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Introduction
Emanuel Derman currently works at the Department of Industrial Engineering and Operations Research, Columbia University, where he is cohead of the financial engineering program. Emanuel does research in Financial Economics, especially on options and volatility. he has also worked previously in Elementary Particle Physics. You can also find many of his papers and other material at emanuel derman.com.
Publications
Publications (102)
Constraints on option prices and the smile from the principle of no riskless arbitrage. The Merton inequalities for option prices. Inequalities for the slope of the smile.
Dynamic replication of exotic options requires frequent and sometimes expensive rebalancing. Weak static replication tries to match the payoffs of an exotic option on all its boundaries using portfolios of standard options. The weights of the static replication portfolio depend on the model used (as does the hedge ratio in dynamic replication). The...
Adding stochastic volatility to the local volatility model. A negative local volatility skew picks up convexity. The partial differential equation for options with stochastic stock volatility. The mixing formula solution to the differential equation.
The binomial model as framework for modeling stock price evolution. The binomial model for option evaluation. Equivalence to the Black-Scholes-Merton model. Extending the binomial model to accommodate more general stock price evolution.
The behavior of the smile when stochastic volatility is mean reverting and uncorrelated with the stock. The effect of nonzero correlations via Monte Carlo simulation. The best stock-only hedge in a stochastic volatility model produces a hedge ratio similar to that of a local volatility model. Stochastic volatility models can produce a rich variety...
Exploring replication. Exact static replication for European options. Approximate static replication for exotic options. Dynamic replication and continuous delta-hedging. What should you pay for convexity? Implied volatility is a parameter, realized volatility is a statistic. Hedging an option means betting on volatility.
When the stock and its stochastic volatility are uncorrelated, the smile is a symmetric function of the log moneyness ln(K/S). When the stock and its stochastic volatility are uncorrelated, the sticky moneyness rule of thumb holds. For small volatility of volatility, the mixing theorem leads to approximate analytic expressions for the smile as a fu...
Transaction costs make a long position worth less, a short position more. The tension between the accuracy and cost of hedging. The effective volatility of a hedged option.
An overview of models consistent with the smile. Local volatility models, stochastic volatility models, jump-diffusion models. In the presence of a smile, the BSM model produces incorrect hedge ratios and exotic option values.
The law of one price: Similar things must have similar prices. Replication: the only reliable way to value a security. A simple up-down model for the risk of stocks, in which expected return μ and volatility σ are all that matter. The law of one price leads to CAPM for stocks. Replicating derivatives via the law of one price.
Financial models in light of the great financial crisis. The difficulties of option valuation. An introduction to the volatility smile. Financial science and financial engineering. The purpose and use of models.
A call option and its underlying stock can be combined to form an instantaneously riskless portfolio. The Black-Scholes-Merton equation. Black-Scholes-Merton options pricing formula. You can hedge the risk of an option in a variety of ways. The profit and loss (P&L) from hedging an option depends on which volatility you use to hedge.
The Dupire equation expresses σ(K, T), the stock's local volatility when the future stock price is K at time T, in terms of the partial derivatives of standard option market prices with respect to expiration T and strike K. These mathematical derivatives represent the market prices of infinitesimal strike spreads, calendar spreads, and butterfly sp...
In a local volatility model that is consistent with the smile defined by standard option prices, the hedge ratios of standard options differ from their BSM values. The values of exotic options differ from their BSM model values, too. The rule of two and the notion that implied volatility is the average of local volatilities provide some intuitive r...
Option values are sensitive to volatility and stock price. A better way to trade pure volatility is through volatility and variance swaps. How to replicate a variance swap out of a portfolio of options that has the payoff of a log contract. How to replicate a variance swap when volatility is stochastic. Valuing the swap. The consequence of errors i...
Local volatility models relate the slope of the current skew, , to the rate of change of volatility, . There are various possible heuristic relationships between and . The sticky strike rule, the sticky delta rule, and the sticky local volatility model are examples. Index option markets do not perfectly satisfy any one of these models or rules.
A local volatility model can fit the smile and produce hedge ratios and option values consistent with the market implied volatilities of standard options. Like all financial models, a local volatility model requires frequent recalibration, which means that it doesn't reflect the behavior of the underlying market in a time-invariant way. For equity...
There are a variety of ways to make volatility stochastic and produce a skew. One can make the BSM volatility stochastic, or make the local volatility stochastic. When volatility is stochastic, an option's volga induces a symmetric smile, and an option's vanna induces an asymmetric skew. Volatility tends to revert to the mean. Risk-neutral valuatio...
In violation of the BSM model, the implied volatilities of options on a single underlier vary with strike and expiration. Just as a bond market is defined by its yield curve, an option market is defined by its smile. It's convenient to plot the smile as a function of delta. A standard measure of the skew is the difference in implied volatility betw...
Stock price jumps can explain the steep short-term skew. Modeling a jump. Calibrating jumps to the stock price distribution. The Poisson distribution of jumps. Option prices from jumps alone.
Hedging perfectly and continuously at no cost is a Platonic ideal. In real life, you can rebalance the hedge only a finite number of times. You are mishedged in the intervals, and the P&L picks up a random component. The more often you hedge, the smaller the deviation from perfection. Transaction costs affect things, too, but that's considered in t...
European call and put prices can be used to determine the implied distribution of the terminal stock price. The implied distribution density is related to the market prices of butterfly spreads. You can replicate any exotic European payoff with a portfolio of zero coupon bonds, a forward, and a portfolio of European puts and calls, even in the pres...
Merton's equation for option prices in a jump-diffusion model. A trinomial version of jump-diffusion, and its calibration. A compensated drift to match the riskless rate. The value of a call in a jump-diffusion model. A qualitative description of the effect of jump-diffusion on the smile. A simple approximate analytic formula for the jump-diffusion...
In a local volatility model, the instantaneous stock volatility σ(S, t) is a function of stock price and future time. How to build and use a binomial tree with variable local volatility. The BSM implied volatility of a standard option in a local volatility model is approximately the average of the local volatilities between the initial stock price...
The author tries to justify a career spent in financial modeling.
In this paper we present an arbitrage pricing framework for valuing and hedging contingent equity index claims in the presence of a stochastic term and strike structure of volatility. Our approach to stochastic volatility is similar to the Heath-Jarrow-Morton (HJM) approach to stochastic interest rates. Starting from an initial set of index options...
Theories deal with the world on its own terms, absolutely. Models are metaphors, relative descriptions of the object of their attention that compare it to something similar already better understood via theories. Models are reductions in dimensionality that always simplify and sweep dirt under the rug. Theories tell you what something is. Models te...
We propose a stochastic difference equation of the form Xn = AnXn-1 + Bn to model the annual returns Xn of a hedge fund relative to other funds in the same strategy group in year n. We fit this model to data from the TASS database over the period 2000 to 2005. We let {An} and {Bn} be independent sequences of independent and identically distributed...
Although the syntax of financial modeling is inspired by and closely resembles the syntax of theoretical physics, the similarities are deceptive. The semantics of the two fields are very different. This article compares modeling approaches in physics and finance, and discusses the appropriate way to use valuation models in financial markets.
A lockup period for investment in a hedge fund is a time period after making the investment during which the investor cannot freely redeem his investment. It is routine to have a one-year lockup period, but recently the requested lockup periods have grown longer. We estimate the premium for such extended lockup, taking the point of view of a manage...
A suggested manifesto for financial modelers.
This article analyzes the methodology of modeling in the physical sciences and in finance. Whereas hobbyists’ models aim for realistic resemblance to the object of the model, physics models aim for accurate divination. Financial models, the article argues, can at best aim to extrapolate or interpolate from the known prices of liquid securities to t...
A stochastic difference equation of the form X_n = A_n X_{n-1}+B_n is proposed to model the annual returns X_n of a hedge fund relative to other funds in the same strategy group in year n, and is fit to data from the TASS database over the period 2000 to 2004. In the proposed model, {A_n} and {B_n} are independent sequences of independent and ident...
What excess return should a fund of funds expect to earn for investing in a hedge fund with an extended lockup? In this paper, we present a simple model for estimating the premium for long-term lockups. Because there is a demonstrated statistical persistence to the quality of hedge fund returns within a particular hedge fund strategy above averag...
Underneath every economic model involving math lies a substrate of great simplification and imagination, says Columbia University's Emanuel Derman.
While modern financial theory holds that options values are derived by dynamic replication, they can be correctly valued far more simply by long familiar static and actuarial arguments that combine stochastic price evolution with the no-arbitrage relation between cash and forward contracts.
For the last few months I have been teaching financial engineering at Columbia University, where I have been struck again by the difference between what can be taught in school and what can be learned on the job. Most of my quant generation arrived on Wall Street ignorant of financial theory; we began to learn its principles under the duress of hav...
Emanuel Derman was one of the first physicists to move to Wall Street, and his career paralleled the growth of quantitative trading over the past twenty years. In My Life as a Quant, he traces his transformation from ambitious young scientist to managing director and head of the renowned Quantitative Strategies group at Goldman, Sachs & Co. Derman'...
There is often an unfortunate strain of pedantry running through the teaching of quantitative finance, one involving an excess of abstraction, formality, rigor and axiomatization that makes the subject unnecessarily daunting and difficult. This article contains a short guide to quantitative finance with a human face.
An essay on distinguishing between the work of cranks, academics and practitioners.
This note traces the past and guesses at the future of concepts in quantitative risk management by looking at the history of its vocabulary and seeing which newly introduced words have survived and prospered. It is adapted from a talk on Future Innovations in Risk Management, presented at the April Risk 2002 Conference in Paris.
Although the language of financial theory resembles the language of
physics, there are actually very few reliable and accurate principles on
which to base a theory of quantitative finance. This talk provides a
short guide to the principles and techniques used to build theories of
quantitative value in the trading world.
What return should you expect when you take on a given amount of risk? How should that return depend upon other people's behaviour? What principles can you use to answer these questions? In this paper, I approach these topics by exploring the consequences of two simple hypotheses about risk.
The first is a common-sense invariance principle: assets...
For a long time I worked in the area of equity derivatives, building models and systems for valuing the desk’s book. I want
to take a backward look at the almost invisible problems you can run into in using the Black-Scholes model and its extensions
to hedge a portfolio of options.
What return should you expect when you take on a given amount of risk? How should that return depend upon other people’s behaviour? What principles can you use to answer these questions? In this paper, I approach these topics by exploring the consequences of two simple hypotheses about risk. The first is a common-sense invariance principle: assets...
Financial theory looks deceptively like physics because of the techniques it uses. But physics models deal with the relatively unchanging parameters of the external world; in contrast, the parameters of financial theory are people's current estimates of, and sentiments about, future behavior. Life as a Ph.D. on Wall St is therefore very different f...
Trading desks at investment banks often have substantial positions in a broad range of long-term or exotic derivative securities which can be marked to market only by means of mathematical models. Verifying the fair value of these securities in all product areas is an increasingly important issue for trading firms, and involves more than just mathe...
This short essay speculates on the causes for the imperfections in financial modeling.
This article presents a practical and useful method for replicating or hedging a target stock option with a portfolio of other options. It shows how to construct a replicating portfolio of standard options with varying strikes and maturities and fixed portfolio weights. Once constructed, this portfolio will replicate the value of the target option...
Quantitative financial modelling seems to employ both the language and techniques of physics, but how similar are the two disciplines? Emanuel Derman comments on the practice of financial modelling and the environment in which it is done.
Investors in equity options experience two problems that compound each other. In contrast to fixed-income and currency markets, there are thousands of underlyers and tens of thousands of options, and each underlyer can have a potentially large volatility skew. How can an options investor gauge which option provides the best relative value? In this...
Trading in derivatives has caused investors, and especially market makers, to be concerned with the volatility of asset returns along with their direction. Uncertain and time-varying volatility imparts risk to an otherwise hedged position, and volatility risk is not easy to manage with ordinary instruments. Volatility swaps are a new class of deriv...
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#cat...
The structure of listed index options prices, examined through the prism of the implied tree model, reveals the local volatility surface of the underlying index. In the same way as fixed-income investors analyze the yield curve in terms of forward rates, so index options investors should analyze the volatility smile in terms of local volatilities....
In this article, the author analyzes financial modeling from the viewpoint of a theoretical physicist. Given his experience in a trading environment, he classifies the modeling principles he has found useful, and discusses bow to develop and use financial models appropriately.
Most real-world barrier option values have no analytic solutions, either because the barrier structure is complex or because of volatility skews in the market. Numerical solutions are therefore a necessity, but options with barriers are notoriously difficult to value numerically on binomial or multinomial trees or on finite-difference lattices. The...
HEQS is a set of tools for numerically solving sets of algebraic equations from their description in a text file. It allows users to compactly define (or alter an already defined) set of equations (a model), and then analyze and solve it with minimal intervention. HEQS automatically checks the algebraic and logical consistency of the equations, rep...
We discuss a simple algorithm for solving sets of simultaneous equations. The algorithm can solve systems of linear and some kinds of non-linear equations, although it has nowhere near the power of a general non-linear equation solver. Its principal advantages over more general algorithms are simplicity and speed. Versions of the algorithm have bee...
A simple dynamical model for the internal structure of the three light lepton and quark generations (νe,e,u,d), (νμ,μ,c,s), and (ντ,τ,t,b) is proposed. Each generation is constructed of only one fundamental massive generation F=(L∘,L-,U,D) with the same (SU3)c×SU2×U1 quantum numbers as the light generations, bound to a core of one or more massive H...
I consider a model in which all leptons and quarks are bound states of one neutral Higgs boson H and only one fundamental lepton and quark generation embedded routinely in the standard gauge model. Relativistic bound calculations suggest MH ~ 15 TeV, and indicate how pointlike form factors, Dirac magnetic moments and suppressed radiative decays for...
We discuss parity violating asymmetries between the scattering of right and left-handed electrons on a variety of targets. Implications for gauge theories from recent SLAC results on deep-inelastic electron-deuterium and electron-proton scattering are examined. A derivation of the asymmetry for electron-electron scattering is given, its advantages...
We unify the n quark and n lepton generations within the standard
SU2 × U1 gauge model by means of the
generation symmetry group Sn. We show that no more than five
generations of quarks and leptons can be incorporated into the theory.
The resultant model always has either one or two exactly conserved
multiplicative quantum numbers with eigenvalue +...
The leptons νe, e, νμ, μ, ντ, τ, and analogously the quarks u, d, c, s, t, b, are unified within the Weinberg-Salam SU2×U1 gauge model without enlarging the gauge group. The result is a theory in which the familiar leptons, quarks, and gauge bosons, plus some extra Higgs bosons necessary for unification, all carry a new multiplicatively conserved q...
Recent SLAC measurements of the polarization asymmetry in inelastic polarized e-d scattering have been executed at a momentum transfer not necessarily large enough for accurate comparison with theory using the parton model. Using isospin symmetry I derive a parton-model-independent relation for calculating the e-d asymmetry in the Weinberg-Salam mo...
Suppose that all differences between the leptons e, μ and τ arise from spontaneous symmetry breaking. Implementation of this idea in the standard gauge model leads to conservation of multiplicative rather than additive lepton number. The consequences of this are analysed.
The similarity between the weak interactions of electron and muon is extended to the principle that all e and mu interactions in gauge models are invariant under e mu exchange. This necessitates the existence of two Higgs doublets phie and phimu, and an extended e mu permutation invariance. After symmetry-breaking, a multiplicatively conserved ``pe...
The dilepton signature of charm production in deep-inelastic nuN scattering allows searches for leptonic correlations odd under momentum and spin reversal. Which correlations truly indicate T non-invariance is discussed. The measurable T-violating asymmetries due to typical maximal T non-invariant charmed weak currents are estimated, and found to b...
If massive charmed particles are being produced in current νN scattering experiments, their semi-leptonic decay leads to dimuon states. The production and decay of such particles in this process is examined in a parton model which has been modified to realistically incorporate the threshold due to the large mass of the charmed particle.The model th...
We consider here the possibility that the recently observed oppositely charged muon pairs produced in high-energy neutrino scattering at the Fermi National Accelerator Laboratory originated from the production of a neutral heavy lepton which then decayed into the dimuon pair and a neutrino. Using V-A coupling we have calculated the total cross sect...
We have analyzed dimuon distributions due to the diffractive production of a new hadronic vector boson in neutrino scattering. Characteristic features that distinguish this mechanism from heavy-lepton-mediated dimuon distributions are presented.
The possibility that dimuons observed in ν(ν[over ¯])-N scattering are due to the production of a neutral heavy lepton is considered. Various distributions are presented.
Tests for a weak neutral current in high-energy inclusive l±-N scattering are discussed, where l is a charged lepton and N a nucleon. If the polarization of the incoming lepton of the target nucleon can be varied, or the outgoing lepton's polarization observed, an unambiguous measurable consequence of such a current is parity violation. Some genera...
A simple field-theoretic model with no ad hoc cutoff is used to obtain an explicit expression to low order in perturbation theory for the cross section for producing a massive muon pair in high-energy proton-neutron collisions. This model provides a specific example of the relation between this process and deep-inelastic electron-nucleon scattering...
We unify the n quark and n lepton generations within the standard SUâ x Uâ gauge model by means of the generation symmetry group S/sub n/. We show that no more than five generations of quarks and leptons can be incorporated into the theory. The resultant model always has either one or two exactly conserved multiplicative quantum numbers with eigen...
Quantitative financial modeling seems to employ both the language and techniques of physics, but how similar are the two disciplines in theory and practice? This talk discusses the move from physics to finance, the nature of financial modeling and its deceptive similarity with theoretical physics, what it's like to work in the financial arena, and...