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Transcript of S TOCHASTIC M ODELS L ECTURE 5 P ART II S TOCHASTIC C ALCULUS Nan Chen MSc Program in Financial...
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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
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Outline1. Generalized Ito Formula2. A Primer on Option Contracts3. Black-Scholes Formula for
Option Pricing
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5.4 GENERALIZED ITO FORMULA
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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:
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Quadratic Variation of an Ito Process• We can show that the quadratic variation of
the above Ito process is
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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:
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Generalized Ito Formula
• Let be an Ito process, and let be a smooth function. Then,
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Generalized Ito Formula
• It is easier to remember and use the result of Ito formula if we recast it in differential form
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Example I: Geometric Brownian Motion• Let
• If we apply Ito formula, we have
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5.2 A PRIMER ON OPTIONS
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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
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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.
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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.
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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.
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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.
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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
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Payoffs for Longing a Call
Call Options
K Stock Price
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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.
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Payoffs for Short Positions in a Call
Payoff Call Options
K Stock Price
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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.
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5.3 BLACK-SCHOLES EQUATION FOR OPTION PRICING
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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.
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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.
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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
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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
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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.
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Black-Scholes Option Pricing Formula
• The above PDE admits a closed-form solution; that is,
where
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Probabilistic Representation
• Under the Feymann-Kac theorem, the solution to the above Black-Scholes PDE has the following probabilistic representation:
where