EExpectation and Price in Incomplete Markets
Paul McCloudDepartment of Mathematics, University College LondonJuly 1, 2020
Abstract
Risk-neutral pricing dictates that the discounted derivative price is amartingale in a measure equivalent to the economic measure. The resid-ual ambiguity for incomplete markets is here resolved by minimising theentropy of the price measure from the economic measure, subject to mark-to-market constraints, following arguments based on the optimisation ofportfolio risk. The approach accounts for market and funding convexi-ties and incorporates available price information, interpolating betweenmethodologies based on expectation and replication.
The principal innovation of the financial derivatives industry is the abilityto transform an arbitrary economic observable into a tradable financial instru-ment. Provided that this is measurable at or prior to settlement, the contractualterms dictate that the measurement of the observable becomes the terminal cashvalue of the corresponding derivative. Failure to deliver on the contract, andany other circumstances that alter the settlement amount, are absorbed in thedefinition of the reference observable, allowing default and external factors tobe incorporated. This contractualisation of economic observables permits thefuture exchange of currencies in amounts that are undetermined at present.The conceptual framework considered here assumes a universe of economicobservables whose values are revealed progressively through time, which are thenutilised as the cash settlement amounts for derivative securities in one or moreidealised currencies. Liquidity constraints are not considered, and settlement ispermitted in arbitrarily large positive or negative amounts. No other propertiesof the securities are investigated, and derivatives that match at settlement inall future scenarios are assumed to be fungible.The price of the derivative must then account for discounting , the adjust-ment due to the funding of future settlements, and convexity , the cost or benefit
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Product Description
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Terms and Conditions
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Figure 1: A sample term sheet for a ‘range accrual’ derivative. The term sheetdescribes the observations (Euribor EUR 6M fixing) and calculations (indicatorfunction for a defined range) used to determine settlement amounts, togetherwith the dates on which the payments will be made. Bilateral arrangements be-tween the counterparties may prescribe further payments, such as margin andcollateral exchanges. The trade is also subject to the terms of the legal juris-diction, covering matters such as regulatory capital requirements and actions inthe event of default. All these settlements related to the derivative potentiallyimpact its valuation.extracted from dynamic hedging in volatile markets. Keeping these contribu-tions in balance is the basis for the fair pricing of the derivative. The profit/lossthat the hedged derivative position accrues over time is decomposed into twocomponents: Hedged P&L = Carry + Gamma P&L (1)where carry is the change in the value of the hedged derivative arising fromfunding costs and the decay in option value, and gamma p&l is the residualprofit/loss due to unhedgeable convexity in the relationship between the deriva-tive price and the prices of liquid underlying securities used for hedging. Efficientmarket dynamics reverts this net profit/loss toward zero, leading to a balancingequation for the equilibrium price of the derivative.2 a l u e State
Hedge (Today) Derivative (Today) Derivative (Tomorrow)
Delta P&L
Gamma P&LCarry
State (Tomorrow)State (Today)
Figure 2: Over a time interval, the profit/loss that accrues on the derivative se-curity is decomposed into the delta p&l , which is hedged using liquid underlyingsecurities, the gamma p&l , the unhedgeable residual from the convexity of theprice function, and carry , the appreciation over time. Fair pricing requires thataccrual from gamma p&l is offset by carry.Terms representing carry and gamma p&l are evident in the Black-Scholesequation for the price c [ t, s ] of a derivative security contingent on the price s ofan underlying security. In this model, the underlying price is assumed to diffuselognormally. Dynamic hedging eliminates the market risk from the volatilityof the underlying price; setting the funded return to zero then leads to theequation: 0 = ∂ c ∂t − r ( c − s ∂ c ∂ s ) + 12 σ s ∂ c ∂ s (2)for the derivative price, where r is the funding rate and σ is the lognormalvolatility of the underlying price. The derivative is delta-hedged by an offsettingposition in the underlying. Gamma p&l accrues for positive convexity whenthe underlying price is volatile, and the Black-Scholes equation balances this‘volatility × convexity’ accrual with the carry from time decay and the fundingcosts of the hedged position.Generalisations of the Black-Scholes model have equivalent terms for thesecontributions, incorporating alternative volatility models for the underlyingprices and other economic variables. The viability of this approach then de-pends on the effectiveness of the market risk transfer from derivative to under-lying provided by dynamic hedging, and the accuracy of the model comparedto realised market volatility over the lifetime of the derivative.3 Economic principles for pricing
The elementary economic principles underpinning this argument are replicabil-ity and the absence of arbitrage, both essentially algebraic constraints on thevaluation map from the observable that represents the settlement amount of thederivative to its price. As with all such principles, they are an approximationto the reality of financial markets, but their validity is assumed throughout thefollowing.The exposition rests on three founding economic principles:
The Principle of Replicability:
A security constructed as a linear combina-tion of underlying securities has price equal to the same linear combinationof underlying prices.
The Principle of No-Arbitrage:
A security that has positive settlement inall future scenarios has positive price.
The Principle of Economic Equivalence:
A security that has zero settle-ment in all possible future scenarios has zero price.The first two principles are consistency conditions that disable the constructionof arbitrages, and dictate that the valuation map is linear and positive. Prohibit-ing unattainable outcomes from impacting price, the final principle connectsthe valuation map with the forecasting model of the economy – expectationsof future outcomes are not necessarily reflected in pricing, but this weaker re-quirement removes from consideration scenarios that have zero measure in theeconomic model, and so are deemed to be impossible. The connection betweenthese economic principles and fundamental mathematical constructions is themain focus of this essay.
The principles of replicability and no-arbitrage connect price with the func-tional calculus of observables.
Let A be the space of economic observables whosevalues can be ascertained at settlement. Following the opening comments, thepayoff of the derivative in a nominated payment currency is represented as anobservable in this space. The price model is then an operation: z : a ∈ A (cid:55)→ z • a ∈ R (3)that maps the derivative payoff a to its price z • a .The principle of replicability imposes the homomorphic relationship withaddition: z • ( a + b ) = ( z • a ) + ( z • b ) (4)thus ensuring that the price of a portfolio can be determined from the pricesof its constituents. The principle of no-arbitrage does not impose the homo-morphic relationship with multiplication, z • ( ab ) = ( z • a )( z • b ), as this deniesthe possibility of extracting value from convexity, an essential property for aviable model of derivative prices. Instead, this principle imposes the weakerrequirement that the price map is positive: a ≥ ⇒ z • a ≥ z • ab ) ≤ ( z • a )( z • b ) (6)4his Cauchy-Schwarz inequality for the price model is satisfied by linear com-binations of ring homomorphisms with positive weights. The principle of economic equivalence connects price with the stochastic cal-culus of observables.
The founding economic principles apply across any timeinterval t i ≤ t j with price model given by a linear and positive operation: z ij : a j ∈ A j (cid:55)→ z ij • a j ∈ A i (7)that maps the derivative payoff a j at time t j to its price z ij • a j at time t i , where A i ⊂ A j are the subspaces of observables whose values can be ascertained at thestart and end of the interval.The principle of economic equivalence relates these operations to the modelof the economy: z ij • a j = ¯z ij • w ij a j (8)where: ¯z ij : a j ∈ A j (cid:55)→ ¯z ij • a j ∈ A i (9)is the expectation of observables at time t j conditional on observables at time t i . The strictly-positive state-price deflator w ij in this expression rescales themeasure of an event in the economic model to the Arrow-Debreu price of thecorresponding digital option. Consistency among the family of price operationsrequires that their compositions satisfy the tower law, a property that followswhen the state-price deflators take the form w ij = w i / w j for a strictly-positivenumeraire process w . The process a is then the price of a tradable security if itsatisfies: a i w i = ¯z ij • a j w j (10)for each time interval t i ≤ t j . The numeraire has the dimension of currency,so that the ratio of price over numeraire is dimensionless, and the price modelstates that this ratio is a martingale in the economic measure.The price model decomposes into two components: the specification of aneconomic model that quantifies the conditional expectations of economic observ-ables; and the identification of a numeraire process that adjusts these expecta-tions to match underlying prices. The price model is not uniquely determined bythe founding economic principles, and further principles are needed to explainthe origin of the numeraire. Completion of the model requires an understandingof the roles that funding and hedging play in the optimisation of trading activity,leading to additional guidelines that crystallise the price of the derivative. The economic principles govern pricing for all market participants, but do notfully determine price. Flexibility in the framework, exemplified by the uniden-tified numeraire w , is only resolved by looking more closely at the activitieswithin the financial institution.Trading activity is funded by issuing bonds and shares, and the costs of thisoperation are charged to the desk via the strictly-positive unsecured fundingprice u representing the unit price of funding for the institution. Benchmarkingagainst the cost of funding allows settlements at different times to be comparedon a consistent basis, and this makes the unsecured funding price the natural5andidate for numeraire in the price model. This hypothesis is overly restrictive,however, as there is no facility in the approach to mark the model to market.The contribution of funding to price is instead revealed by inspecting the fundingsettlements alongside those from the traded security in the context of the pricemodel.Consider the purchase at time t for price a and the subsequent sale at time t + dt for price a + d a of a tradable security transacted over the finite timeincrement dt . The price model relates these initial and terminal prices via thelocal martingale condition: 0 = ¯z • d aw (11)for the conditional expectation from time t + dt to time t of the dimensionlessratio a / w . The purchase of the security is funded with the sale of a / u units ofunsecured funding, returning at time t + dt the proceeds a + d a from the resaleof the security minus the costs ( a / u )( u + d u ) from the repurchase of unsecuredfunding. Feeding these settlements into the price model leads to the alternativelocal martingale condition:0 = ¯z • d a − ( a / u ) d uw + d w = ¯z • u + d uw + d w d au (12)From the perspective of the financial institution, there are now two representa-tions of the local martingale condition for the same transaction. Fortunately,these conditions are aligned when the dimensionless ratio u / w is a martingale:0 = ¯z • d uw (13)The funding of trading activity thus imposes a normalisation constraint on themodel: the numeraire does not necessarily equal the unsecured funding price,but their ratio is required to be driftless in the economic measure.Recognising the elevated status of the unsecured funding price, the pricemeasure ˆz is defined to be the measure equivalent to the economic measure ¯z with Radon-Nikodym kernel given by the martingale u / w . The price model isexpressed in terms of this measure by the local martingale condition:0 = ˆz • d a − ( a / u ) d uu + d u = ˆz • d au (14)for the price a of a tradable security. In this form, the unsecured funding price u replaces the numeraire w as discounting for the security. Further principlesare then needed to identify the price measure from the economic measure.In addition to prescribing settlements, the derivative contract may also stip-ulate terms for its funding. Defining the price of the derivative as the discountedexpectation in the price measure of its terminal settlements neglects these in-cremental funding settlements. More accurately, the derivative should be ac-counted for continuously, including any margin or collateral payments, or anyother costs incurred by the use of financial resources.The crucial consideration is the incremental settlement implied by the termsof the contract, and in this there is commonality across different derivativetypes. In each case, the derivative is associated with a market price a and a6trictly-positive funding price b used to determine the net profit/loss over thetime increment dt for the self-funded position: d a − ab d b = ( b + d b ) d ab (15)This expression, which includes funding costs but neglects additional charges forreserves against potential unwind costs, comprises the profit/loss from derivativeminus the profit/loss from funding, and the owner receives these net settlementsfor as long as they hold the derivative. Price is thus derived from the termi-nal settlements and binding provisions for funding as specified in the contract,varying according to the nature of the derivative. Futures are standardised derivatives traded on exchanges, typically with highliquidity and low transaction costs. Participants use limit and marketorders to submit prices and execute trades, thereafter committing theholder to the series of payments required to maintain the margin account.The net settlement over the time increment is the variation margin: d a (16)on the market price a of the future. In the general expression, this isequivalent to funding with the unit b = 1. Cleared derivatives extend the essential features of futures to a wider rangeof standardised derivatives. Market makers provide firm prices, and exe-cution is intermediated by a clearing house whose exposure to default riskis mitigated through the maintenance of an interest-bearing margin ac-count. The market price a serves as the reference for the variation marginsettlement: d a − ar dt (17)equivalent to funding with the cash account b = (1 + r dt ) t/dt that accruesinterest at the rate r specified by the clearing house. Collateralised derivatives are bilateral transactions commonly used for moreilliquid structures. Default risk is mitigated through the provision of col-lateral, whose terms are dictated by the legal relationship between thetwo parties. Collateral is continuously rebalanced to ensure it covers thederivative valuation. The net settlement over the time increment is then: d a − ab d b (18)being the variation of the derivative valuation a and the collateral valua-tion b in the collateral account. Uncollateralised derivatives offer bespoke derivative features to clients with-out requiring the exchange of collateral. The cost of unsecured funding,covering the anticipated closing price of the derivative, is charged back tothe desk. The net settlement in the trading book is: d a − au d u (19)netting the variation on the derivative valuation a and the incrementalmaintenance cost for unsecured funding with price u .7 + 𝑑𝑡𝑡 Collateralised derivative:
BankCounterparty a + 𝑑 aababa net profit/loss = 𝑑 a − a b 𝑑 b Collateral/FundingCashDerivative 𝑡 + 𝑑𝑡𝑡
Future:
BankExchange 𝑑 a net profit/loss = 𝑑 a 𝑡 + 𝑑𝑡𝑡 Cleared derivative:
BankClearing House 𝑑 a − ar 𝑑𝑡 net profit/loss = 𝑑 a − ar 𝑑𝑡 𝑡 + 𝑑𝑡𝑡 Uncollateralised derivative:
FundingCounterparty auaua net profit/loss = 𝑑 a − a u 𝑑 u Bank au ( u + 𝑑 u ) a + 𝑑 aa Figure 3: The execution and settlement exchanges associated with four types ofderivative. Common to all these is the expression for the net profit/loss over thetime interval, equal to the price change minus the cost of funding as prescribedby the derivative contract. 8or each of these types, holding the derivative becomes a commitment to aseries of self-funded net settlements in the trading book, entered at zero costand returning an amount that nets the variations on the market and fundingvaluations for the derivative. Inserting these initial and terminal prices into theprice model leads to the local martingale condition:0 = ˆz • d a − ( a / b ) d bu + d u = ˆz • b + d bu + d u d ab (20)for the conditional expectation from time t + dt to time t of the dimensionlessprice ratio a / b . Margin settlements are made at the endpoints of each intervalin a series t < · · · < t n of times, and the local condition above is compoundedto the martingale condition: a i b i = ˆz ij • j − (cid:89) k = i b k +1 / u k +1 ˆz k k +1 • b k +1 / u k +1 a j b j (21)The role of numeraire in this expression is taken by the funding price, and theratio of market price over funding price is a martingale in a measure associatedwith the funding that is equivalent to the price measure.Utilising the funding price as numeraire resolves the discounting questionfor the settlements of the derivative. The correction to the price measure isrequired to ensure consistency across the models generated for different fundingarrangements. In each case, the change of measure is driven by the convexitybetween the funding price b of the derivative and the unsecured funding price u ofthe financial institution, with Radon-Nikodym kernel given by the dimensionlessfunding ratio b / u drift-adjusted to be a martingale in the price measure.A natural hierarchy exists in the market, with futures and cleared deriva-tives providing the price transparency and liquidity to mark-to-market bilateralderivatives. This market information is absorbed in the price model throughthe mechanism of hedging, with the price measure emerging from a martingalecondition derived from fair pricing principles. Bridging the gap between the subjective expectations encapsulated in the modelfor the economy and the market expectations expressed by the prices of liquidsecurities is not without ambiguity, but guidance can be found in traditionalmethods of portfolio optimisation. In the following, this ambiguity is resolvedby removing bias in optimal strategies based on the mean and variance of portfo-lio returns, exploiting the market stratification into liquid underlying securities used for hedging and derivative securities marked-to-market following fair pric-ing principles.The performance of the strategy is measured relative to the unsecured fund-ing price u of the financial institution using the statistics from the economicmodel ¯z . For the strategy with price a , performance is quantified by the meanand variance of the increment d ( a / u ), delimiting the expected range for the re-turn on the strategy in proportion to the reference price. Benchmarking againstunsecured funding removes the dependency on the denominating currency andallows the consistent evaluation of settlements at different times.9hile this is an effective method for combining economic and market ex-pectations in the price model, it is not necessarily an accurate representation ofrisk management activities in a financial institution. Both the economic modeland the unsecured funding price depend on the institution, and the resultingprice model is sensitive to these components. Furthermore, using mean andvariance as targets for optimisation may not be suited to the management ofextreme events, and alternative statistics will lead to variations in the derivativeprice. For these reasons, the principle of portfolio optimisation that completesthe price model is here held separate from the three core economic principles. The market provides access to a set of liquid underlying securities whose vectorsof market prices p and funding prices q are directly observed. Normalised usingthe unsecured funding price u , the return from the underlying securities overthe trading interval is: R p = d p − ( p / q ) d qu + d u = q + d qu + d u d pq (22)The return is adjusted to account for the convexity between underlying andunsecured funding, captured in the dimensionless funding ratio q / u . The antic-ipated range for the return vector R p is quantified in terms of the mean vector M p and covariance matrix V p : M p = ¯z • R p (23) V p = ¯z • R p − ( ¯z • R p ) The effectiveness of the investment strategy then depends on its success inidentifying portfolios that minimise risk for a target expected return.The expected performance of the underlying portfolio is measured by themean m = α · M p and variance v = α · V p α of its return, where the compositionof the portfolio is specified by the weights α . The economic model considers thereturn on portfolios with zero variance to be guaranteed, and unless the mean isalso zero this implies the existence of an arbitrage. The technical requirementfor the economic model to avoid arbitrage is:ker[ V p ] · M p = { } (24)For convenience, the stronger condition ker[ V p ] = { } is assumed in the follow-ing. In the more general case, the optimal strategies developed here are validonly when the economic model satisfies the no-arbitrage constraint, and arethen determined only up to the addition of a zero-variance portfolio.Optimal portfolio weights, achieving the minimum possible variance for atarget mean, satisfy the stationarity condition:0 = δ ( α · V p α + λ ( α · M p − m )) (25)= δα · (2 V p α + λ M p )under variations δα of the portfolio α , where the Lagrange multiplier λ mapsto the risk appetite of the investor. The optimal portfolio is thus: α ∝ V − p M p (26)10he mean and variance achieved by this strategy lie on the parabola specifiedby the relation: m √ v = (cid:113) V − p M p · M p (27)where the expression on the right quantifies market sentiment on the expectedreturn that is acceptable for a unit of risk. The optimal investment strategyminimises variance by diversifying risks across the underlying securities, withthe risk appetite determining the position of the strategy on the efficient frontierparabola in mean-variance space. Now consider a stratified market comprising the underlying securities with mar-ket prices p and funding prices q and a derivative security with market price a and funding price b . The underlying securities constitute the hedge mar-ket for the derivative security, with the observed prices of the former used tomark-to-market the latter via fair pricing principles.The strategy is optimised against the joint moments of the underlying return R p and the normalised return R a on the derivative: R a = d a − ( a / b ) d bu + d u = b + d bu + d u d ab (28)In addition to the mean vector M p and covariance matrix V p for the underlyings,the performance of the combined portfolio is quantified in terms of the mean M a and variance V a for the derivative: M a = ¯z • R a (29) V a = ¯z • R a − ( ¯z • R a ) and the cross-covariance vector C pa : C pa = ¯z • R p R a − ( ¯z • R p )( ¯z • R a ) (30)Strategies for optimising the hedged derivative portfolio are determined fromthese statistics.The introduction of the derivative expands the opportunities for diversifica-tion, leading to an incremental improvement in the expected return for a unitof risk from the optimal strategy: (cid:20) V p C pa C t pa V a (cid:21) − (cid:20) M p M a (cid:21) · (cid:20) M p M a (cid:21) = V − p M p · M p + ( M a − V − p M p · C pa ) V a − V − p C pa · C pa (31)Activity in the underlying market uncovers the performance target of partici-pants, and this suggests a principle for marking-to-market the fair price of thederivative. The Principle of Portfolio Optimisation:
The derivative security does notincrease the expected return for a unit of risk.11s can be seen from the expression above, this principle leads to the followingcondition for the fair price of the derivative: M a = V − p M p · C pa (32)Any other price for the derivative could be exploited in a strategy that out-performs the underlying market, and so market efficiency drives the price towardthis equilibrium.An alternative approach looks directly at the optimal portfolio that hedgesthe derivative return. For hedge weights β , the return on the hedged derivativehas mean m and variance v given by: m = M a − β · M p (33) v = V a − β · C pa + β · V p β The optimal hedge weights, achieving the minimum possible variance, then sat-isfy the stationarity condition:0 = δ ( V a − β · C pa + β · V p β ) (34)= 2 δβ · ( V p β − C pa )under variations δβ of the portfolio β . The hedge portfolio is thus: β = V − p C pa (35)The mean and variance achieved by this strategy are: m = M a − V − p M p · C pa (36) v = V a − V − p C pa · C pa The hedge does not eliminate the market risk of the derivative, but this strategyreduces it to the minimum achievable. This suggests an alternative principle fordetermining the fair price of the derivative.
The Principle of Portfolio Optimisation:
The return on a hedged deriva-tive security is unbiased.The implied expression for the fair price of the derivative is then: M a = V − p M p · C pa (37)The efficient market equilibrates at the level that removes bias in the residualhedged return, thereby calibrating the derivative price to the market. Conve-niently, both principles for marking-to-market the derivative lead to the samefair pricing expression.Market completeness – the capacity of the market to replicate the sensi-tivities of the derivative price – impacts the residual variance of the hedgedderivative. Consider the underlying market with two sectors whose returns are R p and R o respectively. The optimal hedge in this case comprises a portfolio β p in the first sector and a portfolio β o in the second sector: (cid:20) β p β o (cid:21) = (cid:20) V − p C pa (cid:21) (38)+ (cid:20) − V − p C po ( V o − C t po V − p C po ) − ( C oa − C t po V − p C pa )( V o − C t po V − p C po ) − ( C oa − C t po V − p C pa ) (cid:21) v = V a − V − p C pa · C pa (39) − ( V o − C t po V − p C po ) − ( C oa − C t po V − p C pa ) · ( C oa − C t po V − p C pa )Each expansion in the range of hedge securities enhances the opportunities foroffsetting the risk of the derivative.By setting the expected return on the hedged derivative to zero, the pricemodel interpolates between two accounting methodologies. At one extreme,when the underlying market is empty price is identified with expectation. Atthe opposite extreme, when the underlying market is complete price is deter-mined by replication. The extent to which the price model relies on expectationversus replication depends on market completeness: the sensitivity of price tothe economic model is mitigated by the calibration to available market prices;conversely, the economic model plugs the gap in pricing when liquidity is limited. For the observables within a time interval, the economic model provides twomeasures: the forecast measure and the empirical measure.
Forecast measure:
The measure ¯z s seen at the start of the time interval,capturing the predicted statistics for the observables. Empirical measure:
The measure ¯z e seen at the end of the time interval,capturing the observed statistics for the observables.Construction of the economic model begins with a reduction to the macroscopicvariables that describe the economy, and proceeds with the consistent assign-ment of probabilities for the values of these observables. Evidence refines theseassignments: the measure starts as informed guess and ends as statistical anal-ysis. Performance of the economic model is then quantified by the gap betweenits forecast and empirical measures.These measures respectively generate the hedge strategies: β s = V s p − C s pa (40) β e = V e p − C e pa The first portfolio is the optimal strategy computed at the start of the interval,based on initial expectations for the future of the economy. The second portfoliois the optimal strategy computed at the end of the interval, hedging with thebenefit of hindsight according to the realised market volatility over the interval.The executed hedge strategy is necessarily derived from the forecast measure.Hedge performance is then quantified by the empirical mean m and variance v of the return on the hedged derivative: m = ( M e a − V e p − M e p · C e pa ) + ( β e − β s ) · M e p (41) v = ( V e a − V e p − C e pa · C e pa ) + ( β e − β s ) · V e p ( β e − β s )13 upport for universal measureSupport for forecast measure Support for empirical measure Anticipated events Unanticipated events
Figure 4: The support for the empirical measure divides into the anticipatedevents shared with the forecast measure and the unanticipated events with zeroforecast measure.The empirical variance decomposes into two components. The market risk com-ponent is the minimum variance achievable in the empirical measure using theempirical hedge strategy. The model risk component is the additional variancedue to the sub-optimality of the forecast hedge strategy.The forecast and empirical measures are not necessarily equivalent: eventsthat are anticipated may not materialise; conversely, and more dangerously,unanticipated ‘black swan’ events accounted in the empirical measure are ne-glected in the forecast measure. Introduce a universal measure z that supportsthe events for both measures, with Radon-Nikodym kernels expressed as: d ¯z s d z = A (42) d ¯z e d z = A + ε D The empirical measure corrects the forecast measure via the term ε D . Consid-ering the case when ε is small, the difference between the empirical and forecasthedges is expanded: β e − β s = V s p − Ω + O [ ε ] (43)where: Ω = z • ε D ( R p − ¯z s • R p )( R a − β s · R p ) (44)Sub-optimality of the forecast hedge is proportional to the deviation of theempirical measure from the forecast measure. It is also proportional to theresidual return from the hedged derivative, confirming that market completenessis a determinant for the performance of the hedge.14nticipated and unanticipated events are separated in the correction term: D = SA + B (45)where B satisfies AB = 0. Relative to the universal measure, the empiricalmeasure deviates from the forecast measure by the kernel ε ( SA + B ) where ε SA corrects the measure for anticipated events and ε B appends the measure forunanticipated events. The expression for the hedge error decomposes as:Ω = Ω a + Ω b (46)The first term in the decomposition:Ω a = ¯z s • ε S ( R p − ¯z s • R p )( R a − β s · R p ) (47)is the hedge error arising from the adjustment S that rescales the probabilitiesof anticipated events following the assessment of actual market volatility overthe interval. The second term in the decomposition:Ω b = ¯z e • ( B (cid:54) = 0)( R p − ¯z s • R p )( R a − β s · R p ) (48)is the hedge error arising from the unanticipated events indicated by the con-dition B (cid:54) = 0. The first term, the ‘known-unknown’ contribution to model risk,is evaluated within the context of the forecast measure by estimating the errorbounds for model parameters. The second term, the ‘unknown-unknown’ con-tribution to model risk, is harder to assess in advance as it is measured againstevents that are initially assumed to be impossible. The founding economic principles dictate that the incremental change in theprice ratio of the derivative security satisfies the local martingale condition:0 = ˆz • b + d bu + d u d ab (49)where the price measure ˆz adjusts the economic measure ¯z , calibrating to theunderlying securities via the local martingale condition:0 = ˆz • q + d qu + d u d pq (50)for the incremental changes in the price ratios of the underlying securities. Thisprice model satisfies the principles of replication and economic equivalence byconstruction; it satisfies the principle of no-arbitrage when the price measure ispositive.While the role of economic expectation in price determination is clear fromthese local martingale conditions, the economic principles and market calibra-tions do not uniquely determine the price measure. Taking its steer from portfo-lio optimisation, the previous section derives an additional condition for the fairprice of the derivative by removing bias in the return from the hedged portfolio.This resolves the ambiguity in the price model.15xpanding the terms in the fair pricing condition that include the return onthe derivative security leads to: ¯z • R a = ¯z • V − p M p · ( R p − ¯z • R p ) R a (51)This is re-arranged to the following relation for the incremental change in thederivative price ratio: 0 = ¯z [ α ] • b + d bu + d u d ab (52)where the dimensionless parameters α are marked-to-market via the relation:0 = ¯z [ α ] • q + d qu + d u d pq (53)for the incremental changes in the underlying price ratios. The price measure ˆz = ¯z [ α ] in these expressions is equivalent to the economic measure ¯z , withRadon-Nikodym kernel adjusting for the returns on the underlying securities: d ¯z [ α ] d ¯z = 1 − α · ( q + d qu + d u d pq − ¯z • q + d qu + d u d pq ) (54)In this representation, the funding price assumes the mantle of numeraire, andthe economic measure is tweaked to calibrate the price measure to market prices.Direct comparison with the general expression confirms that this price modelsatisfies the principles of replication and economic equivalence. It does not,however, satisfy the principle of no-arbitrage: if the underlying price changesdeviate significantly from their expectations, the kernel could be negative. Thishighlights the inadequacies of mean-variance optimisation as a method of pricedetermination. By under-estimating the impact of tail events, these performancemetrics leave open the possibility of arbitrage.The defect in the price model is resolved by substituting the linear kernelwith the following exponential alternative: d ¯z [ α ] d ¯z = exp (cid:20) − α · q + d qu + d u d pq (cid:21) ¯z • exp (cid:20) − α · q + d qu + d u d pq (cid:21) (55)This variant of the price model is justified as it converges to the original whenthe underlying return is small and avoids arbitrage when the underlying return islarge, reweighting the performance measure to avoid arbitrage from tail events.The market calibration is solved by iterating the Newton-Raphson scheme: α (cid:55)→ α + V p [ α ] − M p [ α ] (56)where the mean vector M p [ α ] and covariance matrix V p [ α ] are computed usingthe equivalent measure ¯z [ α ] from the preceding iteration. Starting with α = 0,convergence of this scheme then relies on the invertibility of the covariance ateach iteration. Calibration determines a portfolio of underlying securities whosereturn is exponentiated to generate the kernel for the price measure relative tothe economic measure, with weights α that control the deviation of the under-lying prices from their expectations. 16 (cid:38) (cid:36)(cid:37) MAXIMUM ENTROPY PRICE MODEL
The ingredients of the maximum entropy price model are the economic measure ¯z quantifying the expectations of economic observables, the unsecured fundingprice u of the financial institution, and the observed market prices p and fundingprices q of liquid underlying securities.For a derivative security with market price a and funding price b , the pricemodel is given by the local martingale condition:0 = ¯z • exp (cid:20) − α · q + d qu + d u d pq (cid:21) b + d bu + d u d ab for the change in the derivative price ratio a / b over the time step dt , where thecoefficients α mark the model to the underlying prices via the local martingalecondition: 0 = ¯z • exp (cid:20) − α · q + d qu + d u d pq (cid:21) q + d qu + d u d pq for the change in the underlying price ratios p / q over the time step dt .The price model assumes the expected return on the funded derivative,normalised by the unsecured funding price, is zero in the price measure thatcalibrates the economic measure to underlying prices with minimum relativeentropy.The price expression is compounded on the margin settlement schedule togenerate the martingale condition for the derivative price ratio: a i b i = ¯z ij • j − (cid:89) k = i exp[ − α k · ( q k +1 / u k +1 ) d ( p k / q k )] b k +1 / u k +1 ¯z k k +1 • exp[ − α k · ( q k +1 / u k +1 ) d ( p k / q k )] b k +1 / u k +1 a j b j (57)This expression determines the price of the derivative as the expectation ofits discounted terminal settlement, where discounting matches funding and themeasure is adjusted to calibrate underlying prices and accommodate convexitybetween the funding ratio and the returns on underlying securities.In a Bayesian interpretation of the approach, the economic measure is con-sidered to be the maximum entropy state representing the best assumptions onprice in the absence of market information. Assessed relative to this state, theprice measure with exponential form for the kernel is the measure that max-imises entropy while calibrating to the available prices of underlying securities.The price model, originally derived from the principles of no-arbitrage and port-folio optimisation, is then equivalently derived on the assumption of maximumentropy. The Maximum Entropy Principle:
The price measure minimises entropyrelative to the economic measure subject to calibration to the prices ofliquid underlying securities.Market intelligence on price is incomplete, and this principle proposes that theinformation vacuum is filled by the expectations quantified in the economic17odel. This is achieved by minimising the relative entropy of the price measurefrom the economic measure, subject to the mark-to-market constraints. Thestationarity condition, whose solution is the exponential kernel d ¯z [ α ] /d ¯z definedabove, is: 0 = δ ( ¯z • W log[ W ] + λ ( ¯z • W −
1) + α · ¯z • W q + d qu + d u d pq ) (58)= ¯z • δ W (log[ W ] + (1 + λ ) + α · q + d qu + d u d pq )for variations δ W of the kernel W . The Lagrange multipliers λ and α are solvedfor the normalisation constraint ¯z • W = 1 and the market calibrations.Maximum entropy states arise in thermodynamic applications for the macro-scopic variables of a system whose microscopic variables evolve on a muchshorter timescale. In this perspective, the portfolio weights α are the inverse-temperatures for the underlying securities – ‘hot’ securities discover the equilib-rium of the economic model, while ‘cold’ securities have prices that deviate fromexpectations. This analogy between non-equilibrium thermodynamics and eco-nomic agents discovering price through trading activity suggests an alternativeroute to the price model which will not be pursued further here. The incremental price condition is compounded on the margin settlement sched-ule, and the ratio of market price over funding price for the derivative is a mar-tingale on this schedule in an equivalent measure that calibrates the economicmeasure to underlying returns in each settlement period. If margin is settledfrequently, this can be approximated by a continuous-time model.The technical conditions that enable the perpetual subdivision of a processare captured in the L´evy-Khintchine representation of its stochastic differentialequation. The evolution of the continuous-time vector process s in the measure z is described by its volatility parameters ( µ, ν, φ ), where µ and ν are the ratesof change for the mean and covariance in the continuous diffusion, and φ is thefrequency density for jumps in the discontinuous diffusion. The L´evy-Khintchinerepresentation of the stochastic differential equation is then:log z • exp[ k · d s ] dt = (59) k · µ + 12 k · νk + (cid:90) (exp[ k · j ] − φ [ j ] dj + O [ dt ]The continuous diffusion captures the normal conditions of the evolution, whosebehaviour is accurately described over short intervals by the mean and covari-ance in a Gauss distribution for the increment. The discontinuous diffusionadjusts the skew and kurtosis and other higher moments of the incrementaldistribution, and enables the modelling of regime switches where the normalcorrelations between state variables are inverted.Differentiating the stochastic differential equation repeatedly with respectto the conjugate variable k brings down a power of the increment d s in the inte-grand. This observation is used to identify the stochastic differential equation18 ime = 2Time = 1 Continuous diffusion
Time = 3
Discontinuous diffusion
Figure 5: The evolution of the continuous-time process has a continuous compo-nent described by its mean and covariance rates and a discontinuous componentdescribed by the frequency of jumps of different sizes.19or the derived process f [ t, s ] as a function of time and the state process:log z • exp[ k df [ t, s ]] dt = (60) k ( ˙ f [ t, s ] + f (cid:48) [ t, s ] · µ + 12 tr[ f (cid:48)(cid:48) [ t, s ] ν ]) + 12 k f (cid:48) [ t, s ] · νf (cid:48) [ t, s ]+ (cid:90) (exp[ k ( f [ t, s + j ] − f [ t, s ])] − φ [ j ] dj + O [ dt ]This is the Itˆo lemma in the L´evy-Khintchine representation. Applications ofthe lemma transform the analysis of continuous-time processes into the realmof partial differential-integral equations, and are used in the following to derivehedge strategies and fair pricing conditions for continuously-settled derivatives. The price model is based on the increments d ( p / q ) and d ( a / b ) for the price ratiosof underlying and derivative securities and the increments d ( q / u ) and d ( b / u ) forthe ratios of the corresponding funding prices over the unsecured funding price.These increments are modelled in the economic measure ¯z by their combinedstochastic differential equation in the L´evy-Khintchine representation, enablingthe construction of optimal hedge strategies and the fair pricing condition fromthe volatility parameters. The model then captures the market convexity arisingfrom nonlinearity in the relationships between market prices and the fundingconvexity resulting from differences in funding arrangements.To reduce the dimensionality of the problem, in this section it is assumedthat the underlying and derivative securities have the same funding price: q = b = fu (61)scaling the unsecured funding price by the strictly-positive funding ratio f . Thissimplification puts the focus primarily on market convexity. The neglectedconvexities between underlying and derivative funding can be accommodatedin the framework by extending the volatility parameters to allow decorrelationbetween the funding prices.The joint evolution for the funding ratio f and the price ratios s = p / q and c = a / b of underlying and derivative securities is modelled in the economicmeasure ¯z by the stochastic differential equation:log ¯z • exp[ k f d f / f + k s · d s + k c d c ] dt = (62) k f k s k c · µ f µ s µ c + 12 k f k s k c · ν f ν t fs ν fc ν fs ν s ν sc ν fc ν t sc ν c k f k s k c + (cid:90) (exp[ k f j f + k s · j s + k c j c ] − φ [ j f , j s , j c ] dj f dj s dj c + O [ dt ]The continuous diffusion is described by the mean rates µ f , µ s and µ c and thecovariance rates ν f , ν s and ν c of the funding and price ratios, together with thecross-covariance rates ν fs , ν fc and ν sc between them. The discontinuous diffusion20s described by the frequency density φ [ j f , j s , j c ] for jumps j f , j s and j c in thefunding and price ratios.The model is used to generate statistics in the equivalent measure ¯z [ α ], wherethe parameters α that adjust the economic measure will be used to mark themodel to market. The returns R p and R a are expressed in terms of the increments d f / f , d s and d c : R p = f (1 + d ff ) d s R a = f (1 + d ff ) d c (63)The statistics for the underlying and derivative are encapsulated in the stochas-tic differential equation for the returns as the coefficients in its series expansionwith respect to the conjugate variables. The Itˆo lemma derives the expression:log ¯z [ α ] • exp[ k p · R p + k a R a ] dt = (64) f ( k p · ( µ s [ α ] + ν fs ) + k a ( µ c [ α ] + ν fc ))+ f ( 12 k p · ν s k p + 12 k a ν c + k a k p · ν sc )+ (cid:90) (exp[ f (1 + j f )( k p · j s + k a j c )] − φ [ α ][ j f , j s , j c ] dj f dj s dj c + O [ dt ]with adjusted volatility parameters: µ s [ α ] = µ s − ν s f α (65) µ c [ α ] = µ c − f α · ν sc φ [ α ][ j f , j s , j c ] = exp[ − f (1 + j f ) α · j s ] φ [ j f , j s , j c ]The performance of the hedge strategy and the fair price of the derivative aredetermined using the first and second moments extracted from this equation.The covariances of the returns in the measure ¯z [ α ] are: (cid:20) V p [ α ] C pa [ α ] C t pa [ α ] V a [ α ] (cid:21) = f (cid:20) ˙V s [ α ] ˙C sc [ α ] ˙C t sc [ α ] ˙V c [ α ] (cid:21) dt + O [ dt ] (66)where the variance rates are given by: ˙V s [ α ] = ν s + (cid:90) (1 + j f ) j s φ [ α ][ j f , j s , j c ] dj f dj s dj c (67) ˙V c [ α ] = ν c + (cid:90) (1 + j f ) j c φ [ α ][ j f , j s , j c ] dj f dj s dj c and the covariance rate is given by: ˙C sc [ α ] = ν sc + (cid:90) (1 + j f ) j s j c φ [ α ][ j f , j s , j c ] dj f dj s dj c (68)The optimal hedge strategy minimises the variance for the return on the hedgedderivative. Quantifying performance using the equivalent measure with param-eters α , the optimal hedge weights are: β = ˙V s [ α ] − ˙C sc [ α ] + O [ dt ] (69)21 e r i v a t i v e Underlying
Hedge Derivative ∙ − ∙ Figure 6: The delta hedge matches the continuous price movements for thederivative with an offsetting position in the underlying. Discontinuous pricemovements invalidate the hedge when there is convexity in the relationshipbetween the underlying and derivative prices.This strategy does not necessarily eliminate the market risk of the hedged deriva-tive. The residual variance of the hedged portfolio is calculated as: v = f ( ˙V c [ α ] − ˙V s [ α ] − ˙C sc [ α ] · ˙C sc [ α ]) dt + O [ dt ] (70)including contributions from the continuous and discontinuous components ofthe price diffusion. The hedge strategy strikes a balance between these contri-butions, optimising against both the infinitesimal price moves of the continuousdiffusion and the discrete price moves of the discontinuous diffusion accordingto their relative likelihoods.The continuous contribution dominates when J Φ is small, where J is themaximum size of jumps in the support of the frequency density and Φ is thetotal frequency. The optimal hedge expands as: β = δ + γ + O [ dt ] (71)The contribution δ from the continuous diffusion is the delta hedge againstinfinitesimal changes in the prices, and this is corrected by the adjustment γ that accounts for the discontinuous diffusion: δ = ν − s ν sc (72) γ = (cid:90) (1 + j f ) ( j c − δ · j s )( ν − s j s ) φ [ α ][ j f , j s , j c ] dj f dj s dj c + O [( J Φ) ]22rice discontinuities corrupt the delta hedge strategy when there is convexity inthe relationship between underlying and derivative, requiring a correction to thestrategy in proportion to the deviation ( j c − δ · j s ) from the linear approximation.The delta hedge does not depend on the parameters α that adjust the economicmeasure, nor is it impacted by the volatility of the funding ratio. Sensitivitiesto these elements are only introduced when there is indeterminacy in the sizeand direction of the underlying and derivative price jumps.The means of the returns in the measure ¯z [ α ] are: (cid:20) M p [ α ] M a [ α ] (cid:21) = f (cid:20) ˙M s [ α ] ˙M c [ α ] (cid:21) dt + O [ dt ] (73)where the mean rates are given by: ˙M s [ α ] = µ s [ α ] + ν fs + (cid:90) (1 + j f ) j s φ [ α ][ j f , j s , j c ] dj f dj s dj c (74) ˙M c [ α ] = µ c [ α ] + ν fc + (cid:90) (1 + j f ) j c φ [ α ][ j f , j s , j c ] dj f dj s dj c The price model derived from the maximum entropy principle assumes the meanof the derivative return is zero in the price measure that calibrates the param-eters α so that the means of the underlying returns are also zero: ˙M s [ α ] = 0 (75) ˙M c [ α ] = 0The calibration constraint is solved for the parameters that are then used inthe price equation for the derivative. These expressions simplify when the un-derlying price diffusion is continuous, so that the frequency density takes theform: φ [ j f , j s , j c ] = δ [ j s ] φ [ j f , j c ] (76)In this case, the solution for the parameters is entered into the condition for thederivative return to generate the price equation: µ c + ν fc + (cid:90) (1 + j f ) j c φ [ j f , j c ] dj f dj c = ν − s ν sc · ( µ s + ν fs ) (77)identifying the drift of the derivative price.The assumption of continuous diffusion for the underlying prices can bereasonable in benign market conditions. Liquidity is a desirable property ofmarket hedge instruments, and the price transparency and trading volumes onexchanges facilitate continuous hedging. In stressed conditions, or for hedgingwith less liquid instruments, the economic model should reflect the potential dif-ficulties with continuous hedging by incorporating a discontinuous componentfor the underlying price diffusion. The calibration parameters are then deter-mined from the underlying drift condition via the Newton-Raphson scheme. In the previous section, the drift conditions for the underlying and derivativereturns are derived from the maximum entropy principle on the assumption of23ontinuous margin settlement. The approach is demonstrated in this section byconsidering in detail the economic model:log ¯z • exp[ k s d s + k σ · d σ ] dt = (78) k s µ s + k σ · µ σ + 12 k s ν s + 12 k σ · ν σ k σ + k s k σ · ν s σ + (cid:90) (exp[ k s j s + k σ · j σ ] − φ [ j s , j σ ] dj s dj σ + O [ dt ]describing the evolution in the economic measure of a single underlying price s and an additional state vector σ used to model its volatility. Funding convexityis neglected by setting f = 1, and the mean rates µ s and µ σ , covariance rates ν s , ν σ and ν s σ , and jump frequency φ [ j s , j σ ] are all assumed to be functions of time t and the state variables s and σ . Many popular models for pricing derivativesare included in this setup. Black-Scholes-Merton model:
This model assumes a stationary evolutionfor the underlying price with constant volatility:log ¯z • exp[ k s d s / s ] dt = (79) k s µ + 12 k s σ + (cid:90) ∞ j = − (exp[ k s j ] − ψ [ j ] dj + O [ dt ]The model parameters are the mean rate µ , the volatility rate σ , and thefrequency of jumps ψ [ j ] for the underlying price. Heston model:
This model assumes a stationary evolution for the underlyingprice with stochastic volatility following a correlated square-root process:log ¯z • exp[ k s d s / s + k ν d ν ] dt = (80) k s µ + k ν κ ( θ − ν ) + 12 k s ν + 12 k ν ξ ν + k s k ν ρξ ν + O [ dt ]The additional state variable ν is interpreted as the instantaneous varianceof the underlying price. The model parameters are the mean rate µ of theunderlying price, the mean reversion rate κ and reversion level θ of thevariance, the volatility of variance ξ , and the correlation ρ between theprice and its variance. SABR model:
This model assumes the underlying price has constant elastic-ity of variance with stochastic volatility following a correlated lognormalprocess: log ¯z • exp[ k s d s + k σ d σ + k α d α + k β d β + k ρ d ρ ] dt = (81)12 k s σ s β + 12 k σ α σ + k s k σ ρασ s β + O [ dt ]The additional state variables are the instantaneous volatility σ of theunderlying price, the volatility of volatility α , the elasticity of variance β ,and the correlation ρ between the price and its volatility.24he maximum entropy principle locates the price measure ˆz among theequivalent measures ¯z [ α ] parametrised by the weight α for the underlying return.The evolution of the state variables in this measure is modelled by:log ¯z [ α ] • exp[ k s d s + k σ · d σ ] dt = (82) k s µ s [ α ] + k σ · µ σ [ α ] + 12 k s ν s + 12 k σ · ν σ k σ + k s k σ · ν s σ + (cid:90) (exp[ k s j s + k σ · j σ ] − φ [ α ][ j s , j σ ] dj s dj σ + O [ dt ]with adjusted parameters: µ s [ α ] = µ s − αν s (83) µ σ [ α ] = µ σ − αν s σ φ [ α ][ j s , j σ ] = exp[ − αj s ] φ [ j s , j σ ]These adjustments modify the mean rates and jump frequency of the pricediffusion via the control variable α , which is used to calibrate the price measureto the underlying returns.Market completeness for the derivative depends on the availability of hedgesecurities and the nature of their joint price dynamics. In the following, hedgestrategies are considered for the derivative with price c = c [ t, s , σ ] expressed asa function of time and the state variables. Hedging with only the underlyingneglects the contributions to volatility from the additional state variables, leav-ing residual risk in the portfolio. Hedge performance is improved if the marketincludes options on the underlying that mark the state variables.In the incomplete hedging scenario, the returns on the hedge and derivativesecurities are: R p = d s R a = d c (84)The means of the returns in the equivalent measure ¯z [ α ] are: (cid:20) M p [ α ] M a [ α ] (cid:21) = (cid:20) ˙M s [ α ] ˙M c [ α ] (cid:21) dt + O [ dt ] (85)where the mean rates are given by: ˙M s [ α ] = µ s [ α ] + (cid:90) j s φ [ α ][ j s , j σ ] dj s dj σ (86) ˙M c [ α ] = ∂c∂t + ∂c∂ s µ s [ α ] + ∂c∂ σ · µ σ [ α ] + 12 ∂ c∂ s ν s + 12 tr[ ∂ c∂ σ ν σ ] + ∂ c∂ s ∂ σ · ν s σ + (cid:90) j c φ [ α ][ j s , j σ ] dj s dj σ The jump j c for the derivative price, appearing in the integrand of the discontin-uous contribution, is defined in terms of the jumps j s and j σ for the underlyingprice and state variables: j c = c [ t, s + j s , σ + j σ ] − c [ t, s , σ ] (87)The equation for the fair price of the derivative is obtained by setting themeans of the underlying and derivative returns to zero in the price measure25 z = ¯z [ α ], with the first condition used to calibrate the weight α that is thenapplied in the second condition to determine the derivative price:0 = µ s − αν s + (cid:90) j s exp[ − αj s ] φ [ j s , j σ ] dj s dj σ (88)0 = ∂c∂t + ∂c∂ σ · ( µ σ − αν s σ ) + 12 ∂ c∂ s ν s + 12 tr[ ∂ c∂ σ ν σ ] + ∂ c∂ s ∂ σ · ν s σ + (cid:90) ( j c − ∂c∂ s j s ) exp[ − αj s ] φ [ j s , j σ ] dj s dj σ These expressions simplify when the underlying price diffusion is continuous, sothat the frequency density takes the form: φ [ j s , j σ ] = δ [ j s ] φ [ j σ ] (89)In this case, the calibration constraint is solved by α = µ s /ν s , and the priceequation becomes:0 = ∂c∂t + ∂c∂ σ · ( µ σ − µ s ν s ν s σ ) + 12 ∂ c∂ s ν s + 12 tr[ ∂ c∂ σ ν σ ] + ∂ c∂ s ∂ σ · ν s σ (90)+ (cid:90) j c φ [ j σ ] dj σ The partial differential-integral equation for the fair price of the derivative issolved against the boundary conditions provided by the terminal settlements,as specified in the derivative contract.Now suppose that the hedge market includes options on the underlying, withprices o = o [ t, s , σ ] expressed as functions of time and the state variables. Fur-ther assume that the options mark the additional state variables, with impliedvolatility function σ = σ [ t, s , o ] satisfying: σ [ t, s , o [ t, s , σ ]] = σ (91) o [ t, s , σ [ t, s , o ]] = o These options enable the hedging of all the state variables, thereby reducing theresidual risk of the hedged derivative.In the complete hedging scenario, the returns on the hedge and derivativesecurities are: R p = (cid:20) d s d o (cid:21) R a = d c (92)The covariances of the returns in the economic measure ¯z are: (cid:20) V p C pa C t pa V a (cid:21) = ˙V s ˙C t so ˙C sc ˙C so ˙V o ˙C oc ˙C sc ˙C t oc ˙V c dt + O [ dt ] (93)where the variance rates are given by: ˙V s = ν s + (cid:90) j s φ [ j s , j σ ] dj s dj σ (94) ˙V o = ( ∂o∂ s ) ν s + ∂o∂ σ · ν σ ∂o∂ σ + 2 ∂o∂ s ∂o∂ σ · ν s σ + (cid:90) j o φ [ j s , j σ ] dj s dj σ ˙V c = ( ∂c∂ s ) ν s + ∂c∂ σ · ν σ ∂c∂ σ + 2 ∂c∂ s ∂c∂ σ · ν s σ + (cid:90) j c φ [ j s , j σ ] dj s dj σ ˙C so = ∂o∂ s ν s + ∂o∂ σ · ν s σ + (cid:90) j s j o φ [ j s , j σ ] dj s dj σ (95) ˙C sc = ∂c∂ s ν s + ∂c∂ σ · ν s σ + (cid:90) j s j c φ [ j s , j σ ] dj s dj σ ˙C oc = ∂o∂ s ∂c∂ s ν s + ∂o∂ σ · ν σ ∂c∂ σ + ( ∂o∂ s ∂c∂ σ + ∂o∂ σ ∂c∂ s ) · ν s σ + (cid:90) j o j c φ [ j s , j σ ] dj s dj σ The jumps j o and j c for the option and derivative prices, appearing in theintegrand of the discontinuous contribution, are defined in terms of the jumps j s and j σ for the underlying price and state variables: j o = o [ t, s + j s , σ + j σ ] − o [ t, s , σ ] (96) j c = c [ t, s + j s , σ + j σ ] − c [ t, s , σ ]Hedging with the underlying portfolio β s and the option portfolio β o , theresidual variance for the hedged derivative is: v = (( ∂c∂ s − β s − β o · ∂o∂ s ) ν s (97)+ ( ∂c∂ σ − ∂o∂ σ β o ) · ν σ ( ∂c∂ σ − ∂o∂ σ β o )+ 2( ∂c∂ s − β s − β o · ∂o∂ s )( ∂c∂ σ − ∂o∂ σ β o ) · ν s σ + (cid:90) ( j c − β s j s − β o · j o ) φ [ j s , j σ ] dj s dj σ ) dt + O [ dt ]The choice of hedge portfolio then depends on the objectives of the investor andtheir level of access to the hedge market. Using only the underlying, the deltahedge is: β s = ∂c∂ s β o = 0 (98)This strategy offsets the impact on the derivative price of continuous movesin the underlying price, but neglects the impact from the volatility of statevariables. The residual variance achieved by the strategy is: v = ( ∂c∂ σ · ν σ ∂c∂ σ + (cid:90) ( j c − β s j s ) φ [ j s , j σ ] dj s dj σ ) dt + O [ dt ] (99)Performance of the delta hedge is improved by including options to offset thecontinuous moves in the state variables: β s = ∂c∂ s − ( ∂o∂ σ ) − ∂o∂ s · ∂c∂ σ β o = ( ∂o∂ σ ) − ∂c∂ σ (100)This strategy removes the contribution from the continuous covariance of theunderlying and option prices, leaving only the discontinuous term: v = ( (cid:90) ( j c − β s j s − β o · j o ) φ [ j s , j σ ] dj s dj σ ) dt + O [ dt ] (101)The prize-winning observation is that the continuous contribution to the residualvariance can be completely eliminated when there are sufficient hedge securities27vailable to match against the volatile state variables. Moreover, the discon-tinuous contribution is small if the price jumps broadly align with the linearapproximation, so that the residual ( j c − β s j s − β o · j o ) is negligible. Largemarket disruptions are commonly associated with the breakdown of normal re-lationships between prices, however, and the hedged derivative is exposed to therisk of discontinuities if these moves deviate from the tangent relationship. Vari-ance reduction could be extended to account for such regime switches, adaptingthe hedge strategy to balance the impact from continuous and discontinuousmarket moves, but no strategy can be completely successful at eliminating themarket risks if the direction of price moves is indeterminate.The assumptions that validate the delta hedging strategy, namely that mar-gin is settled continuously and the underlying price moves are themselves con-tinuous, are at best an approximation to real hedging activity. The price modelaccommodates the impact of discrete market moves within the timescale oftrading by admitting discontinuity in the underlying diffusion, substituting riskelimination with portfolio optimisation as the unifying principle for pricing.It should be noted that none of the core economic principles that are thebasis for the price model hold true even in the most elementary markets. Theprinciple of replicability disregards consideration of trading volume that mayimpact price via illiquidity or economy of scale. The principle of no-arbitragedenies the existence of arbitrage, which is certainly possible over short horizonsand has been observed to persist even on the scale of days or months in excep-tional circumstances. Arguably the most pernicious of the three, the principleof economic equivalence has the potential to corrupt the price model throughinvalid economic suppositions, inviting technical assumptions whose mathemat-ical attractiveness can mask the realities of market dynamics. Understanding the relationship between the economic measure and the pricemeasure has been the fundamental question of mathematical finance ever sinceBachelier first applied stochastic calculus in his pioneering thesis [2]. In thisthesis, Bachelier develops a model for the logarithm of the security price asBrownian motion around its equilibrium, deriving expressions for options thatwould not look unfamiliar today. Methods of portfolio optimisation originatedin the work of Markowitz [18] and Sharpe [25] using Gaussian statistics for thereturns, generalised by later authors to allow more sophisticated distributionalassumptions and measures of utility. These approaches are embedded in theeconomic measure, explaining the origin of price as the equilibrium of marketactivity uncovering the expectations of participants.The impact of dynamic hedging on price was first recognised in the articlesby Black and Scholes [3] and Merton [20]. Adopting the framework devised byBachelier, these authors observe that the option return is replicated exactly bya strategy that continuously offsets the delta of the option price to the under-lying price. The theory matured with the work of Harrison and Pliska [10, 11],providing a precise statement of the conditions for market completeness andthe martingale property of price in a measure equivalent to the economic mea-sure. The discipline has since expanded in numerous directions, with significantadvances for term structure and default modelling and numerical methods for28omplex derivative structures.While market evidence for discounting basis existed earlier, the basis widen-ing that occurred as a result of the Global Financial Crisis of 2007-2008 moti-vated research into funding and its impact on discounting. Work by Johannesand Sundaresan [15], Fujii and Takahashi [6], Piterbarg and Antonov [23, 24, 1],Henrard [12], McCloud [19] and others established the theoretical justificationand practical application of collateral discounting.Entropy methods have found application across the range of mathemati-cal finance, including portfolio optimisation [22] and derivative pricing [4, 8]– see also the review essay [26] which includes further references. The article[5] by Frittelli proposes the minimal relative entropy measure as a solution tothe problem of pricing in incomplete markets, and links this solution with themaximisation of expected exponential utility. As a measure of disorder, entropyperforms a role similar to variance but is better suited to the real distributionsof market returns. Proponents of the use of entropy justify the approach by ap-peal to the modelling of information flows in dynamical systems and the analogywith thermodynamics.Most approaches to derivative pricing begin with the assumption of con-tinuous settlement, and stochastic calculus is an essential ingredient for thesedevelopments. The representation of the stochastic differential equation used inthis article follows the discoveries of L´evy [17], Khintchine [16] and Itˆo [14]. Thetechnical requirements of the L´evy-Khintchine representation can be found inthese articles and other standard texts in probability theory. The change fromeconomic measure to price measure implied by the maximum entropy principleextends the result from Girsanov [7] to include the scaling adjustment of thejump density in addition to the drift adjustment.The example models referenced in the section on stochastic volatility arethe Black-Scholes-Merton model [3, 21], the Heston model [13] and the SABRmodel [9], all commonly used models in quantitative finance.
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