Where: C = Call, S = Strike price, PV = Present value, CF = Cash Flow,
r = interest rate, T= Time to expiration of the option
Cash flows for underlying assets are as follows:
- Stocks pay dividends - in formula terms FV (D,O,T) or PV (D,O,T)
- Bonds pay interest - in formula terms FV (CI,O,T) or PV (CI,O,T)
- Currency pays interest
- Commodities have carrying costs
The underlying price is reduced by the PV of the cash flows of the underlying; therefore, the put-call parity relationship is calculated as:
This formula determines the reduction in the price of the underlying assets as related to the present value of the cash flows over the life of the trade.
Interest Rate Changes and Option Prices
Options are priced on a risk-neutral basis, so a long call (for example) would be paired with a short stock. A short-stock position generates interest revenue, which makes the call option more valuable. If interest rates go up, the interest revenue from the short stock position increases, which makes the call worth even more. For put options and dividends, it works in the opposite direction.
When interest rates are high, the prices of calls are higher and the prices of puts are lower. Why? When buying an option, one is essentially using leverage. When rates are high, the option itself is more attractive than the underlying asset. By purchasing an option instead of the underlying asset, an investor saves cash.
Puts are adversely affected by higher rates because investors lose interest while waiting to sell their underlying assets. This works for all underlying assets except when dealing with bonds or interest rates.
Interest rate volatility has a huge effect on option prices. When volatility increases, call and put prices both increase because of the increased possibility that a downside or upside event could occur concerning the option. The upside helps call price and has no effect on puts, while downside helps puts with no effect on calls and is especially true when options are out of the money. Downside does begin to matter when options become in the money.