Econ102 - Econ 102 (Random walks and high finance) Tom...

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Unformatted text preview: Econ 102 (Random walks and high finance) Tom Carter http://astarte.csustan.edu/˜ tom/SFI-CSSS Fall, 2008 1 Our general topics: ← } Financial Modeling } Some random (variable) background } What is a random walk? } Some Intuitive Derivations 2 Financial Modeling ← Let’s use some of these ideas to do some financial modeling. As an example, let’s develop the (in)famous Black-Scholes model for options pricing. • We can start with the simplest financial instrument, the fixed rate bond. If we “buy” amount V of a rate r bond at time t = 0, then at time t = 1 we can redeem the for value V (1 + r ). If we wait longer to redeem the bond, then at some time in the future we can redeem the bond for V ( n,r ) = V (1 + r ) n One can also think of this as a “savings account” with interest rate r . In this case, we can ask the more general question, what is V ( t,r ) for real values of t , rather than just integral values of t ? 3 This will depend on the specifics of the bond (savings account). In its simplest form, the bond will have “coupons” that can be redeemed at specific times in the future, or in the case of a savings account, interest will be “compounded” on specific dates. Let’s look at various possibilities for compounding. Suppose the bond has (annual) interest rate r . If interest is compounded k times during the year ( k would be 4 for quarterly compounding, 12 for monthly compounding, etc.), then the value at time t would be V ( t,r,k ) = V (1 + r k ) kt If we smooth this out, and let k go to infinity (i.e., “continuous compounding”), then we will have V ( t,r ) = V lim k →∞ ((1 + r k ) kt ) = V e rt . We thus know how to set a price for a bond to be redeemed at some time t in the future. 4 • Now let’s generalize. Suppose that the interest rate r , instead of being fixed, varied over time. What price should we be willing to pay for such a financial instrument? We can think of this financial instrument as a stock (share) in a corporation. Our “return on investment” will be uncertain, and will depend on the performance of the corporation (and also on the change in price of the stock). We have the potential to make a large profit (if the price of the stock goes up), but we now also have the potential to lose money (if the price of the stock goes down). There is a difficulty here in that we don’t really have the flexibility to buy the stock at whatever price we want today (depending on our calculation of the future value of the stock), but can only buy at today’s price. One thing we can do (at least potentially, assuming there are sellers willing), is to 5 purchase an “option” to buy the stock at some fixed price at some specific time in the future. Let’s simplify things a bit, and assume that over the time period in question, the stock will not pay any dividends (in other words, our profit/loss will only depend on changes in the price of the stock)....
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This note was uploaded on 01/02/2012 for the course FINANCE 347 taught by Professor Bayou during the Fall '11 term at NYU.

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Econ102 - Econ 102 (Random walks and high finance) Tom...

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