Financial Engineering 6

Lecture 19

Options Pricing with Binomial Lattice

References: Villalobos, Luenberger

Lecture Topics • Option Pricing Review • Lattice Option Pricing • Greek Factors • Examples

Option Pricing • Lattices for a share of stock, an American call option, and a risk-free

investment:

• If the price of the stock goes up with a probability of p, then: – The final price of the stock is S multiplied by u or Su. – The final price of the call option is Max(Su - K, 0) where K is the

strike price. – The final price of the risk-free investment is R = (1 + r), where r is

the risk-free rate.

• If the price of the stock goes down with a probability of 1 – p, then: – The final price of the stock is Sd. – The final price of the call option is Max(Sd - K, 0). – The final price of the risk-free investment is R = (1 + r).

S

Su

Sd

p

1-p C

Max(Su-K,0)

Max(Sd-K,0)

p

1-p 1

R

R

p

1-p

Option Pricing • If we assume that the investor has the alternative to invest in a

combination of: – x units of the underlying stock. – b units of the risk-free investment. – Then for a risk-neutral investor to consider an option, the

return given by an option should match the combined return of the above investment.

• Based on this observation the following relationships follow:

Solving this series of linear equations we get the value of the equivalent call option as:

RbdxC RbuxC

d

u

+= +=

  

  

− −

+ − −

==+ du Cdu Ru

C du dR

R Cbx

1

Option Pricing • Expressed in a different form:

• Under the assumption that u > R > d, we can think of q as the probability of the underlying stock going up in the reference period, and result in the risk-neutral price for the call option.

– Note, q is not really a probability, but it takes on a value between 0 and 1.

• These formulas can be used in combination of binomial lattices to find the price of call options.

( )( )1 1

where

u dC qC q CR R d

q u d

= + −

− =

−

Example • The closing price for Orbital Sciences (ORB) on march 31, 2008

was of $24.10. • Consider the following call option:

– Strike price = $25.00 – Expiration date April (third Friday = April 18) – Days remaining = 14 (trading) days ∼ 3 weeks. – Information provided in Yahoo:

Orbital Sciences Corporation Option Trade with 1X

Position Num OptSym Expire Days Strike Type IV Vol OI Entry

Buy 1 ORBDE 8-Apr 18 25 Call 43.90% 18 565 @ 0.75

Debit Profit Max Profit Max Risk Delta

(Shares) Gamma Vega Theta

$75.00 $-30.00 Unlimited $-75.00 37.7 15.6475 $2.03 ($2.56)

Downside Breakeven Upside Breakeven Max Profit / Max Risk Max Profit / Debit

25.75 25.75 Unlimited% Unlimited%

Implied Volatility

Open Interest (outstanding)

Premium

100 options

Exercise • Let’s calculate the results given by Yahoo for the previous call

option. – Assuming 3 weeks left to expiration. – Assuming 14 days left to expiration. – Data provided in file “Lecture 19dm Lattice Option Pricing

Examples.xlsx”.

1 Week Lattice ORB Call Option

S = 24.10

Su = 24.88

Sd = 23.34

C

Cup = Max(Su - K, 0) = 0

Cdown = Max(Sd - K, 0) = 0

Week 0 1 Stock Value 24.10 24.88Up

23.34Down

Week 0 1 Option Value 0 0Up

0Down

S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% v (annual) 0.324591 σ (annual) 0.22974 Weeks per year 52 Dt 0.019231 u 1.032372 d 0.968643 R 1.000235 q 0.495717 p 0.597965

3 Week Lattice ORB Call Option

S = 24.10

26.52

21.90

24.88

23.34

S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% v (annual) 0.324591 σ (annual) 0.22974 Weeks per year 52 Dt 0.019231 u 1.032372 d 0.968643 R 1.000235 q 0.495717 p 0.597965

Week 0 1 2 3 Stock Value 24.10 24.88 25.69 26.52Up

23.34 24.10 24.88 22.61 23.34

21.90Down

Week 0 1 2 3 Option Value 0.185 0.373 0.752 1.517Up

0 0 0 0 0

0Down

C = 0.185

1.517

0

0

0

Cup = Max(Suuu-K, 0) =1.517

C = (1/R)(qCu+(1-q)Cd) = 0.185

Price per table is 0.75

Is weekly a high enough resolution?

14 Day Lattice ORB Call Option

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

Price per table is 0.75 Why is there a difference? Are we using the same volatility? The implied volatility is what we

determine to satisfy the price that the market is willing to pay.

14 Day Lattice ORB Call Option

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 0.649 0.909 1.248 1.676 2.199 2.817 3.518 4.286 5.101 5.947 6.817 7.712 8.631 9.577 10.549Up

0.395 0.579 0.830 1.165 1.597 2.132 2.769 3.491 4.276 5.099 5.945 6.815 7.709 8.629 0.216 0.333 0.503 0.744 1.075 1.512 2.064 2.725 3.473 4.274 5.097 5.943 6.813

0.102 0.167 0.268 0.421 0.649 0.974 1.418 1.995 2.693 3.471 4.271 5.094 0.040 0.069 0.118 0.199 0.332 0.540 0.856 1.314 1.933 2.690 3.469

0.011 0.021 0.038 0.071 0.128 0.231 0.409 0.709 1.195 1.931 0.002 0.003 0.007 0.014 0.028 0.057 0.116 0.235 0.476

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

Price per table is 0.75

What if we use IV from the table of 0.439?

Exercise Let’s work the “Lecture 19 In Class Example” Excel file!

Terminology

• IV is the Implied Volatility which is the amount market expects the stock to vary to expiration

– The annualized StDev of the ln(Total Returns) estimate the market “backwards” calculated).

• Delta is the amount by which the price of an option changes for every dollar movement in the underlying instrument (derivative of the value function).

• Gamma is the degree by which the delta changes with respect to changes in the underlying instrument's price (second derivative).

• Theta is the Greek measurement of the time decay of an option (derivative with respect to time).

• Vega is the amount by which the price of an option changes when the volatility changes; also referred to as volatility (derivative with respect to σ).

• Rho is the change in price relative to a change in interest.

Orbital Sciences Corporation Option Trade with 1X

Position Num OptSym Expire Days Strike Type IV Vol OI Entry

Buy 1 ORBDE 8-Apr 18 25 Call 43.90% 18 565 @ 0.75

Debit Profit Max Profit Max Risk Delta

(Shares) Gamma Vega Theta

$75.00 $-30.00 Unlimited $-75.00 37.7 15.6475 $2.03 ($2.56)

The Greeks Calculated:

Original Table:

In future lectures we will learn how to estimate these more accurately.

Debit Profit Max Profit Max Risk Delta

(Shares) Gamma Vega Theta

$75.00 $-30.00 Unlimited $-75.00 37.7 15.6475 $2.03 ($2.56)

r 1.22% IV 0.439

Stock Price 23.1 24.1 25.1 Option Value 14 days 0.334 0.649 1.088 Option Value 15 days 0.669

Option Value at IV=0.5 0.788 Option Value at r = 2.22% 0.654

Delta 0.315 0.439 Average Delta 0.377 Gamma 0.125

Theta -0.020 Vega 2.276 Rho 0.005

Exercise Back to our “Lecture 19 In Class Example” Excel file!

ORB Stock and Option Plots

We can create this graph with our data!

Put Options • The value of the European Put option is obtained the same way

as the call option. – Find the expected values of the Put option at expiration and

calculate backwards through the lattice the expected value of the Put option at time zero.

• For the American put option, we need to determine at each node if it is beneficial to exercise the option immediately or wait until expiration.

• Let’s find the value of an American put option for ORB with a strike price of $25 and an expiration date of three weeks from now.

3 Week Lattice ORB Put Option S (stock price) 24.10 K (strike price) 25.00 r (annual rate) 1.22% v (annual) 0.324591 σ (annual) 0.22974 Weeks per year 52 Dt 0.019231 u 1.032372 d 0.968643 R 1.000235 q 0.495717 p 0.597965

Week 0 1 2 3 Option Value 1.074 0.484 0.060 0Up

1.656 0.900 0.120 2.388 1.656

3.097Down

Week 0 1 2 3 Option Value 1.067 0.481 0.060 0Up

1.644 0.894 0.120 2.382 1.656

3.097Down

S = 24.10

26.52

21.90

24.88

23.34P = 1.074

0

3.097

0.120

1.656

Pup = Max(K-Suuu, 0) = 0

P = Max(K-Su, (1/R)(qCu+(1-q)Cd) = 1.074

European Put Option

American Put Option

14 Day Lattice ORB Put Option

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 1.088 0.829 0.597 0.400 0.243 0.129 0.056 0.018 0.003 0 0 0 0 0 0Up

1.346 1.059 0.793 0.556 0.357 0.202 0.094 0.032 0.006 0 0 0 0 0 1.631 1.323 1.028 0.754 0.511 0.308 0.156 0.059 0.012 0 0 0 0

1.936 1.616 1.300 0.996 0.712 0.459 0.252 0.105 0.024 0 0 0 2.255 1.929 1.602 1.278 0.962 0.665 0.399 0.185 0.048 0 0

2.579 2.253 1.924 1.591 1.258 0.928 0.611 0.321 0.095 0 2.902 2.580 2.253 1.922 1.585 1.244 0.898 0.546 0.19

3.222 2.905 2.583 2.256 1.924 1.588 1.246 0.90 3.537 3.225 2.907 2.585 2.258 1.927 1.59

3.848 3.540 3.227 2.910 2.588 2.26 4.154 3.851 3.542 3.229 2.91

4.456 4.157 3.853 3.54 4.754 4.459 4.16

5.047 4.76 5.34Down

Day 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 Option Value 1.092 0.832 0.599 0.401 0.244 0.129 0.056 0.018 0.003 0 0 0 0 0 0Up

1.350 1.062 0.795 0.558 0.358 0.202 0.095 0.032 0.006 0 0 0 0 0 1.636 1.327 1.031 0.756 0.512 0.309 0.156 0.059 0.012 0 0 0 0

1.943 1.621 1.304 0.999 0.714 0.460 0.253 0.105 0.024 0 0 0 2.263 1.935 1.607 1.282 0.965 0.666 0.400 0.185 0.048 0 0

2.589 2.261 1.930 1.596 1.262 0.931 0.612 0.322 0.095 0 2.912 2.589 2.261 1.928 1.590 1.248 0.900 0.547 0.189

3.231 2.912 2.589 2.261 1.928 1.590 1.248 0.900 3.545 3.231 2.912 2.589 2.261 1.928 1.590

Days per year 250 3.854 3.545 3.231 2.912 2.589 2.261 4.159 3.854 3.545 3.231 2.912

4.460 4.159 3.854 3.545 4.756 4.460 4.159

5.048 4.756 5.336Down

European Put Option

American Put Option

Put – Call Parity • Let C and P be the prices of a European Call and Put option,

exercised only at expiration date, both with a strike price of K. • Then the following relationship holds:

Where d is the discount factor in the period to expiration and S the current price of the stock.

C = S - K then we buy a call option

-P = -(K-S) then we sell a put option

C - P + dK = S

S

dK

14 1

0.206 1.088 25.00 24.10 1.00005

C P dK S  − + = → − + =   

Assignments • Luenberger edition 2 Chapter 13 and 14 problems 13.1, 13.6,

13.7, 13.8, 14.6, 14.11, 14.15. • Luenberger edition 1 Chapter 11 and 12 problems 11.1, 11.6,

11.7, 11.8, 12.6, 12.11, 12.15. • Set up your own binomial lattice spreadsheets and using the

Dell Data provided in the file: – “Lecture 19dm lattice Options Pricing Examples.xlsx” – Calculate the ln(Total Return), mean, Stdev, and annual mean and StDev. – Determine the Call option price based on six weeks. – Determine the Call option price based on 39 days. – Determine the American Put option price based on 39 days. – Determine the European Put option price based on 39 days. – Verify the stock price using the European Put-Call Parity equation. – Calculate the Greeks. How did you compare with the Table values? – Recalculate your Call option price based on the Table IV.

• Begin reading Luenberger 2nd edition Chapter 15; 1st edition Chapter 13.