opposite inequality we introduce τ h = inf { t ≥ S t ∈ [ l, u ] | S = u + h } . By assumed regularity, τ h → a.s. as h ↓ . Moreover, by Markov property V ω A ( u ) ≥ E log u (cid:104) e − (cid:82) τh ω ( S w ) dw g ( S τ h ) (cid:105) . Then by (B5) and the space homogeneity of log S t , V ω A ( u + h ) − V ω A ( u ) h ≤ E u + h (cid:104) e − (cid:82) τh ω ( S w ) dw g ( S τ h ) (cid:105) − E u (cid:104) e − (cid:82) τ ω ( S w ) dw g ( S τ ) (cid:105) h ≤ E u + h (cid:104) e − (cid:82) τh ω ( S w ) dw g (( u + h ) S τ h ) (cid:105) − E u (cid:104) e − (cid:82) τ ω ( S w ) dw g ( uS τ ) (cid:105) h and lim sup h ↓ V ω A ( u + h ) − V ω A ( u ) h ≤ g (cid:48) ( u ) , where we use the fact that g is continuously differentiable at u in the last step. This completes the proof. (cid:3) Proof of Theorem 8 . We recall that V ω A Call ( s, K, ζ, σ, Π , l, u ) = E s [ e − (cid:82) τl,u ω ( S w ) dw ( S τ l,u − K ) + ]= E [ e − (cid:82) τl,u η ( X w ) dw ( e X τl,u − K ) + | X = x ] . By our assumption for general Lévy process X t we can define new measure P (1) via d P (1) d P (cid:12)(cid:12)(cid:12)(cid:12) F t = e X t − ψ (1) t ; see also (20) (considered there only for spectrally negative Lévy process). Let x = log S = log s . Then E (cid:104) e − (cid:82) τl,u η ( X w ) dw ( e X τl,u − K ) + | X = x (cid:105) = E (1) (cid:20) e − (cid:82) τ su K, sl K ( ω ( Sw sK ) − ψ (1)) dw ( s − e ˆ X τ su K, sl K ) + | ˆ X = log K (cid:21) , where ˆ S t = e ˆ X t and ˆ X t = − X t is the dual process to X t and from [26, 38, 54] it follows that under P (1) itis again Lévy process with the triple ( − ζ, σ, ˆΠ) for ˆΠ defined in (25). This completes the proof. (cid:3) Proof of Theorem 10 . We prove that for the function h satisfying (27) we have(62) E s (cid:20) h ( S τ l,u ) h ( s ) e − (cid:82) τl,u ω ( S w ) dw (cid:21) = 1 . Since process S t is continuous in Black-Scholes model, S τ l,u equals either to l or u depending on the initialstate of S t . We can distinguish three possible scenarios erpetual American options with asset-dependent discounting (1) s < l : As the process S t is a continuous process and starts below the interval [ l, u ] , then τ l,u = τ + l and S τ l,u = l . Thus, we get(63) v ω A Put ( s, l, u ) = E s (cid:20) e − (cid:82) τ + l ω ( S w ) dw ; S τ + l = l (cid:21) ( K − l )= h ( s ) h ( l ) ( K − l ) . (2) s ∈ [ l, u ] : If the process S t starts inside the interval [ l, u ] which is the optimal stopping region, wedecide to exercise our option immediately, i.e. τ l,u = 0 . Therefore, we have(64) v ω A Put ( s, l, u ) = K − s. (3) s > u : Similarly to the case when s < l , the process S t can enter [ l, u ] only via u and thus τ l,u = τ − u and S τ l,u = u . Therefore,(65) v ω A Put ( s, l, u ) = E s (cid:20) e − (cid:82) τ − u ω ( S w ) dw ; S τ − u = u (cid:21) ( K − u )= h ( s ) h ( u ) ( K − u ) . Identities (63), (64) and (65) give the first part of the assertion of the theorem. Note that boundarycondition (28) follows straightforward from the definition of the value function of the American put option.We are left with the proof of (62). Consider strictly positive and bounded by some C function h ∈ C ( R + ) ⊂ D ( A ) . Then by [54, Prop. 3.2] the process E h ( t ) := h ( S t ) h ( S ) e − (cid:82) t A h )( Sw ) h ( Sw ) dw , is a mean-one local martingale, where in the case of Black-Scholes model A h ( s ) = µsh (cid:48) ( s ) + σ s h (cid:48)(cid:48) ( s ) . Observe that equation (27) is equivalent to ω ( s ) = A h ( s ) h ( s ) . Let τ Ml,u := τ l,u ∧ M for some fixed M > . Applying the optional stopping theorem for bounded stopping time, we derive(66) E s (cid:34) h ( S τ Ml,u ) h ( s ) e − (cid:82) τMl,u ω ( S w ) dw (cid:35) = 1 . We rewrite the left side of (66) as a sum of the following two components I := E s (cid:34) h ( S τ Ml,u ) h ( s ) e − (cid:82) τMl,u ω ( S w ) dw ; τ l,u > M (cid:35) ,I := E s (cid:34) h ( S τ Ml,u ) h ( s ) e − (cid:82) τMl,u ω ( S w ) dw ; τ l,u ≤ M (cid:35) . We prove now that lim M →∞ I = 0 and lim M →∞ I ∈ (0 , + ∞ ) . Let us define the last time when valuefunction (4) is positive by τ last ( K ) := sup { t ≥ S t ≤ K } . J. Al-Hadad — Z. Palmowski
It easy to notice that P ( τ Ml,u ≤ τ last ( K )) = 1 . Then, from the boundedness of h , lower boundedness of ω and Cauchy–Schwarz inequality we obtain I ≤ Ch ( s ) E (cid:104) e − ω ¯ τ last ( K ) ; τ l,u > M (cid:105) = Ch ( s ) E (cid:104) e − ω ¯ τ last ( K ) τ l,u >M (cid:105) ≤ Ch ( s ) (cid:113) E (cid:2) e − ω ¯ τ last ( K ) (cid:3) P ( τ l,u > M ) , where ω ¯ := min s ∈ R + ω ( s ) . Note that (cid:113) E (cid:2) e − ω ¯ τ last ( K ) (cid:3) < ∞ by [10, Thm. 2] because E e − ω ¯ B t < ∞ for any t ≥ . Thus lim M →∞ I = 0 . Moreover, < I ≤ Ch ( s ) E (cid:104) e − ω ¯ τ last ( K ) ; τ l,u < M (cid:105) . Hence by (66) and the dominated convergence we get (62) as long as h is positive and bounded. Finally,since S τ l,u equals either to l or u , the boundedness assumption could be skipped. This completes theproof. (cid:3) Proof of Theorem 12 . From Theorem 2 and Remark 3 it follows that the optimal exercise time is thefirst entrance to the interval [ l ∗ , u ∗ ] and by Theorem 4 the value function V ω A Put ( s ) equals to the maximumover l and u of v ω A Put ( s, l, u ) defined in (4). We recall the observation that if the discounting function ω ispositive, then it is never optimal to wait to exercise option for small asset prices, that is, always l ∗ = 0 inthis case and the stopping region is one-sided. We find now function v ω A Put ( s, l, u ) in the case of (i) and (ii).If σ = 0 , by the lack of memory of exponential random variable, using similar analysis like in the proof ofTheorem 5, we have v ω A Put ( s, , u ) = E ( K − e log u − Y ) + E s (cid:20) e − (cid:82) τ − u ω ( S w ) dw ; τ − u < ∞ (cid:21) = E ( K − e log u − Y ) + E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; σ − < ∞ | X = x − log u (cid:21) = E ( K − e log u − Y ) + (cid:16) Z ( η u ) ( x − log u ) − c Z ( ηu ) / W ( ηu ) W ( η u ) ( x − log u ) (cid:17) = E ( K − e log u − Y ) + (cid:16) Z ( ω u ) (cid:16) su (cid:17) − c Z ( ωu ) / W ( ωu ) W ( ω u ) (cid:16) su (cid:17)(cid:17) . Observing that E ( K − e log u − Y ) + = K − uϕϕ + 1 completes the proof of part (i).If σ > then v ω A Put ( s, , u ) = E ( K − e log u − Y ) + E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; σ − < ∞ , X σ − < | X = x − log u (cid:21) + ( K − u ) E (cid:20) e − (cid:82) σ − η u ( X w ) dw ; σ − < ∞ , X σ − = 0 | X = x − log u (cid:21) . The first increment can be analysed like in the case of σ = 0 . The expression for the second componentfollows from (60).Finally, the smooth fit condition follows straightforward from Theorem 6. (cid:3) erpetual American options with asset-dependent discounting Proof of Theorem 13 . Assume first that σ = 0 . Then(67) W ( x ) = Υ + Υ e γ x . To produce ordinary differential equation for W ( ξ ) ( x ) we start from equation (12). Putting (67) there gives(68) W ( ξ ) ( x ) = Υ + Υ e γ x + Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy + Υ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy. Taking the derivative of both sides gives(69) W ( ξ ) (cid:48) ( x ) = Υ γ e γ x + Υ ξ ( x ) W ( ξ ) ( x ) + Υ (cid:18) ξ ( x ) W ( ξ ) ( x ) + γ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy (cid:19) . From (68) we have (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy = 1Υ (cid:18) W ( ξ ) ( x ) − Υ − Υ e γ x − Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy (cid:19) . We put it into (69) and derive W ( ξ ) (cid:48) ( x ) = ((Υ + Υ ) ξ ( x ) + γ ) W ( ξ ) ( x ) − γ Υ − γ Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy. We take the derivative of both sides again to get equation (34).From (12) and (67) we obtain first boundary condition (35) and from (69) we derive second boundarycondition (36).Similar analysis can be done for the Z ( ξ ) ( x ) scale function producing equation (34) and its boundaryconditions. This completes the proof of the case (i).In the case when σ > observe that(70) W ( x ) = Υ + Υ e γ x + Υ e γ x , thus from (12) W ( ξ ) ( x ) satisfies the following equation W ( ξ ) ( x ) = Υ + Υ e γ x + Υ e γ x + (cid:90) x (Υ + Υ e γ ( x − y ) + Υ e γ ( x − y ) ) ξ ( y ) W ( ξ ) ( y ) dy. We simplify it deriving W ( ξ ) ( x ) = Υ + Υ e γ x + Υ e γ x + Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy + Υ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy + Υ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy. (71)In the next step we take derivative of both sides to get W ( ξ ) (cid:48) ( x ) = Υ γ e γ x + Υ γ e γ x + Υ ξ ( x ) W ( ξ ) ( x ) + Υ (cid:18) ξ ( x ) W ( ξ ) ( x ) + γ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy (cid:19) +Υ (cid:18) ξ ( x ) W ( ξ ) ( x ) + γ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy (cid:19) . (72)From (71) we have (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy = 1Υ (cid:16) W ( ξ ) ( x ) − Υ − Υ e γ x − Υ e γ x − Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy − Υ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy (cid:19) . J. Al-Hadad — Z. Palmowski
We put it into (72) deriving W ( ξ ) (cid:48) ( x ) = Υ ( γ − γ ) e γ x + (Υ + Υ + Υ ) ξ ( x ) W ( ξ ) ( x )+ Υ ( γ − γ ) (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy + γ W ( ξ ) ( x ) − γ Υ − γ Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy. (73)Taking again derivative of both sides produces W ( ξ ) (cid:48)(cid:48) ( x ) = Υ ( γ − γ ) γ e γ x + (Υ + Υ + Υ )( ξ (cid:48) ( x ) W ( ξ ) ( x ) + ξ ( x ) W ( ξ ) (cid:48) ( x ))+ Υ ( γ − γ ) (cid:18) ξ ( x ) W ( ξ ) ( x ) + γ (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy (cid:19) + γ W ( ξ ) (cid:48) ( x ) − γ Υ ξ ( x ) W ( ξ ) ( x ) . (74)From (73) we have (cid:90) x e γ ( x − y ) ξ ( y ) W ( ξ ) ( y ) dy = 1Υ ( γ − γ ) (cid:16) W ( ξ ) (cid:48) ( x ) − Υ ( γ − γ ) e γ x − (Υ + Υ + Υ ) ξ ( x ) W ( ξ ) ( x ) − γ W ( ξ ) ( x ) + γ Υ + γ Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy (cid:19) . We put it into (74) to get W ( ξ ) (cid:48)(cid:48) ( x ) = (Υ + Υ + Υ )( ξ (cid:48) ( x ) W ( ξ ) ( x ) + ξ ( x ) W ( ξ ) (cid:48) ( x ))+ Υ ( γ − γ ) ξ ( x ) W ( ξ ) ( x )+ γ (cid:16) W ( ξ ) (cid:48) ( x ) − (Υ + Υ + Υ ) ξ ( x ) W ( ξ ) ( x ) − γ W ( ξ ) ( x )+ γ Υ + γ Υ (cid:90) x ξ ( y ) W ( ξ ) ( y ) dy (cid:19) + γ W ( ξ ) (cid:48) ( x ) − γ Υ ξ ( x ) W ( ξ ) ( x ) . Taking again derivative and simplifying gives W ( ξ ) (cid:48)(cid:48)(cid:48) ( x ) = ((Υ + Υ + Υ ) ξ ( x ) + γ + γ ) W ( ξ ) (cid:48)(cid:48) ( x )+ (2(Υ + Υ + Υ ) ξ (cid:48) ( x ) + Υ ( γ − γ ) ξ ( x ) − (Υ + Υ + Υ ) γ ξ ( x ) − γ γ − γ Υ ξ ( x )) W ( ξ ) (cid:48) ( x )+ ((Υ + Υ + Υ ) ξ (cid:48)(cid:48) ( x ) + Υ ( γ − γ ) ξ (cid:48) ( x ) − γ (Υ + Υ + Υ ) ξ (cid:48) ( x ) + γ γ Υ ξ ( x ) − γ Υ ξ (cid:48) ( x )) W ( ξ ) ( x ) which is the equation that we wanted to prove.From (70) and (12) we have W ( ξ ) (0) = Υ + Υ + Υ . From (73) it follows that W ( ξ ) (cid:48) (0) = Υ γ + Υ γ + (Υ + Υ + Υ ) ξ (0) . Finally, from (74) we have W ( ξ ) (cid:48)(cid:48) (0) = Υ γ ( γ − γ ) + (Υ + Υ + Υ )( ξ (cid:48) (0) W ( ξ ) (0) + ξ (0) W ( ξ ) (cid:48) (0))+ Υ ( γ − γ ) ξ (0) W ( ξ ) (0) + γ W ( ξ ) (cid:48) (0) − γ Υ ξ (0) W ( ξ ) (0) . erpetual American options with asset-dependent discounting The analysis for Z ( ξ ) ( x ) can be done in the same way. This completes the proof. (cid:3) Appendix
Proof of Lemma 16 . Firstly, we define the function f : R + → R of the form f ( s ) = − s , s ∈ (0 , ,s, s ∈ [2 , + ∞ ) such that f ( s ) ∈ C ( R + ) and f (cid:48) ( s ) > for all s ∈ R + .Taking Y t = f ( S t ) and applying Itô’s lemma on (5), we obtain dY t = ˜ µ ( Y t − , t ) dt + ˜ σ ( Y t − , t ) dB t + (cid:90) R ˜ γ ( Y t − , t, z )˜ v ( dt, dz ) , where ˜ µ ( y, t ) = µ ( f − ( y ) , t ) f (cid:48) ( f − ( y )) + σ ( f − ( y ) , t )2 f (cid:48)(cid:48) ( f − ( y ))+ (cid:90) R (cid:0) ˜ γ ( y, t, z ) − f (cid:48) ( f − ( y )) γ ( f − ( y ) , t, z ) (cid:1) m ( dz ) , ˜ σ ( y, t ) = f (cid:48) ( f − ( y )) σ ( f − ( y ) , t ) , ˜ γ ( y, t, z ) = f ( f − ( y ) + γ ( f − ( y ) , t, z )) − y. We define also the function ˜ ω ( y ) := ω ( f − ( y )) and ˜ g ( y ) := g ( f − ( y )) . We can now verify that the functions ˜ µ ( y, t ) , ˜ σ ( y, t ) , ˜ γ ( y, t, z ) and ˜ g ( y ) satisfy conditions (2 . − (2 . from[58, Section 2]. Let v ( y, t ) := V ω E ( f − ( y ) , t ) . From [58, Theorem 3.1] it follows that v ( y, t ) is a viscosity solution to(75) ˜ L v ( y, t ) = ˜ f ( y, t ) , ( y, t ) ∈ R × [0 , T ) ,v ( y, T ) = ˜ g ( y ) , y ∈ R , where ˜ L v ( y, t ) = − ∂v ( y, t ) ∂t − ˜ σ ( y, t )2 ∂ v ( y, t ) ∂y − ˆ µ ( y, t ) ∂v ( y, t ) ∂y + ˜ ω ( y ) v ( y, t ) with ˆ µ ( y, t ) = ˜ µ ( y, t ) − (cid:90) R ˜ γ ( y, t, z ) m ( dz ) and ˜ f ( y, t ) = − (cid:90) R ( v ( y + ˜ γ ( y, t, z ) , t ) − v ( y, t )) m ( dz ) . In addition, using [58, Prop. 3.3] yields that v ( y, t ) ∈ C ( R × [0 , T ]) and it satisfies(76) | v ( y , t ) − v ( y , t ) | ≤ C ((1 + | y | ) | t − t | + | y − y | ) for some C > and for all t , t ∈ [0 , T ] and y , y ∈ R . Based on (76) and assumptions put on γ we canconclude that ˜ f ( y, t ) ∈ C α ( R × [0 , T ]) ∩ C pol ( R × [0 , T ]) . Then applying [39, Thm. A.14] give us the existenceof a unique classical solution w ( y, t ) to (75) such that w ( y, t ) ∈ C , ( R × [0 , T )) ∩ C pol ( R × [0 , T ]) . In view J. Al-Hadad — Z. Palmowski of the fact that w ( y, t ) is continuous, we can observe that ˜ f ( y, t ) is Lipschitz continuous in y , uniformlyin t . Hence from [58, Lem. 3.1] we know that that w ( y, t ) is also Lipschitz continuous in y , uniformly in t . From the uniqueness result given in [58, Thm. 4.1] we can deduce that v ( y, t ) = w ( y, t ) . Applying [39,Thm. A.18.] we find that v ( y, t ) ∈ C , α ( R × [0 , T ]) . Changing back to the original coordinates, it followsthat V ω E ( s, t ) ∈ C , α ( R + × [0 , T ]) ∩ C pol ( R + × [0 , T ]) and it satisfies (46). (cid:3) Proof of Lemma 17 . The proof follows in the same way as the proof of Lemma 16. However, thistime we apply [39, Thm. A.20] which guarantees the existence of a unique classical solution w ( y, t ) of(75) satisfying w ( y, t ) ∈ C , pol ( R × [0 , T ]) . Hence, coming back to the original coordinates, we have that V ω E ( s, t ) ∈ C , pol ( R + × [0 , T ]) . Therefore, there exist constants n > and K > such that (cid:12)(cid:12)(cid:12)(cid:12) ∂ V ω E ( s, t ) ∂s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ( s − n + s n ) for all ( s, t ) ∈ R + × [0 , T ] . This completes the proof. (cid:3) Proof of Lemma 19 . By the dynamic programming principle formulated e.g. in [33], the value function V ω B Ξ ( s, t ) satsifies(1) At time t = T , the value function V ω B Ξ ( s, t ) is equal to g ( s ) .(2) Given the price V ω B Ξ ( s, t n ) at the time t n = n Ξ T , the price at time t n − = n − Ξ T is V ω B Ξ ( s, t n − ) =max { E s,t n − [ e − (cid:82) tntn − ω ( S w ) dw V ω B Ξ ( S t n , t n )] , g ( s ) } .Thus, the price V ω B Ξ ( s, t n − ) of a Bermudan option at t = t n − can be calculated inductively as the maximumof the payoff function g and the price of a European option with expiry t n and payoff function V ω B Ξ ( S t n , t n ) .From Theorem 15 we know that the value function of European option is convex in s provided the payofffunction is convex, and since the maximum of two convex functions is again a convex function, we concludethat the Bermudan option price V ω B Ξ ( s, t ) is convex in s for all t ∈ [0 , T ] . (cid:3) Concluding remarks
In this paper, we have identified the value function in the optimal stopping problem with functional dis-counting. We have performed a numerical analysis as well.It is tempting to analyse other discounting functions. For example ω might ba a random function or justsimply a random variable dependent on the asset process S t . One can take other processes as a discountrate where the dependence is introduced not only via correlation between gaussian components but viacommon jump structure. This jump-dependence is crucial since crashes in the market affect large portionof business at the same time; see e.g. [22].One can take Poisson version of American options where exercise is possible only at independent Poissonepochs as well. First attempt for classical perpetual American options has been already made in [55]. Webelieve that present analysis can be generalised to this set-up.Obviously, it would be good to work out details for different payoff functions, hence for various options.One could think of barrier options, Russian, Israeli or Swing options. What is maybe even more interestingfor the future analysis is taking into account Markov switching markets and using omega scale matricesintroduced in [25]. We expect that in this setting the optimal exercise time is also the first entrance timeto the interval which ends depend on the governing Markov chain. Data Availability Statement.
Data sharing not applicable – no new data generated, or the article describesentirely theoretical research erpetual American options with asset-dependent discounting References [1] Aase, K.K. (2010). The perpetual american put option for jump-diffusions. In Endre Bjørndal, M. Bjørndal, P. M.Pardalos, and M. Rönnqvist, editors,
Energy, Natural Resources and Environmental Economics , 493–507. Springer BerlinHeidelberg.[2] Alili, L. and Kyprianou, A. (2005). Some remarks on first passage of Lévy processes, the American put and pastingprinciples.
The Annals of Applied Probability , 15(3), 2062–2080.[3] De Angelis, T. and Peskir, G. (2019). Global C Regularity of the Value Function in Optimal Stopping Problems. Toappear in
Annals of Applied Probability .[4] Asmussen, S., Avram, F. and Pistorius, M. (2004). Russian and American put options under exponential phase-type Lévymodels.
Stochastic Processes and their Applications , 109, 79–111.[5] Avram, F., Kyprianou, A. and Pistorius, M. (2004). Exit problems for spectrally negative Lévy processes and applicationsto (Canadized) Russian options.
The Annals of Applied Probability , 14(1), 215–238.[6] Battauz, A., De Donno M., and Sbuelz, A. (2012). Real options with a double continuation region.
Quantitative Finance ,12(3), 465–475.[7] Battauz, A., De Donno M. and Sbuelz, A. (2014). Real options and American derivatives: the double continuation region.
Management Science , 61(5), 1094–1107.[8] Battauz, A., De Donno M. and Sbuelz, A. (2017). On the exercise of American quanto options.
Preprint .[9] Baurdoux, E. and Kyprianou, A. (2008). The McKean stochastic game driven by a spectrally negative Lévy process.
Electronic Journal of Probability , 8, 173–197.[10] Baurdoux, E. (2009). Last exit before an exponential time for spectrally negative Lévy processes.
Journal of AppliedProbability , 46(2), 542–558.[11] Beibel, M. and Lerche, R. (2001). Optimal Stopping of Regular Diffusions under Random Discounting.
Theory of Proba-bility & Its Applications , 45(4), 547–557.[12] Bensoussan, A. and Lions, J.L. (1987).
Impulse Control and Quasi Variational Inequalities . Wiley, New York[13] Bergenthum, J., Ruschendorf, L. (2006). Comparison of option prices in semimartingale models.
Finance and Stochastics ,10, 222–249.[14] Bergman, Y.Z., Grundy, B.D. and Wiener, Z. (1996). General properties of option prices.
Journal of Finance , 51, 1573–1610.[15] Boyarchenko, S.I. and Levendorskii, S.Z. (2002). Perpetual American options under Lévy processes.
SIAM Journal ofControl and Optimization , 40, 1663–1696.[16] Broadie, M. and Detemple, J. (1995). American Capped Call Options on Dividend–Paying Assets.
Review of FinancialStudies , 8(1), 161–191.[17] Carr, P. and Chesney, M. (1996). American put call symmetry.
Preprint .[18] Carr, P., Madan, D., Geman, H. and Yor, M. (2002). The fine structure of asset returns, an empirical investigation.
Journal of Business , 75(2), 305–332.[19] Chan, T. (2005). Pricing Perpetual American options driven by spectrally one–sided Lévy processes.
Exotic OptionPricing and Advanced Lévy Models , ed. Kyprianou. John–Wiley and Sons Inc., England, 195-–216.[20] Chen, X., Chadam, J., Jiang, L. and Zheng, W. (2008). Convexity Of The Exercise Boundary Of The American PutOption On A Zero Dividend Asset.
Mathematical Finance , 18(1), 185–197.[21] Chesney, M. and Jeanblanc, M. (2004). Pricing American currency options in an exponential Lévy model.
Applied Math-ematical Finance , 11, 207–225.[22] Christensen, K., Oomen, R.C.A. and Renó, R. (2018). The drift burst hypothesis. Available at SSRN.[23] Cohen, S., Kuznetsov, A., Kyprianou, A. and Rivero, V. (2013).
Lévy Matters II . Berlin, Heidelberg: Springer.[24] Cont, R. and Tankov, P. (2004).
Financial Modelling with Jump Processes . Boca Raton, FL: Chapman & Hall.[25] Czarna, I., Kaszubowski, A, Li, S. and Palmowski, Z. (2020). Fluctuation identities for omega-killed Markov additiveprocesses and dividend problem.
Advances in Applied Probability , 52(2).[26] De Donno, M., Palmowski, Z. and Tumilewicz, J. (2020). Double continuation regions for American and Swing optionswith negative discount rate in Lévy models.
Mathematical Finance , 30(1), 196–227.[27] Detemple, J. and Kitapbayev, Y. (2018). American Options with Discontinuous Two–Level Caps.
SIAM Journal onFinancial Mathematics , 9(1), 219–250.[28] Detemple, J. (2001). American options: symmetry properties. In Musiela Jouini, Cvitanić, editor.
Option pricing, interestrates and risk management , 67–104. Cambridge University Press.[29] Detemple, J. (2014). Optimal exercise for derivative securities.
Annual Review of Financial Economics , 6, 459–487. J. Al-Hadad — Z. Palmowski [30] Eberlein, E. and Keller, U. (1995). Hyperbolic distributions in finance.
Bernoulli , 1, 281–299.[31] Eberlein, E. and Papapantoleon, A. (2005). Symmetries and pricing of exotic options in Lévy models. In Wilmott Kypri-anou, Schoutens, editor,
Exotic Option Pricing and Advanced Lévy Models . Wiley Finance.[32] Ekström, E. Janson, S. and Tysk, J. (2005). Superreplication of options on several underlying assets.
Journal of AppliedProbability , 42, 27–38.[33] Ekström, E. (2004). Properties of American option prices.
Stochastic Processes and their Applications , 114(2), 265–278.[34] Ekström, E. and Tysk, J. (2007). Properties of option prices in models with jumps.
Mathematical Finance , 17(3), 381–397.[35] Ekström, E. and Tysk, J. (2007). Convexity preserving jump–diffusion models for option pricing.
Journal of MathematicalAnalysis and Applications , 330(1), 715–728.[36] El Karoui, N., Jeanblanc-Picque, M. and Shreve, S. (1998). Robustness of the Black and Scholes formula.
MathematicalFinance , 8, 93–126.[37] Hobson, D. (1998). Volatility misspecification, option pricing and superreplication via coupling.
Annals opf AppliedProbability , 8, 193–205.[38] Fajardo, J. and Mordecki, E. (2006). Symmetry and duality in Lévy markets.
Quantitative Finance , 6(3), 219–227.[39] Janson, S. and Tysk, J. (2004). Preservation of convexity of solutions to parabolic equations.
Journal of DifferentialEquations , 206(1), 182–226.[40] Janson, S. and Tysk, J. (2003). Volatility time and properties of options.
Annals of Applied Probability , 13, 890–913.[41] Kijima, M. (2002). Monotonicity and convexity of option prices revisited.
Mathematical Finance , 12, 411–425.[42] Klimsiak, T. and Rozkosz, A. (2018). The valuation of American options in a multidimensional exponential Lévy model.
Financial Mathematics , 28(4), 1107–1142.[43] Koponen, I. (1995). Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussianstochastic process.
Physical Review E , 52, 1197–1199.[44] Krylov, N. (1980).
Controlled Diffusion Processes . Springer Verlag.[45] Kyprianou, A. (2006).
Introductory Lectures on Fluctuations of Lévy Processes with Applications . Berlin, Heidelberg:Springer–Verlag Berlin Heidelberg.[46] Kyprianou, A. and Surya, B. (2007). Principles of smooth and continuous fit in the determination of endogenous bank-ruptcy levels.
Finance and Stochastics , 11(1), 131–152.[47] Lamberton, D. and Mikou, M. (2012). The smooth-fit property in an exponential Lévy model.
Journal of Applied Prob-ability , 49(1), 137–149.[48] Lamberton, D. (2009). Optimal stopping with irregular reward functions.
Stochastic Processes and their Applications ,119, 3253–3284.[49] Li, B. and Palmowski, Z. (2018). Fluctuations of omega–killed spectrally negative Lévy processes.
Stochastic Processesand Their Applications , 128(10), 3273—3299.[50] Madan, D.B. and Seneta, E. (1990). The variance gamma model for share market returns.
Journal of Business , 63,511–524.[51] Merton, R. (1976). Option pricing when underlying stock returns are discontinuous.
Journal of Financial Economics ,3(1-2), 125–144.[52] Mordecki, E. (2002). Optimal stopping and perpetual options for Lévy processes.
Finance and Stochastics , 6(4), 473–493.[53] Nielssen, O.B. (1998). The McKean stochastic game driven by a spectrally negative Lévy process.
Finance and Stochastics ,1, 41–68.[54] Palmowski, Z., Rolski, T., (2002). A technique for exponential change of measure for Markov processes.
Bernoulli , 8(6),767–785.[55] Palmowski, Z., Pérez, J.L. and Yamazaki, K. (2020). Double continuation regions for American options under Poissonexercise opportunities. Submitted for publication.[56] Peskir, G. and Shiryaev, A. (2006).
Optimal Stopping and Free–Boundary Problems . Basel: Birkhäuser Verlag.[57] Pham, H. (1997). Optimal stopping, free boundary, and American option in a jump-diffusion model.
Applied mathematicsand optimization , 35(2), 145–164.[58] Pham, H. (1998). Optimal Stopping of Controlled Jump Diffusion Processes: A Viscosity Solution Approach.
Journal ofMathematical Systems, Estimation, and Control
8, 1–27.[59] Rodosthenous, N. and Zhang, H. (2018). Beating the omega clock: An optimal stopping problem with random time-horizon under spectrally negative Lévy models.
Annals of Applied Probability , 28(4), 2105–2140.[60] Sato, K. (1999).
Lévy Processes and Infinitely Divisible Distributions.
Cambridge Univeristy Press.[61] Schoutens, W. (2003).
Lévy Processes in Finance: Pricing Financial Derivatives . Wiley. erpetual American options with asset-dependent discounting [62] Strulovici, B. and Szydlowski, M. (2015). On the smoothness of value functions and the existence of optimal strategiesin diffusion models. Journal of Economic Theory
Mathematical Finance , 17(2), 307–317.
Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wyb. Wyspi-ańskiego 27, 50-370 Wrocław, Poland
E-mail address : [email protected] Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, ul. Wyb. Wyspi-ańskiego 27, 50-370 Wrocław, Poland
E-mail address ::