Lec 10: How to Discover Option Prices (Hull, Ch. 10)
Suppose S0 = $50 and r = 25% .
Q: What might be reasonable prices for C0E, C0A, or P0E, P0A
(given K=40, T=1 year)?.
Intuition, or what questions to think about.
▸ Is the stock price expected to ↑ or ↓?
▸ If call is American, I would pay at least $10. Why?
▸ If call is European, why pay anything?
(Exercise ONLY on the Expiration Day!.)
▸ Is it ever possible for C0E = C0A, or P0E = P0A ?
The purpose of this Lecture is to help you develop “good intuition”
about option pricing.
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European Call, Stock pays no dividends: C0E (p.2)
Do these prices make sense?
S0 = $50, C0E(K = $40, T=1yr) = $5, and r = 25%(simple interest)
Intuition. There are two ways to buy stock:
A: Buy the stock right now, CF0 = -50
Or
B: Buy the call and a bond and wait until expiration
{+C, +B(FV=$40, T=1yr)} ➟
CF0 = -5-32 = -$37
At Expiration, for the synthetic stock:
if call is in the money (ST > 40)
➟ CT + 40 = ST
if call is out of the money (ST < 40) ➟ CT + 40 = 40.
Which is the better investment A or B ?
Is it possible to make some “free money”?
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Yes, try the following strategy: {-S, +C, +B(FV=40, T=1yr)}
{Short the stock for $50,
Buy the call for $5, Buy a bond for $32 =40/1.25}
CF0 = +50-5-32 = +$13
At Expiration,
if ST ≥ 40, call is in the money. Bond matures for $40,
use $40 plus call to buy stock.
Use stock to cover short position. CFT=0.
if ST < 40,
call is worthless. Bond matures for $40.
Use some of $40, buy stock and cover the short
position. CFT = 40 - ST > 0.
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This strategy is great! Think about it:
Receive $13 now.
If stock price ↓, make more money (40 - ST).
If ST ↑, lose nothing!
This is known as an ARBITRAGE OPPORTUNITY.
The “Arbitrage Profit” = $13.
Clearly, Call is mis-priced.
To preclude this arbitrage C0E must be at least 5+13 = $18.
In sum, If
S0 = $50, and r = 25%, then C0E(K = $40, T=1yr) > $18
(Compare this answer with initial intuition:
“European Call has little value” ).
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European Call Price C0E, Stock pays a Dividend (p. 3)
Assume stock pays a $5 Dividend (for sure) in 3 months.
How will this affect the Call value?
Do these prices make sense?
S0 = $50, C0E(K = $40, T=1yr)=$6, r=25%, and Div=$5 in 3 months.
There are two ways to buy stock:
A: Buy the stock right now: CF0 = -50,
Or
B: Buy the call.
Buy a bond to mimic the dividend,
and another bond to cover the $40. Wait until expiration.
{+C, +B(FV=5, t=3 months), +B(FV=40, T=1yr) }
CF0 = -6 -4.71 -32 = -$42.71
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Cash Flows for the synthetic stock:
In 3 months,
1st bond matures for $5, just like the $5 Dividend from the stock.
At Expiration,
if call is in the money (ST > 40)
if call is out of the money (ST < 40)
CT + 40 = ST
CT + 40 = 40
Which is the better investment A or B ?
(Synthetic is better: it costs less and has better future outcomes)
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Arbitrage Strategy: {-S, +C, +B(FV=5, T=3 months), +B(FV=40, T=1yr) }
CF0 = +50 - 6 - 32 - 4.71 = +$7.29
In 3 months,
use $5 from the first bond to cover Dividend on short position.
At Expiration,
if ST ≥ 40, Bond matures, receive $40. Call is in the money;
use $40 (from the bond) plus the call to buy stock.
Use stock to cover short position. CFT= 0.
if ST < 40,
Bond matures, receive $40. Call is worthless.
Use some of the $40 from the bond to buy stock
and cover the short. CFT = 40 - ST > 0.
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Thus, we have an arbitrage opportunity.
Receive a CF0 = $7.29 now.
If S↑, lose nothing!
If S ↓, make even more money (40 - ST).
To preclude the arbitrage
C0E must be at least 6 + 7.29 = $13.29.
(Exercise: Assume a $10 Div. in 3 months. Show that C0E > $8.59).
In sum:
if S0 = $50, C0E($40, T=1yr), r = 25%, plus a dividend in 3 months
No Div
$5 Div
$10 Div
C0E ≥
$18
$13.29
$8.59
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American Call Price C0A, no Dividends
(p. 4)
Do these prices make sense?
S0 = $50, C0A(K = $40, T=1yr) = $5
There are two ways to buy the stock:
1. Pay $50 and buy the stock immediately. Or
2. Buy the Call for $5, exercise immediately, Pay only $45
Smell Arbitrage?
Buy the Call for $5, pay $40 to exercise call, sell the stock for + $50,
CF0 = -5 - 40 + 50 =+5. ➟ Arb. profit = $5
To preclude arbitrage we must have:
C0A + 40 > 50; i.e., C0A > $10;
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American Call C0A, Stock Pays a SMALL Dividend
(p. 5)
If the stock paid a dividend, you would want to exercise in order to
collect the dividend. Yes or No?
Suppose:
S0 = $50, C0A(K = $40, T=1yr) = $11
Div = $5 (for sure) in 3 months(t*),
r = 25%/yr ⇒ r for 3 months = 25%/4 = 6¼%.
Must consider:
1) exercise before dividend is paid out
2) forgo dividend, wait till expiration
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1. Create a Synthetic Stock Position for 3 months.
(Exercise just before dividend is paid)
Strategy: {+C, +B(FV = $40, t* = 3 months), -S}.
CF0 = - C0 - PV(K) + S0 = -11 - 40/1.0625 + 50 = $1.35 > 0 ☺
At t* = 3 months, just before ex-dividend day
if S3 > $40 Call is in the money.
Use $40 from bond to exercise call, receive stock,
use it to cover short position before dividend is paid.
CF3= 0.
If S3 < $40 Call out money ∴ do not exercise.
Bond matures for $40; use some of it to buy stock
and cover short .
CF3=40-S3 > 0, and you still own the call!
Clearly, this is an Arb. opp. ⇒ C0 must be > $11.
C0A must be > S0 - PV(K,t*) = $12.35 = 11 + 1.35.
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2. Create a Synthetic Stock Position for 1 year.
(Do not Exercise, give up the dividend)
Synthetic Stock={+C, +B (FV =$5, t*=3 months), +B(FV =$40, T =1yr)},
Synthetic Stock Price = { C0 + D/(1+r/4) + 40/1.25 }
= 11 + 4.71 + 32 = $47.71
Actual Stock Price = S0 = $50 ⇒ Arb. opp.
Set up an arbitrage:
{Buy Synthetic Stock, Short the actual (i.e., physical) stock}
{-S, +C, +Bond(FV=5, t*=3 months) , +Bond(FV=40, T=1) }.
➟
Net CF0 = +$50 - (11 + 4.71 +32) = $2.29
Will it work?
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In 3 months, use the 1st bond (=$5) to cover dividend on short stock.
At T = 1 yr (= Expiration)
if ST > 40 Call in the money,
receive $40 from 2nd bond,
use it to exercise call, receive stock, cover short.
Net CFT = 0.
if ST < 40
Call out of the money, throw it away.
Receive $40 from bond,
use some of it to buy back stock and cover short.
CFT = 40 - ST > 0!
To preclude Arb. C0A > $13.29 (=11+2.29)
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What do we learn?
Go back to original question:
“Is it a good idea to exercise just to receive the dividend?”
If you exercise right before the dividend payment, C0A = $12.35,
If you DO NOT plan to exercise in 3 months,
C0A = $13.29,
It seems that the option is worth more if we forgo the dividend.
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American Call C0A, Stock Pays a LARGE Dividend
(p. 6)
Stock pays a $10 Dividend (for sure) in 3 months (time t*).
Again, consider:
1) Exercise before dividend is paid out, or
2) Wait till expiration
1. Create a Synthetic Stock Position for 3 months.
(Exercise before dividend is paid)
The synthetic position in the stock for 3 months consists of:
{+C, +B (FV = $40, t = 3months)} = { C0 + 40/(1.0625) }
= C0A + $37.65
The real stock costs $50.
➟ C0A > $12.35 ( = 50-37.65)
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2) Create a Synthetic Stock Position for 1 year.
Synthetic stock for 1 yr consists of:
{+C, +B(FV=$10, t=3months), +B(FV=$40, T=1)}
= C0 + $10/(1+r/4) + 40/1.25
= C0 + $41.41
➟ C0A > 50 - 41.41 = $8.59
What is the math telling us?
If you plan to exercise in 3 months, C0A = $12.35,
If you plan to hold call for 1 yr, C0A = $8.59.
Implication: If the dividend is large,
then we should Exercise right before dividend is paid
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The right to exercise at any time: How much is it worth? (p. 7)
Asume S0 = $50, K=$40, T=1 year, Dividend in 3 months, and r = 25%.
No Div $5 Div
$10 Div
C0E >
$18
$13.29
$ 8.59
C0A >
$18
$13.29
$12.35
$0
$0
$ 3.76 Right to Early exercise
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Put Option Prices
European Puts on stocks that pay NO dividends. (p. 7)
Do these prices make sense?
S0 = $75, P0E(K = $100, T=1yr) = $4, and r = 25%
There are two ways to buy a bond:
A: {Buy the stock and the put} and wait until expiration or
B: {Buy the bond right now},
{+S, +P}
➟ CF0 = -75-4 = -$79
{+B(FV=$100, T=1yr)} ➟ CF0 = -$80
Arb Strategy: {+S, +P, -B(FV=100, T=1yr)}. CF0 = -75-4+80 = +$1
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At Expiration, if
ST ≥ 100 Put is worthless.
Sell stock, use some of this cash to pay loan.
CFT = ST - K > 0.
ST < 100
Put is in the money, exercise it.
Hand over stock; receive $100, cover loan. CFT =0
To preclude arbitrage
PE0 > $5 (=4+1). In general, PE0 > max(0, PV(K) - S0)
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European PUT prices for stocks that pay dividends (p. 7)
Assume a $5 Dividend in 3 months. Do these prices make sense?
S0 = $75, $5 Div, P0E(K = $100, T=1yr) = $6, and r = 25%
A synthetic position in a 1-year bond consists of:
{+S, +P, -B(FV=$5, t=3 months) }
➟ Synthetic bond costs: $76.29 (= -6 - 75 + 5/1.0625)
The actual bond costs: 100/1.25 = $80
Is this possible? ➟ There must be an arb. opp.
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Buy cheap: synthetic bond ,
and Sell the expensive one, (actual or physical).
Arb Portfolio
+ Put
+ Stock
- B for Div*
- B for K
ST < 100
ST > 100
-(ST -100)
0
+ ST
+ ST
-5*
-5*
-100
-100
0
+
Arbitrage-free Price: PE0 > 6 + 3.71 = $9.71
CF0
-6
-75
+4.71
+80
$3.71
*in 3 months, receive $5 div, pay off first bond.
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If the Dividend is $10 Dividend in 3 months, then
PE0 ≥ -75 + 10/1.0625 + 100/1.25 = $14.41
Summary:
If S0 = $75, K = $100, T=1 year, and r = 25% and Dividend in 3
months. Then,
No Div $5 Div
$10 Div
PE0 ≥
$5
$9.71
$14.41
P0A ≥
$25
$25
$28.53
$20
$15.29 $14.12 Right to Early exercise
▸ For an American PUT, the right to early exercise is worth quite a
bit.
▸ For an American CALL if S0 = $50, K=$40, T=1 year, r = 25%.
No Div $5 Div
$10 Div
C0E ≥
$18
$13.29
$8.59
C0A ≥ $18
$13.29 $12.35
$0
$0
$3.76
Right to Early Exercise
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Put-Call Parity (p. 11)
European Options on stocks that pay no dividends
Proposition:
For European Options on a stock that pays no dividends
(Call and Put with same K and T),
+S, +P = +C, +B(FV=K,T)
And By the law of one price:
+C0 = + S0 + P0 - B(FV=K,T)
- C0 = - S0 - P0 + B(FV=K,T)
+P0 = - S0 + C0 + B(FV=K,T)
- P0 = + S0 - C0 - B(FV=K,T), etc.
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Put-Call Parity (p. 11)
European Options on stocks that pay dividends
+C0 + B(FV=K,T) + B(FV=Dividend, t*) = +S0 + P0
American Options on stocks with/without Dividends
+C0 + B0(FV=K,T) ≤ +S0 + P0
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Thank You
(A Favara)
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