Basic Option Strategies

An option strategy is a portfolio of options with a single or several objectives. Possible objectives are:

• Hedging. Reduce risks in an existing portfolio of securities.

• Investing. Customized risk and return profile.

• To take advantage of mispriced securities, i.e. arbitrage strategies.

A characteristic of an option strategy is whether the strategy isstatic, i.e. the strategy is setup at initiation and left unchanged over time until maturity, or whether the strategy

isdynamic. Dynamic strategies can be rule based or discretionary or a mixture of them.

For call and put strategies the loss/gain is

• bounded by the exercise price minus the premium for short put,

• unbounded for short call. The gain potential is

• bounded by the premium for short put and short call,

• bounded by the exercise price minus the premium for long put,

• unbounded for long call.

The next more complex strategies are those consisting of an option and the underlying value. We consider:

• Covered call (investment strategy), see Figure2.8;

• Protective put (hedge strategy).

The investment motivation for a covered call is a belief that the underlying value will move sideways or at most increase only moderately. The investor does not search protection for a stress or a crash scenario. The covered call strategy, i.e. long the underlying and short a call option, capture this investment motivation. If the underlying decreases the value of the covered call is always higher than the underlying values since the investor obtained a premium for the sold call option.

125 150 75 50 P&L Underlying short call long Underlying covered call

Figure 2.8: Covered Call.

Table ?? shows: The covered call is always worth more than the underlying if it drops, there is no capital protection and the upside is given up if above a certain level. If the motivation of the investor is to buy protection he goes long the underlying and long a put. This strategy - protective put - reduces loss potential to the put premium keeping the gain potential unbounded.

ST Payoff Call P&L Short Call P&L LongS P&L Covered Call 27 0 3.6 -4 -0.4 28 0 3.6 -3 0.6 29 0 3.6 -2 1.6 30 0 3.6 -1 2.6 31 0 3.6 0 3.6 32 0 3.6 1 4.6 33 0 3.6 2 5.6 34 -1 2.6 3 5.6 35 -2 1.6 4 5.6 36 -3 0.6 5 5.6

Table 2.3: P&L-comparison for covered call, long underlying, short call for the data

K = 33,S(0) = 31. The upside is restricted to 5.6 =K₋S(0) +C(0) = 33₋31 + 3.6.

The next strategy is the so-called 3-options strategy. Consider 3 European or American call options with increasing strikes K1 < K2 < K3 with the same underlying and maturity. If arbitrage is not possible, the following structural,i.e. model-independent, inequality holds: C(K2)≤ K3−K2 K3−K1 C(K1) +K2−K1 K3−K1 C(K3). (2.39)

It exists a combination of strikes such that the option price to the middle strike K2 is never larger than a combination of the minimum price C(K1) and the maximum price

C(K3). This relation is due to the fact that option prices are monotone decreasing with increasing strikes.

We show how the above 3-option relation can be used to test for arbitrage. Assume

S0 = 100and three American calls with strikes and prices: • C1(94) = 8.4.

• C2(100) = 5.5. • C3(108) = 1.4.

Are these prices arbitrage free? To answer this, we write the middle strike K2 = 100 as a convex combination of the two others:

K2 = 100 =a∗K1+ (1−a)K3=a∗94 + (1−a)108 = 108−14a,

a= 4/7 = K3_K3−₋K2_K1 and1₋a= 3/7 = K2_K3−₋K1_K1 follow. Using this, we check (2.39):

K3−K2 K3−K1 | {z } a C(K1) + K2−K1 K3−K1 | {z } 1−a C(K3) = 4/7∗8.4 + 3/7∗1.4 = 5.4.

This price is smaller thanC(K2) = 5.5. Arbitrage is possible. The K2-call is relative to the two other ones to expensive. Can you exploit this?

The next strategy, called conversion, shows how no arbitrage violations measured in the put-call parity can be exploited. We recall the put-call parity for futures:

C−P = (F −K)e−rτ

and consider the example strike of call and put K = 110, futures price F = 100, put priceP = 12, call price C= 4,r= 10% and time to maturityτ = 1 year. Inserting this into the parity gives.

C−P =−8 6= (F −K)e−rτ =−9.05

Since the call is out of the money and the put is in the money it is more likely that the call is mispriced. To exploit the arbitrage opportunity we buy the low priced put and sell the high priced call. At time twe setup the portfolioV:

• Long Put, i.e. P =−12.

• Short Call, i.e. C= 4.

• Long den Future, i.e. F = 0.

• Borrow Cash to finance the strategy, i.e. 8.

This portfolio value is zero at time t. What is the value at time T?

V(T) = PT −CT +F−Xerτ = (K−ST)+ | {z } Put −(ST −K)+ | {z } Call + (ST −F) | {z } Future − Xe_{| {z }}rτ

Pay back loan

The value of the long futures equals the value difference between the underlying at maturityST and the contracted futures priceF at timet. This gives two portfolio values

at T: • ST ≥K V(T) = PT −CT + (ST −F)−Xerτ = 0−(ST −K) + (ST −F)−Xerτ = 0−ST + 110 +ST −100−8e0.1 = 110₋100₋8.84 = 1.16 • ST < K V(T) = PT −CT + (ST −F)−Xerτ = (K₋ST)−0 + (ST −F)−Xerτ = 110₋ST −0 +ST −100−8e0.1 = 110−100−8.84 = 1.16

An arbitrage strategy is found: A strategy which is worth zero at time t and which is worth 1.16 at time T in all possible states. We note that this risk less gain equals the future value of the initial mispricing in the put-call parity:

(9.05₋8)e0.1 = 1.16 .

The next strategies arespread strategies. They are generated by at least two op- tions where the options are identical expect in one or possible two parameters. Variations in strikes are bull and bear spreads, vertical spreads or risk reversals; variation in matu- rity are calendar or horizontal spreads; variations in the option right are straddles and variations in the option right and strike lead to strangles and butterflies.

For a bull spread, the investor believes that the underlying will increase to a certain level and he wants to restrict losses if the belief turns out to be wrong. He invests in a bull spread, i.e. a long call C(KL) and short callC(KH):

Bull Spread=C(KL)−C(KH), whereKH > KL

The loss is restricted to the difference in the premia, see Figure2.9.

125 150

75 50

P&L

Underlying

short call KH=HIGH long call KL =LOW

bull call spread

Figure 2.9: Bull Spread. Verify the following figures for theBull Spread:

P&L = max(S(T)₋KL,0)−max(S(T)−KH,0)−CL(0) +CH(0)

max Profit = KH −KL−CL(0) +CH(0)

max Loss = CH(0)−CL(0)

Break even = KL+CL(0)−CH(0)

If an investor beliefs that events will move the underlying away from its present price but he has no directional view, strangle or straddle allow to invest into this volatility bet, see Figure ??.

A strangle has contrary to the straddle two options with different strikes. A strangle is cheaper than the comparable straddle.

125 150 75 50 Underlying long put long call Straddle = ? Figure 2.10: Straddle.