## Call Option Formula

Although it is usually impossible to find an analytic solution to the Black-Scholes equation, it is possible to find such a solution for a European call option This analytic solution is of great practical and theoretical use The formula uses the function N( v), the standard cumulative normal probability distribution. This is the cumulative distribution of a normal random variable having mean 0 and variance I It can be expressed as (.*) 6'-v 2d.y (13.15) The function N(.x) is illustrated in...

## Simulation

A continuous-time price process can be simulated by taking a series of small time periods and then stepping the process forward period by period There are two natural ways to do this, and they are not exactly equivalent First, consider the process in standard form defined by (11.18). We take a basic period length At and set S( o) So, a given initial price at t to. The corresponding simulation equation is Sfo+i) - S(tk) pS tk)At + aS(tk) tk)VAt where the ( *)'s are uncorrected normal random...

## Multiperiod Options

We now extend the solution method to multiperiod options by working backward one step at a time, A two-stage lattice representing a two-period call option is shown in Figure 12.6. It is assumed as before that the initial price of the stock is S, and this price is modified by the up and down factors u and d while moving through the lattice, The values shown in the lattice are those of the corresponding call option with strike price K and expiration time corresponding to the final point in the...

## Itos Lemma

We saw that the two Ito equations for S( ) and for lnS( ) are different, and that the difference is not exactly what would be expected from the application of ordinary calculus to the transformation of variables from S(t) to In S(t) an additional term o2 is required. This extra term arises because the random variables have order and hence their squares produce first-order, rather than second-order, effects. There is a systematic method for making such transformations in general, and this is...

## The Logoptimal Pricing Formula

The log-optima strategy has an important roie as a universal pricing asset, and the pricing formula is remarkably easy to derive. As before, we assume that there are n risky assets with prices each governed by geometric Brownian motion as (it +dz< , i l, 2,. ., n, Pi Since E(dzi) 0 for all , the covariances a y are defined by E(dzt dz.j) Oij dt, There is also a risk-free asset (asset number 0) with rate of return ly. Any set of weights itfo, W ,w2,- , w with YI'Lq wi I defines a portfolio in...

## Capm As A Pricing Formula

However, the standard CAPM formula does not contain prices explicitly only expected rates of return To see why the CAPM is called a pricing model we must go back to the definition of return Suppose that an asset is purchased at price P and later sold at price Q The rate of return is then r (Q P) P. Here P is known and Q is random. Putting this in the CAPM formula, we have This gives the price of the asset according to the CAPM We highlight this important result...

## Y Qi pi w

For i 1,2, This represents equations. The original budget constraint Yl'UifyPi W is one more equation Altogether, therefore, there are n 4- 1 equations for the n + 1 unknowns 01( 02. - -, ancl It can be shown that X > 0 These equations are very important because they serve two roles. First, and most obviously, they give enough equations to actually solve the optimal portfolio problem An example of such a solution is given soon in Example 9.5. Second, since these equations are valid if there...

## Invariance Theorem

Suppose that you have a sum of money to invest in fixed-income securities, and you will not draw from these funds for n periods (say, n years) You will invest only in Treasury instruments, and there is a current known spot rate curve for these securities You have a multitude of choices for structuring a portfolio using your available money. You may select some bonds with long maturities, some zero-coupon bonds, and some bonds with short maturities. If you select a mix of these securities, then,...

## Net Flows

In conducting a cash flow analysis using either net present value or internal rate of return, it is essential that the net of income minus expense (that is, net profit) be used as the cash flow each period. The net profit usually can be found in a straightforward manner, but the process can be subtle in complex situations In particular, taxes often introduce complexity because certain tax-accounting costs and profits are not always equal to actual cash outflows or inflows Taxes are considered...

## Immunization

We now have the concepts and tools necessary to solve a problem of major practical value, nameiy, the structuring of a bond portfolio to protect against interest rate risk This procedure is termed immunization because it immunizes the portfolio value against interest rate changes. The procedure, as well as its refinements, is in fact one of the most (if not the most) widely used analytical techniques of investment science, shaping portfolios consisting of billions of dollars of fixed-income...

## Investment Implications

The question of interest for the investor is Can the CAPM help with investment decisions There is not a simple answer to this question. The CAPM states (or assumes), based on an equilibrium argument, that the solution to the Markowitz problem is that the market portfolio is the one fund (and only fund) of risky assets that anyone need hold. This fund is supplemented only by the risk-free asset. The investment recommendation that follows this argument is that an investor should simply purchase...

## Qualitative Nature of Price Yield Curves

Would not buy a bond with a yield of 6 when bank CDs are offering 10 , The general interest rate environment exerts a force on every bond, urging its yield to conform to that of other bonds. However, the only way that the yield of a bond can change is for the bond's price to change. So as yields move, prices move correspondingly But the price change required to match a yield change varies with the structure of the bond (its coupon rate and its maturity). So as the yields of various bonds move...

## R ujjii u22 4 wnr

E(r) ui E(n) + u)2E( '2) + + w E ( ,,) This important result shows how the variance of a portfolio's return can be calculated easily from the covariances of the pairs of asset returns and the asset weights used in the portfolio (Recall, an a2 ) Example 6-8 (Two-asset portfolio) Suppose that there are two assets with Fj 12, 2 .15, < 7 20, a2 .18, and a 2 01 (values typical for two stocks) A portfolio is formed with weights tui .25 and w2 75, We can calculate the mean and the variance of the...

## Summary

This chapter is devoted to general theory, and hence it is somewhat more abstract than other chapters, but the tools presented are quite powerful The chapter should be reviewed after reading Part 3 and again after reading Part 4 The first part of the chapter presents the basics of expected utility theory Utility functions account for risk aversion in financial decision making, and provide a more general and more useful approach than does the mean-variance framework. In this new approach, an...

## Evaluating Real Investment Opportunities

Options theory can be used to evaluate investment opportunities that are not pure financial instruments. We shall illustrate this by again considering our gold mine lease problems Now, however, the pr ice of gold is assumed to fluctuate randomly, and this fluctuation must be accounted for in our evaluation of the lease prospect. Example 12.7 Simplico gold mine Recall the Simpfico gold mine from Chapter 2 Gold can be extracted from this mine at a rate of up to 10,000 ounces per year at a cost of...

## Futures Prices

There is, at any one time, only one price associated with a futures contract the delivery price The value of existing contracts is always zero because they are marked to market. The delivery price will in general be different from the spot price of the underlying asset, but the two must bear some relation to each other, In fact, as the maturity date approaches, the futures price and the spot price must approach each other, actually converging to the same value. This effect, termed convergence,...

## Bond Details

Bonds represent by far the greatest monetary value of fixed-income securities and are, as a class, the most liquid of these securities. We devote special attention to bonds, both because of their practical importance as investment vehicles and because of their theoretical value, which will be exploited heavily in Chapter 4 We describe the general structure and trading mechanics of bonds in this section and then discuss in the following few sections some methods by which bonds are analyzed Our...

## Running Present Value

The present value of a cash flow stream is easily calculated in the term structure framework. One simply multiplies each cash flow by the discount factor associated with the period of the flow and then sums these discounted values that is, present value is obtained by appropriately discounting all future cash flows. There is a special, alternative way to arrange the calculations of present value, which is sometimes quite convenient and which has a useful interpretation This different way is...

## Short Rates

Short rates are the forward rates spanning a single time period. The short rate at time k is accordingly rk fkj. t that is, it is the forward rate from k to k 1. The short rates can be considered fundamental just as spot rates, for a complete set of short rates fully specifies a term structure The spot rate sk is found from the short rates from the fact that interest earned from time zero to time k is identical to the interest that would be earned by rolling over an investment each year...

## Exercices On Spot Rate

Gold futures The current price of gold is 412 per ounce The storage cost is 2 per ounce per year, payable quarterly in advance. Assuming a constant interest rate of 9 compounded quarterly, what is the theoretical forward price of gold for delivery in 9 months 2. Proportional carrying charges o Suppose that a forward contract on an asset is written at time zero and there are M periods until delivery Suppose that the carrying charge in period k is qS k , where S k is the spot price of the asset...

## Exercises

I Bull spread An investor who is bullish about a stock believing that it will rise may wish to construct a bull spread for that stock One way to construct such a spread is to buy a call with strike price K and sell a call with the same expiration date but with a strike price of K2 gt K Draw the payoff curve for such a spread Is the initial cost of the spread positive or negative 2. Put-call parity Suppose over the period fO, 7 a certain stock pays a dividend whose present value at interest rate...

## Value of an Interest Rate Swap

Consider a plain vanilla interest rate swap in which party A agrees to make payments of a fixed rate r of interest on a notional principal N while receiving floating rate payments on the same notional principal for M periods The cash flow stream received by A is cq , Cj - r,c2 . , cm times the principal N The cy's are the floating rates We can value the floating portion of this swap with a special trick derived from our knowledge of floating rate bonds For a direct proof using forward pricing...

## Cfi

Procedure, either by recording them at the nodes as the V-values are computed, or by working forward, using the known future V-values. The running dynamic programming method can be written very succinctly by a recurrence relation Define to be the cash flow generated by moving from node kt i to node k 4- 1, a . The recursion procedure is Vki - maximize dkVk 1 An example will make all of this clear Example 5.4 Fishing problem Suppose that you own both a lake and a fishing boat as an investment...

## Pure Investment

Pure investment refers to the objective of obtaining increased future return for present allocation of capital This is the motivation underlying most individual investments in the stock market, for example The investment problem arising from this motivation is referred to as the portfolio selection problem, since the real issue is to determine where to invest available capital. Most approaches to the pure investment problem rely on the risk aversion principle, for in this problem one must...

## Option Concepts

The specifications of an option include, first, a clear description of what can be bought for a call or sold for a put . For options on stock, each option is usually for 100 shares of a specified stock. Thus a call option on IBM is the option to buy 100 shares of IBM. Second, the exercise price, or strike price, must be specified. This is the price at which the asset can be purchased upon exercise of the option. For IBM stock the exercise price might be 70, which means that each share can be...

## Fisher Weil Duration

The details work out most nicely for the case of continuous compounding, and we shall present that case first. Given a cash flow sequence v, 1,,.vf , vf,____,x,n and the FIGURE 4.3 Shifted spot rate curves. The original spot rate curve is the middle curve. This curve is shifted upward and downward by an amount k to obtain the other curves It is possible to immunize a portfolio against such shifts for small values of X spot rate curve s,, r0 lt lt , the present value is The Fisher-Weil duration...

## Explain The Shape Of Feasible Region Of Risky Assets

This problem cannot be reduced to the solution of a set of linear equations It is termed a quadratic program, since the objective is quadratic and the constraints are linear equalities and inequalities. Special computer programs are available for solving such problems, but small to moderate-sized problems of this type can be solved readily with spreadsheet programs. In the financial industry there are a multitude of special-purpose programs designed to solve this problem for hundreds or even...

## Type B Arbitrage

Another form of arbitrage can be identified If an investment has nonpositive cost but has a positive probability of yielding a positive payoff and no probability of yielding a negative payoff, that investment is said to be a type B arbitrage. In other words, a type B arbitrage is a situation where an individual pays nothing or a negative amount and has a chance of getting something An example would be a free lottery ticket you pay nothing for the ticket, but have a chance of winning a prize...

## Determining the Spot Rate

The obvious way to determine a spot rate curve is to find the prices of a series of zero-coupon bonds with various maturity dates. Unfortunately the set of available zero-coupon bonds is typically rather sparse, and, indeed, until recently there were essentially no zeros available with long maturities. Thus it is not always practical to determine a complete set of spot rates this way However, the existence of zero-coupon bonds is not necessary for the concept of spot rates to be useful, nor are...

## Discount Factors

Another important concept is that of a discount factor between two times The discount factors are, of course, fundamental quantities used in present value calculations, it is useful to apply a double indexing system to the discount factors paralleling the system used for forward rates Accordingly, the symbol dj k denotes the discount factor used to discount cash received at time k back to an equivalent amount of cash at time j The normal, time zero, discount factors are d do i, d2 do 2, . ., d...

## YflSS

To use the constant-growth dividend model one must estimate the growth rate g and assign an appropriate value to the discount rate Estimation of g can be based on the history of the firm's dividends and on future prospects, Frequently a value is assigned to r that is larger than the actual risk-free interest rate to reflect the idea that uncertain cash flows should be discounted more heavily than certain cash flows In Chapters 15 and 16, we study better ways to account for uncertainty. Example...

## Interdependent Projects

Sometimes various projects are interdependent, the feasibility of one being dependent on whether others are undertaken We formulate a problem of this type by assuming that there are several independent goals, but each goal has more than one possible method of implementation It is these implementation alternatives that define the projects. This formulation generalizes the problems studied in Chapter 2, where there was only one goal such as buying a new car but several ways to achieve that goal...

## Duration and Sensitivity

Duration is useful because it measures directly the sensitivity of price to changes in yield. This follows from a simple expression for the derivative of the present value expression. In the case where payments are made m times per year and yield is based on those same periods, we have dPV - lt k m dA 1 V 0 1 i A m We now apply this to the expression for price, Here we have used the fact that the price is equal to the total present value at the yield by definition of yield . We find that dPV,...

## Mean Blur

We now show how this amplification effect makes the estimation of expected or mean rates nearly impossible. Let us select a basic period length p such as p 1 12 for a monthly period . We shall try to estimate the mean rate of return for this period That is, we assume that the statistical properties of the returns in each of the periods are identical, with mean value F and standard deviation a. We also assume that the individual returns are mutually uncorrected We wish to estimate the common...

## Qualitative Properties of Duration

The duration of a coupon-paying bond is always less than its maturity, but often it is surprisingly short An appreciation for the relation between a bond's duration and other parameters ol the bond can be obtained by examination of Table 3 6. In this table the yield is held fixed at 5 , but various maturities and coupon rates are considered. This procedure approximates the situation of looking through a list of available bonds at a time when all yields hover near 5 . Within a given class say,...

## Inflation

Inflation is another factor that often causes confusion, arising from the choice between using actual dollar values to describe cash flows and using values expressed in purchasing power, determined by reducing inflated future dollar values back to a nominal level Inflation is characterized by an increase in general prices with time, inflation can be described quantitatively in terms of an inflation rate . Prices 1 year from now will on average be equal to today's prices multiplied by 1...