Risk-Free Interest Rate and Option Price

[Adam]

In Section 11.1 of Hull, it says
"
The risk-free interest rate affects the price of an option in a less clear-cut way. As interest
rates in the economy increase, the expected return required by investors from the stock
tends to increase. In addition, the present value of any future cash flow received by the
holder of the option decreases. The combined impact of these two effects is to increase the
value of call options and decrease the value of put options.
"
Can I interpret this para as follows?

• Expected return on stock increase --> Future share price increases --> call price increases
• Interest rate increase --> discounting increases --> present value decreases --> call price decreases
• The above two are offsetting each other --> Question is then how can one be sure that the first effect is always bigger than the second effect.

 

[mugono]

In Section 11.1 of Hull, it says
"
The risk-free interest rate affects the price of an option in a less clear-cut way. As interest
rates in the economy increase, the expected return required by investors from the stock
tends to increase. In addition, the present value of any future cash flow received by the
holder of the option decreases. The combined impact of these two effects is to increase the
value of call options and decrease the value of put options.
"
Can I interpret this para as follows?

• Expected return on stock increase --> Future share price increases --> call price increases
• Interest rate increase --> discounting increases --> present value decreases --> call price decreases
• The above two are offsetting each other --> Question is then how can one be sure that the first effect is always bigger than the second effect.

I had a look at the relevant passage in Hull: and my personal view is that it could've been articulated slightly differently. Options are priced in relative terms; and we do not need to know anything about an investor's risk preferences. What an investor 'expects' to earn in a particular investment doesn't feature when valuing a derivative - the asset earns risk-free.

Options are priced off of the future.
- Increasing the risk free rate increases the futures price and the value of a call.
- The receipt of a cash flow (e.g. a dividend) reduces the futures price and the value of a call.

Whether the overall effect on the future is an increase / decrease will depend on the relative size of the above (r-q): known as the cost of carry.

This can be 'visualised' algebraically as F(0) = S(0)*exp(r-q)T where
F(0) = value of the future at time 0
S(0) value of the underlying, e.g. shares, at time 0
r = continuously compounded risk free rate
q = continuously compounded dividend / income yield
T = maturity date.