# an introduction to quantitative finance stephen blyth pdf free

The other counterparty agrees to sell the asset, and is short the forward contract. The specified price K is known as the delivery price. The specified time T is known as the maturity. A forward contract is easily defined, yet often causes confusion when first encountered. In particular, it is important to understand the distinction between the terms inherent in the contract the underlying asset, K and T , which are fixed when the contract is agreed upon, and the value of the contract, which will vary over time.

Similarly, the payout at maturity from a short forward contract is K — ST. However, we can at time t enter into a forward contract with any delivery price K, provided we pay VK t, T to do so. However, we can establish now the import- ant distinction between F t, T and VK t, T with the following simple example. Suppose a stock which pays no dividends always has price and interest rates are always zero. Ratios of this form play an important role in Chapter 9.

We prove this result two ways, introducing important concepts that occur throughout quantitative finance. The first method we term the replication proof , which we saw briefly in Chapter 1.

We here define arbitrage to be a situation where we start with an empty portfolio and simply by execut- ing market transactions end up for sure with a portfolio of positive value at time T. A no-arbitrage proof is based on the assumption that there exist no such situations.

In Chapter 6 we more formally define the concept of arbitrage, and show that these two methods of proof are equivalent. In particular, we show that the assumption of no-arbitrage, appropriately defined, allows us to adopt formally proof by replication. Proof I Replication. At current time t, we let portfolio A consist of one unit of stock and portfolio B consist of long one forward contract with delivery price K, plus Ke—r T—t of cash which we deposit at the interest rate r.

At time T portfolio A has value ST. In portfolio B at T we have an amount K of cash, which we use to buy the stock at T via the forward contract. Therefore, portfolio B also has value ST at T. Since the value of these portfolios at time T is the same, and we have neither added nor taken away any assets of non-zero value, their value at t is the same.

At current time t we execute three transactions. We borrow St cash at interest rate r for time T — t, and with the cash we buy the stock at its current market price St. At time T we must sell the stock for F t, T under the terms of the forward contract, and we pay back the loan amount of St er T—t. At time t we can go long one forward contract with delivery price F t, T for no cost, sell the stock for St , and deposit the proceeds at r for time T — t.

You may ask what happens if we do not have the stock to sell. We address this in our review of assumptions. In all other cases we can construct an arbitrage portfolio. Proof II is our first encounter with such an argument and it is important to get comfortable with its construction.

Note that the forward price depends only on the current stock price St , the interest rate r and the time to maturity T — t. It often surprises those encountering forwards for the first time that the determination of the forward price does not depend on the growth rate or standard deviation of the stock, or indeed any distributional assumptions about ST.

The forward price says nothing further about predicting where the stock will be at time T than the spot price does. Furthermore, any two assets which pay no income and which have the same spot price St will have the same forward price regardless of any views about their future movements. To understand this heuristically, suppose one has St of cash. One can either buy the stock today, or invest the cash at rate r and agree today to buy the stock forward at time T.

The invested cash grows to St er T—t ; thus the forward price has to equal this for one to be indif- ferent between the two strategies, which both result simply in a holding of one stock at T. Alternatively, consider the differences and similarities between a going long a forward contract at its forward price and b starting with no cash, borrowing cash to buy the stock, selling the stock at time T and repaying the loan at T.

You should convince yourself that both portfolios have value zero at time t and value ST — St er T—t at time T. Trading a forward contract, however, does allow one to gain exposure to movements in the stock price in a capital-efficient way, that is, without initially having to pay out or bor- row the purchase price at time t. Forwards allow institutions to establish larger exposures for a given amount of initial capital.

Indeed, much of the current regulatory reform is con- cerned with appropriate capital requirements for institutions entering into forwards and other similar derivatives. Note Often the stock is itself used as collateral for the loan taken out to purchase it. Let portfolio A consist of one unit of the asset and —I cash. Note that a negative amount of cash can be thought of simply as a debt or overdraft.

Let portfolio B consist of long one forward contract with delivery price K, plus Ke—r T—t cash. Portfolio B has again value ST by the same argument as before. We assume the equality does not hold, and show that in this case we can create a portfolio with positive value from an initial portfolio of zero value. We go short one forward contract, borrow St of cash and buy one stock, which provides income with value Ier T—t at T. The extent to which these assumptions are valid in practice is a key determining factor of the nature of financial markets.

However, for large market participants, much borrowing and lending nets and is effectively done close to one rate r. This usually holds reasonably well for foreign exchange and interest rate swap markets. We also assume we can buy or sell assets with negligible transaction costs. In some markets—for example, US treasury bonds, bond futures and foreign exchange—bid and offer prices are very close together, and transaction costs are indeed low.

This often represents the largest conceptual hurdle. For ease of understand- ing, we can simply assume we always have the asset we want to sell, or that there is another market participant who holds the asset and can sell it, and for whom the same no-arbitrage arguments apply. In practice, we can usually sell a stock short, without owning it. To do so, we borrow the stock from someone who owns it for example, an asset manager or broker ; sell it; buy it back at later time; and then return the stock to the asset manager.

The owner of the stock keeps the rights to any dividends during the life of the short sale. In reality this may or may not be the case depending on the ability to obtain a stock to borrow, and the regulations surrounding short selling.

In other words, arbitrages cannot persist in the market and arbitrageurs are always present. Many of these assumptions can be questioned and challenged. Indeed, several can be violated to some degree even in normal market conditions. A noteworthy feature of the market turmoil in was gross violations of many of them, viz: 1 Money markets froze as financial market participants were gripped by fear of coun- terparty risk and bankruptcy.

It became very hard to borrow money on an unsecured basis as no one wanted to lend in an environment where major institutions could go bankrupt. Published levels for interest rates bore little relation to the actual rates banks were charging for loans. Question 4 in the exercises explores this.

The sizes one could execute became small, and the effect of any trade executed became outsized. Transaction costs increased as liquidity decreased and as market makers found it far harder to get out of any positions. A general reduction in risk appetite led to wholesale liquid- ation of positions which themselves caused market prices to move dramatically and discontinuously.

In cer- tain jurisdictions, short selling was banned for particular assets or for particular time periods. In addition, the ability to find stock to borrow, given heightened concern about counterparty bankruptcy, was significantly reduced. We encounter some such examples in Chapters 6 and We now show this result holds in general for all assets. At time t we go long a forward contract with delivery price F t, T at no cost , and go short a forward contract with delivery price K.

For the latter we receive VK t, T at t, which is invested at rate r. We go long a forward contract with delivery price K, paying VK t, T at t to do so, and short a forward contract with delivery price F t, T for no cost. Proof Let portfolio A be e—q T—t units of the stock, with dividends all being reinvested in the stock, and let portfolio B again be one forward contract with delivery price K plus Ke—r T—t of cash.

At time T, both portfolios have value ST. There is an advantage to buying the asset spot versus buying it forward, since in the latter case we do not receive any dividends paid before T. The forward price is lower in order to compensate. The forward contract on a foreign currency is similar. Result Suppose Xt is the price at time t in dollars of one unit of foreign currency. Proof I Simply replace q by rf in the result for a stock paying dividends. The foreign cur- rency is analogous to an asset paying a known dividend yield, the foreign interest rate.

See Question 3 in the exercises. However, some forwards are cash settled, meaning one simply receives pays if negative the amount ST — K at T. We encounter both cash and physically settled contracts later in the book. For example, the derivative contract called an interest rate swaption Chapter 12 can upon execution of the contract be specified as physical or cash settlement.

Cash settled forwards are also sometimes known as contracts for difference. However, a cash settled forward has no further exposure to the asset price, whilst a physically settled contract— where one owns the asset at T—continues to have exposure to asset price movements. Thus cash and physical settlement are different in their risk exposure after T. Cash-settled forwards have many similarities with spread bets, and indeed many spread betting companies are populated by former derivative traders.

A key difference is that ST is not the price of a security or tradable asset and so the replication and no-arbitrage arguments for forward pricing do not apply. The price of the spread bet must come from supply and demand, these presumably being generated by some view of how good the Red Sox are.

They are employed extensively by a range of institutions to manage foreign exchange exposure of future cashflows. However, futures contracts—a variant of forwards—on bonds and stock indices are highly liquid with hundreds of thousands of contracts traded each day. We detail futures contracts in Chapter 5. The largest derivative market by far is the interest rate deriv- ative market.

Table 2. Forwards in presence of bid-offer spreads Let St be the current price of a stock that pays no dividends. Both rates are continu- ously compounded. Z t, T What arbitrage is available to you, assuming you can only trade the stock, ZCB and forward contract? Be precise about the transactions you should execute to exploit the arbitrage. FX forwards during the financial crisis FX forwards are among the most liquid derivative contracts in the world and often reveal more about the health of money markets markets for borrowing or lending cash than published short-term interest rates themselves.

Assuming six-month euro interest rates were 5. There are days between 3 October and 3 April What arbitrage oppor- tunity existed? What does a potential arbitrageur need to transact to exploit this opportunity? Explain briefly how these actions may have created the arbitrage opportunity in b , which existed for several months in late For an asset that pays no income, substitute the expression for its forward price into the above equation and give an intuitive explanation for the resulting expression.

How much money have you made or lost? This is sometimes called the carry of the trade. How does your answer change if the asset pays dividends at constant rate q? Dollar-yen and the carry trade A major currency pair is dollar-yen, quoted in yen per dollar. Suppose also that the five-year dollar interest rate is 2. For simplicity use a 0. How could you synthetically go long the forward contract in a?

What is your profit or loss on the forward? Lengthy calculations are not required. What does this imply in practice? Note Does the lack of symmetry in results c and d trouble you? State any assumptions you are making. Consider briefly the cases where: i you own the house and live in it ii you own the house and rent it out iii you own a house in Boston and are prepared to move iv you own no real estate.

At the height of the financial crisis, the difference between the theoretical upper bound and the actual forward price increased significantly. Discuss briefly why this might have occurred. This construct leads naturally to the concept of forward rates. Consider the forward price of this forward contract, denoted F t, T1 , T2 , the delivery price such that the forward contract has zero value at time t.

Equivalently, the forward price is where one can for no upfront cost agree at t to buy at T1 the ZCB with maturity T2. Portfolio A is worth 1 at time T2 by definition. One can use this to buy a ZCB with maturity T2 via the forward contract, and thus portfolio B is also worth 1 at time T2. In this section we use the simplified notation f12 for the forward rate, and we suppose that r1 and r2 are the current zero rates for terms T1 and T2 respectively.

A graphical representation is the best way to picture forward rates. As shown in Figure 3. Alternatively, we can agree at t to lend until T2 at rate r2. A replication argument concludes that the amount accrued at T2 under the two altern- atives must be the same. In general we have to incorporate the current time, the start date of the interest period, the end date of the interest period, plus the compounding frequency of the rate.

Our notation r1 , r2 and f12 suppresses the current time and the compounding frequency, and soon lacks the capacity to distinguish between rates. We introduce more comprehensive notation when we define swaps in Chapter 4. The vast majority of interest rate derivatives have pay- outs that are functions of libor rates, short-dated interest rates of fixed term. All interest is paid at the maturity or term of the deposit, and there is no interim compounding. Most interest rate derivatives typically reference three-month or six-month libor.

The aftermath of the financial crisis of —, however, shone a spot- light on libor and its setting mechanism. Each libor fix is calculated by polling a panel of banks for their rates, discarding the highest and lowest quartiles and averaging the remainder. Although in the past, observable interbank certificate of deposit rates were largely in line with libor submissions, the rates bank submit do not necessarily have to reflect actual transactions. This framework led to two potential abuses.

First, in times of stress, banks could submit artificially low levels so as not to alarm markets by revealing their funding difficulty. Secondly, swap traders, whose underlying derivative contracts we see in Chapter 4 depend heavily on libor levels, could attempt to influence, to their benefit, rate submissions from their own bank.

Both phenomena were components of the libor scandal that broke in In Chapters 4 and 12 we see that libor is integral to the majority of interest rate derivatives.

The only thing related to libor we can trade is the deposit. Note the analogue with the forward price F t, T for a forward contract on an asset.

We repres- ent this expression graphically in Figure 3. This is intuitively clear, since one can invest the unit of cash in a libor deposit, receiving the libor interest payment. This is a key characteristic of forward contracts. Valuing a fixed cashflow is easy. We now have all the machinery required to value interest rate swaps. Give brief reasoning. Table 3. What is the one-year forward one-year rate that is, f11? Assume all rates are annually com- pounded.

Note This is a favourite question for interviewers. Does the borrowing bank need to buy or sell the FRA to do this? What is the fixed rate that the bank locks in? Explain your answers. Floating rate bond Let T0 , T1 ,. The first swap was traded in and explosive growth followed in the s and s.

However, as mentioned in Chapter 1, the values of interest rate derivatives are typically small fractions of the notional. Swaps allow institutions to manage their exposure to interest rate movements or to adjust the nature of their interest rate liabilities.

An excellent and comprehensive summary of interest rate swaps and related derivatives is given by my former colleague at Morgan Stanley, Howard Corb Corb, Note The vast majority of interest rate swaps are over-the-counter contracts. However, the Dodd-Frank Wall Street Reform and Consumer Protection Act, signed into law in , requires—among many other things—that standard interest rate swaps should be cleared through an exchange, once appropriate rules have been established.

Cashflows are calculated on a notional amount, which again we will typically take to be 1. The payment frequency for the floating and fixed legs may differ, but we will assume here they are same.

Question 4 in the exercises shows that the frequency of the floating leg is irrelevant for theoretical valuation. A swap is shown graphically in Figure 4. Our notation for the value of the swap leg for simplicity suppresses information about the dates of the underlying swap. Since the floating leg is a series of regular libor payments, we can value it by replacing each libor with its forward.

Therefore, we have the intuitively appealing result that the value of receiving libor pay- ments on, say, one dollar is equal to the value of receiving one dollar at the beginning of the stream and giving it back at the end.

We can simply take the dollar and repeatedly invest in a series of libor deposits. The forward swap rate at time t for a swap from T0 to Tn is defined to be the value yt [T0 , Tn ] of the fixed rate K such that the value of the swap at t is zero.

Both formulations are useful later, in Chapters 12 and Proof A complete proof is left to the exercises, using the fact that the value at t of the float- ing leg of the swap is by definition equal to the value of a fixed leg with fixed rate yt [T0 , Tn ].

Given par swap rates for all Ti , we can recover Z T0 , Ti for all i, a process known as boot- strapping. An example of this is given in the exercises. Question 2 in the exercises explores this distinction. Let the price of the fixed rate bond at current time t be denoted BFXDc t. We are suppressing information about the maturity of the bond in our price notation. Consider a swap with notional 1 where we pay a fixed rate K and receive libor.

It follows that the value of a swap is bounded, and we prove this in Question 1 b. You should think carefully about the distinction between a fixed rate bond, whose cashflows are known but whose value changes, and a floating rate bond, whose cashflows are unknown but whose value at coupon payment dates is always 1. An exercise provides some more insight into fixed rate bonds. Hint Replace the floating leg of the swap with a fixed leg of equal value, or con- struct a portfolio of two swaps of the same maturity, but with different fixed rates.

Explain the key difference between the value of a swap and a stock forward contract. Bootstrapping and IRR discounting Suppose the current one-year euro swap rate y0 [0, 1] is 1. Calculate the one-year, two-year and three-year zero rates, and compare them to the swap rates.

This is often called IRR internal rate of return discounting. Calculate the error in valuing the annuity in a this way. In yet another Wall Street quirk, euro swaps embedded in certain contracts are occasionally valued for cash settlement using IRR discounting at the current swap rate y0 [0, Tn ], rather than correct valuation using ZCBs.

The logic for this originally was to reduce disagree- ments between banks on cash settlement of swaps. The swap rate is easily observed and IRR discounting is then a deterministic formula, whilst the bootstrapping undertaken in a was deemed too complicated. Suppose all euro swaps suddenly moved to this type of valuation. Given your answers to Question 2 a to c , com- ment briefly on whether you would expect euro swap rates to rise, fall or stay the same.

Carry and rolldown Suppose euro swap rates are as given in Question 2. A hedge fund HF executes the following two trades with a dealer. Assume bid-offer costs are negligible. What is the current value of the remaining part of the HF trade? Discuss briefly the risks of the trade, in particular commenting on which interest rates the HF is exposed to. Swap frequency We have assumed that payment dates for the fixed and floating legs of a swap contract are the same.

Deduce directly whether the annual swap rate for T0 to Tn is higher or lower than the quarterly swap rate for the same period. Assume the same daycount convention. Hint Recall the definition of the swap rate as the fixed rate that gives the swap zero value. However, there are two key differences. Most importantly, a futures contract involves cashflows each day up until the maturity date T and not just at T. Secondly, virtually all futures contracts are traded on exchanges rather than as bilateral over-the-counter contracts between two counterparties.

Futures contracts on major government bonds and equity indices are very liquid, with hundreds of thousands of contracts trading each day. The difference between a future and a forward lies in the cashflows during the life of a contract. In particular, the holder of a futures contract receives or pays changes in the futures price over the life of the contract, and not just at maturity. We below make precise the differences between the cashflows of the two contracts.

Forward contract. At time t, we can go long a forward contract with delivery price F t, T at no cost. There are no payments in between. Futures contract. This amount is also known as the variation mar- gin. There have been many recent mergers amongst exchanges. Futures have standardized monthly or quarterly maturity dates set by established market conventions. Contracts are netted, meaning that if we go long one contract then short one contract then we have no position.

Each market participant deposits initial margin at the exchange, and receives or posts additional variation margin as prices move up or down. It is important to note that margin—and in particular variation margin—accrues interest. This key feature gives rise to the difference between futures and forward contracts which we explore further in the next section.

Note This is not a useful result for interest rate futures! Let the constant continuously compounded interest rate be r, and consider the following trading strategy. At time n — 1 , increase position to 1 contract. With this strategy we receive the following amounts.

This we can invest at rate r, and thus it compounds up by er n—1 to time n. However, consider the portfolio consisting of e—rT F 0, T cash invested at r, plus one forward contract. This is also worth ST at time T. Suppose that the asset price ST is positively correlated with the interest rate and thus tends to increase as interest rates increase.

If we are long a futures contract, then we receive mark-to-market gains earlier than the forward in environments when interest rates are high, and thus the gains can be invested at a higher rate. Similarly, losses from the long position have to be paid early when rates are low and thus the future value of the losses is lower. Thus in this case, we would prefer to hold a long futures position relative to a long forward position.

A more detailed analysis of the futures convexity correction is given in Hunt and Kennedy When the interest rate is random, then the money market account is also random.

We state here the following general result, which makes precise our earlier heuristic argument. Its proof is beyond our scope. The covariance between the asset price and the money market account is proportional to T — t , so typically the convexity correction is greater for futures contracts with longer maturities. The convexity correction tends to zero as t approaches T.

The most liquid futures contracts in the world are futures on US treasuries the bond future and German government bonds the bund future. Here, the bond price and interest rate are almost always negatively correlated. Can you think of a situation when this may not hold? However, since most liquid bond futures have maturities of three months or less, the size of this convexity correction is often small.

Note To create more confusion, Euribor Euro Interbank Offered Rate is a subtly differ- ent interest rate to euro libor. The mechanism for setting euro libor is similar to that for US dollar libor or sterling libor: the rate is set at 11am in London, based on a poll of rates from 15 international contributor banks active in the London interbank market.

Euribor is a complementary rate established by the European Banking Federation, set at 11am in Brussels, one hour earlier than libor. A broader panel of 43 mostly European banks is used. Small differences may occur between Euribor and euro libor, due to the different credit quality of the banks in the two panels.

Most European interest rate derivatives—and in particular Euribor futures—reference Euribor not euro libor. For example, if three-month dollar libor on the maturity date of the futures contract fixes at 1.

Eurodollar futures trade with quarterly maturities out to ten years, maturing on IMM dates see Chapter 1 , the third Wednesdays of March, June, September and December, plus monthly maturities for the first four intermediate months.

That is, if we are long one future and its price moves from Having precise knowledge of the exact contract specification for each futures contract is essential, and many trading mishaps have resulted from incomplete knowledge of the characteristics of a contract. The holder of a long position in a Eurodollar futures contract receives positive variation margin when forward interest rates go down, and the futures price goes up. Forward interest rates and the money market account are usually positively correlated.

Therefore, a long Eurodollar position receives money early when interest rates are low, and pays out money early when rates are high. Thus, the Eurodollar futures price will typically be lower than the forward price. Note As of July , the December futures contract then currently the fourth outstanding December contract has price The forward libor for that date is 1. Which is the better predictor of where libor will be in December ?

Note Eurodollar and other futures contracts on short-dated interest rates are often referred to by a colour code. The first four quarterly contracts are called the whites or fronts , followed by the reds the second four quarterly contracts , blues, greens, golds, purples, oranges, pinks, silvers and coppers.

Contracts beyond the golds are typically less liquid and thus the colour coding beyond the golds is used infrequently. Note Early in my finance career, I tried to persuade my boss that the Eurodollar convexity correction should be zero.

I had a proof that the forward price equals the futures price, I claimed, being eager to show off my mathematical acumen. It was humbling to have it pointed out that the proof depended on interest rates being constant; hardly a valid assumption for Eurodollar futures, which move continually.

We explore the concept of vega further in Chapter Eurodollar futures versus FRAs a Suppose in September we observe that the Eurodollar futures prices for the March , June and September contracts are all Consider FRAs with these three maturities. Which FRA is likely to have the lowest rate? If you are told that the futures price will not move over its life, what trade would you do?

How about if you were told the FRA rate would not move? Futures on bonds and stocks a Is the futures price of a fixed rate bond likely to be higher, lower or the same as its forward price?

These arguments have intuitive appeal, and become a powerful component of the pricing machinery for more complicated derivative contracts such as options. Whilst the definition of arbitrage and arbitrage portfolios in continuous time in particular when infinitely many trades can be made is non-trivial, the key ideas behind no- arbitrage can be well established in the discrete setting, and we proceed accordingly here.

As mentioned, finance is in practice executed in discrete time and value. The assumption of no-arbitrage—that is, there exist no arbitrage portfolios—underpins quantitative finance. Therefore, the no-arbitrage assumption allows pricing of forwards with no assumptions about the distribution of the stock price ST. Assume no-arbitrage. This assumes we can go short—hold negative amounts of assets—at will. We have provided a formal argument for an almost tautological result.

We now can immediately prove an important corollary. Corollary to monotonicity theorem. This corollary is the formal statement of the replication argument from Chapter 2. We refer to the monotonicity theorem and its corollary repeatedly later in the book. Example Let us revisit the FRA, where we considered two portfolios. This strategy can be carried out at zero cost, since the deposit is at the current market libor rate, without any asset of non-zero value being added to or subtracted from the portfolio.

The trading strategy that showed the futures price equalled the forward price was also self- financing, since it involved trading futures at the then current futures price.

Going long a forward at its forward price, paying fixed on a swap at the forward swap rate or borrow- ing money at the current interest rate are all self-financing trades. We formally need to include the self-financing condition in our definition of an arbitrage portfolio. The latter is not self-financing. Assuming no-arbitrage, we are able, for example, to order derivative prices via the mono- tonicity theorem based on knowledge of their payouts at maturity. The edifice of quant- itative finance was constructed upon this assumption and we will adopt it throughout the remainder of the book as we build our theory and results.

However, financial markets in practice have a tendency to challenge foundational assumptions, and key challenges for the field have often arisen from occasions when no- arbitrage arguments were violated. As mentioned in Chapter 2, the period of the financial crisis following the Lehman Brothers bankruptcy in particular contained several examples of extreme price movements and violations of arbitrage bounds that stunned seasoned practitioners.

We examine here one stark and relatively straightforward instance from — A further example is detailed in Chapter Example Consider again the example of two bonds with the same maturity date. Figure 6. We can see these bonds generally traded with a very small yield differential until late The yield differential nor- malized following government policy responses in early Further details of this episode and possible explanations for the observed prices are given in Taliaferro and Blyth Note that the data are yields of coupon bonds, but these can easily be restated in terms of ZCB prices.

Note I like to draw an analogy here with pure mathematics. Underpinning set theory is an assumption about choosing elements from an uncountably infinite number of sets called the axiom of choice.

Without the axiom of choice, set theory becomes far more complex and messy. In finance, the assumption of no-arbitrage plays an analogous role, allowing the establishment of bounds on and ordering of derivative prices. Without the assump- tion, quantitative finance becomes far messier, and it becomes hard to order derivative prices. Further discussion is given in Blyth Arbitrage portfolios Which of the following necessarily imply a violation of the no-arbitrage assumption?

The execution of this right is termed the exercise of the option. The price of a derivative is closely linked to the expected value of its pay-out, and suitably scaled derivative prices are martingales, fundamentally important objects in probability theory.

The prerequisite for mastering the material is an introductory undergraduate course in probability. The book is otherwise self-contained and in particular requires no additional preparation or exposure to finance. It is suitable for a one-semester course, quickly exposing readers to powerful theory and substantive problems.

The book may also appeal to students who have enjoyed probability and have a desire to see how it can be applied. Signposts are given throughout the text to more advanced topics and to different approaches for those looking to take the subject further. ISBN Send-to-Kindle or Email Please login to your account first Need help?