Lec4.pdf

Option Properties

Liuren Wu

Zicklin School of Business, Baruch College

Options Markets

(Hull chapter: 9)

Liuren Wu (Baruch) Option Properties Options Markets 1 / 17

Notation

c : European call option price. C American call price.

p: European put option price. P American put price.

S : stock price. Ft,T time-t forward price with expiry T .

K : strike price.

Today: either time 0 or time t.

T : expiry date (or maturity with t = 0).

: Volatility (annualized standard deviation) of stock return.

r : continuously compounded riskfree rate with maturity T (same as option).

D: present value of discrete dividends paid during options life.

q: continuously compounded dividend yield during options life (for foreigninterest rate for currency options).


Dependence of option values

Variable c p C PSt + - + -K - + - +T ? ? + + + + + +r + - + -D/q/rf - + - +

In examples given below, I use the following benchmark numbers:St = 100;K = 100; = 20%; t = 0;T = 2/12; r = 5%; q = 3%.(unless otherwise specified)


Option variation with spot price

80 85 90 95 100 105 110 115 1200

5

10

15

20

25

Call o

ption

value

, ct

Spot price, St

T=0T=1mT=1y

80 85 90 95 100 105 110 115 1200

2

4

6

8

10

12

14

16

18

20

Put o

ption

value

, pt

Spot price, St

T=0T=1mT=1y

Dependence on spot follows from payoff function (when time to maturity is zero).


Option variation with strike price

80 85 90 95 100 105 110 115 1200

5

10

15

20

25

Call o

ption

value

, ct

Strike price, K

T=0T=1mT=1y

80 85 90 95 100 105 110 115 1200

2

4

6

8

10

12

14

16

18

20

Put o

ptio

n va

lue,

p t

Strike price, K

T=0T=1mT=1y

Dependence on strike follows from payoff function (when time to maturity iszero).


European option variation with time to maturity

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Call o

ption

value

, ct

Maturity, T

K=80K=100K=120

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Put o

ption

value

, pt

Maturity, T

K=80K=100K=120

Normally increase, but can decline.American option value always increases with time, given the option to exerciseearly.


Option variation with volatility

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

Call o

ption

value

, ct

Volatility,

K=80K=100K=120

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

35

40

45

50

Put o

ption

value

, pt

Volatility,

K=80K=100K=120

ATM option value is almost linear in volatility. Options market is mainly a marketfor volatility.


Option variation with interest rates and dividend yields

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

30Ca

ll opti

on va

lue, c t

Interest rate, r

K=80K=100K=120

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

Put o

ption

value

, pt

Interest rate, r

K=80K=100K=120

0 0.02 0.04 0.06 0.08 0.10

5

10

15

20

25

Call o

ption

value

, c t

Dividend yield, q

K=80K=100K=120

0 0.02 0.04 0.06 0.08 0.15

0

5

10

15

20

25

Put o

ption

value

, pt

Dividend yield, q

K=80K=100K=120

Increasing interest rate or reducing dividend yield increases the growth rate of thestock price.


American v. European options

The above graphs are all generated for European options, based a simpleoption pricing model, which well deal with later.

An American option is worth at least as much as the correspondingEuropean option: C c and P p.

I The difference is due to the extra option you get from an Americanoption: You can exercise any time before the expiry date.

I Hence, the difference is also referred to as the early exercise premium.

Pricing American options is a bit harder well need a numerical algorithmto deal with the early exercise premium.


Obvious arbitrage opportunities on call options

Suppose that ct = 2,St = 20,T t = 1,K = 17, r = 5%, q = 0.Is there an arbitrage opportunity?

Suppose that ct = 21,St = 20,T t = 1,K = 1, r = 5%, q = 0.Is there an arbitrage opportunity?


Bounds on a call option

Lower bound:ct max[0,Steq(Tt) Ker(Tt)] = max[0, er(Tt)(Ft,T K )].

I If the call price is negative, you are paid to own an option.I If the call price is lower than the value of the forward at the same

strike, er(Tt)(Ft,T K ), buy the call and short the forward:F Todays value from the transaction is: ct + er(Tt)(Ft,T K) > 0.F At maturity, the payoff is (ST K)+ (ST K) = (K ST )+ > 0.

I Point to remember: An option is worth more than a forward.

Upper bound: ct Steq(Tt) = er(Tt)Ft,T .I If violated, write the call and long a forward at zero strike.

F Todays value from the transaction is: ct er(Tt)Ft,T > 0.F At maturity, the payoff is (ST K)+ + ST = min(ST ,K) > 0.

I A forward at zero strike is the same as a call at zero strike.I A call with positive strike is worth less than a call with zero strike.I More generally, c(K1) c(K2) for all K1 K2. Can you devise an

arbitrage trading when this monotonicity condition is violated?

In the presence of discrete dividends, the bounds are:[St D Ker(Tt),St D].Liuren Wu (Baruch) Option Properties Options Markets 11 / 17

Less obvious arbitrage opportunities on call options

Suppose that ct = 3.2,St = 20,T t = 1,K = 17, r = 5%, q = 3%.Is there an arbitrage opportunity?

I Steq(Tt) Ker(Tt) = 3.238.

Suppose that ct = 19.5,St = 20,T t = 1,K = 1, r = 5%, q = 3%.Is there an arbitrage opportunity?

I Steq(Tt) = 19.41.

Suppose that ct = 19.5,St = 20,T t = 1,K = 1, r = 5%,D = 1.Is there an arbitrage opportunity?

I St D = 19.


Bounds on a put option

Lower bound:pt max[0,Ker(Tt) Steq(Tt)] = max[0, er(Tt)(K Ft,T ).]

I If put price is negative, you are paid to own an option.I If put price is lower than the value of the corresponding short forward,

buy the put and long the forward:F Todays value from the transaction is: pt er(Tt)(Ft,T K) > 0.F At maturity, the payoff is (K ST )+ + (ST K) = (ST K)+ > 0.

I An option to sell is worth more than a forward to sell.I With discrete dividends, the lower bound is: Ker(Tt) St + D.

Upper bound: pt Ker(Tt).I If violated, write the put and save Ker(Tt) in the bank. Spend the

rest at water park.F Todays value: water slides.F At maturity, the return from the bank is K . The worst possible

obligation from writing the put is K when the company goesbankruptcy (ST = 0). So you make even in the worst case, makemoney otherwise. K (K ST )+ = min(ST ,K) > 0.

I Monotonicity condition: p(K1) p(K2) for all K1 K2.Liuren Wu (Baruch) Option Properties Options Markets 13 / 17

Arbitrage opportunities on put options

Suppose that pt = 2,St = 20,T t = 1,K = 23, r = 5%, q = 3%.Is there an arbitrage opportunity?

I Ker(Tt) Steq(Tt) = 2.47.

Suppose that pt = 2,St = 20,T t = 1,K = 23, r = 5%,D = 1.Is there an arbitrage opportunity?

I Ker(Tt) St + D = 2.88.

Suppose the put price at $20 strike is $2 and at $21 strike is $1.9 at thesame maturity. Is there an arbitrage opportunity?


Put-call parity

Recall the put-call parity condition:The difference between a call and a put equals the forward.

ct pt = er(Tt)(Ft,T Kt)= Ste

q(Tt) Kter(Tt)= St D Kter(Tt)

Suppose that ct = 8,St = 100,T t = 1,K = 100, r = 0, q = 0.Is there an arbitrage opportunity if

I pt = 7.I pt = 8.I pt = 9.


Early exercise of American options

Usually there is some chance that an American option will be exercised early.

I Exercise when the exercise value ((St K )+ for call) is higher thanthe option value (Ct).

As a result, the American option is more expensive (valuable) than theEuropean option.

An exception is an American call on a non-dividend paying stock, whichshould never be exercised early.

I Ct ct St Ker(Tt) St K = exercise valueI The call option value for a non-dividend paying stock is always no

lower than the exercise value.


Cheat sheet: Summary of important relations/bounds

Forward pricing:

Ft = e(rq)(Tt)St = er(Tt)(St Dt),

European options bounds:

ct [er(Tt)(Ft,T K )+, er(Tt)Ft,T ]pt [er(Tt)(K Ft,T )+, er(Tt)K ]

Put-call parity for European options:

ct pt = er(Tt)(Ft,T Kt) = eq(Tt)St er(Tt)K .Put-call inequality for American options:

St D K Ct Pt St Ker(Tt)Ste

q(Tt) K Ct Pt St Ker(Tt)

Monotonicity for both European and American options:

p(K1) p(K2), c(K1) c(K2), for all K1 > K2.Most important: Know how to trade against violations of these conditions.


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