Peter Carr Research



Peter’s research interests are primarily in the field of derivative securities, especially Americanstyle and exotic derivatives. In collaboration with others, he has developed several new financial contracts (e.g. variance and corridor variance swaps), hedging strategies (e.g. static and semistatic hedging), and valuation models (e.g. VG, CGMY, and FMLS). He also has previously consulted for several firms and have given numerous talks at both practitioner and academic conferences.
A Simple Robust Link Between American Puts and Credit Protection by Peter Carr and Liuren
Wu Abstract: We develop a simple robust link between deep outofthemoney American put options on a company’s stock and a credit insurance contract on the company’s bond. We assume that the stock price stays above a barrier B before default but drops below a lower barrier A after default, thus generating a default corridor [A,B] that the stock price can never enter. Given the presence of this default corridor, a spread between two coterminal American put options struck within the corridor replicates a pure credit contract paying off when and only when default occurs prior to the option expiry 
Article (pdf) 

Variance Swaps on TimeChanged Levy Processes
by Peter Carr, Roger Lee, and Liuren Wu Abstract: We prove that a multiple of a log contract prices a variance swap, under arbitrary exponential Levy dynamics, stochastically timechanged by an arbitrary continuous clock having arbitrary correlation with the driving Levy process, subject to integrability conditions. We solve for the multiplier, which depends only on the Levy process, not on the clock. In the case of an arbitrary continuous underlying returns process, the multiplier is 2, which recovers the standard nojump variance swap pricing formula as a special case of our framework. In the presence of negatively skewed jump risk, however, we prove that the multiplier exceeds 2, which agrees with calibrations of timechanged Levy processes to equity options data. Finally we show that discrete sampling increases variance swap values, under an independence condition; so if the commonlyquoted 2 multiple undervalues the continuouslysampled variance, then it undervalues furthermore the discretelysampled variance. 
Article (pdf) 

Volatility Derivatives by Peter Carr and Roger
Lee Abstract: Volatility derivatives are a class of derivative securities where the payoff explicitly depends on some measure of the volatility of an underlying asset. Prominent examples of these derivatives include variance swaps and VIX futures and options. We provide an overview of the current market for these derivatives. We also survey the early literature on the subject. Finally, we provide relatively simple proofs of some fundamental results related to variance swaps and volatility swaps. 
Article (pdf) 

Variance Risk
Premiums by Peter Carr and Liuren
Wu Abstract: We propose a direct and robust method for quantifying the variance risk premium on financial assets. We show that the riskneutral expected value of return variance, also known as the variance swap rate, is well approximated by the value of a particular portfolio of options. We propose to use the difference between the realized variance and this synthetic variance swap rate to quantify the variance risk premium. Using a large options data set, we synthesize variance swap rates and investigate the historical behavior of variance risk premia on five stock indexes and 35 individual stocks. 
Article (pdf) 

Options on Realized Variance and Convex Orders
by Peter Carr, Helyette Geman, Dilip Madan, and Marc Yor
Abstract: Realized variance options and options on quadratic variation normalized to unit expectation are analyzed for the property of monotonicity in maturity for call options at a fixed strike. When this condition holds, the risk neutral densities are said to be increasing in the convex order. For Levy processes such prices decrease with maturity. A time series analysis of squared log returns on the S&P 500 index also reveals such a decrease. If options are priced to a slightly increasing level of acceptability, then the resulting risk neutral densities can be increasing in the convex order. Calibrated stochastic volatility models yield possibilities in both directions. Finally, we consider modeling strategies guaranteeing an increase in convex order for the normalized quadratic variation. These strategies model instantaneous variance as a normalized exponential of a Levy process. Simulation studies suggest that other transformations may also deliver an increase in the convex order. 
Article (pdf) 

Pricing swaps and options on quadratic variation under stochastic time change models  discrete observations case by Andrey Itkin and Peter Carr Abstract: We use a forward characteristic function approach to price variance and volatility swaps and options on swaps. The swaps are defined via contingent claims whose payoffs depend on the terminal level of a discretely monitored version of the quadratic variation of some observable reference process. As such a process we consider a class of L\'evy models with stochastic time change. Our analysis reveals a natural small parameter of the problem which allows a general asymptotic method to be developed in order to obtain a closedform expression for the fair price of the above products. As examples, we consider the CIR clock change, general affine models of activity rates and the 3/2 power clock change, and give an analytical expression of the swap price. Comparison of the results obtained with a familiar logcontract approach is provided. 
Article (pdf) 

A Class of Levy Process Models with almost exact calibration of both barrier and vanilla FX options
by Peter Carr and John Crosby
Abstract: Vanilla (standard European) options are actively traded on many underlying asset classes, such as equities, commodities and foreign exchange. The market quotes for these options are typically used by exotic options traders to calibrate the parameters of the (riskneutral) stochastic process for the underlying asset. Barrier options, of many different types, are also widely traded in all these markets but one important of the FX Options market is that barrier options, especially Doublenotouch (DNT) options, are now so activley traded that they are o longer considered, in ay way, exotic options. Instead, traders would, in principle, like ot use them as instruments to which they can calibrate their model. The desirability of doing this has been highlighted by talks at practitioner conferences but, to our best knowledge (at least within the realm of the published literature), there have been no models which are specifically designed to cater for this. In this paper, we indtoruce such a model. It allows for calibration in a twostage process. The first stage fits to DNT options (or other types of double barrier options). The seocnd stage fits to vanilla options. The model allows for jumps (ether finite activity or infinite activity) and also for stochastic volatility. Hence, not only can it give a good fit to the market prices of options, it can also allow for realistic dynamics of the underlying FX rate and realistic future volatility smiles and skews. En route, we significantly extend existing results in the literature by providing closed form (up to Laplace inversion) expressions for the prices of several types of barrier options as well as results related to the distribution of first passage times and of the ``overshoot''. 
Article (pdf) Online Supplement
(pdf) 

Local Volatility Enhanced by a Jump to Default
by Peter Carr and Dilip Madan
Abstract: A local volatility model is enhanced by the possibility of a single jump to default. The jump has a hazard rate that is the product of the stock price raised to a prespecified negative power and a deterministic function of time. The empirical work uses a power of 1.5. It is shown how one may simultaneously recover from the prices of credit default swap contracts and equity option prices both the deterministic component of the hazard rate function and revised local volatility. The procedure is implemented on prices of credit default swaps and equity options for GM and FORD over the period October 2004 to September 2007. 
Article (pdf) 

Stock Options and Credit Default Swaps: A Joint Framework for Valuation and Estimation
by Peter Carr and Liuren
Wu Abstract: We propose a dynamically consistent framework that allows joint valuation and estimation of stock options and credit default swaps written on the same reference company. We model default as controlled by a Poisson process with a stochastic default arrival rate. When default occurs, the stock price drops to zero. Prior to default, the stock price follows a jumpdiffusion process with stochastic volatility. The instantaneous default rate and variance rate follow a bivariate continuous Markov process, with its dynamics specified to capture the empirical evidence on stock option prices and credit default swap spreads. Under this joint specification, we derive tractable pricing solutions for stock options and credit default swaps. We estimate the joint dynamics using data from both markets for eight companies that span five sectors and six major credit rating classes from B to AAA. The estimation highlights the interaction between market risk (return variance) and credit risk (default arrival) in pricing stock options and credit default swaps. 
Article (pdf) 

Multiasset Stochastic Local Variance by Peter Carr and Peter Laurence Abstract: Variance swaps now trade actively overthecounter (OTC) on both stocks and stock indices. Also trading OTC are variations on variance swaps which localize the payoff in time, in the underlying asset price, or both. Given that the price of the underlying asset evolves continuously over time, it is well known that there exists a semirobust hedge for these localized variance contracts. Remarkably, the hedge succeeds even though the stochastic process describing the instantaneous variance is never specified. In this paper, we present a generalization of these results to the case of two or more underlying assets. 
Working Paper (pdf) 

Saddlepoint Methods for Option Pricing by Peter Carr and Dilip Madan Abstract: We show a new way in which saddlepoint methods can be used to determine the arbitragefree price of a Europeanstyle option. 
Working Paper (pdf) 

Time Changed Markov Processes in Unified CreditEquity Modelling by Rafael Mendoza, Peter Carr, and Vadim Linetsky Abstract: This paper develops a novel class of hybrid creditequity models with statedependent jumps, localstochastic volatility and default intensity based on time changes of Markov processes with killing. We model the defaultable stock price process as a time changed Markov diffusion process with statedependent local volatility and killing rate (default intensity). When the time change is a Lévy subordinator, the stock price process exhibits jumps with statedependent Lévy measure. When the time change is a time integral of an activity rate process, the stock price process has localstochastic volatility and default intensity. When the time change process is a Lévy subordinator in turn time changed with a time integral of an activity rate process, the stock price process has statedependent jumps, localstochastic volatility and default intensity. We develop two analytical approaches to the pricing of credit and equity derivatives in this class of models. The two approaches are based on the Laplace transform inversion and the spectral expansion approach, respectively. If the resolvent (the Laplace transform of the transition semigroup) of the Markov process and the Laplace transform of the time change are both available in closed form, the expectation operator of the time changed process is expressed in closed form as a single integral in the complex plane. If the payoff is squareintegrable, the complex integral is further reduced to a spectral expansion. To illustrate our general framework, we time change the jumptodefault extended CEV model (JDCEV) of Carr and Linetsky (2006) and obtain a rich class of analytically tractable models with jumps, localstochastic volatility and default intensity. These models can be used to jointly price and hedge equity and credit derivatives. 
Working Paper (pdf) 

Hedging Variance Options on Continuous Semimartingales by Peter Carr and Roger Lee Abstract: We find robust modelfree hedges and price bounds for options on the realized variance of [the returns on] an underlying price process. Assuming only that the underlying process is a positive continuous semimartingale, we superreplicate and subreplicate variance options and forwardstarting variance options, by dynamically trading the underlying asset, and statically holding European options. We thereby derive upper and lower bounds on values of variance options, in terms of Europeans. 
Article (pdf) 

On the qualitative effect of volatility and duration on prices of Asian options by Peter Carr, ChristianOliver Ewald, and Yajun Xiao Abstract: We show that under the Black Scholes assumption the price of an arithmetic average Asian call option with fixed strike increases with the level of volatility . This statement is not trivial to prove and for other models in general wrong. In fact we demonstrate that in a simple binomial model no such relationship holds. Under the BlackScholes assumption however, we give a proof based on the maximum principle for parabolic partial differential equations. Furthermore we show that an increase in the length of duration over which the average is sampled also increases the price of an arithmetic average Asian call option, if the discounting effect is taken out. To show this, we use the result on volatility and the fact that a reparametrization in time corresponds to a change in volatility in the BlackScholes model. Both results are extremely important for the risk management and risk assessment of portfolios that include Asian options. 
Article (pdf) 

PutCall Symmetry: Extensions and Applications by Peter Carr and Roger Lee Abstract:Classic putcall symmetry relates the prices of puts and calls at strikes on opposite sides of the forward price. We extend putcall symmetry in several directions. Relaxing the assumptions, we generalize to unified local/stochastic volatility models and timechanged \levy processes, under a symmetry condition. Further relaxing the assumptions, we generalize to various \emph{asymmetric} dynamics. Extending the conclusions, we take an arbitrarily given payoff of European style or single/double/sequentialbarrier style, and we construct a conjugate Europeanstyle claim of equal value, and thereby a semistatic hedge of the given payoff. 
Article (pdf) 

Hedging under the Heston Model with JumptoDefault by Peter Carr and Wim Schoutens Abstract: We explain how to perfectly hedge under Heston's stochastic volatility model with jump to default. Besides the usual stock and bond, we use variance swaps and Credit Default Swaps (CDS) as hedge instruments. We explain how to perfectly hedge theoretical payoffs such as power payoffs, gamma payoffs and Dirac payoffs, before turning to the hedge for the vanillas. 
Article (pdf) 

Stochastic
Risk Premiums: Stochastic Skewness in Currency Options, and Stochastic
Discount Factors in International Economics by Gurdip Bakshi, Peter
Carr, and Liuren Wu Abstract: We develop models of stochastic discount factors in international economies that produce stochastic risk premiums and stochastic skewness in currency options. We estimate the model using timeseries returns and options prices on three currency pairs that form a triangular relation. Estimation shows that the average risk premium in Japan is larger than that in the US or the UK, the global risk premium is more persistent and volatile then the country specific risk premiums, and investors respond differently to different shocks. We also identify highfrequency jumps in each economy, but find that only downside jumps are priced. Finally, our analysis shows that the risk premiums are economically compatible with movements in stock and bond market fundamentals. 
Article (pdf) 

Stochastic
Skew for Currency Options by Peter Carr and Liuren
Wu Abstract: We document the behavior of overthecounter currency option prices across moneyness, maturity, and calendar time on two of the most active exchange rates over the past eight years. We find that the riskneutral distribution of currency returns is relatively symmetric on average. However, on any given date, the conditional currency return distribution can show strong asymmetry. Furthermore, all of the standard skewness measures exhibit substantial time variation, switching signs several times over our sample. We design and estimate a class of models that captures this stochastic skewness. We show that our class of models outperform traditional jumpdiffusion stochastic volatility models both insample and outofsample. 
Article (pdf) Overheads
(pdf) 

On the Numerical Valuation of Option Prices in
Jump Diffusion Processes by Peter Carr and Anita Mayo Abstract: The fair price of a financial option on an asset that follows a Poisson jump diffusion process satisfies a partial integrodifferential equation. When numerical methods are used to solve such equations, the integrals are usually evaluated using either quadrature methods or fast Fourier methods. Quadrature methods are expensive since the integrals must be evaluated at every point of the mesh. Though less so, Fourier methods are also computationally intensive since in order to avoid wrap around effects, they require enlargement of the computational domain. They are also slow to converge when the the parameters of the jump process are not smooth and for efficiency require uniform meshes. We present a different and more efficient class of methods which are based on the fact that the integrals often satisy differential equations. Depending on the process the asset follows, the equations are either ordinary differential equations or parabolic partial differential equations. Both types of equations can be accurately solved very rapidly. We discuss the methods and present results of numerical experiments. 
Article (pdf) 

A New Approach for Option Pricing Under Stochastic Volatility by Peter Carr and Jian Sun Abstract: We develop a new approach for pricing Europeanstyle contingent claims written on the time $T$ spot price of an underlying asset whose volatility is stochastic. Like most of the stochastic volatility literature, we assume continuous dynamics for the price of the underlying asset. In contrast to most of the stochastic volatility literature, we do not directly model the dynamics of the instantaneous volatility. Instead, taking advantage of the recent rise of the variance swap market, we directly assume continuous dynamics for the time $T$ variance swap rate. The initial value of this variance swap rate can either be directly observed, or inferred from option prices. We make no assumption concerning the real world drift of this process. We assume that the ratio of the volatility of the variance swap rate to the instantaneous volatility of the underlying asset just depends on the variance swap rate and on the variance swap maturity. Since this ratio is assumed to be independent of calendar time, we term this key assumption the stationary volatility ratio hypothesis (SVRH). The instantaneous volatility of the futures follows an unspecified stochastic process, so both the underlying futures price and the variance swap rate have unspecified stochastic volatility. Despite this, we show that the payoff to a pathindependent contingent claim can be perfectly replicated by dynamic trading in futures contracts and variance swaps of the same maturity. As a result, the contingent claim is uniquely valued relative to its underlying's futures price and the assumed observable variance swap rate. In contrast to standard models of stochastic volatility, our approach does not require specifying the market price of volatility risk or observing the initial level of instantaneous volatility. As a consequence of our SVRH, the partial differential equation (PDE) governing the arbitragefree value of the contingent claim just depends on two state variables rather than the usual three. We then focus on the consistency of our SVRH with the standard assumption that the riskneutral process for the instantaneous variance is a diffusion whose coefficients are independent of the variance swap maturity. We show that the combination of this maturity independent diffusion hypothesis (MIDH) and our SVRH implies a very special form of the riskneutral diffusion process for the instantaneous variance. Fortunately, this process is tractable, wellbehaved, and enjoys empirical support. Finally, we show that our model can also be used to robustly price and hedge volatility derivatives. 
Article (pdf) 

Realized
Volatility and Variance: Options via Swaps by Peter Carr and Roger
Lee Abstract: We present explicit and readily applicable formulas for pricing options on realized variance and volatility. We use variance and volatility swaps  or alternatively vanilla options  as pricing benchmarks and hedging instruments. We also cover the pricing of VIX options. 
Article (pdf) 

Theory and
Evidence on the Dynamic Interactions Between Sovereign Credit Default Swaps
and Currency Options by Peter Carr and Liuren
Wu Abstract: Using sovereign CDS spreads and currency option data for Mexico and Brazil, we document that CDS spreads covary with both the currency option implied volatility and the slope of the implied volatility curve in moneyness. We propose a joint valuation framework, in which currency return variance and sovereign default intensity follow a bivariate diffusion with contemporaneous correlation. Estimation shows that default intensity is much more persistent than currency return variance. The market price estimates on the two risk factors also explain the welldocumented evidence that historical average default probabilities are lower than those implied from credit spreads. 
Article (pdf) 

Self
Decomposability and Option Pricing by Peter Carr, Helyette
Geman, Dilip Madan, and Marc Yor Abstract: The riskneutral process is modeled by a four parameter selfsimilar process of independent increments with a selfdecomposable law for its unit time distribution. Six different processes in this general class are theoretically formulated and empirically investigated. We show that all 6 models are capable of adequately synthesizing European option prices across the spectrum of strikes and maturities at a point of time. Considerations of parameter stability over time suggest a preference for two of these models. Currently, there are several option pricing models with six to ten free parameters that deliver a comparable level of performance in synthesizing option prices. The dimension reduction attained here should prove useful in studying the variation over time of option prices. 
Article (pdf) 

Generating
integrable one dimensional driftless diffusions by Peter Carr, Peter
Laurence, and TaiHo Wang Abstract: A criterion on the diffusion coefficient is formulated that allows the classification of driftless time and state dependent diffusions that are integrable in closed form via point transformations. In the time dependent and state dependent case a remarkable intertwining with the inhomogeneous Burger's equation is exploited. The criterion is constructive. It allows us to devise families of driftless diffusions parametrized by a rich class of several arbitrary functions for which the solution of any initial value problem can be expressed in closed form. We also derive an elegant form for the masters equation for infinitesimal symmetries, previously considered only in the time homogeneous case. 
Article (pdf) 

A Jump to
Default Extended CEV Model: An Application of Bessel Processes by Peter Carr and Vadim
Linetsky Abstract: We develop a flexible and analytically tractable framework which unifies the valuation of corporate liabilities, credit derivatives, and equity derivatives. We assume that the stock price follows a diffusion, punctuated by a possible jump to zero (default). To capture the positive link between default and equity volatility, we assume that the hazard rate of default is an increasing affine function of the instantaneous variance of returns on the underlying stock. To capture the negative link between volatility and stock price, we assume a constant elasticity of variance (CEV) specification for the instantaneous stock volatility prior to default. We show that deterministic changes of time and scale reduce our stock price process to a standard Bessel process with killing. This reduction permits the development of completely explicit closed form solutions for riskneutral survival probabilities, CDS spreads, corporate bond values, and Europeanstyle equity options. Furthermore, our valuation model is sufficiently flexible so that it can be calibrated to exactly match arbitrarily given term structures of CDS spreads, interest rates, dividend yields, and atthemoney implied volatilities. 
Article (pdf) 

A Tale of Two Indices by Peter Carr
and Liuren Wu Abstract: In
1993, the Chicago Board of Options Exchange (CBOE) introduced the COBE
Volatility Index (VIX). This index has become the de facto benchmark for
stock market volatility. On September 22, 2003, the CBOE revamped the
definition and calculation of the VIX, and backcalculated the new VIX up to
1990 based on historical option prices. The CBOE is also planning to launch
futures and options on the new VIX. In this paper, we describe the major
differences between the old and the new VIXs, derive the theoretical
underpinnings for the two indices, and discuss the practical motivation for
the recent switch. We also study the historical behaviors of the two indices.

Article
(pdf) Overheads
(pdf) 

A Note on Sufficient Conditions for No Arbitrage by Peter Carr
and Dilip Madan Abstract: It is
shown that the absence of call spread, butterfly spread, and calendar spread
arbitrages is sufficient to exclude all static arbitrages from a set of
option price quotes across strikes and maturities on a single underlier. 
Article
(pdf) 

Pricing Options on Realized Variance by Peter Carr,
Dilip Madan, Helyette Geman, and Marc Yor Abstract: Models
which hypothesize that returns are pure jump processes with independent
increments have been shown to be capable of capturing the observed variation
of market prices of vanilla stock options across strike and maturity. In this
paper, these models are employed to derive in closed form the prices of
derivatives written on future realized quadratic variation. Alternative work
on pricing derivatives on quadratic variation has alternatively assumed that
the underlying returns process is continuous over time. We compare the model
values of derivatives on quadratic variation for the two types of models and
find substantial differences. 
Article
(pdf) 

The Forward PDE for European Options on Stocks with Fixed Fractional Jumps by Peter Carr
and Alireza Javaheri Abstract: We
derive a partial integro differential equation (PIDE) which relates the price
of a calendar spread to the prices of butterfly spreads and the functions
describing the evolution of the process. These evolution functions are the
forward local variance rate and a new concept called the {\it forward local
default arrival rate}. We then specialize to the case where the only jump
which can occur reduces the underlying stock price by a fixed fraction of its
prejump value. This is a standard assumption when valuing an option written
on a stock which can default. We discuss novel strategies for calibrating to
a term and strike structure of European options prices. In particular using a
few calendar dates, we derive closed form expressions for both the local
variance and the local default arrival rate. 
Article
(pdf) 

From Local Volatility to Local Levy Models by Peter Carr,
Dilip Madan, Helyette Geman, and Marc Yor Abstract: We
define the class of local Levy processes. These are Levy processes time
changed by an inhomogeneous local speed function. The local speed function is
a deterministic function of time and the level of the process itself. We show
how to reverse engineer the local speed function from traded option prices of
all strikes and maturities. The local Levy processes generalize the class of
local volatility models. Closed forms for local speed functions for a variety
of cases are also presented. Numerical methods for recovery are also
described. 
Article
(pdf) 

Corridor Variance Swaps by Peter Carr
and Keith Lewis Abstract: This
paper studies a recent variation of a variance swap called a corridor
variance swap (CVS). For this swap, returns are not counted in the realized
variance calculation if the reference index level is outside some specified
corridor. CVS's allow speculators to bet on both the level of the index and
its realized variance. The authors propose a robust hedge of CVS's which
requires that entry and exit of the corridor be treated asymmetrically in the
contract specification. The main conclusion is that the complications
introduced by the asymmetric treatment of entry and exit are outweighed by
the improved hedge performance. 
Article
(pdf) 

What Type of Process Underlies Options? A Simple Robust Test by Peter Carr
and Liuren Wu 
Article (pdf) 

Frequently Asked Questions in Option Pricing Theory by Peter Carr Abstract: We
consider several Frequently Asked Questions (FAQ’s) in option pricing theory. 
Article
(pdf) Overheads
(pdf) 

Bessel Processes, The Integral of Geometric Brownian motion and Asian options by Peter Carr
and Michael Schröder Abstract: This paper is motivated by questions about
averages of stochastic processes which originate in mathematical finance,
originally in connection with valuing the socalled Asian options. Starting with [Y], these questions about
exponential functionals of Brownian motion have been studied in terms of
Bessel processes using the HartmanWatson theory of [Y80[. Consequences of this approach for valuing
Asian options proper have been spelled out in [GY] whose Laplace transform
results were in fact regarded as a noted advance. Unfortunately, a number of difficulties
with the key results of this last paper have surfaced which are now addressed
in this paper. One of them in
particular is of a principal nature and originates with the HartmanWatson
approach itself: this approach is in general applicable without modifications
only if it does not involve Bessel processes of negative indices. The main mathematical contribution of this
paper is the development of three principal ways to overcome these
restrictions, in particular by merging stochastics and complex analysis in
what seems a novel way, and the discussion for their consequences for the
valuation of Asian options proper. 
Article
(pdf) 

TimeChanged
Levy Processes and Option Pricing by Peter Carr
and Liuren Wu Abstract: The classic BlackScholes option pricing
model assumes that returns follow Brownian motion, but return processes
differ from this benchmark in at least three important ways. First, asset
prices jump, leading to nonnormal return innovations. Second, return
volatilities vary stochastically over time. Third, returns and their
volatilities are correlated, often negatively for equities. Timechanged Lévy
processes can simultaneously address these three issues. We show that our
framework encompasses almost all of the models proposed in the option pricing
literature, and it is straightforward to select and test a particular option
pricing model through the use of characteristic function technology. 
Article
(pdf) Overheads
(pdf) 

Stochastic Volatility for Levy Processes by Peter Carr,
Hélyette Geman, Dilip Madan, and Marc Yor Abstract: Three processes reflecting persistence of
volatility are initially formulated by evaluating three Lévy processes at a
time change given by the integral of a meanreverting square root process.
The model for the meanreverting time change is then generalized to include
nonGaussian models that are solutions to OrnsteinUhlenbeck equations driven
by onesided discontinuous Lévy processes permitting correlation with the
stock. Positive stock price processes are obtained by exponentiating and mean
correcting these processes, or alternatively by stochastically exponentiating
these processes. The characteristic functions for the log price can be used
to yield option prices via the fast Fourier transform. In general,
meancorrected exponentiation performs better than employing the stochastic
exponential. It is observed that the meancorrected exponential model is not
a martingale in the filtration in which it is originally defined. This leads
us to formulate and investigate the important property of martingale
marginals where we seek martingales in altered filtrations consistent with
the onedimensional marginal distributions of the level of the process at
each future date. 
Article
(pdf) 

The Finite Moment Log Stable Process and Option Pricing by Peter Carr
and Liuren Wu Abstract: We document a surprising pattern in S&P
500 option prices. When implied volatilities are graphed against a standard
measure of moneyness, the implied volatility smirk does not flatten out as
maturity increases up to the observable horizon of two years. This behavior
contrasts sharply with the implications of many pricing models and with the
asymptotic behavior implied by the central limit theorem (CLT). We develop a
parsimonious model which deliberately violates the CLT assumptions and thus
captures the observed behavior of the volatility smirk over the maturity
horizon. Calibration exercises demonstrate its superior performance against
several widely used alternatives. 
Article
(pdf) Overheads
(pdf) 

Why Be
Backward? Forward Equations for American Options by Peter Carr
and Ali Hirsa Abstract: Originally developed as a tool for
calibrating smile models, socalled forward methods can also be used to price
options and derive Greeks. Here, Peter Carr and Ali Hirsa apply the technique
to the pricing of continuously exercisable Americanstyle put options,
developing a forward partial integrodifferential equation within a jump
diffusion framework 
Article
(pdf) Working
paper with Appendix (pdf) Overheads
(pdf) 

The Fine Structure of Asset Returns: An Empirical Investigation by
Peter Carr, Hélyette Geman, Dilip B. Madan, and
Marc Yor Abstract: We investigate the importance of diffusion and jumps in a new model for asset returns. In contrast to standard models, we allow for jump components displaying finite or infinite activity and variation. Empirical investigations of time series indicate that index dynamics are devoid of a diffusion component, which may be present in the dynamics of individual stocks. This leads to the conjecture, confirmed on options data, that the riskneutral process should be free of a diffusion component. We conclude that the statistical and riskneutral processes for equity prices are pure jump processes of infinite activity and finite variation. 
Article
(pdf) 

Black Scholes Goes HyperGeometric by Claudio Albanese, Giuseppe Campolieti, Peter Carr, and
Alexander Lipton Abstract: We introduce a general pricing formula that extends BlackScholes and contains as particular cases most analytically solvable models in the literature, including the quadratic and the constant elasticity of variance models for Europeanstyle and barrier options. In addition, large families of new solutions are found, containing as many as seven free parameters 
Article
(pdf) 

Pricing and Hedging in Incomplete Markets by Peter Carr, Hélyette Geman, and Dilip B. Madan Abstract: We present a new approach for positioning, pricing, and hedging in incomplete markets that bridges standard arbitrage pricing and expected utility maximization. Our approach for determining whether an investor should undertake a particular position involves specifying a set of probability measures and associated floors which expected payoffs must exceed in order for the investor to consider the hedged and financed investment to be acceptable. By assuming that the liquid assets are priced so that each portfolio of assets has negative expected return under at least one measure, we derive a counterpart to the first fundamental theorem of asset pricing. We also derive a counterpart to the second fundamental theorem, which leads to unique derivative security pricing and hedging even though markets are incomplete. For products that are not spanned by the liquid assets of the economy, we show how our methodology provides more realistic bid–ask spreads. 
Article
(pdf) Overheads (pdf) 

Commodity Covariance Contracting by Peter Carr
and Anthony Corso Abstract: ER presents a solution to the problems in
developing the replication of variance and covariance swaps. 
Article
(pdf) Overheads
(pdf) 

Optimal Positioning in Derivative Securities by Peter Carr
and Dilip Madan Abstract: We
consider a simple single period economy in which agents invest so as to
maximize expected utility of terminal wealth. We assume the existence of
three asset classes, namely a riskless asset (the bond), a single risky asset
(the stock), and European options of all strikes (derivatives). In this
setting, the inability
to trade continuously potentially induces investment in all three asset
classes. We consider both a partial equilibrium where all asset prices are
initially given, and a more general equilibrium where all asset prices are
endogenously determined. By restricting investor beliefs and preferences in
each case, we solve for the optimal position for each investor in the three
asset classes. We find that in partial or general equilibrium, heterogeneity
in preferences or beliefs induces investors to hold derivatives individually,
even though
derivatives are not held in aggregate. 
Article
(pdf) Overheads
(pdf) 

Optimal Investment in Derivative Securities by Peter Carr,
Xing Jin, and Dilip Madan Abstract: We
consider the problem of optimal investment in a risky asset, and in
derivatives written on the price process of this asset, when the underlying
asset price process is a pure jump L´evy process. The duality approach of
Karatzas and Shreve is used to derive the optimal consumption and investment
plans. In our economy, the optimal derivative payoff can be constructed from
dynamic trading in the risky asset and in European options of all strikes.
Specific closed forms illustrate the optimal derivative contracts when the
utility function is in the HARA class and when the statistical and
riskneutral price processes are in the variance gamma (VG) class. In this
case, we observe that the optimal derivative contract pays a function of the
price relatives continuously through time. 
Article
(pdf) 

Deriving Derivatives of Derivative Securities by Peter Carr Abstract: Various
techniques are used to simplify the derivations of “greeks” of
pathindependent claims in the BlackMertonScholes model. First, delta,
gamma, speed, and other higherorder spatial derivatives of these claims are
interpreted as the values of certain quantoed contingent claims. It is then shown that all partial
derivatives of such claims can be represented in terms of these spatial
derivatives. These observations permit
the rapid deployment of higherorder Taylor series expansions, and this is
illustrated for the case of European options. 
Article
(pdf) Overheads
(pdf) 

Going with the Flow: Alternative Approach for Valuing Continuous Cash Flows by Peter Carr,
Alex Lipton, and Dilip Madan Abstract: 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. 
Article
(pdf) Working Paper (pdf) 

The Valuation of Executive Stock Options in an Intensity Based Framework by Peter Carr and Vadim Linetsky Abstract: This
paper presents a general intensitybased framework to value executive stock
options (ESOs). It builds upon the recent advances in the credit risk
modeling arena. The early exercise or forfeiture due to voluntary or
involuntary employment termination and the early exercise due to the executive’s
desire for liquidity or diversification are modeled as an exogenous point
process with random intensity dependent on the stock price. Two analytically
tractable specifications are given where the ESO value, expected time of
exercise or forfeiture, and the expected stock price at the time of exercise
or forfeiture are calculated in closedform. 
Article
(pdf)
Correction
(pdf)


Option Pricing and the Fast Fourier Transform by Peter Carr and Dilip Madan Abstract: In this paper the authors show how the fast Fourier transform may be used to value options when the characteristic function of the return is known analytically. 
Article (pdf) Overheads
(pdf) 

Static Hedging of Timing Risk by Peter Carr
and Jean Picron Abstract: Many exotic options involve a payoff which
occurs at the first passage time to a constant barrier. Although the amount
to be paid is known, the time at which it is paid is not. This paper shows
how a static position in vanilla European options can be used to hedge
against this timing risk. We also show how the results can be used to price
any barrier options. 
Article
(pdf) 

Currency Covariance Contracting by Peter Carr
and Dilip Madan Abstract:
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. 
Article (pdf) 

Static Hedging of Exotic Options by Peter Carr, Katrina Ellis, and Vishal Gupta Abstract: This paper develops static hedges for several exotic options using standard options. The method relies on a relationship between European puts and calls with different strike prices. The analysis allows for constant volatility or for volatility smiles and frowns. 
Article (pdf) 

The Variance Gamma Process and Option Pricing by Dilip Madan,
Peter Carr, and Eric Chang Abstract: A three
parameter stochastic process, termed the variance gamma process, that
generalizes Brownian
motion is developed as a model for the dynamics of log stock prices. The
process is obtained
by evaluating Brownian motion with drift at a random time given by a gamma
process. The two
additional parameters are the drift of the Brownian motion and the volatility
of the time change.
These additional parameters provide control over the skewness and kurtosis of
the return distribution.
Closed forms are obtained for the return density and the prices of European
options. The
statistical and risk neutral densities are estimated for data on the
S&P500 Index and the prices of
options on this Index. It is observed that the statistical density is
symmetric with some kurtosis, while
the risk neutral density is negatively skewed with a larger kurtosis. The
additional parameters also
correct for pricing biases of the Black Scholes model that is a parametric
special case of the option
pricing model developed here. 
Article
(pdf) 

Randomization and the American Put by Peter Carr Abstract: While American calls on nondividendpaying stocks may be valued as European, there is no completely explicit exact solution for the values of American puts. We use a technique called randomization to value American puts and calls on dividendpaying stocks. This technique yields a new semiexplicit approximation for American option values in the BlackScholes model. Numerical results indicate that the approximation is both accurate and computationally efficient. 
Article (pdf) Overheads
(pdf) 

Breaking
Barriers by Peter Carr
and Andrew Chou 
Article
(pdf) 

Two Extensions to Barrier Option Valuation by Peter Carr 
Article
(pdf) 

Static
Simplicity by Peter Carr
and Jonathan Bowie 
Article
(pdf) Overheads
(pdf) 

Alternative
Characterizations of American Put Options by Peter Carr, Robert Jarrow, and Ravi Myneni Abstract: We derive alternative representations of the
McKean equation for the value of the American put option. Our main result
decomposes the value of an American put option into the corresponding
European put price and the early exercise premium. We then represent the
European put price in a new manner. This representation allows us to
alternatively decompose the price of an American put option into its
intrinsic value and time value, and to demonstrate the equivalence of our
results to the McKean equation. 
Article
(pdf) 

The StopLoss
StartGain Strategy and Option Valuation by Peter Carr and Robert Jarrow Abstract: The downside risk in a leveraged stock
position can be eliminated by using stoploss orders. The upside potential of
such a position can be captured using contingent buy orders. The terminal
payoff of this stoploss startgain strategy is identical to that of a call
option, but the strategy costs less initially. This article resolves this
paradox by showing that the strategy is not selffinancing for continuous
stockprice processes of unbounded variation
The resolution of the paradox leads to a new decomposition of an
option’s price into its intrinsic and time value. When the stock price
follows geometric Brownian motion, this decomposition is proven to be
mathematically equivalent to the BlackScholes (1973) formula. 
Article
(pdf) 

The Valuation
of Sequential Exchange Opportunities by Peter Carr Journal of Finance, December 1988, pp. 123556 Abstract: Sequential exchange opportunities are valued
using the techniques of modern optionpricing theory. The vehicle for
analysis is the concept of a compound exchange option. This security is shown
to exist implicitly in several contractual settings. A valuation formula for
this option is derived. The formula is
shown to generalize much previous work in option pricing. Several applications of the formula are
presented. 
Article
(pdf) 

A Calculator Program for Option Values and Implied Standard Deviations by Peter Carr Journal of
Financial Education, Fall 1988 
Article
(pdf) 

A Note on the
Pricing of CommodityLinked Bonds by Peter Carr Journal of Finance, September 1987, pp.
107176 Abstract: The owner of a commoditylinked bond can exchange the bond’s face value for the maturity value of a commodity. In an interesting application of optionpricing theory, Schwartz provides a general framework for valuing commoditylinked bonds. This very general valuation framework allows for commodity price risk, default risk, and interest rate risk, along with interest payments and dividends. A partial differential equation and its associated boundary condition are derived but not solved. Schwartz states that the solution to the general problem is difficult even by numerical methods. In order to find closedform solutions, Schwartz considers commodity price risk in conjunction with default risk or interest rate risk but not both. This note presents a closedform solution for the value of a commoditylinked bond when all three types of risk are present. 
Article
(pdf) 
Forward Evolution Equations for KnockOut Options by
Peter Carr and Ali Hirsa Advances in Mathematical Finance (Festschrift in honor of Dilip Madan’s 60th birthday)
M.C. Fu, R.A. Jarrow, J.Y. Yen, and R.J. Elliott, editors, Birkhäuser Boston, (July 2007),
We derive forward partial integro differential equations (PIDEs) for
upandout and downandout call options when the underlying is a jump diffusion.
We assume that the jump part of the returns process is an additive process.
This framework includes the variance gamma, finite moment logstable, Merton jump diffusion,
Kou jump diffusion, Dupire, CEV, arcsinhnormal, displaced diffusion, and Black Scholes models as special cases. 
Article
(pdf) 
The Value of Purchasing Information to Reduce Risk in Capital Investment Projects by
Larry Chorn and Peter Carr Risk Options and Business Strategy, Trigeorgis, ed. Chapter
12, p. 279294 RiskWaters
Publishers (1999) Abstract: Successful investment management of
capitalintensive, longlived energy projects requires an understanding of
the economic uncertainties, or risks, as well as the mechanisms to resolve
them. Industry traditionally manages these risks by purchasing information
(seismic, well testing, appraisal drilling, reservoir simulation, market
capacity and price studies, etc.) about the project and making incremental
investments as new information reduces the uncertainties to acceptable
levels. Traditional discounted cash flow
analyses cannot readily deal with valuing information. We suggest that the
purchase of information about a project has considerable value and can be
treated as purchasing an option on the project. As with options on equities,
if the information leads to the expectation of a positive investment outcome,
the project should be funded. Similarly, options on capital investment
projects also have a time factor dictating value and the proper time to
undertake the investment. This article discusses the
application of option pricing techniques (OPT) to valuing information. We
show how OPT is used to value the information surrounding a production
capacity decision for an offshore gas field development. In the example, we
value the field development alternatives and the acquisition of incremental
information for the alternative selection process. As a further extension of
OPT to capital investment projects, we create a dynamic model to identify
investment alternatives to capture additional value over the project's
lifetime. The dynamic model uses information acquired in development drilling
and field operations to maximize the investment outcome. 
Article
(pdf) 
Real Options and the Timing and Implementation of Emission Limits Under Ecological Uncertainty by Peter Carr and
JeanDaniel Saphores Project Flexibility,
Agency, and Competition, M. J. Brennan and L. Trigeorgis, ed. Oxford University Press,
1999 
Article
(pdf) 
Determining Volatility Surfaces and Option Values From an Implied Volatility Smile by Peter Carr and Dilip
Madan October 2, 1998 Quantitative Analysis in Financial
Markets, Vol II, M. Avellaneda, ed, pp. 163191 Abstract: Using only the implied volatility smile of
a single maturity T and an assumption of pathindependence, we analytically
determine the riskneutral stock price process and the local volatility
surface up to an arbitrary horizon T’ >= T. Our pathindependence
assumption requires that each positive future stock price St is a function of
only time t and the level Wt of the driving standard Brownian motion (SBM)
for all tЄ(0; T’). Using the Tmaturity option prices, we identify this
stock pricing function and thereby analytically determine the riskneutral
process for stock prices. Our pathindependence assumption also implies that
local volatility is a function of the stock price and time which can be
explicitly represented in terms of the known stock pricing function. Finally,
we derive analytic valuation formulae for standard and exotic options which
are consistent with the observed Tmaturity smile. 
Article
(pdf) Overheads
(pdf) 
Simulating American Bond Options in an HJM Framework by Peter Carr and Guang
Yang February 26, 1998 Quantitative Analysis in
Financial Markets, Vol II, M. Avellaneda, ed. Abstract: This paper develops a method called Markov
Chain Approximation (MCA) to approximate the value of American bond options
in a general multifactor HeathJarrowMorton (HJM) framework. Our approach
is based on the methodology of Barraquand and Martineau (1995), which was
developed for processes which are Markovian in a finite number of state
variables. We have extended their methodology to the HJM setting, which in
general is only Markovian in an infinite number of state variables. The
numerical results obtained from our MCA compare quite closely with closed
form or tree solutions. 
Article
(pdf) 
Simulating Bermudan Interest Rate Derivatives by Peter Carr and Guang
Yang December 3, 1997 Quantitative Analysis in Financial Markets, Vol II, M. Avellaneda, ed, pp. 295316 Abstract:
We
use simulation to develop a Markov chain approximation for the value of caplets
and Bermudan interest rate derivatives in the Market Model developed by
Brace, Gatarek, and Musiela (1995) and Jamshidian (1996a,b). One and two
factor versions of the Market Model were numerically studied. Our approach
yields numerical values for caplets which are in close agreement with
analytic solutions. We also provide numerical solutions for several Bermudan
swaptions. 
Article
(pdf) 
American Options: A Comparison of Numerical Methods by Farid AitSahlia and
Peter Carr Numerical Methods in
Finance, L.C.G. Rogers and D. Talay ed., pp. 6787 Cambridge University
Press, 1997 
Article
(pdf) 
The Valuation of American Exchange Options with Application to Real Options by Peter Carr Real Options in Capital
Investment: New Contributions, Lenos Trigeorgis ed., 10920 Abstract: An American exchange option gives its owner
the right to exchange one asset for another at any time prior to expiration.
A model for valuing these options is developed using the GeskeJohnson
approach for valuing American put options. The formula is shown to generalize
much previous work in option pricing. Application of the general valuation
formula to the timing option in capital investment theory and other real
options is presented. 
Article
(pdf) 
Towards a Theory of Volatility Trading by Peter Carr and Dilip
Madan 
Article
(pdf) Overheads
(pdf) 
A Discrete Time Synthesis of Derivative Security Valuation Using a Term Structure of Futures Prices by Peter Carr and Robert
Jarrow, in the finance volume of Handbooks in Operations Research and Management Science (finance volume), R.
Jarrow, V. Maksimovic, & B. Ziemba, ed., pp. 225–249 
Article
(pdf) 
Valuing Bonds with
Detachable Warrants by Peter Carr Japanese Financial Market
Research,
W. Bailey, Y. Hamao, & B. Ziemba, eds., 46779 
Article
(pdf) 
Static Hedging
of Standard Options by Peter Carr and Liuren
Wu Abstract: We consider the hedging of derivative securities when the price movement of the underlying asset can exhibit jumps of random size. Working in a single factor Markovian setting, we derive a new spanning relation between a given option and a continuum of shorter term options written on the same asset. In the portfolio of shorter term options, the portfolio weights do not vary with changes in stock price or time. We then implement this static relation using a finite set of shorter term options based on a quadrature rule and use Monte Carlo simulation to determine the hedging error thereby introduced. We compare this hedging error to that of a delta hedging strategy based on daily rebalancing in the underlying futures. The simulation results indicate that the two types of hedging strategies exhibit comparable performance in the classic BlackScholes environment, but that our static hedge strongly outperforms deltahedging when the underlying asset price is governed by Merton (1976)’s jumpdiffusion model. Further simulation exercises indicate that these results are robust to model misspecification, so long as one performs ad hoc adjustments based on the observed implied volatility. We also compare the hedging effectiveness of the two types of strategies using more than six years of data on S&P 500 index options. We find that in all cases considered, a static hedge using just five calls outperforms daily delta hedging with the underlying futures. The consistency of this result with our jump model simulations lends empirical support for the existence of jumps of random size in the movement of the S&P 500 index. We also find that the performance of our static hedge deteriorates moderately as we increase the time between the maturity date of the target call and the common maturity of the calls in the hedge. We interpret this result as evidence of the existence of additional random factors such as stochastic volatility. 
Working Paper (pdf) Overheads
(pdf) 
Put Call
Reversal by Peter Carr
and Jesper Andreasen Abstract: Assuming that the stock price process is a jump diffusion, we derive a new relation between puts and calls termed Put Call Reversal (PCR). We show how PCR gives simple new probabilistic interpretations of deltas and gammas. We also show how PCR simplifies semistatic hedging of long dated options. 
Working
Paper (pdf) Overheads
(pdf) 
The Reduction
Method for Valuing Derivative Securities by Peter Carr,
Alex Lipton, and Dilip Madan Abstract: It is
well known that derivative security valuation often reduces to solving a
certain linear partial differential equation with variable coefficients. We
derive a complicated expression which the three coefficients must satisfy in
order that this PDE can be transformed into the heat equation. We also
present a technique for constructing a triplet of coefficients which solve
this expression. We thereby exhibit a technique for generating closed form
solutions for derivative security values in a wide array of models. 
Working
Paper (pdf) Overheads
(pdf) 
Factor Models for Option Pricing by Peter Carr
and Dilip Madan Abstract: Options on stocks are priced using
information on index options and viewing stocks in a factor model as
indirectly holding index risk. The method is particularly suited to
developing quotations on stock options when these markets are relatively
illiquid and one has a liquid index options market to judge the index risk.
The pricing strategy is illustrated on IBM and Sony options viewed as holding
SPX and Nikkei risk respectively. 
Working
Paper (pdf) Overheads
(pdf) 
On the Nature of Options by Peter Carr,
Keith Lewis, and Dilip Madan Abstract: We consider the role of
options when markets in its underlying asset are frictionless and when this
underlying has a volatility process and jump arrival rates which are
arbitrarily stochastic. By combining a static option position with a
particular dynamic hedging strategy, we characterize the option's time value
as the (riskneutral) expected benefit from being able to buy or sell one
share of the underlying at the option's strike whenever the strike price is
crossed. The buy/sell decision can be based on the post jump price, so that a
rational investor buys on rises and sells on drops. Thus, an option provides
liquidity at its strike even when the market doesn't. We next present two
methods for extending this local liquidity to every price between the pre and
post jump level. The first method involves holding a continuum of options of
all strikes. The second method holds one option, but adjusts the dynamic
hedging strategy. We discuss the advantages and disadvantages of each
approach and consider the benefits of combining them. 
Working
Paper (pdf) Overheads
(pdf) 
Closed Form Option Valuation with Smiles by Peter Carr, Michael
Tari, and Thaleia Zariphopoulou Abstract: Assuming that the
underlying local volatility is a function of stock price and time, we develop
an approach for generating closed form solutions for option values for a
certain class of volatility functions. The class is the set of volatility
functions which solve the same partial differential equation as derivative
security values in the Black Scholes model. We illustrate our results with
three examples. 
Working
Paper (pdf) Overheads
(pdf) 
Hedging Complex Barrier Options by Peter Carr and Andrew
Chou April 1, 1997 Abstract: We show how several complex barrier
options can be hedged using a portfolio of standard European options. These
hedging strategies only involve trading at a few times during the option's
life. Since rolling, ratchet, and lookback options can be decomposed into a
portfolio of barrier options, our hedging results also apply to them. 
Working
Paper (pdf) 
American Put Call Symmetry by Peter Carr and Marc
Chesney November 13, 1996 Abstract: We
derive a simple relationship between the values and exercise boundaries of
American puts and calls. The relationship holds for options with the same
“moneyness", although the absolute level of the strike price and
underlying may differ. The result holds in both the Black Scholes model and
in a more general diffusion setting. 
Working
Paper (pdf) 
On the valuation of
arithmetic–average Asian options: the Geman–Yor Laplace transform revisited by Peter Carr and M.
Schröder 
Working
Paper (pdf) 
Delta Hedging
With Stochastic Volatility by Peter Carr September 2005 
Overheads (pdf) 
Hedging With
Options by Peter Carr February 2005 
Overheads (pdf) 
A Tale of Two
Indices by Peter Carr and Liuren
Wu February 2004 
Overheads (pdf) 
Trading
Autocorrelation by Peter Carr and Roger
Lee May 2003 
Overheads (pdf) 
Robust Replication of Volatility Derivatives by Peter Carr and Roger
Lee April 2003 
Peter's
Old Talk (pdf) Roger's New Talk (pdf) 
Option Pricing Using Integral Transforms by Peter Carr, (joint work with Helyette G´eman, Dilip Madan, Liuren Wu, and Marc Yor) 
Overheads
(pdf) 
Survey of Preference Free Option Pricing with Stochastic Volatility by Peter Carr 
Overheads
(pdf) 
Financial Interpretations of Probabilistic Concepts by Peter Carr October 21, 1998 
Overheads
(pdf) 