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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