Risk Management Applications of Options Flashcards
Notation
Value of a covered call at time T
Profit of a covered call a time T
Covered call summary
Value of a protective put at time T
Profit of a protective put at time T
Protective put summary
Bull spread (bull vertical call & bull vertical put)
- Long call + short call with a higher exercise price
- Short put + long put with a lower exercise price
Value of a bull spread at time T
Profit of a bull spread at time T
Bull spread summary
Value of a bear spread at time T
Profit of a bear spread at time T
Bear spread (bear vertical put & bear vertical call)
- Long put + short put with a lower exercise price
- Short call + long call with a higher exercise price
Bear spread summary
Butterfly spread
- An option strategy that combines a bull and a bear spreads and has three exercise prices
- Long butterfly spead (the wings are long) →
- Long call at X1, short 2 calls at X2 and long call at X3
- Long put at X1, short 2 puts at X2 and long put at X3
Value of a butterfly spread (long bull spread + short bull spread) at time T
Profit of a butterfly spread at time T
Butterfly spread summary
Black-Scholes-Merton
- rc is the continuously compounded risk-free rate
- N(d1) and N(d2) are normal probabilities associated with the values d1 and d2
Value of a collar at time T
Profit of a collar at time T
Collar summary
- Sometimes called a range forward or a risk reversal
Box spread
- Is composed of a bull spread and a bear spread
- Long call and short put at X1
- Short call and short put at X2
- If the present value of the payoff exceeds the initial value, the box spread is underpriced and should be purchased
- (X2 − X1) / (1 + r)T > c1 − c2 + p2 − p1
Value of a straddle at time T
Profit of a straddle at time T
Straddle summary
Value of a box spread at time T
Profit of a box spread at time T
PV of the payoff on a box spread
- If the present value of the payoff exceeds the initial value, the box spread is underpriced and sould be purchased
Box spread summary
Straddle variant
- Strap
- Strip
- Strangle
- Straddle + call
- Straddle + put
- Straddle with a put and call that have a different exercise price
Collar
Long put at X1 and short call at X2
Risk reversal
Short put at X1 and long call at X2
- Straddle (long)
- Strangle (long)
- Collar (long)
- Risk reversal (short collar)
- Butterfly (long)
- Seagull (short)
- Long ATM put and long ATM call
- Long OTM put and long OTM call
- Long OTM put and short OTM call
- Short OTM put and long OTM call
- Long OTM call at X1, short 2 ATM calls at X2 and long OTM call at X3
- Short OTM put at X1, long ATM put at X2 and short OTM call at X3
Payoff of an interest rate call option
Payoff of an interest rate put option
Caplets payoff (based on libor)
- The payoff for the caplets is made at the end of the period
Floorlets payoff (based on libor)
Interest call options effective rate
- Effective loan proceeds = notional amount - value of the option at the initiation of the loan
- Call payoff = notional amount * [Max(0, rate at time t – strike rate)(n/360)]
- Effective interest = interest on the loan - call payoff
- Effective annualized loan rate = [(notional amount + effective interest) / effective proceeds]365/n -1
Interest rate collar
- Long position in a cap with a short position in a floor
- The borrower will benefit from falling rates and be hurt by rising rates within that range. Any rate increases above the cap exercise rate will have no net effect, and any rate decreases below the floor exercise rate will have no net effect
The ratio of calls to shares for Delta hedging
- Is the negative of 1 over the delta
Delta hedge with two options
Delta hedging asymmetric effects
- For calls, the delta underestimates the effects of increases in the underlying and overestimates the effects of decreases in the underlying
Gamma
= Change in delta / Change in underlying price
Vega
Change in the option price / Change in volatility
Delta of a spread
- The delta is the sum of the delta and not the average
- Intuition → Close to the expiration, if one option is in-the-money and the other is out-the-money, the first one will have a delta of 1 and the second a delta of 0. The result will be that the position moves 1-to-1 with the underlying. Hence, a delta of 1.
