STOCHASTIC MODELS LECTURE 5 PART II

STOCHASTIC CALCULUS

Nan ChenMSc Program in Financial EngineeringThe Chinese University of Hong Kong

(Shenzhen)Dec. 9, 2015

Outline1. Generalized Ito Formula2. A Primer on Option Contracts3. Black-Scholes Formula for

Option Pricing

5.4 GENERALIZED ITO FORMULA

Ito Processes• Let be a Brownian motion. An Ito

process is a stochastic process of the following form

where is nonrandom, and and are adaptive stochastic processes.• We often denote the above integral form by

the following differential form:

Quadratic Variation of an Ito Process• We can show that the quadratic variation of

the above Ito process is

Stochastic Integral on an Ito Process

• Let be an Ito process and let be an adaptive process. Define the integral with respect to an Ito process as follows:

Generalized Ito Formula

• Let be an Ito process, and let be a smooth function. Then,

Generalized Ito Formula

• It is easier to remember and use the result of Ito formula if we recast it in differential form

Example I: Geometric Brownian Motion• Let

• If we apply Ito formula, we have

5.2 A PRIMER ON OPTIONS

Option Contract• Options give the holders a right to buy or sell

the underlying asset by a certain date for a certain price.

• Four key components of an option contract:– Underlying asset– Exercise price/strike price– Expiration date/maturity– Long position and short position

Call and Put

• There are two basic types of option: – A call option gives the option holder the right to

buy an asset by a certain date for a certain price. – A put option gives the option holder the right to

sell an asset by a certain date for a certain price.

An Example of a Call Option

• Consider a 3-month European call option on Intel’s stock. Suppose that the strike price is $20 per share and the maturity is Mar 9, 2016.

• The long position is entitled a right to buy Intel shares at the price of $20 per share on Mar 9, 2016.

Payoffs of Long Position in Call Options• Suppose that Intel stock price turns out to be

$25 per share on Mar 9, 2016.• The long position buys shares at the price of

$20 per share by exercising the option. He/she buys shares at lower price than the spot price. The gain he/she realizes is 25-20 = $5 per share.

Payoffs of Long Position in Call Options (Continued)• Suppose that Intel stock price turns out to be

$15 per share on Mar 9, 2016.– Options are rights. The holders are not required

to exercise them if they do not want to. • The contract charges a higher price than the

spot market. Of course, the holder will choose not to exercise it. The contract does not generate any economic outcomes to the holder.

Payoffs of Long Position in Call Options (Continued)• In general, suppose that the strike price is , and

the underlying asset price at the maturity is . Then, the payoff of the long position of the call option should be

Payoffs for Longing a Call

Call Options

K Stock Price

Payoffs for Shorting a Call• The writer of a call option has liability to satisfy

the requirement of the long position if he/she asks to exercise options.

• In the previous example, – If = $25 per share, the option is exercised. The writer

loses $5 per share.– If = $15 per share, the option is not exercised.

Payoffs for Short Positions in a Call

Payoff Call Options

K Stock Price

Options Premium (Option Price)

• The long position of an option always receive non-negative payoffs in the future while the writer always has non-positive payoffs.

• The long position must pay a compensation to the writer of an options. The compensation is known as the options premiums or options prices.

5.3 BLACK-SCHOLES EQUATION FOR OPTION PRICING

Option Pricing Problem• Suppose that a stock in the market follows

the geometric Brownian motion

• Suppose that there is a bank account in the market offering as risk free interest rate; that is, the wealth will grow

if you invest all your money in this account.

Option Pricing Problem• Consider a European call option that pays

at What is the fair value of this option at time • Our idea is to create a portfolio with known

value to “replicate” the performance of the option. Then, we can use the value of the portfolio to evaluate the option.

Evolution of Portfolio Value• Consider at each time the investor holds

shares of stock, and the remainder of the portfolio value is invested in the bank account.

• Then, the portfolio value will evolve as

Evolution of Option Value

• On the other hand, let denote the value of the call option at time if the stock price at that time is

• Computing the differential of , we have

Black-Scholes Partial Differential Equation• Compare the evolutions of portfolio value

and option value. If we want them to agree at any time, we need

• Together with

we have the Black-Scholes partial differential equation for option pricing.

Black-Scholes Option Pricing Formula

• The above PDE admits a closed-form solution; that is,

where

Probabilistic Representation

• Under the Feymann-Kac theorem, the solution to the above Black-Scholes PDE has the following probabilistic representation:

where