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-rTN(d2), where the symbols carry their usual meanings
• Also, Rho (put) = -KTe-rTN(-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

Example

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.

1. dÎ”18 < dÎ”20
2. dÎ”18 > dÎ”20
3. dÎ”18 = dÎ”20
4. dÎ”18 > dÎ”20 and d Î”20 = 0

Solution

A.

Example

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

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

Solution

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

Example

Which of the following statements is false?

1. European-styled call and put options are most affected by changes in Vega when they are at the money
2. The delta of a European-styled put option on an underlying stock would move towards zero as the price of the underlying stock rises
3. The gamma of an at-the-money European-styled option tends to increase as the remaining maturity of the option decreases
4. 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

Solution

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