5.1
Option pricing:
pre-analytics
Lecture 5
5.2
•
•
•
•
•
•
Notation
• C : American Call option
c : European call
option price
p : European put
option price
S0 : Stock price today
X : Strike price
T : Life of option
: Volatility of stock
price
•
•
•
•
price
P : American Put option
price
ST :Stock price at time T
D : Present value of
dividends during option’s
life
r : Risk-free rate for
maturity T with cont comp
Calls: An Arbitrage
Opportunity?
• Suppose that
c =3
T =1
X = 18
• Is there an arbitrage
S0 = 20
r = 10%
D=0
opportunity?
5.3
5.4
Lower Bound for European Call
Option Prices; No Dividends
consider 2 portfolios:
a. buy a call and a ZCB worth X at T
b. buy the stock
c + Xe
-rT
 S0
in fact, at maturity:
if S>X, identical result
if S<X, superior result
5.5
Lower Bound for European Call
Option Prices; No Dividends
c + Xe  S0
-rT
c  S0 -Xe
c  20 -18*0.9048
c  3.71
-rT
5.6
arbitrage
if c<3.71
TODAY
• sell the stock at 20, and with the proceeds:
• buy the call at 3 and a ZCB maturing in 1 year time for 16,28
today (i.e. 18=X)
• invest the diff=Y=20-3-16,28=0.72 in a ZCB maturing in 1 year
AT MATURITY
• if S<18, you have 18+0.79 (so better off than keeping the stock
• if S>18, you get:
•
-
S-X from the call (=S-18)
X from the ZCB (=18)
0.79 from the residual
sum up the terms and you will be happier than having kept the
stock
5.7
Puts: An Arbitrage Opportunity?
• Suppose that
p =1
T = 0.5
X = 40
• Is there an arbitrage
opportunity?
S0 = 37
r =5%
D =0
Lower Bound for European Put
Prices; No Dividends
consider 2 portfolios:
a. buy a put and a stock
b. buy the ZCB worth X at maturity
p + S0  Xe
-rT
p  Xe - S0
-rT
5.8
5.9
Put-Call Parity; No Dividends
•
•
•
Consider the following 2 portfolios:
- Portfolio A: European call on a stock + PV of the
strike price in cash
- Portfolio B: European put on the stock + the stock
Both are worth MAX(ST , X ) at the maturity of the
options
They must therefore be worth the same today
- This means that
c + Xe -rT = p + S0
5.10
Arbitrage Opportunities
• Suppose that
•
c =3
S0 = 31
T = 0.25
r = 10%
X =30
D =0
What are the arbitrage
possibilities when
p = 2.25 ?
p =1?
arbitrage
•
c + Xe -rT = p + S0
3+30*0.97531=p+31
3+29.2593=p+31
p=1.2593
• if p=1, then
• TODAY:
-
•
-
sell the call at 3 and buy the put at 1. You have 2 euro remaining.
buy the stock at 31, by borrowing 30.23 and using 0.77 from the
options proceeds
invest 1.23 (out of the original 2) in a ZCB
in 3 months time:
-
if S=33, for example, put=0, call=3. So you are at -3.
sell the stock at 33 and repay your debt. So you are at +2
so you are at -1, but in reality you still have 1.23*exp(0.25*10%)
5.11
The Impact of Dividends on
Lower Bounds to Option Prices
c  S 0  D  Xe
p  D  Xe
 rT
 rT
 S0
5.12
Effect of Variables on Option
Pricing
Variable
S0
X
T

r
D
c
+
–
+
+
+
–
p
–
++
+
–
+
C
+
–
+
+
+
–
P
–
+
+
+
–
+
5.13
5.14
American vs European Options
An American option is worth
at least as much as the
corresponding European
option
C c
P p
5.15
The Black-Scholes
Model
5.16
The Stock Price Assumption
• Consider a stock whose price is S
• In a short period of time of length Dt the
change in then stock price is assumed
to be normal with mean mSdt and
standard deviation
S Dt
 m is expected return and  is volatility
5.17
The Lognormal Distribution
E ( ST )  S0 e mT
2 2 mT
var ( ST )  S0 e
(e
2T
 1)
5.18
The Concepts Underlying
Black-Scholes
•
•
•
•
The option price & the stock price depend on the
same underlying source of uncertainty
We can form a portfolio consisting of the stock
and the option which eliminates this source of
uncertainty
The portfolio is instantaneously riskless and must
instantaneously earn the risk-free rate
This leads to the Black-Scholes differential
equation
5.19
The Derivation of the
Black-Scholes Differential Equation
1 of 3:
DS  mS Dt  S Dz
ƒ  f (S , t )
ƒ
ƒ
 2ƒ 2
Dƒ 
DS 
Dt  ½ 2 DS  o(t 2 )
S
t
S
 ƒ
ƒ
 2ƒ 2 2
ƒ
D ƒ  
mS 
 ½ 2  S Dt 
 S Dz
t
S
S
 S

We set up a portfolio consisting of
 1 : derivative
ƒ
+
: shares
S
5.20
The Derivation of the
Black-Scholes Differential Equation
2 of 3:
The value of the portfolio  is given by
ƒ
  ƒ 
S
S
The change in its value in time Dt is given by
ƒ
D   D ƒ 
DS
S
•
there are no stochastic terms inside
5.21
The Derivation of the
Black-Scholes Differential Equation
3 of 3:
The return on the portfolio must be the risk - free rate. Hence
D  r Dt
We substitute for D ƒ and DS in these equations to get the
Black - Scholes differenti al equation :
2
ƒ
ƒ

ƒ
2 2
 rS
½ S
 rƒ
2
t
S
S
5.22
Risk-Neutral Valuation
• The variable m does not
•
•
•
appear in the Black-
Scholes equation
The equation is independent of all variables
affected by risk preference
The solution to the differential equation is
therefore the same in a risk-free world as it
is in the real world
This leads to the principle of risk-neutral
valuation
5.23
Applying Risk-Neutral
Valuation
1. Assume that the expected
return from the stock price is
the risk-free rate
2. Calculate the expected
payoff from the option
3. Discount at the risk-free
rate
5.24
The Black-Scholes Formulas
c  S 0 N ( d1 )  X e
p Xe
 rT
 rT
N (d 2 )
N (  d 2 )  S 0 N (  d1 )
2
ln( S0 / X )  (r   / 2)T
where d1 
 T
2
ln( S0 / X )  (r   / 2)T
d2 
 d1   T
 T
5.25
Implied Volatility
• The implied volatility of an option is the
•
•
volatility for which the Black-Scholes
price equals the market price
The is a one-to-one correspondence
between prices and implied volatilities
Traders and brokers often quote implied
volatilities rather than dollar prices
5.26
Dividends
• European options on dividend-paying
•
•
stocks are valued by substituting the
stock price less the present value of
dividends into Black-Scholes
Only dividends with ex-dividend dates
during life of option should be included
The “dividend” should be the expected
reduction in the stock price expected
European Options on Stocks
Paying Continuous Dividends
continued
We can value European options by
reducing the stock price to S0e–q T and
then behaving as though there is no
dividend
5.27
5.28
Formulas for European Options
c  S 0 e  qT N ( d1 )  Xe  rT N ( d 2 )
p  Xe  rT N ( d 2 )  S 0 e  qT N (  d1 )
ln( S 0 / X )  ( r  q   2 / 2)T
where d1 
 T
ln( S 0 / X )  ( r  q   2 / 2)T
d2 
 T
5.29
The Foreign Interest Rate
• We denote the foreign interest rate by rf
• When a European company buys one
•
•
unit of the foreign currency it has an
investment of S0 euro
The return from investing at the foreign
rate is rf S0 euro
This shows that the foreign currency
provides a “dividend yield” at rate rf
Valuing European Currency
Options
• A foreign currency is an asset that
•
provides a continuous “dividend yield”
equal to rf
We can use the formula for an option
on a stock paying a continuous
dividend yield :
Set S0 = current exchange rate
Set q = rƒ
5.30
Formulas for European Currency
Options
c  S0 e
 rf T
p  Xe
 rT
N (d1 )  Xe  rT N (d 2 )
N (  d 2 )  S0 e
where d1 
d2 
 rf T
N (  d1 )
ln( S0 / X )  (r  r
f
  2 / 2) T
 T
ln( S0 / X )  (r  r
 T
f
  2 / 2) T
5.31