<Text-field style="Heading 1" layout="Heading 1"><Font italic="true">Perpetual Call</Font></Text-field>

JSFH

QyQ+SSJDRzYiZio2I0kiU0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZJIllHRiUiIiIpOSRJIndHRiVGLkYlRiVGJUYu

Zio2I0kiU0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZJIllHRiUiIiIpRiRJIndHRiVGK0YlRiVGJQ==

This is the generic Perpetual Call Option formula with SSJ3RzYi as a generic exponent in the Call Option formula above. Since the call is American, we can exercise it any time. We can reasonably assume that the holder of this option will exercise it when the stock price in the future JkkiU0c2IjYjSSJ0R0Yk will rise above level SSJTRzYi with 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Our task is to determine the value of JkkiU0c2IjYjSSJ0R0Yk We know that when LyZJIlNHNiI2I0kidEdGJUYkand the holder exercises the option, the option's value will become Ly1JIkNHNiI2IyZJIlNHRiU2I0kidEdGJSwmRiciIiJJIktHRiUhIiI=where SSJLRzYi is the strike price in the standard option notation. From this equation we can determine the value of coefficient Y

QyQ+SSVZU29sRzYiLUkmc29sdmVHRiU2JC8tSSJDR0YlNiMmSSJTR0YlNiNJInRHRiUsJkYtIiIiSSJLR0YlISIiSSJZR0YlRjI=

LCQqJiwmJkkiU0c2IjYjSSJ0R0YnISIiSSJLR0YnIiIiRiwpRiVJIndHRidGKkYq

We can use this result to determine the value of LUkiQ0c2IjYjSSJTR0Yk

QyQ+SSVDU29sRzYiLUkpc2ltcGxpZnlHRiU2Iy1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJZR0YlSSVZU29sR0YlLUkiQ0dGJTYjSSJTR0YlIiIi

LCQqKCwmJkkiU0c2IjYjSSJ0R0YnISIiSSJLR0YnIiIiRiwpRiUsJEkid0dGJ0YqRiwpRiZGL0YsRio=

Our objective is to select the value JkkiU0c2IjYjSSJ0R0Yksuch that the option value is maximized before its exercise.

Take the derivative LUklZGlmZkclKnByb3RlY3RlZEc2JC1JIkNHNiI2I0kiU0dGKCZGKjYjSSJ0R0Yo

Set it to zero and solve for JkkiU0c2IjYjSSJ0R0Yk

QyQ+SSZTdFNvbEc2Ii8mSSJTR0YlNiNJInRHRiUtSSZzb2x2ZUdGJTYkLy1JJWRpZmZHJSpwcm90ZWN0ZWRHNiRJJUNTb2xHRiVGJyIiIUYnIiIi

LyZJIlNHNiI2I0kidEdGJSooSSJ3R0YlIiIiSSJLR0YlRiosJkYqISIiRilGKkYt

This is the desired value of JkkiU0c2IjYjSSJ0R0Yk. The reader can see that the exercise value JkkiU0c2IjYjSSJ0R0Yk is stock price-independent and is the function of option's strike SSJLRzYi and the exponent.

Qyg+SSV3dmFsRzYiLUkpc2ltcGxpZnlHRiU2JComLUkkbG9nRzYkJSpwcm90ZWN0ZWRHSShfc3lzbGliR0YlNiNJI3IxR0YlIiIiLUYrNiNJIlNHRiUhIiJJKXN5bWJvbGljR0YlRjU+SSNTdEdGJS1GJzYjLUklc3Vic0dGLTYkL0kid0dGJUYkLUkkcmhzR0YtNiNJJlN0U29sR0YlRjEtSSVwbG90R0YlNiUtSSVldmFsR0YtNiRGODclL0kickdGJSQiIiYhIiMvSSJxR0YlJCIiJEZPL0kmc2lnbWFHRiUkIiIjRjUvSSJLR0YlO0YxIiM1L0kmdGl0bGVHRiU3JFFCVmFsdWV+b2Z+U3R+YXN+ZnVuY3Rpb25+b2Z+c3RyaWtlRiUvSSVmb250R0YlNyVJJlRJTUVTR0YlSSVCT0xER0YlIiM3RjE=

KigsKiomIiIjIiIiSSJyRzYiRiYhIiIqJkYlRiZJInFHRihGJkYmKiQpSSZzaWdtYUdGKEYlRiZGJiokKSwuKiYiIiVGJilGJ0YlRiZGJiooIiIpRiZGJ0YmRitGJkYpKihGM0YmRidGJkYtRiZGJiomRjNGJilGK0YlRiZGJiooRjNGJkYrRiZGLUYmRiYqJClGLkYzRiZGJiNGJkYlRiZGJkYmSSJLR0YoRiYsKkYsRilGJEYpRipGJkYvRiZGKQ==

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

Once JkkiU0c2IjYjSSJ0R0Yk has been determined, we can substitute it into SSVZU29sRzYi

QyQ+SSVZUmVzRzYiLUkpc2ltcGxpZnlHRiU2Iy1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvJkkiU0dGJTYjSSJ0R0YlLUkkcmhzR0YrNiNJJlN0U29sR0YlSSVZU29sR0YlIiIi

KihJIktHNiIiIiIsJkYlISIiSSJ3R0YkRiVGJykqKEYoRiVGI0YlRiZGJywkRihGJ0Yl

This is the value of coefficient SSJZRzYi from the generic Perpetual Call Option formula. With this substitution, the call option will now become:

QyQtSSlzaW1wbGlmeUc2IjYkLUklc3Vic0clKnByb3RlY3RlZEc2JC9JIllHRiVJJVlSZXNHRiUtSSJDR0YlNiNJIlNHRiVJKXN5bWJvbGljR0YlIiIi

KiopSSJLRzYiLCYiIiJGJ0kid0dGJSEiIkYnKSwmRidGKUYoRidGK0YnKUYoLCRGKEYpRicpSSJTR0YlRihGJw==

This is the optimal value of JkkiU0c2IjYjSSJ0R0Yk that maximizes the value of the Perpetual Call Option for all LUkjPj1HJSpwcm90ZWN0ZWRHNiQmSSJTRzYiNiNJInRHRihGJw==. With substitutions, the final formula then becomes:

QyQ+SSZDUGVycEc2Ii1JKXNpbXBsaWZ5R0YlNiQqJi1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ3R0YlSSV3dmFsR0YlSSVZUmVzR0YlIiIiSSNyMUdGJUYySSlzeW1ib2xpY0dGJUYy

LCQqLiIiIyIiIilJIktHNiIsJCooI0YlRiRGJSwqKiQpSSZzaWdtYUdGKEYkRiUhIiIqJkYkRiVJInJHRihGJUYwKiZGJEYlSSJxR0YoRiVGJSokKSwuKiYiIiVGJSlGMkYkRiVGJSooIiIpRiVGMkYlRjRGJUYwKihGOUYlRjJGJUYuRiVGJSomRjlGJSlGNEYkRiVGJSooRjlGJUY0RiVGLkYlRiUqJClGL0Y5RiVGJUYrRiVGJUYlRi5GMEYwRiUpLCpGMUYwRjNGJUYtRiVGNUYlLCQqKEYrRiVGLkYwRkRGJUYwRiUpRiwsJEYqRiVGJSlJIlNHRigsJEZGRiVGJUYuRiVGJQ==

QyQtSSdwbG90M2RHNiI2Jy1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJJkNQZXJwR0YlNyUvSSJyR0YlJCIiJiEiIy9JInFHRiUkIiIkRjEvSSZzaWdtYUdGJSQiIiMhIiIvSSJTR0YlOyIiIiIjNS9JIktHRiU7Rj9GPi9JJnRpdGxlR0YlNyRRUVBlcnBldHVhbH5DYWxsfk9wdGlvbjp+U3RvY2t+L35TdHJpa2V+ZGVwZW5kZW5jeUYlL0klZm9udEdGJTclSSZUSU1FU0dGJUklQk9MREdGJSIjNy9JJWF4ZXNHRiVRJmJveGVkRiVGPg==

6'-%*AXESSTYLEG6#%$BOXG-%+AXESLABELSG6%Q"S6"Q"K6"Q!6"-%%GRIDG6&;$"""""!$"#5""!;$"""""!$"#5""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm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erpetual~Call~Option:~Stock~/~Strike~dependency6"-I%FONTG6"6%I&TIMESG6"I%BOLDG6""#7-%,ORIENTATIONG6$$"%S<!""$"$5(!""

How about the exercise strategy in practice? Is is worth exercising Perpetual Call Option?

When option is exercised, the holder will receive LCZJIlNHNiIiIiJJIktHRiQhIiI=

The value of the option will always be equal to SSZDUGVycEc2Ig==

We can compare two strategies in dynamic setting - by varying SSJTRzYi and SSJLRzYi

QyY+SSVDUGF5RzYiLCZJIlNHRiUiIiJJIktHRiUhIiJGKi1JJ3Bsb3QzZEdGJTYnPCRGJC1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJJkNQZXJwR0YlNyUvSSJyR0YlJCIiJiEiIy9JInFHRiUkIiIkRjkvSSZzaWdtYUdGJSQiIiNGKi9GJztGKCIjNS9GKUZDL0kmdGl0bGVHRiU3JFFFUGVycGV0dWFsfkNhbGx+T3B0aW9ufnZzfkNhbGx+UGF5b2ZmRiUvSSVmb250R0YlNyVJJlRJTUVTR0YlSSVCT0xER0YlIiM3L0klYXhlc0dGJVEnZnJhbWVkRiVGKA==

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000004015000000000000401380000000000040120000000000004010800000000000400E000000000000400B0000000000004008000000000000400500000000000040020000000000003FFE0000000000003FF80000000000003FF20000000000003FE80000000000003FD80000000000000000000000000000-%&COLORG6#%*XYSHADINGG-%&TITLEG6$QEPerpetual~Call~Option~vs~Call~Payoff6"-I%FONTG6"6%I&TIMESG6"I%BOLDG6""#7-%,ORIENTATIONG6$$!%q:!""$"$5(!""

As option's payoff is linear, one can easily see that the value of the option will always be worth at the least the same as the immediate exercise: LUkjPj1HJSpwcm90ZWN0ZWRHNiRJJkNQZXJwRzYiLCZJIlNHRiciIiJJIktHRichIiI=

This simply confirms that it is never optimal to exercise the Perpetual Call Option

<Text-field style="Heading 1" layout="Heading 1"><Font italic="true">Perpetual Put</Font></Text-field>

JSFH

QyQ+SSJQRzYiZio2I0kiU0dGJUYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZJIlpHRiUiIiIpOSRJIndHRiVGLkYlRiVGJUYu

Zio2I0kiU0c2IkYlNiRJKW9wZXJhdG9yR0YlSSZhcnJvd0dGJUYlKiZJIlpHRiUiIiIpRiRJIndHRiVGK0YlRiVGJQ==

This is the generic Perpetual Put Option formula. Again, we want to find the value JkkiU0c2IjYjSSJ0R0Yk that will trigger an exercise when the stock price falls below that level: MSZJIlNHNiI2I0kidEdGJUYk

QyQ+SSVaU29sRzYiLUkmc29sdmVHRiU2JC8tSSJQR0YlNiMmSSJTR0YlNiNJInRHRiUsJkkiS0dGJSIiIkYtISIiSSJaR0YlRjM=

KiYsJiZJIlNHNiI2I0kidEdGJiEiIkkiS0dGJiIiIkYrKUYkSSJ3R0YmRik=

QyQ+SSVQU29sRzYiLUkpc2ltcGxpZnlHRiU2Iy1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJaR0YlSSVaU29sR0YlLUkiUEdGJTYjSSJTR0YlIiIi

KigsJiZJIlNHNiI2I0kidEdGJiEiIkkiS0dGJiIiIkYrKUYkLCRJIndHRiZGKUYrKUYlRi5GKw==

The level of JkkiU0c2IjYjSSJ0R0Yk has to be determined s.t. the option value is maximized prior to exercise.

QyQ+SSdTdFNvbDJHNiIvJkkiU0dGJTYjSSJ0R0YlLUkmc29sdmVHRiU2JC8tSSVkaWZmRyUqcHJvdGVjdGVkRzYkSSVQU29sR0YlRiciIiFGJyIiIg==

LyZJIlNHNiI2I0kidEdGJSooSSJ3R0YlIiIiSSJLR0YlRiosJkYqISIiRilGKkYt

QyQ+SSVaUmVzRzYiLUkpc2ltcGxpZnlHRiU2Iy1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvJkkiU0dGJTYjSSJ0R0YlLUkkcmhzR0YrNiNJJ1N0U29sMkdGJUklWlNvbEdGJSIiIg==

LCQqKEkiS0c2IiIiIiwmRiYhIiJJIndHRiVGJkYoKSooRilGJkYkRiZGJ0YoLCRGKUYoRiZGKA==

Similar to the Call Option case, this is the the value of JkkiU0c2IjYjSSJ0R0Yk that maximizes the value of Perpetual Put Option for LUkjPj1HJSpwcm90ZWN0ZWRHNiRJIlNHNiImRiY2I0kidEdGJw==

QyQ+SSZQUGVycEc2Ii1JKXNpbXBsaWZ5R0YlNiQqJi1JJXN1YnNHJSpwcm90ZWN0ZWRHNiQvSSJ3R0YlLUYnNiQqJi1JJGxvZ0c2JEYsSShfc3lzbGliR0YlNiNJI3IyR0YlIiIiLUY0NiNJIlNHRiUhIiJJKXN5bWJvbGljR0YlSSVaUmVzR0YlRjlGOEY5Rj5GOQ==

LCQqLiIiIyIiIilJIktHNiIsJCooI0YlRiRGJSwqKiQpSSZzaWdtYUdGKEYkRiVGJSomRiRGJUkickdGKEYlRiUqJkYkRiVJInFHRihGJSEiIiokKSwuKiYiIiVGJSlGMUYkRiVGJSooIiIpRiVGMUYlRjNGJUY0KihGOUYlRjFGJUYuRiVGJSomRjlGJSlGM0YkRiVGJSooRjlGJUYzRiVGLkYlRiUqJClGL0Y5RiVGJUYrRiVGJUYlRi5GNEYlRiUpLCpGMEYlRjJGNEYtRjRGNUYlLCQqKEYrRiVGREYlRi5GNEYlRiUpRiwsJEYqRjRGJSlJIlNHRigsJEZGRjRGJUYuRiVGJQ==

QyQtSSdwbG90M2RHNiI2Jy1JJWV2YWxHJSpwcm90ZWN0ZWRHNiRJJlBQZXJwR0YlNyUvSSJyR0YlJCIiJiEiIy9JInFHRiUkIiIkRjEvSSZzaWdtYUdGJSQiIiMhIiIvSSJTR0YlOyIiIiIjNS9JIktHRiU7Rj9GPi9JJnRpdGxlR0YlNyRRUFBlcnBldHVhbH5QdXR+T3B0aW9uOn5TdG9ja34vflN0cmlrZX5kZXBlbmRlbmN5RiUvSSVmb250R0YlNyVJJlRJTUVTR0YlSSVCT0xER0YlIiM3L0klYXhlc0dGJVEmYm94ZWRGJUY+

6'-%+AXESLABELSG6%Q"S6"Q"K6"Q!6"-%*AXESSTYLEG6#%$BOXG-%%GRIDG6&;$"""""!$"#5""!;$"""""!$"#5""!X,I)anythingG%*protectedG6"6"[gl'!%"!!#\bm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erpetual~Put~Option:~Stock~/~Strike~dependency6"-I%FONTG6"6%I&TIMESG6"I%BOLDG6""#7-%,ORIENTATIONG6$$!$](!""$"$](!""

This is the Perpetual Put Option value as function of SSJTRzYiand SSJLRzYi. As expected, the put value is the highest when strike is high and stock price is low.

Closed-from solution allows sensitivities to be computed analytically. Let Maple calculate the Perpetual Put Delta:

QyY+SShQUGRlbHRhRzYiLUklZGlmZkclKnByb3RlY3RlZEc2JEkmUFBlcnBHRiVJIlNHRiUiIiItSSVwbG90R0YlNiUtSSVldmFsR0YoNiRGJDcmL0kiS0dGJSIiKC9JInJHRiUkIiImISIjL0kicUdGJSQiIiRGOy9JJnNpZ21hR0YlJCIiIyEiIi9GKztGLCIjNS9JJnRpdGxlR0YlNyRRNFBlcnBldHVhbH5QdXR+RGVsdGFGJS9JJWZvbnRHRiU3JUkmVGltZXNHRiVJJUJvbGRHRiUiIzZGLA==

LCQqLilJIktHNiIsJCooIyIiIiIiI0YqLCoqJClJJnNpZ21hR0YmRitGKkYqKiZGK0YqSSJyR0YmRipGKiomRitGKkkicUdGJkYqISIiKiQpLC4qJiIiJUYqKUYxRitGKkYqKigiIilGKkYxRipGM0YqRjQqKEY5RipGMUYqRi5GKkYqKiZGOUYqKUYzRitGKkYqKihGOUYqRjNGKkYuRipGKiokKUYvRjlGKkYqRilGKkYqRipGLkY0RipGKiksKkYwRipGMkY0Ri1GNEY1RiosJCooRilGKkZERipGLkY0RipGKilGLCwkRihGNEYqKUkiU0dGJiwkRkZGNEYqRkRGKkZKRjRGNA==

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

We can also compare the option value and the option payoff when L0kiU0c2IiZGIzYjSSJ0R0Yk

Take the difference between the two

Differentiate it w.r.t JkkiU0c2IjYjSSJ0R0Yk

Substitute it back into JkkiU0c2IjYjSSJ0R0Yk definition

Qyg+SShWYWxEaWZmRzYiLUkiK0clKnByb3RlY3RlZEc2JEklUFNvbEdGJSwmSSJLR0YlISIiSSJTR0YlIiIiRi8+SSZERGlmZkdGJS1JKXNpbXBsaWZ5R0YlNiMtSSVkaWZmR0YoNiRGJCZGLjYjSSJ0R0YlRi8tRjM2Iy1JJXN1YnNHRig2JC9GOC1JJHJoc0dGKDYjSSdTdFNvbDJHRiVGMUYv

LCgqKCwmJkkiU0c2IjYjSSJ0R0YnISIiSSJLR0YnIiIiRiwpRiUsJEkid0dGJ0YqRiwpRiZGL0YsRixGJkYsRitGKg==

LCQqJilJIlNHNiJJIndHRiYiIiIsKCkmRiU2I0kidEdGJiwkRichIiJGKComRipGKEYnRihGLyooKUYrLCZGJ0YvRihGL0YoRidGKEkiS0dGJkYoRihGKEYv

IiIh

What is the point of this?

When LyZJIlNHNiI2I0kidEdGJUYk, the option's delta and option's payoff delta are identical. This implies that the Perpetual Put Option value is maximized by the exercise strategy where delta at JkkiU0c2IjYjSSJ0R0Yk exists and is continuous.

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

NiktJSdDVVJWRVNHNiQ3VzckJCIjNSEiIiQhMi0rKysrKysrIiEjOzckJCItdiQ+KDMpNCIhIzYkITJiNnlgLU83OSIhIzs3JCQiLHZRdWg+IiEjNSQhMig0YVtKIjQkZjchIzs3JCQiL0RKSSY+OkciISM4JCEyaiIpZVEhKVF0TSIhIzs3JCQiLkRKbmtvTyIhIzckITI7Tl1RZHZWVSIhIzs3JCQiL3Y9N1UlR1kiISM4JCEydUstUWxzLV0iISM7NyQkIi1EXlAjKWU6ISM2JCEyIylcKlwpXEJvYyIhIzs3JCQiKzAmUmFsIiEiKiQhMmNCQyZ5Oi1FOyEjOzckJCItdmVfMF88ISM2JCEyc205Q3ohcHk7ISM7NyQkIi5EMVBvViU+ISM3JCEyT04nWy0qXHp3IiEjOzckJCIuREpDbkU3IyEjNyQhMUFOXip5YmkkPSEjOjckJCIuRCIpUiZHMkIhIzckITJAQWprJHkmZSo9ISM7NyQkIi5EImV4QClcIyEjNyQhMmN0XDM1SyNbPiEjOzckJCIuRGN6UCYpbyMhIzckITI0UC5FP05JKj4hIzs3JCQiMi0rK3ZNMFYpRyEjOyQhMmpyeUx5XkguIyEjOzckJCItdjh2dGNJISM2JCEyOTh1QEl2UTEjISM7NyQkIjIuKytESGIzRCQhIzskITI4WWVGZGlaNCMhIzs3JCQiLHY4cWRXJCEjNSQhMil6dE1KXkZBQCEjOzckJCIyLisrdnExT2okISM7JCExOFhsWGQqZjkjISM6NyQkIi5EIjMuPS9RISM3JCExOEgvJCpvXWxAISM6NyQkIix2IikzcSslISM1JCExWGJnSWJhJz0jISM6NyQkIjIwKysrPEopeVQhIzskITFzUiozOnFGPyMhIzo3JCQiLkRKVCwoeVYhIzckITEjXFM0PFQrQSMhIzo3JCQiK1gvaWJYISIqJCExMiJwJD1YMU1BISM6NyQkIi5ESm1HKFxaISM3JCEyeDIqWyJbWyNbQSEjOzckJCIyLyt2JCplbFgkXCEjOyQhMS8tNl54cmdBISM6NyQkIjEsK0RKTlVGXiEjOiQhMXRpPWYocEZGIyEjOjckJCIuRGNLRlhJJiEjNyQhMk8nMyJwT2xJRyMhIzs3JCQiMiYqKipcUHBkYlwmISM7JCEyJCo0JyopZm1VJEgjISM7NyQkIi5EImVYKVJwJiEjNyQhMkYvYScqKkhYLkIhIzs3JCQiLnZvczpuJ2UhIzckITFiIj5LeUc7SiMhIzo3JCQiLURALEZgZyEjNiQhMih6OyFRdk0qPkIhIzs3JCQiMiYqKioqKlwxKipmQychIzskITEqSCxxYyUqeksjISM6NyQkIi1ET25hTWshIzYkITE7aCE9STdhTCMhIzo3JCQiLkRKSXdwaCchIzckITJiJj51a3A9VUIhIzs3JCQiLXZMS2A+byEjNiQhMnQoZmRCV0dcQiEjOzckJCIxKioqKipca1I6KyghIzokITF6LCVbXzZgTiMhIzo3JCQiMSwrXVAuKGU+KCEjOiQhMTIwQic9NTlPIyEjOjckJCIuRCJHRyc+UCghIzckITIxSikzKHplbU8jISM7NyQkIjIwKytEVnlXYyghIzskITJ2KGZRLnQ2c0IhIzs3JCQiMSoqXDc4MWlYeCEjOiQhMlBSIlFrYit4QiEjOzckJCIuRGNRbVwkeiEjNyQhMlZNWE9qdz1RIyEjOzckJCIsdlsnMz8iKSEjNSQhMCY9P3QkPmtRIyEjOTckJCIxKipcN3krKlFKKSEjOiQhMk5xL0xXZTRSIyEjOzckJCIxLCsrcWZhKyYpISM6JCEyLnZbV1dNXlIjISM7NyQkIi12eSZHOXApISM2JCEyJUh3KVFdPiMqUiMhIzs3JCQiLkRKZUkyKSkpISM3JCEwaFFNSSg0LkMhIzk3JCQiK2wlelkwKiEiKiQhMkxSKjNbd14xQyEjOzckJCItdjhYL2EjKiEjNiQhMUJnczQnei1UIyEjOjckJCIqKiplQlYqISIpJCEyJClmSGZpNE5UIyEjOzckJCIuREpUekNpKiEjNyQhMUxkZk07I29UIyEjOjckJCIudm8iKltXISkqISM3JCExdTMlUU1yKT5DISM6NyQkIiQrIiEiIiQhMjArK3ElXC1CQyEjOy0lJkNPTE9SRzYmJSRSR0JHJCIjNSEiIiQiIiEhIiIkIiIhISIiLSUnQ1VSVkVTRzYkN1c3JCQiIzUhIiIkIipLYlhLJiEiKTckJCItdiQ+KDMpNCIhIzYkIjFDJTRKPnYkPVkhIzo3JCQiLHZRdWg+IiEjNSQiMWBIZGknNCFHUyEjOjckJCIvREpJJj46RyIhIzgkIjE+ZnErOCd5ZSQhIzo3JCQiLkRKbmtvTyIhIzckIjJGQ1syWHhFPyQhIzs3JCQiL3Y9N1UlR1kiISM4JCIyTk8pKTQwIz5CRyEjOzckJCItRF5QIyllOiEjNiQiMjBeLXYjeVYhXCMhIzs3JCQiKzAmUmFsIiEiKiQiMUIpeXRVWlc+IyEjOjckJCItdmVfMF88ISM2JCIyVm1Fek4sNiQ+ISM7NyQkIi5EMVBvViU+ISM3JCIyRkJvdiFlIVtbIiEjOzckJCIuREpDbkU3IyEjNyQiMi1SS0NQd0s5IiEjOzckJCIuRCIpUiZHMkIhIzckIjEnKilRb1BoRVgpISM7NyQkIi5EImV4QClcIyEjNyQiMEE4djpbUiRlISM6NyQkIi5EY3pQJilvIyEjNyQiMmpZSilwSXokZiQhIzw3JCQiMi0rK3ZNMFYpRyEjOyQiMil5VDFKUyd6ZiIhIzw3JCQiLXY4dnRjSSEjNiQiMSN5ViRIIjQpeV4hIz03JCQiMi4rK0RIYjNEJCEjOyQhMjAuQnpWdkRcIiEjPDckJCIsdjhxZFckISM1JCExYypvdE8uI29HISM7NyQkIjIuKyt2cTFPaiQhIzskITIlW2NzIzNNVTAlISM8NyQkIi5EIjMuPS9RISM3JCExc2M5Xzl6SF0hIzs3JCQiLHYiKTNxKyUhIzUkITFNc0ZJTHMiMychIzs3JCQiMjArKys8Sil5VCEjOyQhMXUmKXBXViZIKm8hIzs3JCQiLkRKVCwoeVYhIzckITBkQ3FNMGx2KCEjOjckJCIrWC9pYlghIiokITFBYFg9Rm5kJSkhIzs3JCQiLkRKbUcoXFohIzckITAoUVh1MyhvOyohIzo3JCQiMi8rdiQqZWxYJFwhIzskITEoPjViTk0ueiohIzs3JCQiMSwrREpOVUZeISM6JCEybU9KZlokSFI1ISM7NyQkIi5EY0tGWEkmISM3JCExPFZiOTp4ITQiISM6NyQkIjImKioqXFBwZGJcJiEjOyQhMm5cIVt6emRVNiEjOzckJCIuRCJlWClScCYhIzckITJKQHEjeSc0Rj4iISM7NyQkIi52b3M6bidlISM3JCEyYHgmNCdmKWVMNyEjOzckJCItREAsRmBnISM2JCEyKilSMyFcJT1eRiIhIzs3JCQiMiYqKioqKlwxKipmQychIzskITJiXDFdXjxhSiIhIzs3JCQiLURPbmFNayEjNiQhMil6MC4qPTFETiIhIzs3JCQiLkRKSXdwaCchIzckITJreDRQXXpqUSIhIzs3JCQiLXZMS2A+byEjNiQhMSkpKXp5em09VSIhIzo3JCQiMSoqKioqXGtSOisoISM6JCExJiozPy9CK185ISM6NyQkIjEsK11QLihlPighIzokITFNRDo2Y1wjWyIhIzo3JCQiLkQiR0cnPlAoISM3JCEyT2JUYW1RKDM6ISM7NyQkIjIwKytEVnlXYyghIzskITJqKSlIcD5KZ2AiISM7NyQkIjEqKlw3ODFpWHghIzokITJzJ3AhPl1zL2MiISM7NyQkIi5EY1FtXCR6ISM3JCEyQ3NFI1t5I1tlIiEjOzckJCIsdlsnMz8iKSEjNSQhMilbIzRnYVR2ZyIhIzs3JCQiMSoqXDd5KypRSikhIzokITE9Tl8nKm9CSTshIzo3JCQiMSwrK3FmYSsmKSEjOiQhMkV2VkEhcDZeOyEjOzckJCItdnkmRzlwKSEjNiQhMmo5USUqPlU6biIhIzs3JCQiLkRKZUkyKSkpISM3JCExXEk+KD5KNHAiISM6NyQkIitsJXpZMCohIiokITJrJ3BXP0guMzwhIzs3JCQiLXY4WC9hIyohIzYkITE6LGpHRiVvcyIhIzo3JCQiKioqZUJWKiEiKSQhMjIqems0RypIdSIhIzs3JCQiLkRKVHpDaSohIzckITJObXlIJkdiZjwhIzs3JCQiLnZvIipbVyEpKiEjNyQhMihvVj8qUixbeCIhIzs3JCQiJCsiISIiJCEyLysrXVRwMHoiISM7LSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIiISEiIiQiIzUhIiItJSdDVVJWRVNHNiQ3VzckJCIjNSEiIiQiK1pcLUI4ISIpNyQkIi12JD4oMyk0IiEjNiQiMlspPWlAKil5Ij0iISM6NyQkIix2UXVoPiIhIzUkIjIzZjliImVyajUhIzo3JCQiL0RKSSY+OkciISM4JCIwJT1USjknb3YqISM5NyQkIi5ESm5rb08iISM3JCIxJ1snXEpQXCcpKikhIzo3JCQiL3Y9N1UlR1kiISM4JCIxRm4oPiRIX0YjKSEjOjckJCItRF5QIyllOiEjNiQiMjotMF1bOT9jKCEjOzckJCIrMCZSYWwiISIqJCIxWHd2JW9MKyhwISM6NyQkIi12ZV8wXzwhIzYkIjFITCZlYVRMVychIzo3JCQiLkQxUG9WJT4hIzckIjFsazhYL3ZdYiEjOjckJCIuREpDbkU3IyEjNyQiMHlrW2QicG5bISM5NyQkIi5EIilSJkcyQiEjNyQiMHluYDVyO0YlISM5NyQkIi5EImV4QClcIyEjNyQiMVdFXWglR3p1JCEjOjckJCIuRGN6UCYpbyMhIzckIjEkSG1SVygqKSpIJCEjOjckJCIyLSsrdk0wVilHISM7JCIyZCRHQE87dCtIISM7NyQkIi12OHZ0Y0khIzYkIjJ4b2UjW2tcImYjISM7NyQkIjIuKytESGIzRCQhIzskIjJZUjpDdUJFRyMhIzs3JCQiLHY4cWRXJCEjNSQiMTRpX2MiKVwyPyEjOjckJCIyLisrdnExT2okISM7JCIyLihbWDg/SHE8ISM7NyQkIi5EIjMuPS9RISM3JCIyZSczZFIwPXY6ISM7NyQkIix2IikzcSslISM1JCIyT2JXUjslems4ISM7NyQkIjIwKysrPEopeVQhIzskIjJjR2c1J3phLTchIzs3JCQiLkRKVCwoeVYhIzckIjJoMyZmZ3gkKUg1ISM7NyQkIitYL2liWCEiKiQiMWQkKjNqRy8nKikpISM7NyQkIi5ESm1HKFxaISM3JCIxakE0XmxreHUhIzs3JCQiMi8rdiQqZWxYJFwhIzskIjEyJ3oqKWU+MkInISM7NyQkIjEsK0RKTlVGXiEjOiQiMG5zOHk9Yi0mISM6NyQkIi5EY0tGWEkmISM3JCIyL204KjMhZWYqUiEjPDckJCIyJioqKlxQcGRiXCYhIzskIjIkbytSNShHKWZIISM8NyQkIi5EImVYKVJwJiEjNyQiMiVSZGZNWj5kPiEjPDckJCIudm9zOm4nZSEjNyQiMlhcJTN5amhSNiEjPDckJCItREAsRmBnISM2JCIyN0I/JCk+JD4hNCQhIz03JCQiMiYqKioqKlwxKipmQychIzskITIlZSEqSCwraXBcISM9NyQkIi1ET25hTWshIzYkITEnZjYxW04oUTchIzs3JCQiLkRKSXdwaCchIzckITFGYj51PD87PiEjOzckJCItdkxLYD5vISM2JCExaHhmZHclZmkjISM7NyQkIjEqKioqKlxrUjorKCEjOiQhMSUqeSwleWQnR0shIzs3JCQiMSwrXVAuKGU+KCEjOiQhMllvXUkjUl9RUSEjPDckJCIuRCJHRyc+UCghIzckITIlcDUkKTNdUWpWISM8NyQkIjIwKytEVnlXYyghIzskITJhcyhmUWNCNFwhIzw3JCQiMSoqXDc4MWlYeCEjOiQhMVYkUiJRPDEpUiYhIzs3JCQiLkRjUW1cJHohIzckITFZV2BrJ29eKWUhIzs3JCQiLHZbJzM/IikhIzUkITF2XD0/RVdSaiEjOzckJCIxKipcN3krKlFKKSEjOiQhMWkuWkknXEx6JyEjOzckJCIxLCsrcWZhKyYpISM6JCExYV0oW3VcNEAoISM7NyQkIi12eSZHOXApISM2JCExRkh3KW9iJT53ISM7NyQkIi5ESmVJMikpKSEjNyQhMSMpNCdRa05zKykhIzs3JCQiK2wlelkwKiEiKiQhMUckUiozLEZcJCkhIzs3JCQiLXY4WC9hIyohIzYkITEnSC1FRm1hcykhIzs3JCQiKioqZUJWKiEiKSQhMTgpZkgqeVlbISohIzs3JCQiLkRKVHpDaSohIzckITBGdCZmKG8neiQqISM6NyQkIi52byIqW1chKSohIzckITBQKDMlb1JZbyohIzo3JCQiJCsiISIiJCEyLysrKysrKysiISM7LSUmQ09MT1JHNiYlJFJHQkckIiIhISIiJCIjNSEiIiQiIiEhIiItJSVWSUVXRzYkOyQiIzUhIiIkIiQrIiEiIiUoREVGQVVMVEctJStBWEVTTEFCRUxTRzYnLUkjbWlHNiMvSSttb2R1bGVuYW1lRzYiSSxUeXBlc2V0dGluZ0dJKF9zeXNsaWJHNiI2NVEiUzYiLyUnZmFtaWx5R1EoREVGQVVMVDYiLyUlc2l6ZUdRIzEwNiIvJSVib2xkR1EmZmFsc2U2Ii8lJ2l0YWxpY0dRJXRydWU2Ii8lKnVuZGVybGluZUdRJmZhbHNlNiIvJSpzdWJzY3JpcHRHUSZmYWxzZTYiLyUsc3VwZXJzY3JpcHRHUSZmYWxzZTYiLyUrZm9yZWdyb3VuZEdRKFswLDAsMF02Ii8lK2JhY2tncm91bmRHUS5bMjU1LDI1NSwyNTVdNiIvJSdvcGFxdWVHUSZmYWxzZTYiLyUrZXhlY3V0YWJsZUdRJmZhbHNlNiIvJSlyZWFkb25seUdRJmZhbHNlNiIvJSljb21wb3NlZEdRJmZhbHNlNiIvJSpjb252ZXJ0ZWRHUSZmYWxzZTYiLyUraW1zZWxlY3RlZEdRJmZhbHNlNiIvJSxwbGFjZWhvbGRlckdRJmZhbHNlNiIvJTZzZWxlY3Rpb24tcGxhY2Vob2xkZXJHUSZmYWxzZTYiLyUsbWF0aHZhcmlhbnRHUSdpdGFsaWM2IlEhNiItJSVGT05URzYlJShERUZBVUxURyUoREVGQVVMVEciIzUlK0hPUklaT05UQUxHJStIT1JJWk9OVEFMRy0lJlRJVExFRzYlLSUpX1RZUEVTRVRHNiNRR0RlbHRhfmNvbnRpbnVpdHl+Zm9yfmRpZmZlcmVudH5zdHJpa2VzNiItJSVGT05URzYlJTBUaW1lc35OZXd+Um9tYW5HJSVCT0xERyIjNy0lLVRSQU5TUEFSRU5DWUc2IyQiIiEhIiItJSVST09URzYnLSUpQk9VTkRTX1hHNiMkIiRJJSEiIi0lKUJPVU5EU19ZRzYjJCIkXSMhIiItJS1CT1VORFNfV0lEVEhHNiMkIiVnTSEiIi0lLkJPVU5EU19IRUlHSFRHNiMkIiVxTyEiIi0lKUNISUxEUkVORzYi

As we can see, the delta is continuous at various strike levels, so the conditions for optimal strategy are satisfied.