Binomial Option Pricing Model (BOPM) References: Neftci, Chapter 11.6 Cuthbertson & Nitzsche, Chapter 8 1.

Binomial Option Pricing Model Binomial Option Pricing Model (BOPM)(BOPM)References:

Neftci, Chapter 11.6

Cuthbertson & Nitzsche, Chapter 8

1

Linear State PricingLinear State Pricing

A 3-month call option on the stock has a strike price of 21. Can we price this option?

◦ Can we find a complete set of traded securities to price the option payoffs?

◦ If we make the simplifying assumption that there are only 2 states of the world (up and down), then we only need the prices of two independently distributed traded assets, e.g. the underlying stock and the risk-free asset

Linear State PricingLinear State Pricing

Algebraically,

If S = 2, this is a system of equations in two unknowns

To get a unique solution for it, we need at least 2 independent equations

S

ss

f

sS

sq

sff

sS

s

s

R

dRR

dP

s

10

11 00

1

1

)1(1

1

The Binomial ModelThe Binomial Model

A stock price is currently S0 = $20

In three months it will be either S0u = $22 or S0d = $18

Stock Price = $22

Stock Price = $18

Stock price = $20

Stock Price = $22Option Price = $1

Stock Price = $18Option Price = $0

Stock price = $20Option Price=?

A One-Period Call OptionA One-Period Call Option

Option tree:

6

Risk-Neutral Valuation Risk-Neutral Valuation ReminderReminderUnder the RN measure the stock price

earns the risk-free rateThat is, the expected stock price at time T

is S0erT

When we are valuing an option in terms of the underlying the risk premium on the underlying is irrelevant

See handout on RNV

Risk-Neutral TreeRisk-Neutral Tree

f = [ q f u + (1 – q ) f d ]e-rT

The variables q and (1– q ) are the risk-neutral probabilities of up and down movements

The value of a derivative is its expected payoff in a risk-neutral world discounted at the risk-free rate

S0u ƒu

S0d ƒd

S0

ƒ

q

(1– q )

Risk-Neutral ProbabilitiesRisk-Neutral ProbabilitiesS0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0

ƒ

q

(1– q )

Since q is a risk-neutral probability,

22q + 18(1 – q) = 20e0.12 (0.25)

q = 0.6523

RN Binomial Probabilities RN Binomial Probabilities FormulaFormula

RN probability of up move:

RN probability of down move:

1 – qWith this probabilities, the underling grows

at the risk-free rate (check it out)

du

deq

rT

6523.09.01.1

9.00.250.12

e

du

deq

rT

Using the RN Binomial Using the RN Binomial Probabilities FormulaProbabilities Formula

In above example, u = 22/20 = 1.1 and d = 18/20 = 0.9

So, assuming r = 12% p.a.,

Valuing the OptionValuing the OptionS0u = 22 ƒu = 1

S0d = 18 ƒd = 0

S0

ƒ

0.6523

0.3477

The value of the option is

e–0.12(0.25) [0.6523 1 + 0.3477 0]

= 0.633

A Two-Step ExampleA Two-Step Example

Each time step is 3 months

20

22

18

24.2

19.8

16.2

Valuing a Call OptionValuing a Call Option

Value at node B = e–0.120.25(0.65233.2 + 0.34770) = 2.0257

Value at node A = e–0.120.25(0.65232.0257 + 0.34770)

= 1.2823

201.2823

22

18

24.23.2

19.80.0

16.20.0

2.0257

0.0

A

B

C

D

E

F

A Put Option Example; K=52A Put Option Example; K=52

504.1923

60

40

720

484

3220

1.4147

9.4636

A

B

C

D

E

F

15

What Happens When an Option What Happens When an Option is American is American (see spreadsheet)(see spreadsheet)

505.0894

60

40

720

484

3220

1.4147

12.0

A

B

C

D

E

F

And if we did not have u and d?And if we did not have u and d?

One way of matching the volatility of log-returns is to set

where is the volatility andΔt is the length of the time step. This is the approach used by Cox, Ross, and Rubinstein ◦ Handout on Asset Price Dynamics

t

t

ed

eu

17

The Probability of an Up MoveThe Probability of an Up Move

contract futures afor

rate free-risk foreign the is wherecurrency afor

index the on yielddividend the is whereindex stock afor

stock paying dnondividen afor

1

)(

)(

a

rea

qea

ea

du

dap

ftrr

tqr

tr

f

EXOTICS PRICING (examples)EXOTICS PRICING (examples)

a) Average price ASIAN CALL

payoff = max {0, Sav – K}

Remember: cheaper than an ‘ordinary’ option

b) Barrier Options (e.g. up and out put)pension fund holds stocks and is worried about fall in price but does not think price will rise by a very large amount◦ Ordinary put? - expensive◦ Up and out put - cheaper

04/10/23

Pricing an Asian Option (BOPM)

Average price ASIAN CALL(T = 3)

1. Calculate stock price at each node of tree2. Calculate the average stock price Sav,i at expiry, for each of

the 8 possible paths (i = 1, 2, …, 8).3. Calculate the option payoff for each path, that is

max[Sav,i – K, 0] (for i = 1, 2, …, 8)

The risk neutral probability for a particular path is

qi* = qk(1 – q)n-k

q = risk neutral probability of an ‘up’ movek = number of ‘up’ moves(n – k) = the number of ‘down’ moves

04/10/23

Pricing an Asian Option (BOPM), cont’d

4. Weight each of the 8 outcomes for the call payoff max[Sav,i – K, 0] by the qi* to give the expected payoff:

5. The call premium is then the PV of ES*, discounted at the risk free rate, hence:

8

1,

* ]0,max[)(ˆi

iavi KSqSE

3)(ˆ rAsian eSEC

04/10/23

Pricing Barrier Options (BOPM)

Down-and-out callS0 = 100. Choose K = 100 and H = 90 (barrier)

Construct lattice for SPayoff at T is max {0, ST – K }

Follow every ‘path’ (ie DUU is different from UUD) If on say path DUU we have any value of S < 90 , then the value at T is set to ZERO (even if ST – K > 0).

Use BOPM risk neutral probabilities for each path and each payoff at T

04/10/23

Example: Down-and-out call

S0 =100, K= 100, q = 0.857, (1 – q) = 0.143

H = 90

UUU ={115, 132.25, 152.09} Payoff = 52.09 (q* = 0.8573, 0.629)

DUU ={80, 92,105.8} Payoff = 0 NOT 5.08 (q* = 0.105)

C = e-rT ‘Sum of [q* payoffs at T]’

where qi* = qk(1 – q)n-k,