Later in this series of posts, we will study option price sensitivities in more detail. These sensitivity measures have Greek names:

Delta is the sensitivity of the option price to a change in the price of the underlying. Gamma is a measure of how well the delta sensitivity measure will approximate the option price’s response to a change in the price of the underlying.

Rho is the sensitivity of the option price to the risk-free rate.

Theta is the rate at which the time value decays as the option approaches expiration.

Vega is the sensitivity of the option price to volatility.

## The effect of interest rates and volatility

It is important to know that interest rates and volatility exert an influence on option prices. When interest rates are highel; call option prices are higher and put option prices are lower. This effect is not obvious and strains the intuition somewhat. When investors buy call options instead of the underlying, they are effectively buying an indirect leveraged position in the underlying. When interest rates are higher, buying the call instead of a direct leveraged position in the underlying is more attractive. Moreover, by using call options, investors save more money by not paying for the underlying until a later date. For put options, however, higher interest rates are disadvantageous. When interest rates are higher, investors lose more interest while waiting to sell the underlying when using puts. Thus, the opportunity cost of waiting is higher when interest rates are higher. Although these points may not seem completely clear, fortunately they are not critical. Except when the underlying is a bond or interest rate, interest rates do not have a very strong effect on option prices. Volatility, however, has an extremely strong effect on option prices. Higher volatility increases call and put option prices because it increases possible upside values and increases possible downside values of the underlying. The upside effect helps calls and does not hurt puts. The downside effect does not hurt calls and helps puts. The reasons calls are not hurt on the downside and puts are not hurt on the upside is that when options are out-of-the-money, it does not matter if they end up more out-of-the-money. But when options are in-the-money, it does matter if they end up more in-the-money.

Volatility is a critical variable in pricing options. It is the only variable that affects option prices that is not directly observable either in the option contract or in the market. It must be estimated.

## The effect of cash flows on the underlying asset

Both the lower bounds on puts and calls and the put-call parity relationship must be modified to account for cash flows on the underlying asset. In earlier posts we discussed situations in which the underlying has cash flows. Stocks pay dividends, bonds pay interest, foreign currencies pay interest, and commodities have carrying costs. As we have done in the previous posts, we shall assume that these cash flows are either known or can be expressed as a percentage of the asset price. Moreover, as we did previously, we can remove the present value of those cash flows from the price of the underlying and use this adjusted underlying price in the results we have obtained above.

In the previous posts, we specified these cash flows in the form of the accumulated value at T of all cash flows incurred on the underlying over the life of the derivative contract. When the underlying is a stock, we specified these cash flows more precisely in the form of dividends, using the notation FV(D,O,T) as the future value, or alternatively PV(D,O,T) as the present value, of these dividends. When the underlying was a bond, we used the notation FV(CI,O,T) or PV(CI,O,T), where CI stands for “coupon interest.” When the cash flows can be specified in terms of a yield or rate, we used the notation 6 where Sd(1 + is the underlying price reduced by the present value of the cash flows.

Using continuous compounding, the rate can be specified as 6″ so that S, ~C”~ is the underlying price reduced by the present value of the dividends. For our purposes in this series of posts on options, let us just write this specification as PV(CF,O,T), which represents the present value of the cash flows on the underlying over the life of the options. Therefore, we can restate the lower bounds for European options as and put-call parity as co + Xl(1 + r)T = po + [So – PV(CF,O,T)] which reflects the fact that, as we said, we simply reduce the underlying price by the present value of its cash flows over the life of the option.