1. Chapter 3: Insurance, Collars, and Other Strategies
FINA0301 Derivatives Faculty of Business and Economics University of Hong Kong
Dr. Huiyan Qiu

2. Chapter Outline Basic insurance strategies:
insuring a long position: floors;
insuring a short position: caps;
selling insurance
Synthetic forwards: put-call parity
Spreads and collars: bull and bear spreads; box spreads; ratio spreads; collars
Speculating on volatility: straddles; strangles; butterfly spreads; asymmetric butterfly spreads

3. Long / Short Call / Put Options

4. Basic Insurance Strategies
Insurance strategies using options:
Used to insure long positions
Buying put options
Used to insure short positions
Buying call options
Written against asset positions (selling insurance)
Covered call
Covered put

5. Insuring a Long Position
A long position in the underlying asset combined with a put option
Goal: to insure against a fall in the price of the underlying asset
At time 0
Buy one stock at cost S0 (long position in the asset)
Buy a put on the stock with a premium p
An insured long position (buy an asset and a put) looks like a call!

6. Example: S&R index and a S&R put option with a strike price of $1,000 together

7. Combined Payoff / Profit

8. Protective Puts
The portfolio consisting of a long asset position and a long put position is often called “Protective Put”.
Protective puts are the classic “insurance” use of options.
The protective put in the portfolio ensures a floor value (strike price of put) for the portfolio. That is, the asset can be sold for at least the strike price at expiration.
Varying the strike price varies the insurance cost.

9. Insuring a Short Position
A call option is combined with a short position in the underlying asset
Goal: to insure against an increase in the price of the underlying asset
At time 0
Short one stock at price S0
Buy a call on the stock with a premium c
An insured short position (short an asset and buy a call) looks like a put

10. Example: short-selling the S&R index and holding a S&R call option with a strike price of $1,000

11. Combined Payoff / Profit

12. Selling Insurance
For every insurance buyer there must be an insurance seller
Naked writing is writing an option when the writer does not have a position in the asset
Covered writing is writing an option when there is a corresponding position in the underlying asset
Write a call and long the asset
Write a put and short the asset

13. Covered Writing
Covered calls: write a call option and hold the underlying asset. (The long asset position “covers” the writer of the call if the option is exercised.)
A covered call looks like a short put
Covered puts: write a put option and short the underlying asset
A covered put looks like a short call

14. Covered Writing: Covered Calls
Example: holding the S&R index and writing a S&R call option with a strike price of $1,000

15. Combined Payoff / Profit
Writing a covered call generates the same profit as selling a put!

16. Covered Writing: Covered Puts
Example: shorting the S&R index and writing a S&R put option with a strike price of $1,000

17. Combined Payoff / Profit
Writing a covered put generates the same profit as writing a call!

18. Insurance vs. Pure Option Position
Buying an asset and a put generates the same profit as buying a call
Short-selling an asset and buying a call generates the same profit as buying a put
Writing a covered call generates the same profit as selling a put
Writing a covered put generates the same profit as selling a call
How to make the positions equivalent?

19. Insurance vs. Pure Option Position
To make positions equivalent, borrowing or lending has to be involved. Following table summarizes the equivalent positions.

20. Options Combined Underlying asset: S&R Index, spot price = $1,000
6-month Forward: forward price = $1,020
6-month 1,000-strike call: call premium = $93.81
6-month 1,000-strike put: put premium = $74.20
Effective interest rate over 6 month = 2%
Positions: long call + short put
Time-0 cash flow: – = – 19.61
What happens 6 months later?

21. Long Call + Short Put ST >1000 ST < 1000 Long Call Short Put
Outcome at expiration: pay the strike price of $1,000 and own the asset
ST >1000
ST < 1000
Long Call
Pay 1000, get asset
(ST – 1000)
Nothing (0)
Short Put
Total

22. Synthetic Forwards
A synthetic long forward contract: buying a call and selling a put on the same underlying asset, with each option having the same strike price and time to expiration
Example: buy the $1,000- strike S&R call and sell the $1,000-strike S&R put, each with 6 months to expiration

23. Synthetic Forwards (cont’d)
Both synthetic long forward contract and actual forward contract result in owning the asset at the expiration.
Differences
The forward contract has a zero premium, while the synthetic forward requires that we pay the net option premium
With the forward contract, we pay the forward price, while with the synthetic forward we pay the strike price

24. Put-Call Parity
The net cost of buying the index using options (synthetic forward contract) must equal the net cost of buying the index using a forward contract
Call (K, t) – Put (K, t) = PV (F0,t – K)
Call (K, t) and Put (K, t) denote the premiums of options with strike price K and time t until expiration
PV (F0,t ) is the present value of the forward price
This is one of the most important relations in options!

25. More Option Strategies
Combined option positions can be taken to speculate on price direction or on volatility.
Speculating on direction: bull and bear spreads; ratio spreads; collars
Speculating on volatility: straddles; strangles; butterfly spreads; asymmetric butterfly spreads
Box spread

26. Underlying Asset and Options
Underlying asset: XYZ stock with current stock price of $40
8% continuous compounding annual interest rate
Prices of XYZ stock options with 91 days to expiration:
Strike
Call
Put 35
6.13
0.44 40
2.78
1.99 45
0.97
5.08

27. Bull Spreads
A bull spread is a position with the following profit shape.
It is a bet that the price of the underlying asset will increase,
but not too much

28. Bull Spreads (cont’d)
A bull spread is to buy a call/put and sell an otherwise identical call/put with a higher strike price
Bull spread using call options:
Long a call with no downside risk, and
Short a call with higher strike price to eliminate the upside potential
Bull spread using put options:
Short a put to sacrifice upward potential, and
Long a put with lower strike price to eliminate the downside risk

29. Bull Spread with Calls Value K2
Long a call (strike price K1, premium c1)
Value = when ST ≤ K1
= – K1 + [1]ST when ST > K1
Value K2
Short a call (K2 > K1, c2 < c1)
Value = when ST ≤ K2 = K2 + [-1]ST when ST > K2
K2-K1
Portfolio
Value = when ST ≤ K1
= –K1+ [1]ST when K1<ST ≤ K2
= K2 – K1 when ST >K2 K1 K2
FV(-c1+c2) ST
-K1
c1 > c2
Initial cash flows = – c1 + c2 <0

30. Bear Spreads
A bear spread is a position in which one sells a call (or a put) and buys an otherwise identical call (or put) with a higher strike price. Opposite of a bull spread.
Example: short 40-strike call and long 45-strike put
It is a bet that the price of the underlying asset will decrease, but not too much
Option traders trading bear spreads are moderately bearish on the underlying asset

31. Ratio Spreads
A ratio spread is constructed by buying a number of calls ( puts) and selling a different number of calls (puts) with different strike price
Figure: profit diagram of a ratio spread constructed by buying a low-strike call and selling two higher-strike calls.
Limited profit and unlimited risk. To bet that the stock will experience little volatility.

32. Collars
A collar is a long put combined with a short call with higher strike price
It resembles a short forward with a flat middle
To bet that the price of the underlying asset will decrease significantly
A zero-cost collar can be created when the premiums of the call and put exactly offset one another
Long 40-strike put and
short 45-strike call

33. Box Spreads
A box spread is accomplished by using options to create a synthetic long forward at one price and a synthetic short forward at a different price
Synthetic long forward: long a call and short a put with the same strike price
The combination of payoff diagrams of a synthetic long forward and a synthetic short forward is a horizontal line.
A box spread is a means of borrowing or lending money. It has no stock price risk!

34. Speculating on Volatility
Non-directional speculations:
Straddles
Strangles
Butterfly spreads
Asymmetric butterfly spreads
Who would use non-directional positions?
Investors who have a view on volatility but are neutral on price direction
Speculating on volatility

35. Straddles
Buying a call and a put with the same strike price and time to expiration
A straddle is a bet that volatility will be high relative to the market’s assessment
Figure Combined profit diagram for a purchased 40-strike straddle.

36. Strangles
Buying an out-of-the-money call and put with the same time to expiration
A strangle can be used to reduce the high premium cost, associated with a straddle
Figure 40-strike straddle and strangle composed of 35-strike put and 45-strike call.

37. Written Straddles
Selling a call and put with the same strike price and time to maturity
A written straddle is a bet that volatility will be low relative to the market’s assessment
Figure Profit at expiration from a written straddle: selling a 40-strike call and a 40-strike put.

38. Butterfly Spreads
A butterfly spread is = write a straddle + add a strangle = insured written straddle
Figure Written 40-strike straddle, purchased 45-strike call, and purchased 35-strike put.

39. Butterfly Spread Value Sell a call (Strike price K2, premium c2)
Sell a put (Strike price K2, premium p2)
Written straddle
Value = –K2 + [1]ST when ST ≤ K2
= K2 + [-1]ST when ST >K2
Long a put (Strike price K1<K2,premium p1) K1 K3
Long a call (Strike price K3>K2>k1, c3) K2 ST
Long strangle
Value = K1 + [-1]ST when ST ≤ K1
= when K1<ST ≤ K3
= –K3 + [1]ST when ST > K3
Butterfly spread (K3 – K2 = K2 – K1)
Value = –K2 + K1 when ST ≤ K1
= –K2 + [1]ST when K1<ST ≤ K2
= K2 + [-1]ST when K2<ST ≤K3
= –K3 + K2 when ST >K3

40. Butterfly Spread
Value
Initial option costs = c2 + p2 – p1 – c3
= (c2 – c3) + (p2 – p1)
> 0 K1 K3
A butterfly spread is an insured written straddle. Can be used to bet for low volatility. K2

41. Asymmetric Butterfly Spreads
By trading unequal units of options

42. Summary of Various Strategies
Option strategy positions driven by the view on the stock price and volatility directions.

43. End of the Notes!