# Rho

- Rho of a portfolio of options is the rate of change of its value with respect to changes in the interest rate
- Rho = , where Î is the value of the portfolio, and r is the rate of interest
- For European options on non dividend paying stocks, we have;
- Rho (call) = KTe
^{-rT}N(d2), where the symbols carry their usual meanings - Also, Rho (put) = -KTe
^{-rT}N(-d2), the symbols carrying their usual meanings

Summary

- Traders may combine naked and covered positions to evolve a stop-loss strategy
- Stop loss strategy is impractical given the realities of trade and transaction costs involved

- Delta hedging is an improvement
- Involves that delta of the portfolio is maintained at zero
- Requires frequent rebalancing as delta change
- Dynamic hedging strategy

- Hedging can also be attempted with respect to changes in:
- Time (theta)
- Option delta itself (Gamma)
- Volatility (Vega)
- Interest rate (Rho)

- In addition to option Greeks, traders also rely on scenario analysis
- Involves evaluating the option value for simultaneous changes in:
- Time (Theta)
- Volatility (Vega)
- Interest rate (Rho)
- Other factors

A stock trading at $20 has call options available on it with exercise prices $18 and $20. For $1 increase in the stock price how will the delta of the two options change? Choose the most appropriate answer. Change in deltas for the two options are denoted by d Î”_{18} and d Î”_{20}.

- dÎ”
_{18}< dÎ”_{20} - dÎ”
_{18}> dÎ”_{20} - dÎ”
_{18}= dÎ”_{20} - dÎ”
_{18}> dÎ”_{20}and d Î”_{20}= 0

**A.**

Which of the following statements is true regarding optionsâ€™ Greeks?

- Theta tends to be large and positive for at-the-money options
- Gamma is greatest for in-the-money options with long times remaining to expiration
- Vega is greatest for at-the-money options with long times remaining to expiration
- Delta of deep in-the-money put options tends towards +1

**C.**

Vega is the rate of change in the price of an option with respect to changes in the volatility of the underlying asset. Vega is greatest for at-the-money options with long times remaining to expiration

Which of the following statements is false?

- European-styled call and put options are most affected by changes in Vega when they are at the money
- The delta of a European-styled put option on an underlying stock would move towards zero as the price of the underlying stock rises
- The gamma of an at-the-money European-styled option tends to increase as the remaining maturity of the option decreases
- Compared to an at-the-money European-styled call option, an out-of-the money European option with the same strike price and remaining maturity would have a greater negative value for theta

**D.**

Theta is large and negative for an atâ€“the-money European-styled option, whilst theta is close to zero

when the price for the underlying stock is very low. Therefore the theta for an out-of theâ€“money European styled call option would have a lower negative value compared to that of an at-the-money European-styled call option