Yes, certificate must be installed with private key for it work with OWIN. I had to go through the same pain as most(all?) CA issue certificates without private keys. However you must have received the private key before hand. You must have your certificate in .crt format. This does not include the private key in it. So you need to create a certificate of .pfx format with private key in it.
If your private key is in plain text, then create .key file with plain text in as its content.Note that your .key file should have the standard first and last lines of private key else it'll complain about invalid key.
-----BEGIN PRIVATE KEY-----
<key-content>
-----END PRIVATE KEY-----
Now create a .pfx certificate using OpenSSL tool.
openssl pkcs12 -export -out servername.pfx -inkey servername.key -in servername.crt
To import this certificate just double click on .pfx file. In the import wizard choose 'Local Computer - Personal' as certificate store. After successful import you'd also notice a small (lock)key icon appearing on top of installed certificate icon.
Assuming you have bounded the server port with your OWIN application using netsh http add sslcert
, it should start working!