Greeks options - Rho

By roma, 16 August, 2020

ρ = δV/δr

V - option price
r - risk-free interest rate

Rho measures the sensitivity of the option price to a one percentage point change in interest rates. It is not unusual that Rho is at the end of the list of risks associated with option trading, as usually the interest rates have very little effect on the option price and are therefore practically ignored by traders.

However, in the case of high yielding currencies such as the rouble, Brazilian real and Turkish lira, interest rates can have a significant impact on a trader's income, especially if the trader or investor intends to hold an option position for several months.

Figure 1 illustrates the behavior of the at-the-money call option price as interest rates go up. Rho is a tangent line at a certain point. In this case, the value of the rho is equal to $0.20, i.e. a 1% increase in the interest rate will increase the value of the option by $0.20, all other parameters being equal.

Call price
Call price

Put price
Put price

 

Rho for call option

When pricing an option it is assumed that short option position can be hedged by buying or selling the delta. In the case of a short call option, the trader needs to buy shares to make the delta zero. The Black-Scholes model assumes that the purchase of shares is financed by a loan and dividend payments from the shares. Therefore, an increase in interest rates will increase the value of call options, as the call option writer will want to incorporate expenses to obtain credit (to buy shares to hedge the delta) to the option buyer.

 

Rho for put option

On the other hand, rising interest rates will serve as a catalyst for falling prices of put options. The seller of put options must sell short the shares to neutralize the portfolio delta. As the rates go up, the trader who sells put options will be able to invest the proceeds of selling the shares at higher interest rates. Therefore, the trader (seller of put options) is ready to lower the put option price.

 

Delta

Gamma

Theta

Vega

Rho

Long Call

+

+

Pay

+

+

Short Call

Receive

Long Put

+

Pay

+

Short Put

+

Receive

+

 

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