equity/bond calls in relation to interest rates understand, interest, relation, decrease, increase

Elliotbay

I am trying to understand the differences in call values between the 'calls in equities' and 'call-embedded options in bonds'.

Bonds:
We know that an increase in interest rates decrease the values of both call and put options of an option-embedded bond. Similarly, a risk free rate rise in the bond will decrease the value of both options. Increase in price volatility, however, will increase the value of both the put and the call.

Equity:
If risk free rate rises, value of put on equities will decrease and value of call will increase. This can be determined using a put-call parity equation. And a rise on price volatility will increase the value of both the put and the call.

The call/put relationships between equities and bonds seem to be similar except for the calls: during a rate rise, a value of a call in a callable bond will decrease while a call option on an equity will increase. Can someone please explain the relationship that makes them different? Is there something else that I am missing?

ogoluwa

Thanks for articulating the concepts...I do have follow up questions on the applications of rho and delta but will wait after the exam.