Hi everyone, I was wondering why rho is positive for call options. Intuitively, if interest rates go down, the price of the underlying security would increase (lower discount rate -> higher present value -> higher share price), so the price of the call option would also increase. However, rho is positive for call options, and negative for put options, but one would expect the opposite. Thanks.
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rates fall >> cost of carrying position falls so the opportunity cost of buying the underlying instead of the option falls >> holding call is relatively less attractive than holding the underlying >> rho, partial to options value wrt rates is positive
rates fall >> present value of strike increases, makes call option less valuable since ur effectively paying more than pv terms when/if exercised in future >> rho then is positive
how come you're mentioning the opportunity cost of buying the underlying instead of the option when in the first part you mention cost of carry [of holding the option]? and why would holding the call be less attractive when the cost of carry falls?
your thinking is correct, however you're thinking too much CAPM and not enough put call parity. 1) the PV of the strike price also decreases when rates rise 2) opportunity cost increases when rates rise
1) rates rise --> higher discount rate --> lower PV of the strike price --> cheaper to exercise the options --> more attractive options you're holding (for calls) --> option value rises --> profit
essentially: impact of rates on the strike price > impact of rates on the underlying asset price
2) opportunity cost - when rates rise, holding cash doesn't make sense because you can put that money elsewhere and earn % (aka more $). Buying a call is cheaper than buying the underlying (yet you can still get the same upside exposure) so you're gonna buy the call over the underlying. And remember, every extra dollar you spend on the underlying over the cheaper call option, that's more cash you could've parked somewhere and earned interest on. More demand for call = call increases in value. ||||| Say I have $10,000 to invest. I can 1) buy $10,000 worth of stock via shares or 2) buy a "$1,000" call and book the rest ($9,000) into an interest bearing account.
re: 2, doesn't it make sense to earn interest on cash when rates are high? or do you mean holding the underlying doesn't make sense.
Don’t overcomplicate. The underlying of an option is a forward. With a call, you’re long a forward. Given an increase in rates, forwards increase in value. Thus, the call increases in value too. Hence, rho is positive.
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