In case you are talking about a One Time Password
there is a simple reason, why TOTP-Algorithms are using a symmetric key on the client/device and the server side.
If you take a look at HOTP-TOTP (RFC6238), it is derived fro HOTP-OTP (RFC4226), the event based OTP algorithm.
This was specified in 2005. The first smartphone arived in 2007.
So what? The OTP algorithms are designed to work with hardware devices. These hardware devices have a display, so that the user can punch in the one time password. During the generation of the one time password the RFC defines a truncation method, so that the user is actually able to punch in a limited set of characters.
The output of HMAC-SHA1 is 20 bytes hexcode. Noone wants to enter this as a one time password.
To be able to do assymmetric cryptography you can not use any truncation function. Otherwise the public key would not be able to verify the signature. The private key would not be able to decrypt the message.
The data to transfer with assymmetric keys would be much larger - 1024 bit, 2048 bit... Again, the user would have to enter manually a very long one time password.
Well, if the client transmits this data electonically you would not have to truncate it. But then we are again not talking about one time passwords but about publy key crypto or client certificate based authentication.
So in the end it is all about human readability and truncation.