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13
Towards a Theory of Volatility Trading
 Reprinted in Option Pricing, Interest Rates, and Risk Management, Musiella, Jouini, Cvitanic
, 1998
"... Introduction ffl Three methods have evolved for trading vol: 1. static positions in options eg. straddles 2. deltahedged option positions 3. volatility swaps ffl The purpose of this talk is to explore the advantages and disadvantages of each approach. ffl I'll show how the first two methods can ..."
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Cited by 60 (12 self)
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Introduction ffl Three methods have evolved for trading vol: 1. static positions in options eg. straddles 2. deltahedged option positions 3. volatility swaps ffl The purpose of this talk is to explore the advantages and disadvantages of each approach. ffl I'll show how the first two methods can be combined to create the third. ffl I'll also show the link between some "exotic" volatility swaps and some recent work by Dupire[3] and Derman, Kani, and Kamal[2]. Part I Static Positions in Options Trading Vol via Static Positions in Options ffl The classic position for trading vol is an atthemoney straddle. ffl Unfortunately, the position loses sensitivity to vol as the underlying moves away from the strike. ffl Is there a static options position which maintains its sensitivity to vol as the underlying moves? ffl To answer this q
Spanning and DerivativeSecurity Valuation
, 1999
"... This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the char ..."
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Cited by 57 (5 self)
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This paper proposes a methodology for the valuation of contingent securities. In particular, it establishes how the characteristic function (of the future uncertainty) is basis augmenting and spans the payoff universe of most, if not all, derivative assets. In one specific application, from the characteristic function of the stateprice density, it is possible to analytically price options on any arbitrary transformation of the underlying uncertainty. By differentiating (or translating) the characteristic function, limitless pricing and/or spanning opportunities can be designed. As made lucid via example contingent claims, by exploiting the unifying spanning concept, the valuation approach affords substantial analytical tractability. The strength and versatility of the methodology is inherent when valuing (1) Averageinterest options; (2) Correlation options; and (3) Discretelymonitored knockout options. For each optionlike security, the characteristic function is strikingly simple (although the corresponding density is unmanageable/indeterminate). This article provides the economic foundations for valuing derivative securities.
The Second Fundamental Theorem of Asset Pricing

, 1998
"... This paper presents a resolution of the paradox proposed by the example of an economy with complete markets and a multiplicity of martingale measures constructed by Artzner and Heath (1995). The resolution lies in noting that completeness is with respect to a topology on the space of cash flows and ..."
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Cited by 13 (4 self)
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This paper presents a resolution of the paradox proposed by the example of an economy with complete markets and a multiplicity of martingale measures constructed by Artzner and Heath (1995). The resolution lies in noting that completeness is with respect to a topology on the space of cash flows and is connected with uniqueness of the price functional in the topological dual space. Uniqueness may be lost outside the dual and this is what occurs in the counterexample of Artzner and Heath.
Breaking Barriers: Static Hedging of Barrier Securities
, 1996
"... this paper, we focus on single barrier securities for simplicity, leaving multiple barrier options for future research. Single barrier securities allow for an arbitrary payoff at maturity provided that the barrier has been touched (inbarrier securities) or not touched (outbarrier securities). They ..."
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Cited by 3 (2 self)
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this paper, we focus on single barrier securities for simplicity, leaving multiple barrier options for future research. Single barrier securities allow for an arbitrary payoff at maturity provided that the barrier has been touched (inbarrier securities) or not touched (outbarrier securities). They are usually further classified into "down securities" (barrier below spot) and "up securities" (barrier above spot). The standard methodology for hedging and valuing barrier securities applies dynamic replication strategies in the underlying assets. In this paper, we hope to add insight into these structures by looking at them in a nontraditional manner. In particular, we show how barrier securities can be broken up into more fundamental securities, which in turn can be created out of vanilla European options. This allows us to hedge pathdependent barrier securities with pathindependent vanilla options, with trading in the latter options occurring only at the initiation and the expiration
Who Buys and Who Sells Options: The Role and Pricing of Options in an Economy with Background Risk
 JOURNAL OF ECONOMIC THEORY
, 1997
"... In this paper, we derive an equilibrium in which some investors buy call#put options on the market portfolio while others sell them. Since investors are assumed to have similar riskaverse preferences, the demand for these contracts is not explained by di#erences in the shape of utility functions. R ..."
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Cited by 2 (0 self)
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In this paper, we derive an equilibrium in which some investors buy call#put options on the market portfolio while others sell them. Since investors are assumed to have similar riskaverse preferences, the demand for these contracts is not explained by di#erences in the shape of utility functions. Rather, it is the degree to which agents face other, nonhedgeable, background risks that determines their risktaking behavior in the model. We show that investors with lowornobackground risk have a concave sharing rule, i.e., they sell options on the market portfolio, whereas investors with high background risk have a convex sharing rule and buy these options. A general increase in the background risk in the economy reduces the forward price of the market portfolio. Furthermore, the prices of put options rise and the prices of call options fall. 1 INTRODUCTION The spectacular growth in the use of derivatives to manage risks has been one of the most signi#cant recent developments in the #n...
Currency Covariance Contracting
, 1999
"... We show how contracts paying the realized covariance between two currencies can be constructed by combining static positions in a continuum of options with continuous trading in underlying futures or forward contracts. The construction is general in that the volatilities and correlations are arbitra ..."
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Cited by 1 (1 self)
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We show how contracts paying the realized covariance between two currencies can be constructed by combining static positions in a continuum of options with continuous trading in underlying futures or forward contracts. The construction is general in that the volatilities and correlations are arbitrary. We thank Mark Broadie, Tony Corso, Jose Lopez, and Nedia Miller for comments. Any errors are our own. I Introduction Volatility swaps have recently emerged on several overthecounter markets (see [5] for example). These contracts pay the difference between the realized volatility over a specified time interval and a constant 1 agreed upon at the outset of the contract. The motivation for contracts whose payoffs are tied to volatility has been discussed by several authors. For example, Gastineau[13] and Galai[11] propose the development of option indices which can be used as the underlying for derivative contracts. Brenner and Galai[2] propose the development of realized volatility in...
An Alternative Approach for Valuing Continuous Cash Flows
, 2000
"... We consider the problem of replicating the payoffs from variable annuities with a continuous cash flow given by a function of some traded asset's price. The standard approaches involve either dynamic trading in this underlying asset or a static position in a continuum of options of all strikes and m ..."
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We consider the problem of replicating the payoffs from variable annuities with a continuous cash flow given by a function of some traded asset's price. The standard approaches involve either dynamic trading in this underlying asset or a static position in a continuum of options of all strikes and maturities. We present an alternative approach which combines dynamic trading in the underlying asset with a static position in options of a single maturity. In many instances, our approach yields explicit valuation formulas and hedging strategies when the volatility of the underlying is an arbitrary function of its price.
Commodity Covariance Contracting
, 2000
"... this paper is to propose a solution which addresses all three drawbacks. By changing the definition of variance in the swap from the realized variance of returns to the realized variance of price changes, we show that the new payo# can be perfectly replicated in the presence of jumps, discrete monit ..."
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this paper is to propose a solution which addresses all three drawbacks. By changing the definition of variance in the swap from the realized variance of returns to the realized variance of price changes, we show that the new payo# can be perfectly replicated in the presence of jumps, discrete monitoring, and discrete trading opportunities. We further show that a contract paying the realized covariance of price changes can also be synthesized in this setting. We illustrate our results in the context of commodity options as the markets for these structures have many of the features which we require. In particular, to synthesize covariance swaps, we use spread options, which represent one of the few options written on two assets and listed on an organized exchange. Papers on the valuation of spread options include Broadie and Detemple[2], Grabbe [8], Heenk, Kemna, and Vorst[10], Pearson[13], Ravindran[14], and Shimko[16]. All of these papers assume that the covariance between the two commmodities in the spread is constant. In contrast, this paper assumes that the covariance between the two commodities is random, and furthermore that the stochastic process governing covariance 1 is unknown. Rather than price spread options in terms of a fixed covariance, we turn the problem around and show how the covariance between the price changes in two commodity futures can be traded, given the ability to trade dynamically in the futures and to take static positions in spread options and in options written on each component of the spread. The outline of this paper is as follows. The next section reviews the theory of static replication using options. The following section shows how this theory can be combined with the standard theory of dynamic replication using futures to create co...
Towards a Theory of Volatility Trading
 Reprinted in Option Pricing, Interest Rates, and Risk Management, Musiella, Jouini, Cvitanic
, 1998
"... this article, the term #volatility" refers to either the variance or the standard deviation of the return on an investment. and Morton#17##HJM#, Dupire modelled the evolution of the term structure of this forward variance, thereby developing the #rst stochastic volatility model in which the market p ..."
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this article, the term #volatility" refers to either the variance or the standard deviation of the return on an investment. and Morton#17##HJM#, Dupire modelled the evolution of the term structure of this forward variance, thereby developing the #rst stochastic volatility model in which the market price of volatility risk does not require speci#cation, even though volatility is imperfectly correlated with the price of the underlying