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# Black Scholes Option Pricing by ulf16328

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```									Finance with Dr. John Elder
1
1        Black Scholes Option Pricing
•     Read Hull chapters 12 and 13.
•     Assumptions
•     Model and Derivation
•     Estimating Volatility
•     Implied Volatility
•     Dividends (see section 13.1 and 13.2)
•     Currency Options (see section 13.5)

• Black-Scholes Formula
c0 = So e–qT N(d1) – K e–rT N(d2)
d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2)

• Put-Call parity
– Put-Call parity with no dividends c0 – p0 = S0 – Ke –rfT
– Put-Call parity with known dollar div c0 – p0 = S0 – Ke –rT – PV(Div)
– Put-Call parity with constant div yield c0 – p0 = S0e –qT – Ke –rT
Finance with Dr. John Elder
2
2       Binomial Model Review
• Step 1: Consider a stock (S0=??) and call (K=??)
– compute payoff for call at each node.

• Step 2: Construct portfolio replicating risk-free
investment.
– Hedge ratio “D” for riskless hedge portfolio
D = (fu–fd) / (Su–Sd).

• Step 3: Price call by setting cost (PV) of RHP
equal to discounted payoff.
– Find fair price of call C t at each node:
PV of riskless hedge portfolio = FVe –rdt
DS0 – f = (DS0u – fu)*e –r(δ t)
OR (DS0d – fd)*e –r(δ t)

• If we choose up/down moves carefully, as we
decrease time between nodes, possible future
stock prices are distributed log-normal!
(+/– tree moves not absolute!)
Finance with Dr. John Elder
3
3       Assumptions for Black-Scholes
• Black-Scholes (B-S) model – option pricing formula derived under no-arbitrage
conditions with set of formal assumptions, e.g., no taxes, transaction costs, margin:
– Underlying is risky; no div; trades continuously in fractional units.
– Risk-free rate, r, constant for all maturities. Volatility (σ) constant over time.
– Option is euro; exercise price is K; expiration is T; ∆ denotes small change.

• Distributional assumptions
– C.c. return normally distributed R0,t = ln(St/S0) ~ N([µ–½σ2]∆t , σ√(∆t)).
– Then St is distributed log-normal, since St = S0 exp(R0,t ).
– Note µ∆t is analogous to arithmetic avg, while [µ–½σ2]∆t is geometric avg.
Normal Distribution                                       Log-Normal Distribution
0.45                                                          0.35
0.40                                                                                                     s=1
0.30
0.35                                                                                                     s=2
0.25
0.30
0.25                                                          0.20

0.20                                                          0.15
0.15
0.10
0.10                                  s=1
0.05                                  s=1.5                   0.05

0.00                                                          0.00
-4    -3       -2       -1    0        1   2   3   4          -4   -3   -2   -1   0   1   2   3    4    5   6   7   8   9   10
Finance with Dr. John Elder
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4       Assumptions – Are they Reasonable?
• Actual Stock Returns – histogram of index daily returns, with normal overlay.
– Normality means 95% of daily returns between 0.04%–2% and 0.04%+2%
– Stock returns tend to have big negatives, higher peak and fatter tails.
– Technically, neg skew (–0.3), excess kurtosis (4.1); and heteroskedastic.
–
–
–

Stock Index (daily for 9 yrs) and Random Walk          Histogram of Stock Index Daily Returns(cc)
0.16
3500                          Which is which?                             Mean=0.04% Std=1.00%
0.14
3000                                                                Annualized Mean =8.90% Std=15.85%
0.12
2500
0.1
2000
0.08
1500
0.06
1000                                            ???       0.04

500                                            ???
0.02

0                                                        0
1001
1101
1201
1301
1401
1501
1601
1701
1801
1901
2001
2101
-32201
101
201
301
401
501
601
701
801
901
1

0%

0%

0%
0%
%

%

%

%

%

0%

0%

0%
%

%

0%
.50

.00

00

.50

50
50

00

0.5

2.5

3.5
1.5
0.0

2.0

3.0
1.0
-2.

-0.
-2.

-1.
-3

-1
Finance with Dr. John Elder
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5       Black-Scholes Derivation
• Derivation - The objective is to derive an equation that expresses value of call
option as function of its strike price and other variables.

• Step 1. Set-up riskless hedge portfolio long ∆ shares and short one call option.
– Denoting Π as current market value of portfolio, Π = ∆S – C.
– Change in value of portfolio is w.r.t to change in S is dΠ = ∆dS – dC.
– To ensure riskless hedge portfolio, we find value of ∆ such that dΠ = 0.
– ∆ = dC/dS.

•      Step 2. Construct portfolio that replicates a risk-free investment.
–

• Step 3: Price call option by setting cost (PV) of RHP equal to discounted payoff.
– Price of call must satisfy second order partial differential equation.
– Seldom easy to solve, but B-S-M redefined variables to give familiar form.
–
Finance with Dr. John Elder
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6       Black-Scholes for Euro Options (with dividends)
c= S0 e–qT N(d1) – K e–rT N(d2)
p= –S0e–qT N(– d1) + K e–rT N(– d2)

S0         σ2 
ln 
 K  + r −q+
         T d = d − σ T
             2 
   2   1
d1 =
σ T
where
c0 = current value of European call.
p0 = current value of European put.
S0 = current stock price
N(d) = probability that random draw from normal distribn is less than d.
K = Exercise price.
q = annualized div yield of underlying stock (text ignores div yld for now).
e = 2.71828, base of natural log.
r = Risk-free interest rate matching maturity of option, c.c.
T = time to maturity of option in years.
σ= Std deviation of return on stock, c.c.
Note: Black-Scholes is derived for European calls! No value for early exercise!
Finance with Dr. John Elder
7
7       Black-Scholes Characteristics 1
• What is Black-Scholes? c = S0 e–qT N(d1) – K e–rT N(d2)
– N(di) interpreted as risk-adjusted probability option will expire in-the-money
– Recall that value of call is discounted expected payoff, Max(0,ST - K).
– Suppose for below that q=0 (no dividends).

• Suppose there is low probability call option will be exercised (S0 << K).
–

• Suppose there is high probability call option will be exercised (S0 >> K).
– Then N(d1) and N(d2) are close to one, and value of call is c0 = S0 – Ke –rT
–

• Using Black-Scholes – derived for European options
– American calls on non-div stocks not exercised prior to expiration.
–
Finance with Dr. John Elder
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12
8             Black-Scholes Prices (Euro, no div)
Call (X=100, T=1-mo, r=5%, Volatility=15%)
10                                                                                              Parity for Euro:
c–p = S0 – K e–rT
8
Option Value

6                                                                                        Black-Scholes:
4                                                                                        c=S0N(d1)–K e–rT N(d2)
2
N(di): risk-adj prob
0                                                                                        option expires
90         92     94     96    98       100       102    104    106    108    110
-2                                                                                            in-the-money
Stock Price
12
Put (X=100, T=1-mo, r=5%, Volatility=15%)                          Euro Put: possible
10
early exercise
Option Value

8
exceeds value if deep
6                                                                                    in-the-money.
4

2

0
90     92      94    96    98       100       102    104    106    108    110
-2                                   Stock Price
Finance with Dr. John Elder
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9        Black-Scholes Example
•       Suppose S0 = \$50, K= \$45, T = 6 months, r = 10% c.c., and s=28%.
Calculate the value of a call and a put option.

•       Black-Scholes c= S0 e–qT N(d1) – K e–rT N(d2)
d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2)

(     )  
ln 50 45 +  0.10 − 0 +

0.282 
2 
 0.50
d1 =                                    = 0.884
0.28 0.50
d 2 = 0.884 − 0.28 0.50 = 0.686
c0             = S0 e–qT N(d1) – K e–rT N(d2)
c0             =
=
=

p0      =
=
Note: Excel worksheet fn =NORMSDIST(x), but need to load Analysis Toolpack.
Finance with Dr. John Elder
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10        Black-Scholes – Calculating interest rate
• Compute interest rate from T-bills, based on avg of bid/ask
Maturity               Days Maturity Bid         Ask

•       Compute average of bid and ask
–

• Compute price of 100 par value bill based on bank discount rate.

• Compute the EAR
– EAR =

• Compute the continuously compounded, annual rate of return.
– (1+EAR) =
–

•
Finance with Dr. John Elder
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11        Black-Scholes – Estimating Volatility
• c= S0 e–qT N(d1) – K e–rT N(d2); d1=[ln(S0/K)+(r–q+σ2/2)T]/(σ T1/2); d2=d1–σ T1/2

• Estimating volatility - Let St be daily price and let ut=ln(St/St–1) be daily return.
– Use 90-180 trading days, or maturity. Discard data around ex-div date (tax).
– Find daily variance of ut; multiply by 252 days; take sqrt(σ2) for std dev.

∑i =1 (ut − E (ut ))
1    N
σ 2 annual = 252 *σ 2 daily = 252 *
2

N −1
• Implied Volatility – level of volatility implied by B-S or binomial model.
– May avg from several liquid options on same asset; used to price less liquid.
– Issues: non-simultaneity; bid-ask; model mispecification.
– VIX –
– Volatility skew -                                                                                       Volatility Skews for IBM Options
60
Implied Standard Deviation (%)

58

56

54

52
Put ISDs
50

48

46
Call ISDs
44

42

40
110   115      120         125       130         135   140   145

Strikes (\$)
Finance with Dr. John Elder
12
12        Black-Scholes with Dividends
• Black-Scholes
– c= S0 e–qT N(d1) – K e–rT N(d2)
– d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2)

• Dividend Yield – basic Black-Scholes formula does not include dividends.
– Accounting for div yield – discount current stock price by div yield.
–

• Dividends paid as known dollar values –
– Recall that stock price (and call value) drops on ex-div date.
–
–

• Black adjustment for dividends – devised by Fischer Black before PC era.
– Price option as greater of
(1) Euro option
(2) Euro maturing just prior to latest ex-div date.
Finance with Dr. John Elder
13
13        Binomial Model vs Black Scholes
• Binomial Model vs. Black Scholes
– Euro call no div –
– American calls on div paying stocks –
– Euro and American puts –

• In practice – computing power is now cheap.
– Analysts use binomial model to price American and other options.
– Consider American call/put with K=100; S0=100; T=6-mo; r=5%; σ=25%.
Call Prices       Put Prices
Time    Binomial Black- Binomial Black
Steps              Scholes          Scholes
5      8.601      8.26   6.334     5.791
10      8.087     8.26    5.932    5.791
15      8.373     8.26    6.128    5.791
20      8.173     8.26    5.977    5.791
30      8.202     8.26    5.992    5.791
50      8.225     8.26    6.004    5.791
99      8.277     8.26    6.039    5.791
Finance with Dr. John Elder
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14        Tips on Pricing Options for the Savvy Investor
• Black-Scholes is powerful formula for pricing Euro options
– Derived before binomial model.
– Some assumptions are strong (continuous trading, normally distributed cc
returns), but model works well.
– No arbitrage assumption cornerstone in finance and revolutionized field.
– B-S can be viewed as special binomial model with many steps.
– B-S less useful for pricing Amer puts and calls on div paying stocks.

• There are many adjustments to Black-Scholes, but still very useful benchmark.
– Options on stock indexes used with dividend yield
– Options on indiv stocks often adjusted by subtracting PV of div for S0.

• Black-Scholes Formula: c0 = So e–qT N(d1) – K e–rT N(d2)
d1 = [ln(S0/K) + (r–q+σ2/2)T] / (σ T1/2); d2 = d1 – (σ T1/2)
• Put-Call parity
– Put-Call parity with known dollar div c0 – p0 = S0 – Ke –rT – PV(Div)
– Put-Call parity with constant div yield c0 – p0 = S0e –qT – Ke –rT
Finance with Dr. John Elder
15
15        Standard Normal Probabilities

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