Transcript THE INSURANCE INDUSTRY - University of Pennsylvania

VALUING STOCK OPTIONS
HAKAN BASTURK
Capital Markets Board of
Turkey
April 22, 2003
PRESENTATION PLAN
Understanding Stock Options (Definition,
classifications & related terms)
Trading of Stock Options & Option
Strategies
Restrictions & Determinants of Stock Option
Valuation
Methods for Valuing Stock Options
Understanding Stock Options
A stock option is a contract which conveys to its holder
the right, but not the obligation, to buy or sell shares
of the underlying stock at a specified price on or before
a given date.
Stock options primarily can help investors:




Protect their stock holdings against a decline in market prices
Increase their income on current or new stock holdings
Buy stock at a lower price
Benefit from a stock price’s rise or fall without owning the stock
or selling it outright
Understanding Stock Options
An option contract is defined by the following
elements:






Type (Call or Put)
Style ( American, European or capped)
Underlying Security
Unit of Trade
Strike Price
Expiration Date
Understanding Stock Options
Tradable Stock Options (exchanges & over the
counter)
Non – tradable Stock Options (Employee
Stock Option Plan)
LEAPS (Long Term Equity Anticipation
Securities)
Trading of Stock Options & Option Strategies
The relationship between the Strike Price (X)
and stock price (St)



In – the – money (For call option St > X)
At – the – money (St = X)
Out – of – the – money (St < X)
Trading of Stock Options & Option Strategies
 Example :
 St
= $100
 X = $95
Trading of Stock Options & Option Strategies
The Payoffs of Stock Options at Expiration Date
Call option
X
X
ST
ST
(a)
holder
(b)
writer
Put option
X
(c)
holder
X
ST
ST
(d)
writer
Trading of Stock Options & Option Strategies
Option Strategies
a) Protective Put
profit
stock
Protective put
portfolio
St
A
-P
-S0
S0 = X
Put option
price
Trading of Stock Options & Option Strategies
b) Covered Calls
Stock only
Profit
Covered call
X
ST
Naked writing
position
Trading of Stock Options & Option Strategies
c) Straddle :
Buying call
Profit
stock
X
ST
Buying
put
Trading of Stock Options & Option Strategies
d) Spreads :
Profit
ST
X1
X2
Determinants and Restrictions of Stock Option
Valuation
Main difference of stock options from the
other assets:
Since the risk of stock option changes
every time with changing stock prices,
finding the opportunity cost of capital is
impossible, so we cannot use discounted
cash flow analysis to stock options.
Determinants and Restrictions of Stock Option
Valuation
Restrictions
 C (St,X,t,T) > 0
 C > S0 – PV (X) – PV (D)
 C < S0
Determinants and Restrictions of Stock
Option Valuation
Value of
call
Upper bound: = S0
B
Lower bound =
adjusted intrinsic
value = S0 – PV (X) –
PV(D)
C
A
Share price
Determinants and Restrictions of Stock
Option Valuation
Difference of American & European Call



c > C ( before the maturity date)
Longer time to maturity, more valuable American
call options,
Call price at least equal to exercise price before
maturity date.
Determinants and Restrictions of Stock
Option Valuation
Factors on the value of a stock option:






Stock Price
Exercise Price
Volatility of the stock price
Time to expiration
Interest rate
Dividend payouts
Determinants and Restrictions of Stock
Option Valuation
If this variable increases
The Value of a Call Option:
Stock Price (S)
Increases
Exercise Price (X)
Decreases
Volatility (б)
Increases
Time to expiration (t)
Increases
Interest rate (rf)
Increases
Dividend payouts (D)
Decreases
Determinants and Restrictions of Stock Option Valuation
Put – Call Parity:
Investor’s payoff
Investor’s payoff
Buying
share
Investor’s payoff
Protection
Buying put
option
a)
=
+
50
Future
stock
price
50
Investor’s payoff
Investor’s payoff
Bank deposit
paying $50
Buying
call option
b)
Future
stock
price
Future
stock
price
Future
stock
price
Investor’s payoff
Protection
=
+
50
50
50
Future
stock
price
50
Future
stock
price
Value of call + present value of exercise price = value of put + share price or;
C + PV (X) = P + S0
Methods for Valuing Stock Options
Methods for Valuing Stock Options:
 Binomial Method
 Black – Scholes Option Valuation Model
 Other Models
Methods for Valuing Stock Options
Binomial Method is based on a simple approach:
In a single time period, underlying stock price
can only move to two possible levels (up & down)
Basic Assumptions of Binomial Method
 No riskless arbitrage opportunities
 Perfect market conditions
 Risk – neutral valuation principle.
Methods for Valuing Stock Options
Single Period Model for Binomial Method
EXAMPLE:
For XYZ stock;
S0 = $80 at Jan 1,
S1 = $160 or S1 = $40 for Dec 31
X = $ 120 at the end of the year
rf = 6%
Methods for Valuing Stock Options
1) Buying Call Option
$160
$80
$40
$80
$40
(stock price possibilities)
0
(call option payoffs)
2) Buying stock with borrowing
Value of stock at year end
$160
$40
- Repayment of loan with interest
- $40
-$40
Total
$120
0
Methods for Valuing Stock Options
Solving 1 &2:
$120
3C = 42.26
C = 14.09
$42.26
0
Strategy 3 ; buy stock (no loan) & write 3 call
Value of stock at year end
- Obligations from 3 calls written
Total
$160
$40
- $120
-0
$40
$40
PV of Strategy 3 Portfolio = $80 – 3C = 37.74
C = 14.09
Methods for Valuing Stock Options
The hedge ratio equals the ratio of ranges because
the option and the stock are perfectly correlated, a
perfect hedge requires that the option and stock be
held in a fraction determined only by relative
volatility.
H = C+ - C- / S+ - SIn the example hedge ratio equal to 0.33
The year end value of portfolio (0.33 shares and 1 written call) equal
to $13.33
PV of $13.33 from 6% int. rate equal to $12.57 If we use this
amount at the portfolio function we hold C = $14.09 (Same
solution)
Methods for Valuing Stock Options
General Binomial Method
Binomial Tree
Methods for Valuing Stock Options
General Binomial Method
EXAMPLE:
Methods for Valuing Stock Options
Black – Scholes Option Valuation Model
Formula:
The variables are:
S = stock price
X = strike price
t = time remaining until expiration,
expressed as a percent of a year
r = current continuously
compounded risk-free interest rate
v = annual volatility of stock price
(the standard deviation of the shortterm returns over one year).
ln = natural logarithm
N(x) = standard normal cumulative
distribution function
e = the exponential function
Methods for Valuing Stock Options
Basic Assumptions of BS Model
 The stock pays no dividends during the option's
life
 European exercise terms are used
 Markets are efficient
 No commissions are charged
 Interest rates remain constant and known
 Returns are lognormally distributed
Methods for Valuing Stock Options
Single Period Model for Binomial Method
EXAMPLE:
For XYZ stock;
S0 = $80 at Jan 1,
S1 = $160 or S1 = $40 for Dec 31
X = $ 120 at the end of the year
rf = 6%
Methods for Valuing Stock Options
BS Formula EXAMPLE:
For XYZ stock;
S0 = $100 , X = $ 95, rf = 10%,
T = 0.25 year, ∂ = 0.50
 First of all we can calculate d1 and d2 values.
d1 = [ln (100 / 95) + (0.10 + (0.5)2/2) 0.25] / 0.5 √0.25 = 0.43
d2 = 0.43 - 0.5 √0.25 = 0.18
 Next we will find N (d1) and N (d2) from the tables about the values of the
normal distribution. We can from the table that:
 N(0.43) = 0.6664
 N(0.18) = 0.5714
 Thus the value of the call option is:
 C = 100 * 0.6664 – 95 ℮-0.10*25 * 0.5714 = 66.64 – 52.94 = $13.70
Methods for Valuing Stock Options
Advantages of Binomial Method and BS Formula
 Binomial Model can be used to American options
 Binomial Model isn’t related with the returns of stocks.
 BS Formula is easy to use
 BS Formula gives a rapid solution
Disadvantages of Binomial Method and BS Formula
 Binomial Method relatively has a slow speed. Not easy
to calculate with lots of sub period.
 BS Formula is only related with European type options.
 Two models have lots of assumptions.
Methods for Valuing Stock Options
Modified BS Formulas and Other Models





Modified BS European Model : equities with dividend and for
other securities,
Modified BS American Model : minimum value for option at
intrinsic value
Modified BS French Model : trading days instead of calendar
year,
Whaley (Quadratic Approximation) Method: It gives a value for
American options which equals to the value of European option
plus American options’ early exercise option
Pseudo – American Method : cash flows such as cash dividends
Methods for Valuing Stock Options
The Greek Letters
These are related with the degrees of change
in option price according to the change in the
variables. They give the sensitivity of option
price to the change in variables.
Methods for Valuing Stock Options
Delta (∆) : The sensitivity of current option value
to its current underlying asset price. It is easily
calculated from a binomial tree.
∆ = ∂C / ∂St = N (d1) < 1 (for call option non-dividend)
∆ = N (d1) – 1 < 0 (for put option non-dividend)
Option
price
Slope = D
B
A
Stock
price
Methods for Valuing Stock Options
Gamma (Γ) measures the rate at which the delta
changes as the underlying asset price changes.
Γ = ∂2C / ∂St2 = N’ (d1) > 1 (for call option non-dividend)
Theta (Θ) measures the sensitivity of the current
option value to a reduction in time-to-expiration.
Θ = - ∂C / ∂(T-t) < 0
Methods for Valuing Stock Options
Vega (ν) is the rate of change of the value of
a derivatives portfolio with respect to
volatility
ν = S0 √T N’ (d1)
Rho (ρ) is the rate of change of the value of
a derivative with respect to the interest rate
ρ = X T e –rT N (d2)
Methods for Valuing Stock Options
Using of Option Pricing Models
Delta hedging
Portfolio Insurance (dynamic hedging)
Methods for Valuing Stock Options
Rules & Comments of Regulatory Authorities
Financial Accounting Standards Board (FASB)
Statement No. 123
SEC, NASD & stock exchanges