Introduction

The op-amp astable oscillator covered in Section 6.5 was our first example of an oscillator – a circuit that produces a periodic output signal without an input signal. These types of circuits have some kind of feedback mechanism that allows them to oscillate spontaneously. We can categorize oscillators into two broad groups: relaxation oscillators and sinusoidal oscillators. Each of these groups has common features. The relaxation oscillators are characterized by non-sinusoidal output waveforms, timing that is set by capacitor charging and discharging, and the non-linear operation of its active components. The analysis of relaxation oscillator circuits is done in the time domain (i.e., by determining the circuit response as a function of time). For example, our op-amp astable oscillator has a square wave output with a period set by the charging/discharging of capacitor C through resistor R f, and the op-amp is operating non-linearly, switching back and forth between its saturation voltages. On the other hand, sinusoidal oscillators, as the name implies, have sinusoidal output waveforms and linear operation of the active components, and the analysis is done in the frequency domain (i.e., by considering how the circuit responds to different frequencies). We will now examine examples of each type of oscillator.

Relaxation oscillators

SCR sawtooth oscillator

Our first relaxation oscillator is shown in Fig. 7.1. It uses two components we have studied previously: the SCR and the bipolar transistor.

Introduction

In this chapter we introduce the second major type of transistor: the field-effect transistor. Like the bipolar junction transistors (BJTs) we studied in Chapter 4, field-effect transistors (FETs) allow the user to control a current with another signal. The key difference is that the FET control signal is a voltage while the BJT control signal is a current. Also, the FET control input (called the gate) has a much higher input impedance than the base of a BJT. Indeed, the DC gate impedance for FETs varies from a few megaohms to astounding values in excess of 1014 Ω. High input impedance is a highly desirable feature that greatly simplifies circuit analysis.

The BJT has three connections: the collector, base, and emitter. The corresponding connections on an FET are called the drain, gate, and source. Some versions of the FET have a fourth connection called the bulk connection. Bipolar transistors come in just two types with opposite polarities: the npn and the pnp. Field-effect transistors have greater variety. In addition to the polarity pairs (termed n-channel and p-channel), there are differences in gate construction (junction and metal oxide), and doping (depletion and enhancement). In terms of analysis, however, they are all very similar, so we will not have to consider each variety separately. Also, as we did with the bipolar transistor, we will focus on one of the polarities (n-channel) since the other polarity simply involves swapping the labels for n and p and changing the sign of the voltages and the direction of the currents.

Basics

We start our study of electronics with definitions and the basic laws that apply to all circuits. This is followed by an introduction to our first circuit element – the resistor.

In electronics, we are interested in keeping track of two basic quantities: the currents and voltages in a circuit. If you can make these quantities behave like you want, you have succeeded.

Current measures the flow of charge past a point in the circuit. The units of current are thus coulombs per second or amperes, abbreviated as A. In this text we will use the symbol I or i for current.

As charges move in circuits, they undergo collisions with atoms and lose some of their energy. It thus takes some work to move charges around a circuit. The work per unit charge required to move some charge between two points is called the voltage between those points. (In physics, this work per unit charge is equivalent to the difference in electrostatic potential between the two points, so the term potential difference is sometimes used for voltage.) The units of voltage are thus joules per coulomb or volts, abbreviated V. In this text we will use the symbol V or v for voltage.

In a circuit, there are sources and sinks of energy. Some sources of energy (or voltage) include batteries (which convert chemical energy to electrical energy), generators (mechanical to electrical energy), solar cells (radiant to electrical energy), and power supplies and signal generators (electrical to electrical energy).

Introduction

We now turn to an examination of the properties and uses of the operational amplifier or op-amp. A detailed analysis of this multi-stage amplifier circuit is beyond the scope of this text, so we will treat it as a black box device as we did earlier with the voltage regulator. Thus, to use the device, we need only learn and apply some simple rules and, later, the real-world limitations of the device.

In current usage, the operational amplifier is usually packaged as an integrated circuit with multiple leads or pins. While there are hundreds of different op-amps with different specifications, they all follow the same usage rules. To be specific, we will focus on a “classic” version: the 741 op-amp.

The circuit symbol for the op-amp is shown in Fig. 6.1. There are inputs for two power supply voltages (one positive and one negative relative to ground, labeled V+ cc and V- cc, respectively). There are also two signal inputs: the inverting input, labeled with a minus sign, and the non-inverting input, labeled with a plus sign. Happily, there is only one output.

As we know, voltages are always between two points, but our description of the op-amp inputs seems to refer to voltages at one point, the various input pins. It is thus important to note that all of the voltages for the op-amp are referenced to ground (i.e., the second point is ground).

Introduction

In this chapter we look at some extensions of the basic call option model. In Section 8.2 we consider European call options on dividend-paying securities under three different scenarios for how the dividend is paid. In Section 8.2.1 we suppose that the dividend for each share owned is paid continuously in time at a rate equal to a fixed fraction of the price of the security. In Sections 8.2.2 and 8.2.3 we suppose that the dividend is to be paid at a specified time, with the amount paid equal to a fixed fraction of the price of the security (Section 8.2.2) or to a fixed amount (Section 8.2.3). In Section 8.3 we show how to determine the no-arbitrage price of an American put option. In Section 8.4 we introduce a model that allows for the possibilities of jumps in the price of a security. This model supposes that the security's price changes according to a geometric Brownian motion, with the exception that at random times the price is assumed to change by a random multiplicative factor. In Section 8.4.1 we derive an exact formula for the no-arbitrage cost of a call option when the multiplicative jumps have a lognormal probability distribution. In Section 8.4.2 we suppose that the multiplicative jumps have an arbitrary probability distribution; we show that the no-arbitrage cost is always at least as large as the Black–Scholes formula when there are no jumps, and we then present an approximation for the no-arbitrage cost.

Brownian Motion

A Brownian motion is a collection of random variables X(t), t ≥ 0 that satisfy certain properties that we will momentarily present. We imagine that we are observing some process as it evolves over time. The index parameter t represents time, and X(t) is interpreted as the state of the process at time t. Here is a formal definition.

Definition The collection of random variables X(t), t ≥ 0 is said to be a Brownian motion with drift parameter μ and variance parameter σ2 if the following hold:

1. (a) X(0) is a given constant.

2. (b) For all positive y and t, the random variable X(t + y) – X(y) is independent of the the process values up to time y and has a normal distribution with mean μt and variance tσ2.

Assumption (b) says that, for any history of the process up to the present time y, the change in the value of the process over the next t time units is a normal random with mean μt and variance tσ2. Because any future value X(t + y) is equal to the present value X(y) plus the change in value X(t + y) – X(y), the assumption implies that it is only the present value of the process, and not any past values, that determines probabilities about future values.

An important property of Brownian motion is that X(t) will, with probability 1, be a continuous function of t.

An Example in Options Pricing

Suppose that the nominal interest rate is r, and consider the following model for pricing an option to purchase a stock at a future time at a fixed price. Let the present price (in dollars) of the stock be 100 per share, and suppose we know that, after one time period, its price will be either 200 or 50 (see Figure 5.1). Suppose further that, for any y, at a cost of Cy you can purchase at time 0 the option to buy y shares of the stock at time 1 at a price of 150 per share. Thus, for instance, if you purchase this option and the stock rises to 200, then you would exercise the option at time 1 and realize a gain of 200 – 150 = 50 for each of the y options purchased. On the other hand, if the price of the stock at time 1 is 50 then the option would be worthless. In addition to the options, you may also purchase x shares of the stock at time 0 at a cost of 100x, and each share would be worth either 200 or 50 at time 1.

We will suppose that both x and y can be positive, negative, or zero. That is, you can either buy or sell both the stock and the option.

Introduction

The options we have so far considered are sometimes called “vanilla” options to distinguish them from the more exotic options, whose prevalence has increased in recent years. Generally speaking, the value of these options at the exercise time depends not only on the security's price at that time but also on the price path leading to it. In this chapter we introduce three of these exotic-type options – barrier options, Asian options, and lookback options – and show how to use Monte Carlo simulation methods efficently to determine their geometric Brownian motion risk-neutral valuations. In the final section of this chapter we present an explicit formula for the risk-neutral valuation of a “power” call option, whose payoff when exercised is the amount by which a specified power of the security's price at that time exceeds the exercise price.

Barrier Options

To define a European barrier call option with strike price K and exercise time t, a barrier value v is specified; depending on the type of barrier option, the option either becomes alive or is killed when this barrier is crossed. A down-and-in barrier option becomes alive only if the security's price goes below v before time t, whereas a down-and-out barrier option is killed if the security's price goes below v before time t. In both cases, v is a specified value that is less than the initial price s of the security.

Introduction

In this chapter we derive the celebrated Black–Scholes formula, which gives – under the assumption that the price of a security evolves according to a geometric Brownian motion – the unique no-arbitrage cost of a call option on this security. Section 7.2 gives the derivation of the no-arbitrage cost, which is a function of five variables, and Section 7.3 discusses some of the properties of this function. Section 7.4 gives the strategy that can, in theory, be used to obtain an abitrage when the cost of the security is not as specified by the formula. Section 7.5, which is more theoretical than other sections of the text, presents simplified derivations of (1) the computational form of the Black–Scholes formula and (2) the partial derivatives of the no-arbitrage cost with respect to each of its five parameters.

The Black–Scholes Formula

Consider a call option having strike price K and expiration time t. That is, the option allows one to purchase a single unit of an underlying security at time t for the price K. Suppose further that the nominal interest rate is r, compounded continuously, and also that the price of the security follows a geometric Brownian motion with drift parameter μ and volatility parameter σ. Under these assumptions, we will find the unique cost of the option that does not give rise to an arbitrage.

Limitations of Arbitrage Pricing

Although arbitrage can be a powerful tool in determining the appropriate cost of an investment, it is more the exception than the rule that it will result in a unique cost. Indeed, as the following example indicates, a unique no-arbitrage option cost will not even result in simple one-period option problems if there are more than two possible next-period security prices.

Example 9.1a Consider the call option example given in Section 5.1. Again, let the initial price of the security be 100, but now suppose that the price at time 1 can be any of the values 50, 200, and 100. That is, we now allow for the possibility that the price of the stock at time 1 is unchanged from its initial price (see Figure 9.1). As in Section 5.1, suppose that we want to price an option to purchase the stock at time 1 for the fixed price of 150.

For simplicity, let the interest rate r equal zero. The arbitrage theorem states that there will be no guaranteed win if there are nonnegative numbers p50, p100, p200 that (a) sum to 1 and (b) are such that the expected gains if one purchases either the stock or the option are zero when pi is the probability that the stock's price at time 1 is i (i = 50, 100, 200).

Introduction

In this chapter we consider some optimization problems involving onetime investments not necessarily tied to the movement of a publicly traded security. Section 11.2 introduces a deterministic optimization problem where the objective is to determine an efficient algorithm for finding the optimal investment strategy when a fixed amount of money is to be invested in integral amounts among n projects, each having its own return function. Section 11.2.1 presents a dynamic programming algorithm that can always be used to solve the preceding problem; Section 11.2.2 gives a more efficient algorithm that can be employed when all the project return functions are concave; and Section 11.2.3 analyzes the special case, known as the knapsack problem, where project investments are made by purchasing integral numbers of shares, with each project return being a linear function of the number of shares purchased. Models in which probability is a key factor are considered in Section 11.3. Section 11.3.1 is concerned with a gambling model having an unknown win probability, and Section 11.3.2 examines a sequential investment allocation model where the number of investment opportunities is a random quantity.

A Deterministic Optimization Model

Suppose that you have m dollars to invest among n projects and that investing x in project i yields a (present value) return of fi(x), i = 1, …, n. The problem is to determine the integer amounts to invest in each project so as to maximize the sum of the returns.