Financial Engineering 6

Lecture 20

Option Pricing with Black-Scholes

References: Villalobos, Luenberger

Lecture Topics • Option Pricing Review • History and Terminology • Basic Assumptions • Probability of Being in the Money • Black-Scholes Equation for a Call Option • Examples

• Suppose Microsoft stock is currently trading at $30 per share. You think the stock price will rise and you have $300,000 to invest.

• If you invest all of your money ($300,000) in the underlying stock, you can buy at most: 300,000 / 30 = 10,000 shares.

• Let’s suppose the stock price rises by 15% to $34.5 per share. You return of investment is:

10,000 * ($34.5 - $30) = $45,000

Option vs. Underlying Stock (a Speculator Point of View)

Instead of stock, you can invest in call options of a strike price of $33.3 these call options are expected to sell for $0.40 per contract.

The amount of contracts you can buy is: $300,000 / $0.40 = 750,000.

If the stock price does rise to $34.5 per share, each of your call option worth ($34.5 - $33.3) = $1.2. Your return buy selling these options is:

750,000 * ($1.2 – $0.4) = $600,000! (vs. $45,000)

Instead of 15%, you just earned 200%. Options provided you with a leverage of about 14 times over purchasing stock.

Option vs. Underlying Stock (a Speculator Point of View)

• One of the main questions asked in Financial Engineering is the “fair” or market price for an option.

• Essentially there are two types of analytical models to answer this question:

– Discrete (binomial lattices) Single period vs. Multiple periods

– Continuous (Black-Scholes Equation).

Option Pricing Review

Option Pricing Review

S

Su

Sd

p

1-p C

Max(Su-K,0)

Max(Sd-K,0)

p

1-p 1

R

R

p

1-p

Suu

S Sud

Sdd

Su

Sd

Cuu

C Cud

Cdd

Cu

Cd

Single period: Lattices for a share of stock.

Multiple period: If more than one period is involved we need to extend the lattice to include the extra periods.

• The Black-Scholes model for pricing stock options was developed by Fischer Black, Myron Scholes and Robert Merton in the early 1970’s.

• It is arguably the most important result in financial engineering, and is certainly a rich source of interview questions in the financial services industry.

• This isn’t necessarily all that easy as the formula involves some relatively complex mathematics. However, it is possible to get an intuitive understanding of what the various parts of the formula mean.

Introduction to Black-Scholes Equation

(Jan 11th, 1938 – Aug 30th, 1995)

Education: Mathematics; Harvard University

Best known for his contributions to the famous Black-Scholes equations

Fischer Sheffey Black

(July 1st, 1941 – )

Nobel Prize winner in Economics in 1997.

Education: Economics from The University of Chicago.

Best known for his contributions to the famous Black-Scholes equations.

Myron S. Scholes

(July 31st, 1944 – ), Nobel Prize winner in Economics in 1997.

Education: Engineering and Economics; Columbia University, California Institute of Technology, Massachusetts Institute of Technology.

Had a direct influence on the development of the Black-Scholes formula and generalized it in important ways. By devising another way of deriving the formula, he applied it to other financial instruments, such as mortgages and student loans. The work generated new financial instruments and has facilitated more effective risk management in society.

Robert C. Merton

1. Efficient Market: Market movements cannot be predicted

• This assumption of the Black-Scholes model suggests that people cannot consistently predict the direction of the market or an individual stock.

• The Black-Scholes model assumes stocks move in a manner referred to as a Random Walk.

• Random walk means that at any given moment in time, the price of the underlying stock can go up or down with the same probability.

• The price of a stock in time t+1 is independent from the price in time t.

Basic Assumptions

2. Lognormal Distribution Returns • The stock price follows a Geometric Brownian Motion with constant drift

and volatility. • The Black-Scholes model assumes that returns on the underlying stock

are lognormally distributed. • This assumption is reasonable in the real world.

3. Constant volatility • The most significant assumption is that volatility, a measure of how

much a stock can be expected to move in the near-term, is a constant over time.

• While volatility can be relatively constant in very short term, it is never constant in longer term.

• Some advanced option valuation models substitute Black-Schole's constant volatility with stochastic-process generated estimates.

Basic Assumptions

4. No dividends. • No dividends are paid out on the underlying stock during the option life. • In the real world, most companies pay dividends to their share holders (this

assumption has been removed in subsequent extension of the basic model). • A common way of adjusting the Black-Scholes model for dividends is to

subtract the discounted value of a future dividend from the stock price.

5. Constant risk-free interest rate. • It is possible to borrow and lend cash at a known constant risk-free interest

rate. • The Black-Scholes model uses the risk-free rate to represent this constant and

known rate. • In the real world, there is no such thing as a risk-free rate, but it is possible to

use the U.S. Government Treasury Bills 30-day rate since the U. S. government is considered to be credible enough.

Basic Assumptions

6. No commissions, taxes or transaction costs. • Transactions do not incur any fees or costs (frictionless market).

7. Market has no arbitrage. • It is impossible to secure a risk free profit.

8. Liquidity. • The market is perfectly liquid. • It is possible to purchase or sell any amount of stock or options or their

fractions at any given time.

Comments: • Many of the assumptions are invalid in the real world; therefore,

applying the Black-Scholes formula to real-world situations can possibly lead to incorrect numbers.

• Modern versions of the Black-Scholes formula account for changing interest rates, transaction costs and taxes, and dividend payout.

Basic Assumptions

The Black–Scholes Equation • It is in a continuous-time framework.

• The Black–Scholes equation is a partial differential equation, which describes the price of the option over time.

• At any time we form a portfolio with portions of the stock and the bond so that the portfolio exactly matches the (instantaneous) return characteristics of the derivative security.

• The option price and the stock price depend on the same underlying source of uncertainty.

• A portfolio consisting of the stock and the option eliminates this source of uncertainty.

• Thus, the portfolio is instantaneously riskless and must instantaneously earn the risk-free rate.

• The price S of an underlying security (stock) follows a geometric Brownian motion process over time interval [ 0, T ]:

where z is standard Brownian motion. Discrete analogue for z is a simple random walk.

• The infinitesimal increment dz represents the only source of uncertainty in the price history of the stock.

• 𝝁𝝁 is the annualized drift rate of S.

• 𝝈𝝈 is the volatility of the stock’s returns.

• The 𝐝𝐝𝐝𝐝 𝐝𝐝

is the infinitesimal rate of return on the stock.

d𝑆𝑆 = 𝜇𝜇𝑆𝑆d𝜇𝜇 + 𝜎𝜎𝑆𝑆d𝜎𝜎

The Black–Scholes Equation

• Suppose there is a risk-free asset (a bond) carrying an interest rate of r over [ 0, T ].

• The value B of this bond satisfy:

• Consider a security that is derivative to S, thus the price is a function of S and t.

• Let f(S, t) denote the price of this security at time t when the stock price is S.

• An equation for the function f(S, t) will give the price of the derivative explicitly.

d𝐵𝐵 = 𝑟𝑟𝐵𝐵d𝜇𝜇

The Black–Scholes Equation

• A derivative of this security has a price of f(S, t) satisfies the partial differential equation:

• Proof is shown in the Luenberger textbook.

𝜕𝜕𝜕𝜕 𝜕𝜕𝜇𝜇

+ 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆

𝑟𝑟𝑆𝑆 + 1 2 𝜕𝜕2𝜕𝜕 𝜕𝜕𝑆𝑆2

𝜎𝜎2𝑆𝑆2 = 𝑟𝑟𝜕𝜕

The Black–Scholes Equation

• Consider the stock itself, it is a derivative of S. • So f(S, t) = S should satisfy the B-S equation.

• Therefore, f(S, t) = S is a solution.

𝜕𝜕𝜕𝜕 𝜕𝜕𝜇𝜇

+ 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆

𝑟𝑟𝑆𝑆 + 1 2 𝜕𝜕2𝜕𝜕 𝜕𝜕𝑆𝑆2

𝜎𝜎2𝑆𝑆2 = 𝑟𝑟𝜕𝜕

𝜕𝜕𝜕𝜕 𝜕𝜕𝜇𝜇

= 0 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆

= 1 𝜕𝜕2𝜕𝜕 𝜕𝜕2𝑆𝑆

= 0

The Black–Scholes Equation

• Consider a simple bond as a derivative of S. • So f(S, t) = 𝒆𝒆𝒓𝒓𝒓𝒓 should satisfy the B-S equation.

• Thus, f(S, t) = 𝒆𝒆𝒓𝒓𝒓𝒓 is also a solution. • There are many many more solutions . . . .

𝜕𝜕𝜕𝜕 𝜕𝜕𝜇𝜇

+ 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆

𝑟𝑟𝑆𝑆 + 1 2 𝜕𝜕2𝜕𝜕 𝜕𝜕𝑆𝑆2

𝜎𝜎2𝑆𝑆2 = 𝑟𝑟𝜕𝜕

𝜕𝜕𝜕𝜕 𝜕𝜕𝜇𝜇

= 𝑟𝑟𝑒𝑒𝑟𝑟𝑟𝑟 𝜕𝜕𝜕𝜕 𝜕𝜕𝑆𝑆

= 0 𝜕𝜕2𝜕𝜕 𝜕𝜕2𝑆𝑆

= 0

The Black–Scholes Equation

Anticipated value for a given investment.

In statistics and probability analysis, expected value is calculated by multiplying each of the possible outcomes by the likelihood that each outcome will occur, and summing all of those values.

If a game has a 50% chance of winning $1,000,000, a 20% chance of winning $200,000 and a 30% chance of losing –$500,000 we can say the ‘expected value’ of playing is:

50% x 1,000,000 + 20% x 200,000 + 30% x (–500,000) = $390,000

By calculating expected values, investors can choose the scenario that is most likely to give them their desired outcome.

Expected Value

Lognormal Model • The Lognormal model for stock prices states that in (small)

amount of time t the returns of the stock prices follow the distribution:

• Where µ is the annual rate of return, σ is the annualized standard deviation of the returns and

• In a small time ∆t, the natural logarithm ln(S) of the current stock price will change (Ito’s Lemma) by an amount that is normally distributed with parameters:

( ) ( ) ( ) ( )0 0

ln Nor , ln ln Nor ,t t S

t t S S t t S

µ σ µ σ  

≈ → ≈ +   

0

E ln t S

t S

µ   

=     

2

2

0

E t

tS e S

σ µ  

+    

  = 

 

( )tS 20 5.lnMean σµ ++= StDev tσ=

Pricing a European Call Option • From the previous information, we have almost everything we

need to price a European call option. • Remember that a European call option can be exercised only at

maturity. • The option is worth something if the price of the underlying

stock is greater than the strike value of the option. • In this case, the value of the option is the difference between

the stock price and the strike value. • Thus the expected value of the option is:

• Where K is the strike price of the option and Ct is the value of the call option at maturity.

[ ] ( )[ ]0,maxEE KSC Tt −=

Probability of Being in the Money • By using conditional expectation:

• To get its present value we discount this value and get:

• We transform the probability term to:

[ ] [ ] [ ]( ) [ ]( )E Pr E | 1 Pr 0t T T T TC S K S S K K S K= ≥ ≥ − + − ≥

[ ] [ ] [ ]( )KKSSKSeC TTTrtt −≥≥= − |EPrE

[ ] 0 0

* Pr Pr ln 1 N

*T

K SK

S K Z S

µ

σ

 −     ≥ = ≥ = −          

[ ] [ ]( )Pr E |T T TS K S S K K= ≥ ≥ −

Probability of Being in the Money • The expected change in stock prices is:

2

2

0

E t

tS e S

σ µ  

+    

  = 

  2

2r σ

µ= + rtt e s s

= 

  



0

E

2

0

E ln 2

tS t r S

σ µ

   = = −  

  

[ ]

2 2

0 00

* 2 2

Pr 1 N 1 N N *T

K KK r t r t S SS

S K t t

σ σ µ

σ σ σ

        − − + −−                ≥ = − = − =

         

     

• If we define r as we get:

• We can state to obtain:

Expected Value if in the Money • The expected value if we are in the money is given by:

• Which results in:

[ ] 0 0 0

0 0 0 0

E | E ln | ln ln

ln ln ln ln

T T T T

T T T

K

S S K S S K

S S S

S S SK f d

S S S S

∞

       ≥ = ≥      

               

= ≥                   ∫

[ ] ( ) ( )2

1 0|E dN

dN eSKSS rtTT =≥

t

tr K S

d σ

σ  

  

 ++

 

  

= 2

ln 2

0

1

2 0

2 1

ln 2

S r t

K d d t

t

σ

σ σ

  + −      = = −

0

ln T S

f S

        

0

ln K S

     

Value of a Call Option • Combining the previous expressions we get:

[ ] ( ) ( )

( ) [ ]

1 2 0

2

0 1 2

N N

N

N

rt rt

rt

d C d e S e K

d

S N d Ke d

−

−

  = −  

  = −

( ) ( ) ( ) ( )1 2N N r T tC S,t S d Ke d− −= −

( )

tT

tTr K S

d −

− 

  

 ++

 

  

= σ

σ 2

ln 2

1

( ) 2

2 1

ln 2

S r T t

K d d T t

T t

σ

σ σ

  + − −       = = − −

−

• Which is the Black-Scholes model for a European call option valuation with a strike price of K

• According to Luenberger:

Standard Normal Distribution

d1 or d2

( ) ( ) ( ) ( )1 2N N r T tC S,t S d Ke d− −= −

• Normal distribution with mean = 0 and StDev = 1 • Cumulative distribution function in Excel:

=NORM.DIST(d,0,1,TRUE)

KO Example • The closing price for KO on April 3, 2007 was of $48.99. • Consider the following call options for the Coca Cola:

– KODJ, strike price = $50, expiration date April; third Friday is April 20)

– Trading days remaining = 13; 17 days total. – Annual interest rate of 4%. – Information provided by Yahoo:

Position Num OptSym Expire Days Strike Type IV Vol OI Buy 1 KODJ 7-Apr 17 50 Call 15.30% 3286 10370 @ 0.3

Coca-Cola Company (The) Option Trade with 1X Entry

Debit Profit Max Profit Max Risk Delta (Shares) Gamma Vega Theta $30.00 $-5.00 Unlimited $-30.00 27.9 20.7278 $3.55 $-1.64

50.30 50.30 Unlimited% Unlimited% Downside Breakeven Upside Breakeven Max Profit/Max Risk Max Profit/Debit

KO Example • Using the Black-Scholes model, get the price of the option

assuming: – S =48.99 – K = 50.00 – r = 4% – σ = 0.153 – Days per year = 250 – Days to expiration = 13

( ) ( )( )1 2Nr T tC SN d Ke d− −= −

( ) 2

1 2 1

ln 2

S r T t

K d d d T t

T t

σ

σ σ

  + + −       = = − −

−

KO Example

( )2

1

0.15348.99 13 ln 0.04

50 2 250 0.5078

13 0.153

250

d

    + +         = = −      

( ) ( ) 13

0.04 25048.99N 0.5078 50 N 0.5427 0.3279C e

 −    = − − − =

S (stock price) 48.99 K (strike price) 50.00 r (annual rate) 4.00% σ (annual) 0.153 Days per year 250 Days to expiration 13

T-t 0.0520 d1 -0.5078 d2 -0.5427 N(d1) cum norm dist 0.3058 N(d2) cum norm dist 0.2937 C 0.3279

2 13

0.5078 0.153 0.5427 250

d  = − − = −   

ORB 14 Day Lattice

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 0.206 0.298 0.423 0.588 0.798 1.056 1.361 1.707 2.081 2.473 2.874 3.280 3.693 4.112 4.537Up

0.114 0.174 0.259 0.379 0.541 0.754 1.019 1.335 1.692 2.076 2.471 2.871 3.278 3.691 0.055 0.089 0.140 0.218 0.331 0.490 0.705 0.981 1.312 1.684 2.073 2.468 2.869

0.022 0.037 0.063 0.106 0.173 0.277 0.432 0.652 0.944 1.298 1.681 2.071 0.006 0.012 0.021 0.039 0.070 0.123 0.214 0.363 0.592 0.917 1.295

0.001 0.002 0.004 0.008 0.017 0.033 0.067 0.134 0.270 0.542 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0

0 0 0Down

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Stock Value 24.10 24.45 24.81 25.17 25.54 25.92 26.30 26.68 27.07 27.47 27.87 28.28 28.69 29.11 29.54Up

23.75 24.10 24.45 24.81 25.17 25.54 25.92 26.30 26.68 27.07 27.47 27.87 28.28 28.69 23.41 23.75 24.10 24.45 24.81 25.17 25.54 25.92 26.30 26.68 27.07 27.47 27.87

23.07 23.41 23.75 24.10 24.45 24.81 25.17 25.54 25.92 26.30 26.68 27.07 22.74 23.07 23.41 23.75 24.10 24.45 24.81 25.17 25.54 25.92 26.30

22.41 22.74 23.07 23.41 23.75 24.10 24.45 24.81 25.17 25.54 22.09 22.41 22.74 23.07 23.41 23.75 24.10 24.45 24.81

21.77 22.09 22.41 22.74 23.07 23.41 23.75 24.10 21.46 21.77 22.09 22.41 22.74 23.07 23.41

Days per year 250 21.15 21.46 21.77 22.09 22.41 22.74 20.84 21.15 21.46 21.77 22.09

20.54 20.84 21.15 21.46 20.24 20.54 20.84

19.95 20.24 19.66Down

ORB 14 Day Black-Scholes

2 14

0.6346 0.2297 0.6890 250

d  = − − = −   

( ) ( ) 14

0.0122 25024.10N 0.6346 50 N 0.6890 0.2032C e

 −    = − − − =

S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% σ (annual) 0.2297 Days per year 250 Days to expiration 14

T-t 0.0560 d1 -0.6346 d2 -0.6890 N(d1) cum norm dist 0.2628 N(d2) cum norm dist 0.2454 C 0.2032

( )2

1

0.229724.10 14 ln 0.0122

25 2 250 0.6346

14 0.2297

250

d

    + +         = = −      

How do we compare with the lattice?

• Call option price from the 14 day lattice is:

0.206

• The option price from Black-Scholes is:

0.203

Assignments • Set up your own Black-Scholes call option pricing spreadsheet.

• Use the Black-Scholes equation to solve for the Dell option price; data can be found in

– “Lecture 20dm Black-Scholes Call Option Pricing Examples.xlsx”

– How did this compare to the lattice solution that you solved earlier?