Introduction

Rho:

- represents the sensitivity of an option’s price to a movement in IRs.

- positive for Call options

- negative for Put options

 

The prices of vanilla options are almost linear in IRs --> only has a first-order effect. 

This effect comes from the impact of IRs on: 

- cost of the delta-hedge

- discounting the option price

 

The second effect is generally smaller than the first one. 

In the case of call options --> the two effects work in opposite direction

In the case of put options --> the two effects work in the same direction. 

 

A trader sells a call option and delta hedges by buying delta shares.

To buy those shares, he must borrow money at the bank. He will have to pay interest on his loan.

The higher the IR --> the more interests --> the higher the cost of the hedge --> the higher the call price. 

The discounting effect slightly offsets this delta hedge impact though.

 

--> If you make more money on your delta hedge, you are going to pay more (receive less) for it.​

 

In the BS model: 

the Rho of a call option is given by:

and the Rho of a put option is the negative of this. 

 

Moneyness:

Rho is larger for ITM options and decreases steadily as the option changes to become OTM. 

 

Time to Maturity:

Rho increases as time to expiration increases. 

Long-dated options are far more sensitive to changes in IR than short-dated options --> larger rho. 

 

These two factors are explained by the effect that IRs have on the cost of carry of an option.

long-dated ITM options will have higher premiums --> require more cash to hold the option until the expiration date.

 

Though rho is a primary input in the BS model, a change in IRs generally has a minor overall impact on the pricing of options.

Because of this, rho is usually considered to be the least important of all the option Greeks.

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