3. Los teoremas de Markov y de Birkhoff-Kakutani 29
3.2. Un salto de axioma de separaci´ on
0
Stock price S
Figure 2: Price of a call option vs. stock price S0
rT { Ke European American E B A r , decrease in ¾ or T Increase 0 Intrinsic value = K - S Put option price
K Stock price S0
Early exercise: Puts on a non-dividend-paying stock
• It can be optimal to exercise an American put option on a non-dividend-paying stock early: – A put option should always be exercised early if it is sufficiently deep in the money • Like a call option, a put option can be viewed as providing insurance:
– Put option, when held with stock, insures against stock price falling below certain level
– However, a put option is different from a call option in that it may be optimal for an investor to forgo this insurance and exercise early in order to realize the strike price immediately
• Early exercise of put option more attractive as S0 decreases, as r increases, and as volatility decreases
• Eq. (2): p ≥ Ke−rT − S
0. American put P : Stronger condition P > K − S0 holds (immediate exercise)
• Fig. 3 shows the general way in which the price of an American put varies with S0
– When early exercise is optimal, the value of the option is K − S0
– Value of put merges into put’s intrinsic value, K − S0, for small value of S0 (point A)
– The line relating the put price to the stock price moves in the direction indicated by the arrows when r decreases, when the volatility increases, and when T increases
• Because there are some circumstances when it is desirable to exercise an American put option early, it follows that American put option always worth more than corresponding European put option
– Point B (price of option = intrinsic value) represents higher value of stock price than A
– Point E is where S0= 0 and the European put price is Ke−rT
Effect of dividends
• In the US most exchange-traded stock options have a life of less than 1 year and dividends payable during the life of the option can usually be predicted with reasonable accuracy
– Let D to denote the PV of the dividends during the life of the option
– In the calculation of D, a dividend is assumed to occur at the time of its ex-dividend date • Lower bound for calls and puts
– Redefine portfolios A and B as follows:
∗ Portfolio A: One European call option plus amount of cash = D + Ke−rT
∗ Portfolio B: One share
– A similar argument to the one used to derive Eq. (1) shows that:
c ≥ S0− D − Ke−rT (5)
– Also redefine portfolios C and D as follows:
∗ Portfolio C: One European put option plus one share
∗ Portfolio D: An amount of cash = D + Ke−rT
– A similar argument to the one used to derive Eq. (2) shows that:
p ≥ D + Ke−rT − S0 (6)
• Early exercise
– When dividends expected, no longer true that American call option not exercised early – Sometimes, optimal to exercise American call immediately prior to an ex-dividend date – It is never optimal to exercise a call at other times
• Put-call parity
– With dividends, the put call parity result in Eq. (3) becomes:
c + D + Ke−rT = p + S0 (7)
– Dividends cause Eq. (4) to be modified to:
Hull - Ch. 10: Trading strategies involving options
Strategies involving a single option and a stock
Short Stock Short Stock Long Call Short Call Long Stock Long Stock Protective put Covered call Profit d) c) b) a) Short Put Long Put T S K T S T S T S K K K
Figure 1: Profit patterns involving a single option and a stock
• In all figures: Dashed line = relationship between profit/stock price for individual securities. Solid line = relationship between profit/stock price for the whole portfolio
• Writing a covered call
– Fig. 1(a): Portfolio consists of long position in stock plus short position in call option
– The long stock position “covers” the investor from the payoff on the short call that becomes necessary if there is a sharp rise in the stock price
• Reverse of writing a covered call
– In Fig. 1(b), a short position in a stock is combined with a long position in a call option • Protective put
– In Fig. 1(c), the investment strategy involves buying a put option on a stock and the stock itself • Reverse of a protective put
– In Fig. 1(d), a short position in a put option is combined with a short position in the stock • The profit patterns in Fig. 1 have the same general shape as the profit patterns discussed in Ch. 8 for
short put, long put, long call, and short call, respectively
– Put-call parity provides a way of understanding why this is so:
p + S0 = c + Ke−rT + D (1)
– Eq. (1) shows that a long position in a put combined with a long position in the stock is equivalent
to a long call position plus a certain amount (= Ke−rT + D) of cash
⇒ The profit pattern in Fig. 1(c) is similar to the profit pattern from a long call position – The position in Fig. 1(d) is the reverse of that in Fig. 1(c)
⇒ The profit pattern similar to that from a short call position
– Eq. (1) can be rearranged to become: S0− c = Ke−rT + D − p
– This shows that a long position in a stock combined with a short position in a call is equivalent to
a short put position plus a certain amount (= Ke−rT + D) of cash
⇒ The profit pattern in Fig. 1(a) is similar to the profit pattern from a short put position – The position in Fig. 1(b) is the reverse of that in Fig. 1(a)
Spreads
• Spread strategy: Taking a position in two/more options of same type (e.g., two or more calls) • Bull spreads
– Created by buying a call option on a stock with a certain strike price and selling a call option on the same stock with a higher strike price (both options have same expiration date)
) 1 Long call option (K
)
2
Short call option (K
Bull spread from 2 call options
2 K 1 K T S Profit
Figure 2: Profit from bull spread using calls
Profit T S 1 K 2 K Bull spread from 2 put options
)
2
Short put option (K
) 1 Long put option (K
Figure 3: Profit from bull spread using puts – Call price always decreases as strike price increases ⇒ Value of option sold < value of option bought
⇒ A bull spread, when created from calls, requires an initial investment
– Bull spreads also created by buying put (low strike price) and selling put (high strike price) ∗ Unlike bull spread created from calls, bull spreads from puts involve positive up-front CF to
investor (ignoring margin requirements) and payoff ≤ 0 – Payoff from a bull spread created using calls
Stock price Long call payoff Short call payoff Total payoff
ST ≤ K1 0 0 0
K1< ST < K2 ST− K1 0 ST− K1
ST ≥ K2 ST− K1 −(ST − K2) K2− K1
– A bull spread strategy limits the investor’s upside as well as downside risk:
∗ Strategy: The investor has a call option with a strike price equal to K1 and has chosen to give up some upside potential by selling a call option with strike price K2 (K2> K1)
∗ In return for giving up upside potential, investor gets the price of option with strike price K2 – Three types of bull spreads can be distinguished:
1. Both calls are initially out of the money (most aggresive)
2. One call is initially in the money, the other call is initially out of the money 3. Both calls are initially in the money
∗ Type 1: Cost very little to set up/small probability of a high payoff (= K2− K1) ∗ As we move from type 1 → 2 and from 2 → 3, spreads become more conservative • Bear spreads
– Bear spreads created by buying put with one strike price/selling put with another strike price ∗ The strike price of the option purchased is greater than the strike price of the option sold ∗ Payoff from a bear spread created with put options
Stock price Long put payoff Short put payoff Total payoff ST ≤ K1 K2− ST −(K1− ST) K2− K1
K1< ST < K2 K2− ST 0 K2− ST
ST ≥ K2 0 0 0
– Bear spread from puts involves initial cash outflow (price of put sold < price of put purchased) ∗ In essence, the investor has bought a put with a certain strike price and chosen to give up some
of the profit potential by selling a put with a lower strike price
Profit T S 1 K 2 K Bear spread from 2 put options
) 2 Long put option (K
)
1
Short put option (K
Figure 4: Profit from bear spread using puts
)
1
Short call option (K
) 2 Long call option (K
Bear spread from 2 call options
2 K 1 K T S Profit
Figure 5: Profit from bear spread using calls – Bear spreads can be created using calls instead of puts:
∗ Investor buys call with high strike price/sells call with low strike price (Fig. 5) ∗ Bear spreads from calls involve an initial cash inflow (ignoring margin requirements) – Comparison with bull spread
∗ Like bull spreads, bear spreads limit both upside profit potential/downside risk ∗ An investor who enters into a bull spread is hoping that the stock price will increase ∗ By contrast, an investor who enters into a bear spread hopes that stock price will decline • Box spreads
– A box spread is a combination of a bull call spread with strike prices K1 and K2 and a bear put
spread with the same two strike prices. The payoff from a box spread is always K2− K1
Stock price Bull call spread payoff Bear put spread payoff Total payoff
ST ≤ K1 0 K2− K1 K2− K1
K1< ST < K2 ST − K1 K2− ST K2− K1
ST ≥ K2 K2− K1 0 K2− K1
– The value of a box spread is therefore always the PV of this payoff or (K2− K1)e−rT – If it has a different value there is an arbitrage opportunity:
∗ If the market price of the box spread is too high, it is profitable to sell the box
∗ This involves buying a call with strike price K2, buying a put with strike price K1, selling a call with strike price K1, and selling a put with strike price K2
– A box-spread arbitrage only works with European options • Butterfly spreads
– A butterfly spread involves positions in options with three different strike prices:
∗ Buy call option with low strike price K1, buy call option with high strike price K3, and sell two call options with strike price K2 halfway between K1 and K3
)
2
2 short call options (K
1
K
T
S
Profit
Butterfly spread using call options
2 K 3 K ) 1
Long call option (K
)
3
Long call option (K
Figure 6: Profit from butterfly spread using calls
) 3 Long put option (K
) 1 Long put option (K
)
2
2 short put options (K
Butterfly spread using put options
3 K 2 K Profit T S 1 K
∗ Profit if stock price stays close to K2, but small loss if significant price move in either direction ⇒ Appropriate strategy for an investor who feels that large stock price moves are unlikely ∗ The strategy requires a small investment initially
∗ Payoff from a butterfly spread (assuming K2 = (K1+ K3)/2)
Stock price 1st long call payoff 2nd long call payoff Short calls payoff Total payoff
ST ≤ K1 0 0 0 0
K1< ST < K2 ST− K1 0 0 ST − K1
K2< ST < K3 ST− K1 0 −2(ST − K2) K3− ST
ST ≥ K3 ST− K1 ST − K3 −2(ST − K2) 0
– Butterfly spreads can be created using put options:
∗ The investor buys one put with a low strike price, another with a high strike price, and sells two puts with an intermediate strike price, see Fig. 7
∗ If all options are European, use of puts results in same spread as with calls
∗ Put-call parity can be used to show that the initial investment is the same in both cases – A butterfly spread can be sold or shorted by following the reverse strategy:
∗ Options sold with strike prices K1/K3, and two options purchased with middle strike price K2
∗ This strategy produces a modest profit if there is a significant movement in stock price