Lecture no43 option pricing

Elements of Financial Option Pricing ModelStock price Strike price Time until expiration Volatility Dividends Interest rates Intrinsic value of The option Probability of a profitable mov

Option Pricing

Lecture No 43 Chapter 13 Contemporary Engineering Economics

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Elements of Financial Option Pricing Model

Stock

price

Strike price

Time until expiration Volatility Dividends Interest rates

Intrinsic value of

The option Probability of a profitable move

Adjustment

To share price Cost of money

Option Pricing Model

Option Valuation Approaches

Two approaches to value options

oDiscrete-time

oBinomial Lattices

oContinuous-time

oBlack-Shores Model

Discrete-Time Approach

Assumptions

o The underlying asset follows a discrete, binomial,

multiplicative stochastic process throughout

time.

o Arbitrage-free pricing

o The law of one price, which states that if two

portfolios are equal in value at time T, then they

must have equivalent values today

Option Pricing: Mathematical Symbols

• Δ = Number of shares to purchase

• b = the amount cash borrowed

• R = (1 + r), where r = risk-free rate

• S 0 = value of the underlying asset today

• uS = upward movement in the value of S

• dS = downward movement in the value of S

• K = strike (exercise) price of the option

• C = value of the call option

• C u = upward movement in the value of the call option

How Would You Price a One-Day Call Option?

Three Different Approaches to Valuing a

Financial Option

o Replicating-Portfolio

o Risk-Free Financing

o Risk-Neutral Probability

 All three approaches

lead to the same valuation, but the risk- neutral probability

approach is most commonly adopted.

Approach 1: Replicating-Portfolio Concept

with a Call Option

o Create an arbitrage

portfolio that contains two

risky assets: the share of

stock and the call option on

the stock.

o An arbitrage (replicating)

portfolio is a portfolio that

earns a sure return.

Creating an Arbitrage Portfolio

• Arbitrage portfolio o Objective : Select the value of Δ

such that the total value of the portfolio is the same regardless of the value of the share of stock at option expiration.

o Mathematical expression

o What it means

o Long: 0.5 shares

o Short: 1 call option

o The value of portfolio at day 1

315 15 285

0.5

 

Pricing Option Value at Day 0

 Option values between day 0 and day 1

 Establishing equivalence between two

options values by using discounting factor

(r), a risk-free rate:

 Option value calculation at r = 6% per

year or 0.016% per day

Approach 2: Risk-Free Financing Approach

Creating a Hedge Portfolio

 ΔS = $150, b = −$142.48

A portfolio needs to be

formed with $150 worth of

stock financed in part by

$142.48 at the risk-free rate

of 6%.

 Option value on day 0

C = ΔS + b = $150 − $142.48

= $7.52

Approach 3: Risk-Neutral Probability

Approach

 Value the option in a

risk-free world by calculating

a risk-neutral probability

 The objective probability

(p) never enters into the

option value calculation

In other words, the

probability of a stock

price moving a typical

direction will not affect

the option value.

 This risk-neutral property

permits us to use a

risk-free interest rate in

valuing an option

Risk-Neutral Probability Concept

Example 13.5: A Put Option Valuation with

Two-Period Binomial Lattice Option

Valuation

o Step 1 : Calculate q.

o Step 2 : Determine the call

option value at day 1.

o Step 3 : Determine the call

option value at day 0.

Multi-period Binomial Lattice Option

Valuation

 At issue : As we increase

the number of steps in a

year, what would happen

to the resulting price

Important Relationship

A smaller Δt for the binomial lattice will provide option

values closer to its continuous-time counterpart (the Black-Scholes equation)

Example 13.6: Construction of a Binomial

Black-Scholes Option Model: Continuous

Model

Black, Scholes, and Merton were the first to derive the option value using the replicating portfolio concept Some of the key assumptions in deriving their model are:

o Constant interest rate

o A continuously operating market, where asset values'

returns are normal , which implies that the distribution of

terminal asset values is lognormal ; this process is known as a geometric Brownian motion

o No arbitrage opportunities exist, which implies a risk-neutral

world.

The B-S Call and Put Equations

1

20

The Black-Scholes equation for European calls and puts are:

S0 = Underlying asset price today

K = Exercise price

T = Time to expiration

where

Example 13.4: Option Valuation under a

Continuous-Time Process

 Given : S 0 = $40, K = $44,

r = 6%, T = 2 years, and σ =

40%

for both call and put

options

 Comments : The call

option value is greater

than the put option value,

indicating the upside

potential is higher than

Excel Worksheet to Evaluate the B-S Formulas

American Options and B-S Model

o Generally speaking, the Black-Scholes

formula cannot value American call or put options

o For our purpose, we will use the binomial

lattice approach to value American options.

Key Point

• As long as a portfolio consisting of 0.25 shares of stock

plus a short position in one call option is set up, the value of this portfolio at expiration will equal $10 in both the up-state and the down-state

• In essence, this portfolio mitigates all risk associated

with the underlying asset’s price movement.

• Because all risk has been ‘hedged’ away, the

appropriate discount rate to account for the time

value of money is the risk-free rate.