Modern Stochastics: Theory and Applications

The financial derivatives are one of the most important financial instruments against the market risk. By their very nature, they are a major indicator of the investor expectations for the future market behavior. On the other hand, the options are one of the most traded derivatives. They preserve their holder from the market fluctuations when he wants to buy (calls) or sell (puts) some asset at a price no larger (calls) or lower (puts) than a predefined level known as the strike price or simply the strike. Thus the options can be viewed as an insurance instrument. Furthermore, the price structure they generate w.r.t. the strike is very informative for the investors beliefs. On the other hand, two main types – European and American – can be distinguished depending on when the contracts expire. For the European style options, the exercise can be done only at a predefined maturity date. Alternatively, the American options give to their owner the right to chose the moment for exercising. There are further modifications known as exotic options – barrier, Asian, look-back, digital, straddle, strangle, and many others. Although the options exhibit such variety, the American ones have a largest segment amongst all traded options namely due to the property of early exercising preferred by investors. However, there is some difference between the holder and the writer of these options since the writer has only obligations. To overcome this, a new class of financial instruments has been designed, known as the cancellable American options. The main feature distinguishing these derivatives from the regular American options is the writer’s right to cancel the contract prematurely paying some penalty above the usual payoff – a traditional assumption is that it is fixed during the option’s life, but we shall examine more complicated cases.

These derivatives are first introduced in the scientific literature by Kifer (2000) under the name game options. Later, the term Israeli is also used, see Kifer (2013). Regardless of the penalty structure, these financial instruments fall in the field of the so-called Dynkin stochastic games (see Dynkin (1969)). Thus their pricing problem turns into finding of the optimal strategies (if they exist) for both of option’s writer and holder. In the stochastic terms, these strategies have to form a saddle point in the field of the stopping times w.r.t. the natural filtration. Some existence results for the models based on diffusion processes can be found in Friedman (1973), Bensoussan and Friedman (1974), Bensoussan and Friedman (1977), Ekström (2006), Karatzas and Sudderth (2006), Ekström and Villeneuve (2006), Gapeev and Lerche (2011). The results for significantly larger classes of process are derived in Ekström and Peskir (2008) and Peskir (2009) – it turns out that the right continuity leads to the Stackelberg equilibrium, whereas a left continuity w.r.t. the stopping times (quasi-left continuity) is necessary for the Nash equilibrium. Note that some of the most applied stochastic processes in financial modeling exhibit both of these requirements, for example, the Lévy processes, the stochastic differential equations they generate, particularly the diffusuions, etc.

Several important works devoted to the game options are published after Kifer (2000). The call style instruments are examined first in Kunita and Seko (2004). These results are refined later by Emmerling (2012) and Yam et al. (2014). The put style options are explored in Kyprianou (2004), Ekström (2006), Suzuki and Sawaki (2007), and Kühn and Kyprianou (2007). Some exotic game options are investigated in Kyprianou (2004) (Russian), Baurdoux and Kyprianou (2004) (integral options, related to the Asian ones), Gapeev (2005) (spread options), Ekström (2006) (capped options), Guo et al. (2014) (look-back), Guo et al. (2020) (Asian). The cancellable options under some generalized assumptions are examined in Kallsen and Kühn (2004), Hamadène (2006), Kühn et al. (2007), Dumitrescu et al. (2017), Guo and Rutkowski (2017), Guo (2020), Dolinsky (2020), and Palmowski and Stȩpniak (2023).

We position the present study in the framework by Black and Scholes (1973) – the underlying asset is driven by a log-normal process. In addition to the usual assumption for a fixed penalty, we consider a three-component structure – a proportion of the usual payoff, shares of the underlying asset, and a fixed amount. These instruments are investigated with and without maturity constraints in Zaevski (2023, 2025b), see also Ekström and Villeneuve (2006) and Zaevski (2020a) for cancellable options with proportional penalties. Regardless of the penalty structure, it turns out that these instruments may exhibit three different behaviors. In all of them, the holder’s exercise set contains all points below/above some boundary for the put/call style options. The distinction comes from the shape of the writer’s optimal set. For some large enough penalties, the premature cancellation is never optimal – the canceling price is larger than the expected losses. In such a way, the option is regular American. On the other hand, the relatively middle values make the first hitting to the strike the unique writer’s optimal strategy. In this case, the option is rather American than game, since it can be viewed as a derivative that gives an early exercise right to its holder and expires at the maturity or when the underlying asset hits the strike – in the last case the holder receives a predefined amount. If it is L, then we shall entitle the option L-American. Finally, the canceling right has a real impact for the small enough penalties – we shall use the name real game or real cancellable in this case. The writer’s optimal set is an interval with the strike for the right/left endpoint for the puts/calls. The options without maturity horizon determine which of these three cases holds in the sense that they contain the whole information. First, if the perpetual option is regular American, then all finite maturity options are regular American too. Second, if the option is of the L-American kind, then there exists a critical value for the time to maturity above which the option is L-American too, but it is regular American for the lower maturities. Third, if the perpetual option is real cancellable, then there exist two critical values for the time to maturity where the finite maturity option changes its behavior. This importance of the perpetual case motivates the present research. For a survey of such instruments, we refer to Kyprianou (2004), Kunita and Seko (2004), Ekström (2006), Suzuki and Sawaki (2007), Emmerling (2012), Yam et al. (2014), Zaevski (2020a,b,c, 2023), and Gapeev et al. (2021).

The main results of this paper are in the recognition which values of the penalty triple (proportion, shares, and fixed amount) to which case lead. We derive the critical values that distinguish the three possible types. It is interessting to be mentioned that the first component (the proportion of the usual payoff) does not influence the transition between the regular American and L-American type. On the other hand, if the payoff taken at the strike (it depends only on the penalty components related to the number of shares and the fixed amount) is less than the price of the at-the-money regular American option (the initial asset price is the strike), then the option is either L-American or real cancellable. Here appears the impact of the first penalty component. Next we derive iteratively all critical values in this order: proportion of the payoff, number of shares, and finally, the fixed amount. Based on these results, we provide an algorithm for recognizing the option’s type. These relations are examined in detail for the put options, whereas the calls are examined through some symmetry arguments. Several numerical experiments are provided to illustrate and validate the theoretical findings.

It is worth to mention that along with this, we investigate the optimal sets of the L-American options as well as their pricing rules.

The paper is organized as follows. The base we use later is given in Section 2. The L-American options are discussed in Section 3. The results for the put options are obtained in Section 4, whereas the calls are investigated in Section 5. Some numerical examples are provided in Section 6.

Let the underlying asset be driven by the geometric Brownian motion under the filtered probability space $\left(\Omega ,\mathcal{F},{\mathcal{F}_{t}},\mathbb{Q}\right)$. We shall use a superscript to mark the initial value, i.e. ${\mathbb{E}^{t,x}}$ means the expectation under the assumption ${S_{t}}=x$. Also, if $t=0$, we shall mark only the dependence on x. On the other hand, somewhere it is more appropriate if we mark the dependence directly in the process, i.e. ${S_{t}^{x}}$ notates the process if ${S_{0}}=x$. The measure $\mathbb{Q}$ is risk-neutral w.r.t. to the risk-free rate r. We assume that it is a constant during the option’s life – note that it can take negative values. We introduce an extra discount factor with rate λ, assuming $\lambda \ge 0$ and $r+\lambda \gt 0$. Let the option’s strike price be denoted by K. The holder of a regular American option (put or call) receives the amount of if he exercises in a moment t at the spot price ${S_{t}}=x$. Additionally, the cancellable American option gives its writer the right to cancel prematurely paying some amount above the usual payoff. We assume that it consists of three parts: ${\eta _{1}}\ge 1$ being proportion of the payoff, ${\eta _{2}}\ge 0$ shares of the underlying asset, and a fixed amount of ${\eta _{3}}\ge 0$. Thus the holder owes the total amount of if he cancels the option. As a consequence, we have a stochastic game (see Dynkin (1969)) between two players – the option’s holder and writer.

Let us denote by ${n_{1}}\left(x\right)$ and ${n_{2}}\left(x\right)$ the respective undiscounted payoffs, i.e. for the puts and for the calls.

We assume now that $t=0$ since the model is time-homogeneous in the perpetual case. Suppose that the buyer’s (holder’s) strategy is to exercise in the stopping time ${\tau ^{b}}$ and the seller’s (writer’s) one is another stopping time ${\tau ^{s}}$. The financial result of these strategies at the point x is The option’s holder/writer has to maximize/minimize the value of (6) w.r.t. all stopping times. Based on Theorem 2.1 from Ekström and Peskir (2008), it is proven in Zaevski (2023) that this stochastic game exhibits a Nash equilibrium (see also Peskir (2009)). We shall denote its value function by where $\mathcal{T}$ is the set of all stopping times. Hence, ${n_{1}}\left(x\right)\le V\left(x\right)\le {n_{2}}\left(x\right)$. The following lemma gives the time-relations for the price function.

Having in mind Lemma 2.1, we define the holder’s and writer’s optimal sets as the points $\left(t,x\right)$ for which $V\left(x\right)={n_{1}}\left(x\right)$ or $V\left(x\right)={n_{2}}\left(x\right)$, respectively.1 Thus the optimal strategies ${\tau ^{b}}$ and ${\tau ^{s}}$ can be defined as We shall denote the optimal sets by ${\Upsilon ^{b}}$ and ${\Upsilon ^{s}}$. The boundaries of these sets are known as early exercise or optimal boundaries. We shall discuss in detail the particular form of the optimal sets and their boundaries in the corresponding sections devoted to the put and call options.

The existence of two optimal boundaries (holder’s and writer’s) leads to a problem for the first exit from a strip of the underlying asset or, equivalently, of a Brownian motion. We need the following well-known result for diffusion processes – for the proof see Darling and Siegert (1953) or Lehoczky (1977).

If diffusion (10) is a Brownian motion with drift, then the functions $\mu \left(\cdot \right)$ and $\sigma \left(\cdot \right)$ are constants, and the last one is equal to one. Thus ODE (11) turns into It is characterized by the quadratic equation the solutions of which are Note that they are real since $y\ge 0$. Thus a pair of fundamental solutions is ${f_{1,2}}\left(u\right)={e^{{u_{1,2}}u}}$. Therefore, the Laplace transforms (12) can be written as Suppose that the underlying asset starts from the point ${S_{0}}=x$. Note that the first exit of process (1) from a strip $0\lt A\lt {S_{0}}\lt B$ is equivalent to the exit of a Brownian motion with drift from the strip $\left(a,b\right)$, where $a=\frac{1}{\sigma }\ln \frac{A}{x}$ and $b=\frac{1}{\sigma }\ln \frac{B}{x}$. Thus the Laplace transforms (16) taken at the point of the total discount rate, $y=r+\lambda $, turn into where the constants p and q are expressed in roots (15) as We have $p\ge q+1$ and the equality holds only when $\lambda =0$. We can derive the Laplace transforms of the one-sided hits taking $B=\infty $ and $A=0$, respectively:

We need the following relations between the constants p and q.

Let us define a new American-style financial instrument – we name it an L-American option.

We can view the L-American options as financial instruments with stochastic maturity (the moment when the asset reaches the strike) with final payout L. In this light, the payoff ${N_{1}}\left(t,x\right)$ that the holder can receive is continuous whereas the payout at this stochastic maturity is a specification of the option contract. Note that the points not below (not above) the strike are never optimal for the puts (calls) since the payoff is zero-valued in this region.

We shall denote the price of an L-American option by ${V_{L}}\left(\cdot \right)$. Thus, if we denote by ${\tau ^{y,x}}$ the first hitting moment of an asset starting at y to the value x, then

Let $\mathcal{B}$ be the differential operator We need the following well-known result for the optimal stopping problems – see for example van Moerbeke (1973) or Jacka (1992).

We shall prove a proposition that characterizes the optimal sets of L-American options.

Proposition 3.3 means that the optimal set of an L-American put option contains all points below some boundary. Furthermore, this boundary is a constant during the time, since there are no maturity constraints. Under the dividend parametrization, the optimal boundary of a put style L-American option as well as its price are discussed in Theorem 2 of Kyprianou (2004), in Section 3.1 of Ekström (2006), and in Theorem 3.1 of Suzuki and Sawaki (2007); for the calls, see formulas (3.1) from Emmerling (2012) and (2.12) from Yam et al. (2014). We need the following lemma prior to providing the related results under the discount parametrization.

Some symmetrical arguments lead to the result for the call style L-American options.

We need the following definition for further distinction.

For our further purposes, we need the price of the regular American options under the perpetual assumptions. The optimal boundary and the price can be obtained in a closed form since the boundary is time independent. Under the dividend parametrization, this is made in many studies, for example, see formula (52) from Merton (1973), formulas (9) and (15) from Kim (1990), Proposition 2.3 from Jacka (1991), Theorem 7.2 from Karatzas and Shreve (1998), or formula (5.1.10) from Kwok (2008). Under the discount parametrization, these results can be found in Theorems 1 and 2 from Shiryaev et al. (1995) and in Theorems 6.1 and 6.2 of Zaevski (2021). If we denote by ${V_{a}}\left(\cdot \right)$ the price function, we can write for a put-styled option Particularly, for $x=K$, we define the important value $\overline{\eta }$,

We consider now the cancellable call options through some symmetrical arguments. Some proofs will be omitted since they are similar to the put versions. The shape of the optimal sets for the calls is symmetric w.r.t. the strike to those for the puts. The respective results for the fixed penalties under the dividend parametrization are obtained in Kunita and Seko (2004), Ekström and Villeneuve (2006), Emmerling (2012), and Yam et al. (2014) whereas proportional to the usual payoff penalties are considered in Ekström and Villeneuve (2006). The options with three-component penalties are examined in Zaevski (2023). Note that the case $\lambda =0$ is special – the early exercise is never optimal for the option’s holder. All necessary results in this case are obtained in Theorem 3.9 from the same work.

Suppose now that $\lambda \gt 0$ or equivalently $p\gt q+1$. Note that we have to consider the related functions in the interval $\left(1,\infty \right)$ instead of $\left(0,1\right)$ since the possible exercise boundaries are above the strike. The price of the perpetual American call option when ${S_{0}}=K$ is $\overline{\eta }:=K{\overline{\xi }_{2}}$, where The holder’s optimal set is an interval $\left(A,\infty \right)$ for some constant A not below the strike. The writer’s one can be the empty set, the singleton $\left\{K\right\}$, or an interval $\left[K,B\right]$, $K\lt B\lt A$. The constants ${\xi _{1}}$ and ${\xi _{2}}$ are defined now as Note that the constant L keeps its value. It is proven in Proposition 3.2 of Zaevski (2023) that canceling is never optimal for the writer if ${\eta _{3}}\ge {\eta _{1}}K$. Thus we consider only the values of ${\xi _{1}}$ and ${\xi _{2}}$ such that ${\xi _{1}}\gt {\xi _{2}}$. The restriction presented in Proposition 4.1 also holds but with the actual value of $\overline{\eta }$. Furthermore, the analogue of Theorem 4.2 gives the criteria for the option to be real cancellable. We summarize these results in the following theorem.

Having in mind that the function $h\left(\cdot \right)$ is taken for $\xi =-\frac{L}{K}$, we see that Proposition 4.3 still holds. Theorem 5.1 shows that we have to find the right derivative of the price function (36) at the strike: We shall proceed further using the method presented in the beginning of Section 4.2. Proposition 4.4 is true for the function We need to know when the function $g\left(\cdot \right)$ has a positive domain larger than one, i.e. when inputs larger than one make the function positive. The function related to its derivative $m\left(a;{\xi _{1}},{\xi _{2}}\right)$ now takes the form Its endpoints are always negative except in the limiting case since $\xi \ge 1$ and ${\xi _{2}}\ge 0$. The behavior of function (74) is similar to the put case considered in Lemma 4.5 – we can recognize the following three cases:

The sign of the risk free rate is again important. The analogue of Proposition 4.6 is as follows.

In addition to this proposition, we can provide financial arguments similar to those in Remark 7 why the option cannot be real cancellable when $r\lt 0$. The important term $\left(\mathcal{B}{n_{2}}\right)\left(K\right)$ now is $\left(\mathcal{B}{n_{2}}\right)\left(K\right)=K\left(r{\eta _{1}}-\lambda {\eta _{2}}\right)-\left(r+\lambda \right){\eta _{3}}$ and it can be positive only when $r\gt 0$.

Suppose now that $r\gt 0$ or equivalently $p\lt 2q+1$. The condition obtained in Proposition 4.7 can be rewritten as follows.

The next step is to prove that if the function $g\left(\cdot \right)$ has a positive domain, then the point ${a^{\ast }}$ belongs to it. Let us keep the meaning of ${\overline{a}_{1}}\left({\xi _{1}},{\xi _{2}}\right)\gt {\overline{a}_{2}}\left({\xi _{1}},{\xi _{2}}\right)$, i.e. the function $g\left(a;{\xi _{1}},{\xi _{2}}\right)$ achieves its maximum and minimum at these points, respectively. If $g\left({\overline{a}_{1}}\left({\xi _{1}},{\xi _{2}}\right);{\xi _{1}},{\xi _{2}}\right)\gt 0$, then the function $g\left(a;{\xi _{1}},{\xi _{2}}\right)$ has two roots: ${a_{1}}\left({\xi _{1}},{\xi _{2}}\right)$ and ${a_{2}}\left({\xi _{1}},{\xi _{2}}\right)$. The analogues of Propositions 4.9 and 4.10 and their proofs are identical to the original ones and we omit them. The important Proposition 4.11 still holds – the unique difference in the proof is in the presentation (98) of the function $h\left(a;{\xi _{2}}\right)$. In the call case, it is

Thus we reach the corresponding results for the call options.

We present now some examples. Let us consider first put style options with parameters $r=-0.02$, $\lambda =0.03$, $\sigma =0.3$, $K=1$. We chose these values because $r+\lambda \gt 0$ and $r\lt 0$, see point one from Theorem 4.16. The value one for the strike is chosen this way to ignore its impact since it can be viewed as a scaling parameter. The results are visualized in Figure 1a. Theorem 4.16 shows that the triples $\left\{{\eta _{1}},{\eta _{2}},{\eta _{3}}\right\}$ that lead to real cancellable options are in the pyramid formed by the green points and the point $\left(1,0,0\right)$. The green points are obtained as follows:

The value for $L={\eta _{2}}K+{\eta _{3}}$ that distinguishes the L-American options from the regular ones is given by formula (39). Its value is 0.6537, see the yellow points. Thus the triples $\left\{{\eta _{1}},{\eta _{2}},{\eta _{3}}\right\}$ that lead to an L-American option are in the prism between the plains $\left\{{\eta _{1}}=1\right\}$, $\left\{{\eta _{2}}=0\right\}$, $\left\{{\eta _{3}}=0\right\}$, and the blue one, cut by the above-mentioned pyramid for the real cancellable options (the red plain). The triples that lead to the regular American options are above the blue plain – they are

Let us consider the call style options. We use the same parameters except the risk-free rate – we assume now that $r=0.02$ due to the first point of Theorem 5.4. The results are presented in Figure 1b. The critical values that form the pyramid for the real cancellable options are ${\xi _{1}^{\ast }}=1.0843$, ${\eta _{2}^{\ast }}\left(1\right)=0.0374$, and ${\xi _{2}^{\ast }}\left(1\right)=0.0698$. Critical value (70) for $L={\eta _{2}}K+{\eta _{3}}$ above which the option is real cancellable is 0.4510.

Some particular values are presented for put and call options in Tables 1 and 2, respectively. We give the critical value ${\xi _{1}^{\ast }}$ for the coefficient ${\eta _{1}}$ in the head of the tables. Above this level, the option turns from real cancellable into L-American. The critical values for ${\eta _{2}}$ given ${\eta _{1}}$ are presented in a separate column. The rest of the tables contain the critical values for ${\eta _{3}}$ given ${\eta _{1}}$ and ${\eta _{2}}$. The value for L above which the option is regular American, $\overline{\eta }$, is given again in the head of the tables.