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What is Rho?

Another state variable that can change is the risk free rate. The Greek associated with this is referred to as 'rho'. Rho is the rate of change in the value of the option, given a change in the risk free rate.

It is positive for calls and negative for puts.

This result may strike you as being a little odd because you know that if interest rates increase the predicted effect on the underlying stock prices is negative. For example, consider the large negative stock market reactions that take place in response to unanticipated interest rate increases. This news is not exactly bullish for call option prices; so why would option pricing theory predict otherwise?

On the surface this may appear as a contradiction. In reality it serves as a nice way of illustrating the care that must be exercised when interpreting and applying the Greeks (delta, gamma, vega, theta and rho).

In this current case careful consideration of the “ceterus paribus” conditions resolves this apparent contradiction. Mathematically, each Greek is a partial derivative. This means that only two variables are permitted to change: the option value and the driver variable in question (i.e. underlying asset price, volatility, time, and interest rate).

Thus, because all other variables are held constant this implies, for the case of rho, that the underlying asset price is held constant; whereas, in reality, the option trader has to consider the relative magnitude of the impact of multiple variables. For the case of rho, the relative magnitude arising from an increase in interest rates is swamped by the option price effects induced from a change in the underlying price (i.e. delta).

This leaves open, however, the interesting question: why do increasing interest rates have a positive relative effect upon a call option's price?

Understanding the Impact of Interest Rates upon Option Prices

As is always the case, the best way of understanding arbitrage free option pricing is to view the problem through the eyes of a market maker who attempts to earn the bid ask spread. To earn the bid/ask spread when order flows are irregular over time requires hedging.

More formally, the most efficient theoretical hedge is a position that exactly mimics the option contract. This is known as the synthetic equivalent of the option contract.

If a perfect synthetic equivalent can be constructed (not involving the option contract being mimicked), then a riskless hedge can be constructed by going long (short) the option contract, and short (long) the synthetic.

But now suppose further that we can construct this synthetic equivalent using securities with observable spot prices. Now we have identified the arbitrage free price of the option contract because any deviation from the cost of the synthetic equivalent lets one simultaneously take a long/short position that makes a riskless gain.

For the case of an option, this synthetic equivalent must be maintained over time, i.e. it is a dynamic hedge (see earlier lessons on delta, gamma, and theta). The cost of carrying this synthetic position equals the arbitrage free price of the option. This is the essence of the very important cost of carry model that underlies all modern derivative pricing theory.

Impact of Interest Rates upon the Cost of Carry

Although the formal details are outside the objectives of this lesson, the synthetic equivalent of a call option can be constructed from a long position in the underlying (i.e. delta units of the underlying), plus some borrowings.

The cost of carry, therefore, increases if interest rates increase (holding all other things equal). This implies that the arbitrage free price of the call option must also increase if we hold all other things equal. In particular, we hold the impact of interest rate increases upon the underlying asset price constant.

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