Risk management application of option strategies


1. Option strategies: Bull spread, bear spread, butterfly spread, collar, straddle, box spread.

A bull spread consists of a long call and a short call. The short call has a higher exercise price and its premium subsidizes the long call.

It offers gains if the underlying asset’s price goes up, but the upside is limited.

A bear spread strategy is the opposite side of a bull spread. It offers a limited upside gains if the underlying asset’s price declines.

A butterfly spread consists of two long and two short call positions. It offers a return, with a limited upside if the underlying asset does not move very much.

– A collar strategy is simply a covered call and a protective put combinesd to limit the down and upside value of the position.

– A long straddle is a long call and long put with the same exercise price. The greater the mvoe in the stock price, the greater the payoff from a straddle.

– A box spread strategy combines a long put and a short put with a long call and a short call to produce a guarateed return. That return should be the risk- free rate.




2. Calculate the effective annual rate for a given interest rate outcome when a borrower (lender) manages the risk of an anticipated loan using an interest rate call (put) option.

– To hedge a future borrowing, purchase a call on interest rates for protecting from increasing rates.

– To hedge a future lending, purchase a put on interest rates for protection from declining interest rates.

– Assume that at time 0, the option to hedge the risk is purchased, and the purchase price is financed by borrowing at a rate reflecting the primary loan spread for a net CF of zero at time 0.

– At time t when the primary loan occurs, net the cash flow of the primary loan with the option premium financing repayment to determine a net CF at time t.

– At time T, when the primary loan is repaid, net the repayment of primary loan cash flow and any payoff on the option for a net CF at time T.

– Calculate the EAR between T t and T T net CFs


3. Calculate the payoffs for a series of interest rate outcomes when a floating rate loan is combined with 1. interest rate cap, 2. interest rate floor, 3. an interest rate collar.

– An interest rate cap is a series of interest rate calls with the same strike rate but different expiration dates. Settlements are at the end of each period but are based on rates at the beginning of each period.

– An interest rate floor is a series of interest rate puts with the same strike rate but different expiration dates. As with the cap, settlements are at the end of each period based on rates at the beginning of each period.

– An interest rate collar is a combination of cap and floor where the investor is long one and short the other. A short cap and long floor would be of use to a lender of floating- rate loans. The collar will guarantee a floating rate borrower a range of interest costs on the loan.


4. Explain why and how a dealer delta hedges an option position, why delta changes and how the dealer adjusts to maintain the delta hedge.

– Delta hedging generally refers to immunizing the value of an option position from changes in the value of the underlying asset.

– Delta will change and the manager should periodically adjust the asset’s position and invest in or borrow at the risk-free asset to keep the hedge.


Delta call = ΔC / ΔS

ΔC: change in the price of the call over a short time interval

ΔS: change in the price of the underlying stock over a short time interval.


5. Interpret the gamma of a delta-hedged portfolio and explain how gamma changes as in-the-money and out-of-the-money options move toward expiration.


Gamma = Δdelta / Δstock

– As an at- or near-the-money option approaches expiration, its delta will tend to move quickly to either one or zero, depending on the direction of the stock price movement.

– Gamma of an at-the-money option is greatest near the expiration date. When option values are subject to changes, the position faces the most risk, and a delta hedger is more likely to gamma hedge.

– The hedge entails combining the underlying stock position with two options position in such a manner that both delta and gamma are equal to zero.