Modern Stochastics: Theory and Applications

The following restriction on the penalty coefficients appears: if ${\eta _{3}}\ge {\eta _{1}}K$, then early canceling is never optimal for the writer. Hence, the option would be a pure American.

Hereafter we assume that ${\eta _{3}}\lt {\eta _{1}}K$. The following propositions hold.

Propositions 3.1, 3.3, and 3.5 indicate that the holder’s exercise region has the form ${\Upsilon ^{b}}=[B,\infty )$ for some constant $B\gt K$, whereas the writer’s one is the interval ${\Upsilon ^{s}}=[K,A]$ (A is a constant less than B, $K\lt A\lt B$), the singleton ${\Upsilon ^{s}}=\{K\}$, or the empty set ${\Upsilon ^{s}}\equiv \varnothing $.

Suppose that the starting point x is between the boundaries A and B, $A\lt x\lt B$, and let us denote the option price as $f(A,B,x)$ under these assumptions. In such a way the pricing problem turns to a problem of first exit of a Brownian motion with drift $\psi =\frac{r}{\sigma }-\frac{\sigma }{2}$ from the strip $({A_{1}},{B_{1}})$ for If we denote by ${\tau ^{A}}$ and ${\tau ^{B}}$ the first hitting moments to the values ${A_{1}}$ and ${B_{1}}$, then the option price can be derived as Let the constants p and q be defined as We have that $p\ge q+1$, equality holds only in the undiscounted case $\lambda =0$. Assume now that discounting really exists, i.e. $\lambda \gt 0$ or equivalently $p\gt q+1$. Using equations (45) and (46) from Appendix A we convert call price (14) to Let us fix the upper boundary B. After the substitution $a=\frac{A}{B}$, $k=\frac{K}{B}$, $\xi =\frac{{\eta _{3}}}{B}$, and $y=\frac{x}{B}$, formula (16) turns to We have the order $0\le \frac{\xi }{{\eta _{1}}}\lt k\lt a\lt 1$, because ${\eta _{3}}\lt {\eta _{1}}K$. Since the option’s writer minimizes his financial result, we have to derive for which value of a the function is smallest in the interval $(0,1)$. Its derivative is We prove in Proposition B.2 that function (19) has a unique root in the interval $(0,1)$ – we denote it by $a(B)$. It leads to the minimum of price function (17).

Let now fix the writer’s boundary A. We have to find which value B maximizes function (16), since the option’s holder maximizes his financial utility. Let us change the variables as $b=\frac{B}{A}$, $k=\frac{K}{A}$, $\xi =\frac{{\eta _{3}}}{A}$, and $y=\frac{x}{A}$. Now the order is $0\le \frac{\xi }{{\eta _{1}}}\lt k\le 1\lt b$. Thus option’s price function (16) turns to Hence, we have to derive the maximum of the function Its derivative is We show in Proposition B.3 that this function has just one root in the interval $(1,\infty )$ except in the marginal case ${\eta _{2}}={\eta _{3}}=0$, which is examined in [65]. We shall denote the root by $b(A)$. Thus pricing function (20) has a maximum for $B=b(A)A$.

Let us denote the true boundaries (if they exist) by ${A^{\ast }}$ and ${B^{\ast }}$. We search the potential writer’s optimal boundary as the unique solution $\overline{A}$ of the equation $yb(y)a(yb(y))=y$ or equivalently If $\overline{A}\ge K$, then this is the true boundary, i.e. ${A^{\ast }}=\overline{A}$ and ${B^{\ast }}=b(\overline{A})\overline{A}$. It may happen that $\overline{A}\lt K$, because we have changed the original payment functions in formula (14) from ${N_{1}}(t,x)={e^{-\lambda t}}{(x-K)^{+}}$ and ${N_{2}}(t,x)={e^{-\lambda t}}({\eta _{1}}{(x-K)^{+}}+{\eta _{2}}x+{\eta _{3}})$ to ${N_{1}}(t,x)={e^{-\lambda t}}(x-K)$ and ${N_{2}}(t,x)={e^{-\lambda t}}({\eta _{1}}(x-K)+{\eta _{2}}x+{\eta _{3}})$, respectively. Note that we are just in this case when $r\lt 0$ due to Proposition 3.7. Hence, if $\overline{A}\lt K$ we have to recognize whether the writer’s exercise region is the singleton $\{K\}$ or the empty set. Suppose that the initial asset price is the strike, $x=K$. The writer has the alternatives to cancel immediately or to do nothing. In the first case he has to pay amount of ${\eta _{2}}K+{\eta _{3}}$, whereas in the second one the option turns to ordinary American. It is shown in [68], Proposition 6.1, that its price is for $\gamma =p-q$. We conclude that if ${\eta _{2}}K+{\eta _{3}}\le \overline{\eta }$, then the writer would prefer to cancel the option immediately, i.e. ${\Upsilon ^{s}}=\{K\}$ and thus ${A^{\ast }}=K$ and ${B^{\ast }}=b(K)K$. Otherwise, if ${\eta _{2}}K+{\eta _{3}}\gt \overline{\eta }$, then ${\Upsilon ^{s}}=\varnothing $, which means that the option is ordinary American, ${A^{\ast }}$ does not exist, and ${B^{\ast }}=\frac{\gamma }{\gamma -1}K$, see Proposition 6.1 from [68].

Note at last that if the writer’s exercise region is not empty and the asset starts below the strike, $x\lt K$, then the writer cancels when the asset hits the strike and this strategy leads to the option price due to Proposition A.1.

We summarize the derived results in the following theorem.

Let us discuss now the smooth fit principle, i.e. when the derivative of the value function $V(x)$ is continuous at the optimal boundaries. We have that ${V^{\prime }}(x)=1$ for $x\in {\Upsilon ^{b}}$ and ${V^{\prime }}(x)={\eta _{1}}+{\eta _{2}}$ for $x\in {\Upsilon ^{s}}$. If $x\gt K$ and $x\in \overline{\Upsilon }$, then we derive ${V^{\prime }}(x)$ differentiating formula (16): Let us check the smooth fit at the holder’s boundary. Using again the change of variables ${b^{\ast }}=\frac{{B^{\ast }}}{{A^{\ast }}}$, $k=\frac{K}{{A^{\ast }}}$, $\xi =\frac{{\eta _{3}}}{{A^{\ast }}}$, and $y=\frac{x}{{A^{\ast }}}$, we derive for derivative (25) at the point ${b^{\ast }}$ We can easily check that ${V^{\prime }}({b^{\ast }})=1$, because ${b^{\ast }}$ is the root of function (49). Hence, there is a smooth fit at the holder’s boundary. Analogously, we can establish the smooth fit at the writer’s boundary ${A^{\ast }}$ when ${A^{\ast }}=\overline{A}\ge K$ having in mind that we use the root of function (48) to derive the value of $\overline{A}$. Otherwise, if $\overline{A}\lt K$, then we have not smooth fitting at the writer’s boundary, namely the strike, because it is not ${B^{\ast }}$-writer optimal. Note that ${B^{\ast }}$ is K-holder optimal which confirms the smooth fit at ${B^{\ast }}$.

On the other hand, all points below the strike belong to the continuation region. Suppose that the writer’s optimal region is not empty. We cannot expect lower smooth fit at the strike because the writer’s payoff function ${N_{2}}(t,x)$ is not smooth namely at the strike.

We can summarize: we have always smooth fit at the holder’s boundary, but only when $\overline{A}\ge K$ at the writer’s one.

Suppose now, that $\lambda =0$ or equivalently $p=q+1$. We have $r\gt 0$, because the total discount factor is positive. Proposition 3.6 shows that it is never optimal for the holder to exercise the option. Hence, his boundary is infinitely large. This conclusion is supported by the fact that derivative (21) is always positive, which is proven in Proposition B.3. Thus the holder maximizes his utility for $B=\infty $. Suppose that the writer’s optimal boundary is a constant $A\ge K$ and the underling asset starts above it, $x\gt A$. Let us denote the first hitting moment of the asset to the level A by ζ and the price of a down-and-out barrier option with strike K and barrier A by ${C_{DO}}(x,A,K)$. Therefore, the option price can be presented as the following dependent on A function Using Proposition A.1 and equation (10.45) from [71] we obtain Substituting equations (28) into (27) we derive for the option price Its derivative is ${F^{\prime }}(A)=\frac{1}{A{\sigma ^{2}}}{(\frac{A}{x})^{\frac{2r}{{\sigma ^{2}}}}}h(A)$, where the function $h(A)$ is It is linear and increasing with root Note that it is positive. We have two cases to examine – suppose first that $M\le K$. Thus derivative ${F^{\prime }}(A)$ is always positive for $A\gt K$. Hence, the price function (29) is increasing and therefore its minimum is for $A=K$. We have to recognize whether the writer’s exercise region is the singleton $\{K\}$ or the empty set. Suppose that the option is at-the-money, i.e. the initial asset price is the strike. The writer has the alternatives to cancel the option immediately or to do nothing. If he chooses the first one, then he has to pay ${\eta _{2}}K+{\eta _{3}}$. The second alternative turns the option to European, since the holder will never exercise earlier, too. Its price is just the initial asset value – we can see this if we take the limit $T\to \infty $ in the Black–Scholes formula. Hence, $\Upsilon \equiv \{K\}$ when ${\eta _{2}}K+{\eta _{3}}\le K$, and it is the empty set otherwise. Thus, if ${\eta _{2}}K+{\eta _{3}}\le K$, the option price (29) takes the form when $x\ge K$. Note that function (32) is increasing, since $K({\eta _{2}}-1)+{\eta _{3}}\lt 0$ and therefore its minimum is for $x=K$ and it is ${\eta _{2}}K+{\eta _{3}}$. When $x\lt K$ we use Proposition A.1 to derive the option price as If ${\eta _{2}}K+{\eta _{3}}\gt K$, early exercising is never optimal neither for the writer nor for the holder. Hence, the option turns to a European one and its price is the initial asset price x.

Suppose now that $M\gt K$ and therefore function (30) is negative for $A\in (K,M)$ and positive for $A\gt M$. Therefore price function (29) has a minimum for $A=M$. This means that the exercise boundary is given by formula (31) and the writer’s exercise region is the interval $(K,M)$. Hence, the option price is given by formula (33) when $x\lt K$. Also, if $x\ge M$, then option price formula (29) turns to

We can summarize the results above in the following theorem.

As we have seen above, the penalty coefficients ${\eta _{1}}$, ${\eta _{2}}$, and ${\eta _{3}}$ influence significantly the option behavior. Roughly said, the option looks more like the corresponding ordinary American call when they are larger. We shall see for which values the cancellable feature has its impact. First, Proposition 3.2 says that ${\eta _{3}}$ is limited by the inequality ${\eta _{3}}\lt {\eta _{1}}K$. It turns out that this evaluation is too weak. We know that if the writer’s optimal set is not empty, then the strike belongs to it. Hence, ${\eta _{2}}K+{\eta _{3}}\lt \overline{\eta }$, where $\overline{\eta }$ is given by equation (23). Obviously, this equation is stronger due to $\overline{\eta }\lt K$. Otherwise, the assumption ${\Upsilon ^{s}}=\varnothing $ means that the nonuse of the canceling right is the best writer’s strategy, particularly better than the immediate exercise. So, the inequality ${\eta _{2}}K+{\eta _{3}}\ge \overline{\eta }$ holds and hence it determines whether the option is ordinary American or cancellable. Also, we can see that if the number of shares, ${\eta _{2}}$, is larger than $\frac{1}{\gamma }{(\frac{\gamma -1}{\gamma })^{\gamma -1}}$, then the option is ordinary American ($\gamma =p-q$). Otherwise, the value $K[\frac{1}{\gamma }{(\frac{\gamma -1}{\gamma })^{\gamma -1}}-{\eta _{2}}]$ is critical for the fixed amount ${\eta _{3}}$: if it is larger, then the option is ordinary American; otherwise it is a real cancellable option. We have to mention an important fact that the option’s essence does not depend on the coefficient ${\eta _{1}}$ – it influences whether the writer’s optimal set is only the strike once we know that the option is real cancellable.

Having in mind the previous restrictions, we examine call options with the following parameters: the risk free rate $r=0.05$, the discount factor $\lambda =0.01$, the volatility $\sigma =0.3$, the strike $K=\mathrm{\$ }5$, and the initial asset value $x=\mathrm{\$ }20$. We vary the penalty parts as ${\eta _{1}}\in (1,1.1)$, ${\eta _{2}}\in (0,0.3)$, and ${\eta _{3}}\in (0,1)$. When we fix some of these penalties, we use the values ${\eta _{1}}=1.05$, ${\eta _{2}}=0.2$, and ${\eta _{3}}=0.5$.

The behavior of the optimal boundaries w.r.t. the three components of the penalty is presented in Figure 1. We can see that the writer’s boundary decreases to the strike when the penalty coefficients increase – we mark by red color the critical values. Note that this boundary vanishes when the penalties are large enough. Also, the holder’s boundary is an increasing function and it tends to the American optimal boundary. We present the call prices at Figures 3a, 3b, and 3c varying the three different penalty coefficients.

The results for some particular options are reported in Table 1. There can be seen the option prices – the second line – as well as the optimal boundaries. The writer’s boundary is the first value at the first line, whereas the holder’s one is given at the second place. We vary the three parts of the penalty among ${\eta _{1}}\in \{1;1.05;1.1;1.2\}$, ${\eta _{2}}\in \{0.05;0.1;0.15,0.2\}$, and ${\eta _{3}}\in \{0.25;0.5;0.75;1\}$.

Analogously to the call case, we can see that the inequality ${\eta _{2}}K+{\eta _{3}}\lt \overline{\eta }$ determines when the option is ordinary American or cancellable – we have a cancellable option when it holds and pure American otherwise. Note that $\overline{\eta }$ is given by equation (42). We conclude that we have a real cancellable option if (A) the number of shares is less than $\frac{1}{q+1}{(\frac{q}{q+1})^{q}}$ and (B) the fixed amount ${\eta _{3}}$ is less than $K[\frac{1}{q+1}{(\frac{q}{q+1})^{q}}-{\eta _{2}}]$. If one of these conditions does not hold, then we have a noncancellable American option. Let us discuss the role of the penalty coefficient ${\eta _{1}}$. Proposition 4.2 says that a necessary condition the option to be real cancellable is ${\eta _{2}}\lt {\eta _{1}}$. On the other hand, inequality ${\eta _{2}}K+{\eta _{3}}\lt \overline{\eta }$ is stronger, since $\overline{\eta }\lt K$ and therefore ${\eta _{2}}\lt 1\le {\eta _{1}}$. Hence, as in the call case, the coefficient ${\eta _{1}}$ influences whether the writer’s optimal set is the strike, but not whether we have a real cancellable option or pure American.

Taking into account the previous limitations, we consider put options with the following parameters: the risk free rate $r=-0.03$, the discount factor $\lambda =0.05$, the volatility $\sigma =0.3$, the strike $K=\mathrm{\$ }10$, and the initial asset value $x=\mathrm{\$ }5$. The penalties are taken as before: ${\eta _{1}}\in (1,1.1)$, ${\eta _{2}}\in (0,0.3)$, and ${\eta _{3}}\in (0,1)$. When we fix some of them, we use the values ${\eta _{1}}=1.05$, ${\eta _{2}}=0.2$, and ${\eta _{3}}=0.5$.

We present the optimal boundaries in Figure 2 fixing one of the penalty parts and varying the others. As we expected, the writer’s boundary increases w.r.t. the penalties and goes to the strike. The meaning of the red points is preserved – they mark namely the values for which the writer’s boundary turns to the strike. Also, we can see that the holder’s boundaries are decreasing functions. The resulting price behavior is presented in Figures 3d, 3e, and 3f.

We report some results for option prices and the related exercise boundaries in Table 1. The optimal boundaries are placed at the first line – the holder’s boundary is first; the writer’s one is second. The obtained prices are presented at the second line. The three parts of the penalties are again among ${\eta _{1}}\in \{1;1.05;1.1;1.2\}$, ${\eta _{2}}\in \{0.05;0.1;0.15,0.2\}$, and ${\eta _{3}}\in \{0.25;0.5;0.75;1\}$.