28C00400_lecture_2 - Aalto University Derivatives LECTURE 2 Matti Suominen STOCK PRICES AS A RANDOM WALK In efficient markets prices reflect all past

28C00400_lecture_2 - Aalto University Derivatives LECTURE 2...

This preview shows page 1 - 9 out of 20 pages.

Aalto University Derivatives LECTURE 2 Matti Suominen
Image of page 1
2 STOCK PRICES AS A RANDOM WALK In efficient markets prices reflect all past public information. ð༏ The expected returns depend only on beta ð༏ Prices move when unexpected news come to the market Normal distribution (with a mean that depends on beta) is a useful approximation for the distribution of stock returns.
Image of page 2
3 Examples of stock price paths: real and simulated 50 60 70 80 90 100 110 0 50 100 150 200 250 50 60 70 80 90 100 110 0 50 100 150 200 250 50 70 90 110 130 150 170 0 50 100 150 200 250 50 70 90 110 130 150 0 50 100 150 200 250 50 70 90 110 130 150 0 50 100 150 200 250 50 60 70 80 90 100 110 0 50 100 150 200 250
Image of page 3
4 BINOMIAL MODEL OF STOCK PRICES We model stock prices using a binomial tree: Each period the stock price can go up by a factor u or down by a factor of d . If we select u and d “correctly”, then as Δ t 0 this discrete approximation approaches a continuous model of stock prices where the stock returns are normally distributed. To begin with, we will assume that there is only one period before the option expires. In the next lecture we will extend the analysis to multiple periods. u ² S uS S udS dS d ² S Δ t T
Image of page 4
5 EXACT OPTION PRICING Basic idea: Options (and other derivatives) are priced by no arbitrage. Find a portfolio replicating the payoff of the option. A simple case: Δ t = 1 S 0 = 50, u = 1.1, d = 0.9 50 What is the price of a call option, expiring at t = 1, with EX = 50 when r f = 5% ? 1.1 x 50 = 55 0.9 x 50 = 45
Image of page 5
6 Assume traders can borrow and lend at the risk free rate. Find a portfolio composed of the underlying stock (buy h 0 stocks) combined with riskless lending/borrowing (borrow B 0 ) that replicates the payoff of the call. t = 0 t = 1 Value of call at t = 1 Value of replicating portfolio at t = 1 55 C u = 50 45 C d = h 0 and B 0 must satisfy the following two conditions: * * Value of call: This is the basic idea in derivatives pricing: The price of the derivative must equal the value of the replicating portfolio.
Image of page 6
7 More generally: Number of shares bought: Ratio Hedge or Delta Option 45 55 0 5 0 = = = d u d u S S C C h Amount borrowed: f d d r C S h B + = 1 0 0 Value of the call: C 0 = h 0 S 0 - B 0 Note that call price does not depend on the probability that the stock price goes up; nor on the traders’ risk preferences !!!
Image of page 7
8 VALUE OF PUT OPTION With similar procedure, consider a put option with EX = 50 with expiration date at t = 1: t = 0 t = 1 Value of put at t = 1 Value of replicating portfolio at t = 1 55 P u = max{EX-S u ,0} = 0 55h 0 + 1.05B 0 50 45 P d = max{EX-S d ,0} = 5 45h 0 + 1.05B 0 Hedge ratio: 5 . 0
Image of page 8
Image of page 9

You've reached the end of your free preview.

Want to read all 20 pages?

What students are saying

  • Left Quote Icon

    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

    Student Picture

    Kiran Temple University Fox School of Business ‘17, Course Hero Intern

  • Left Quote Icon

    I cannot even describe how much Course Hero helped me this summer. It’s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

    Student Picture

    Dana University of Pennsylvania ‘17, Course Hero Intern

  • Left Quote Icon

    The ability to access any university’s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLA’s materials to help me move forward and get everything together on time.

    Student Picture

    Jill Tulane University ‘16, Course Hero Intern

Stuck? We have tutors online 24/7 who can help you get unstuck.
A+ icon
Ask Expert Tutors You can ask You can ask You can ask (will expire )
Answers in as fast as 15 minutes