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This is a masters-level overview of the mathematical concepts needed to fully grasp the art of derivatives pricing, and a must-have for anyone considering a career in quantitative finance in industry or academia. Starting from the foundations of probability, this textbook allows students with limited technical background to build a solid knowledge of the most important principles. It offers a unique compromise between intuition and mathematics, even when discussing abstract ideas such as change of measure. Mathematical concepts are introduced initially using toy examples, before moving on to examples of finance cases, both in discrete and continuous time. Throughout, numerical applications and simulations illuminate the analytical results. The end-of-chapter exercises test students' understanding, with solved exercises at the end of each part to aid self-study. Additional resources are available online, including slides, code and an interactive app.
Narrow bracketers who are myopic in specific decisions would fail to consider preexisting risks in investment and neglect hedging opportunities. Growing evidence has demonstrated the relevance of narrow bracketing. We take a step further in empirical investigation and study individual heterogeneity in narrow bracketing. Specifically, we use a lab experiment in investment and hedging that elicits subjects’ preferences on rich occasions to uncover the individual degree of narrow bracketing without imposing distributional assumptions. Combining prospect theory and narrow bracketing can explain our findings: Subjects who invest more also insure more, and subjects insure significantly less in the loss domain than in the gain domain. More importantly, we show that the distribution of the individual degree of narrow bracketing is skewed at two extremes, yet with a substantial share of people in the middle who partially suffer from narrow bracketing. Neglecting this aspect, we would overestimate the severity of narrow bracketing and misinterpret its relation with individual characteristics.
The proliferating Sino-US peer competition is increasingly impacting Latin American states and triggering uncertainty. As China’s expanding influence in the region challenges longstanding US supremacy in the western hemisphere and reshapes the strategic calculus for regional states, hedging behaviour becomes increasingly opportune. This most notably includes Brazil, the largest state in Latin America both politically and economically, whose hedging behaviour oscillated between governments, a characteristic normally associated with states facing higher systemic pressures. As such, how does the Sino-US peer competition impact Brazil’s hedging strategy? And why do coping behaviours differ on various indices between different administrations, from Lula to Bolsonaro? Findings suggest that depending on whether the incumbent government was left- or right-wing, Brazil’s hedge was recalibrated as either pro- or anti-US regional supremacy.
Belief-elicitation experiments usually reward accuracy of stated beliefs in addition to payments for other decisions. But this allows risk-averse subjects to hedge with their stated beliefs against adverse outcomes of the other decisions. So can we trust the existing belief-elicitation results? And can we avoid potential hedging confounds? We propose an experimental design that theoretically eliminates hedging opportunities. Using this design, we test for the empirical relevance of hedging effects in the lab. Our results suggest that hedging confounds are not a major problem unless hedging opportunities are very prominent. If hedging opportunities are transparent, and incentives to hedge are strong, many subjects do spot hedging opportunities and respond to them. The bias can go beyond players actually hedging themselves, because some expect others to hedge and best respond to this.
Despite the growing interest in secondary state efforts to avoid choosing sides in great power competition, International Relations scholars have paid scant attention to the question of how great powers respond to secondary state ‘hedging’. We offer a first approximation for this important question by focusing on ‘high-value’ hedgers, i.e. secondary states whose location or capabilities afford them the potential to tip the scales in a great power war. We posit that great powers are likely to accommodate high-value hedgers and refrain from trying to manipulate their alignment choice. This is because the likelihood and costs of losing a high-value hedger are such that competing great powers would rather be safe than sorry. Concretely, we expect established and rising great powers to (re)assure high-value hedgers: the former by demonstrating their commitment to a regional balance of power, and the latter by showing they harbour no ill intent towards the hedging secondary state. To probe our argument, we examine how Great Britain and Germany responded to Dutch hedging in the early 20th century, and how the United States and China are responding to Singapore’s hedging today.
Is it possible to achieve almost riskless, nonfluctuating investment payoffs in the long run, at a fraction of the traditional funding requirement, using equity investments? What is their shortfall risk? These questions are motivated by the need to increase yields, while limiting the variability of investment results. We show how to use contingent claims, denominated in units of a stock index, to achieve an almost riskless investment outcome. To control the risk of the proposed hedge portfolios, we introduce an overfunded scheme and show its reliability using bootstrapping. Results show that a modest amount of overfunding is an effective risk-management approach that brings the probability of not achieving the target to less than 1 percent. Our results are based on the use of the minimal market model and a change of numeraire. Robustness tests support their validity under different market specifications.
All firms have to deal with risks and uncertainty, but international firms face additional risks and uncertainties, as analysed in this chapter. The first part of the chapter deals with largely manageable risks that international firms are familiar with, can quantify, and know how to confront, such as foreign exchange risk and spreading risks via diversification. The second part of the chapter asks how international firms should act in the face of fundamental uncertainty when the external environment in which they operate changes because of a systemic shock or global crisis. We will show that risk assessment by firms plays an important role, both in terms of the causes and the consequences of crises. We use various insights of previous chapters to understand how firms deal with uncertainties. We conclude with our view of how the rise in risk and uncertainty might affect the global economy.
Although potentially useful for financially hedging systemic weather-related risks, weather contracts/derivatives (also referred to as parametric insurance) have not seen wide adoption in agriculture outside of applications in developing countries, frequently supported by governments and non-governmental organizations (NGOs). A significant impediment is the lack of financial firms willing to stand ready to sell weather derivatives to individual agricultural producers in the over-the-counter market who, due to the localized nature of weather, face idiosyncratic weather-related risks. In particular, the administrative and reinsurance costs of supplying relatively small contracts with specific terms to many different producers are often prohibitive. The current study considers the potential use of weather derivatives in hedging the aggregate yield/revenues of viticulture producers represented by an industry association located in the province of Ontario, Canada. We examine the sensitivity of aggregate industry yields to several relevant weather-related risks employing copula function analysis. We then consider the potential of a weather derivative in hedging the financial risk associated with cold winter temperatures, which pose the greatest risk to aggregate vinifera yields. The issue of attributing costs and payouts to individual association members remains unresolved, and several alternatives are suggested.
The strategic rivalry between the United States and China has heightened since COVID-19. Secondary states face increasing difficulties maintaining a 'hedging' strategy between the United States and China. This Element introduces a preference-for-change model to explain the policy variations of states during the order transition. It suggests that policymakers will perceive a potential change in the international order through a cost–benefit prism. The interplays between the perceived costs and the perception of benefits from the order transition will shape states' policy choices among four strategic options: (1) hedging to bet on uncertainties; (2) bandwagoning with rising powers to support changes; (3) balancing against rising powers to resist changes; and (4) buck-passing to ignore changes. Four case studies (Australia, New Zealand, Singapore, and Thailand) are conducted to explore the policy choices of regional powers during the international order transition. This title is also available as Open Access on Cambridge Core.
Chapter 4. In a fiat standard, the money is not useful for any non-monetary purpose, or redeemable for any commodity with a non-monetary use. Fiat monies historically emerged not from market forces but from default on gold-redeemable central-bank or Treasury liabilities. The quantity and purchasing power of fiat money obeys the logic of supply and demand in the special form of the Quantity Theory of Money. Central banks control the growth rate of the quantity of money and thereby the rate of inflation. In principle fiat standards could produce lower inflation, but in practice have produced higher inflation than silver or gold standards. Higher inflation imposes real burdens on the public. These burdens, especially when we include the expenditure of labor and capital to produce hedges against inflation, has exceeded the resource burden of a gold standard.
Frederick, Levis, Malliaris & Meyer (2018) report a package of laboratory studies where participants underestimate the value of “hedges”: Risky bets which cancel out the risk of another presently-held bet. However, it might be questioned to what extent laboratory findings predict field behavior. People might better understand hedges when more money is at stake, or when they have more time to reflect. We discuss three gamblers who, instead of hedging, used a costly “cash-out” option to eliminate the risk of their bets on Leicester FC’s improbable victory in the 2015/2016 English Premier League soccer season. The decision to cash-out rather than to hedge led to individual losses of up to £8,000, and did not seem plausibly explained by rational economic factors. High-stakes hedges are misunderstood too.
The chapter focuses on the pragmatic valence of the parenthetical clause ut mihi (quidem) uidetur, a formulation which, by underscoring the subjective value of a statement, often conveys important interactional functions. A close survey of the use of this expression in Latin authors from Cicero to Augustinus highlights the essentially redressive value of the formula, which mainly acts as a mitigating hedge in stance taking, aimed at avoiding the negative effects of potential face-threatening acts and at managing self-presentation. Special attention is paid to Cicero’s usage of the formula as a refined conversational marker, through which the speaker showing modesty and awareness of others’ value reveals his superior moral and social standing. Hence, the chapter explores the role of speaker-oriented strategies of politeness in Rome, as part of an interactional style aimed at indexing a precise social identity.
This paper proposes a shift in the valuation and production of long-term annuities, away from the classical risk-neutral methodology towards a methodology using the real-world probability measure. The proposed production method is applied to three examples of annuity products, one having annual payments linked to a mortality index and the savings account and the others having annual payments linked to a mortality index and an equity index with a guarantee that is linked to the same mortality index and the savings account. Out-of-sample hedge simulations demonstrate the effectiveness of the proposed less-expensive production method. In contrast to classical risk-neutral production, which revolves around the savings account as reference unit, the long-term best-performing portfolio, the numéraire portfolio of the equity market, is employed as the fundamental reference unit in the production of the annuity. The numéraire portfolio is the strictly positive, tradable portfolio that when used as denominator or benchmark makes all benchmarked non-negative portfolios supermartingales. Under real-world valuation, the initial benchmarked value of a benchmarked contingent claim equals its real-world conditional expectation. The proposed real-world valuation and production can lead to significantly lower values of long-term annuities and their less-expensive production than suggested by the risk-neutral approach.
Here we study a fairly general jump–diffusion price process. We investigate the existence of equivalent martingale meaures, derive the Hansen–Jagannathan bounds, and extend the theory to include dividends. Completeness questions are discussed in some detail, and we also develop the theory for change of numeraire.
This chapter contains an introduction to financial economics, giving the reader the necessary background for the rest of the text. It coversportfolio theory, arbitrage theory, martingale measures, change of numeraire, stochastic discount factors, Hansen–Jagannathan bounds, dividends and consumption.
We present an introductory example featuring a stock price process driven by a Poisson process. This is analyzed in some detail using both the classical Black–Scholes and the modern martingale technique.
This paper revisits and fine-tunes a spin-off from Knight's (1921, Risk, Uncertainty and Profit) influential distinction between risk (‘probability of unknown events’) and uncertainty (‘unquantifiable randomness’): the contrast between actuarial institutions and entrepreneurship. It contends that this opposition is not exclusive and argues that the act of insurance does not automatically reduce entrepreneurial profits. To maintain this claim, the paper emphasizes hedging and, more specifically, draws on an innovative actuarial scheme – parametric (or index-based) insurance – which, unlike indemnity-based insurance, does not rely on a damage assessment but indemnifies the policyholder according to the variation of an index. This argument sheds new light on the function habitually assigned to actuarial institutions, amends the theory of entrepreneurial profits, and integrates hedging within entrepreneurial judgment.
This article examines evidentiality in the frame of inferential adverbs in written interaction from the perspective of Finnish, a language that does not have evidentiality as a grammatical category. The analysis focuses on six adverbs, such as käsittääkseni ‘as far as I understand’ and tietääkseni ‘to my knowledge, as far as I know’. Evidentiality and epistemic modality intertwine in their semantics, as these adverbs represent a writer’s access to information, but also indicate her evaluation of its reliability. First, this article offers a description of the interactional functions of these adverbs such as marking a writer’s opinion in contrasts, expressing slight hedging in order to anticipate corrections, to allow space for other opinions, or to create irony. Second, in the framework of cognitive grammar, the analysis focuses on the meaning of the evidential verb+kseni construction and the effect of different verb stems on it. These adverbs share similar functions in texts, which is due to their flexible constructional meaning. While varying from lexeme to lexeme, specific evidential and epistemic dimensions can either be foregrounded and relevant in a situation or remain backgrounded and not activated.
Commodity exchanges have proliferated greatly since 2000, both in terms of places where the trade is conducted, and in terms of the products that are subject to trade. The main aim of this chapter is to describe these commodity exchanges, as well as the commodities traded there. The main instruments used, including their functioning, is presented as well as the main actors and their objectives. The chapter ends with discussing commodity exchanges impact on price formation, including the role of speculation.