This additive price model has constant absolute volatility and can take negative stock values. With constant cash interest rate , its market price of risk is . Under an equivalent risk-neutral measure, its drift becomes , so its conditional law is Gaussian with mean and variance . It is distinct from a multiplicative Black–Scholes diffusion.
Let and , where . The displayed discounted Gaussian positive-part expectation prices a call in the arithmetic stock model with constant volatility. Its delta is . The stock holding and bank-account holding replicate the payoff with nonnegative wealth. A nonnegative discounted-wealth supermartingale bound proves this is the least replication capital.
The delta of the call price in an arithmetic stock model with interest is the normal distribution function at its standardized discounted moneyness. It lies strictly between zero and one before maturity. Its maturity limit is the call payoff derivative away from the strike, an exceptional event of probability zero under the Gaussian pricing law.
A local volatility model uses a deterministic function of current time and spot as its diffusion volatility, . It can reproduce a surface of European call option prices through the Dupire equation.
The stock diffusion has spot-dependent local volatility. With positive volatility and the Brownian filtration, Brownian martingale representation theorem converts discounted payoff martingales into stock gains.
For a non-dividend-paying stock and constant interest rate , discounted call prices satisfy . Strike differentiation recovers the discounted terminal density as ; differentiating the discounted payoff identity supplies the maturity derivative. Put-call parity implies the same equation for put prices.
Where , rearranging the Dupire equation gives . The strike curvature represents discounted density, while the adjusted maturity derivative gives the local diffusion contribution.
For a square-integrable discounted claim martingale and discounted stock , choose stock holdings . The remaining wealth lies in the bond. A nonnegative claim makes this replication admissible; discounted admissible wealth is a supermartingale, establishing the minimal initial cost.
Failure to meet a financial obligation, modeled by a random default time. A zero-recovery default model sets the affected asset value to zero after this time.
The random time at which credit default occurs. In an observed default model it is a stopping time for the market filtration.
A collection of holdings in traded assets. Its value is the dot product of the holdings with the asset prices. A self-financing portfolio is an investment portfolio whose trading requires no external funding.
The value of an investment portfolio, obtained by summing each holding times its current asset price. Consumption removes money from portfolio wealth; a self-financing portfolio changes its value only through asset gains. Nonnegative portfolio wealth is a common admissibility restriction in an investment-consumption problem.
Brownian portfolio exposures are the coefficients of independent Brownian motions in the wealth equation. An invertible asset volatility matrix lets one optimize directly over these exposures. The drift risk premium is then the dot product of exposures with the market price of risk vector.
The expected excess return of a portfolio divided by its return standard deviation. Under the capital asset pricing model, the ratio for one asset equals its correlation with the market times the market's ratio.
In a one-period normal-return model with common beliefs and unrestricted exponential utility investors, their risky holdings are proportional to , where is the expected excess-gain vector and its positive definite covariance matrix. The aggregate risky holding defines the common market portfolio.
The expected excess return obeys in the mean-variance market model. The beta of an asset is a covariance-to-market-variance ratio.
For a nondegenerate market return , . It measures the asset's market-related exposure and enters the capital asset pricing model.
A stock is a traded ownership interest in a company. In a mathematical finance model, its price is a stochastic process and a self-financing portfolio may hold positive or negative numbers of its shares.
A stock model whose price can be killed or reduced at a default time. The zero-recovery default model combines continuous price dynamics before default with a jump to zero.
For an independent exponential distribution default time of rate and a continuous nonnegative martingale , this process is a martingale. The compensating pre-default factor balances the survival probability . The strict survival inequality gives a càdlàg price process.
A numéraire is a strictly positive traded asset used as the unit in which all other asset prices are expressed. Changing the numéraire changes the associated martingale measure while preserving no-arbitrage prices.
In a one-period market, a numéraire portfolio has strictly positive value at both dates and can therefore be used to convert initial consumption into terminal wealth.
A numéraire strategy is a self-financing portfolio with zero consumption whose wealth is strictly positive at every date. Its wealth process can carry consumption received at an earlier date forward to a later date.
An arbitrage is a zero-cost self-financing portfolio whose terminal payoff is nonnegative in every state and strictly positive with positive probability.
An investment-consumption arbitrage starts with zero capital, has nonnegative consumption at every date, and has strictly positive consumption at some date with positive probability.
A pure-investment arbitrage has zero initial cost, no intervening consumption, and a nonnegative terminal payoff that is strictly positive with positive probability.
Consumption in a self-financing portfolio model is wealth withdrawn for use outside the portfolio. With no initial capital, a one-period position of nonpositive initial cost and nonnegative terminal payoff permits nonnegative consumption at both dates.
A terminal-consumption arbitrage has zero initial cost and a nonnegative terminal payoff that is strictly positive with positive probability. Allowing negative initial cost gives the broader one-period notion in which some consumption may occur immediately.
If the terminal price vector is normal with covariance matrix , every portfolio outside has a terminal value with full support on the real line. Arbitrage can therefore arise only from portfolios in , whose terminal values are deterministic.
A portfolio is self-financing when every change in its asset holdings is paid for entirely by selling or buying assets within the portfolio, with no external cash added or removed.
A self-financing portfolio is admissible when its wealth satisfies the prescribed lower-bound restriction, typically a deterministic bound below. Nonnegative wealth is sufficient. This condition permits supermartingale bounds on discounted gains and excludes doubling strategies.
Relative to a positive numéraire, a predictable self-financing strategy with risky holdings and discounted risky prices has discounted valueThus its gains are the discrete stochastic integral of its holdings against discounted price increments.
In expected-utility portfolio choice, an increasing utility function prefers greater wealth, while concavity models aversion to risk.
An affine utility on probability measures functional satisfies on mixtures of probability measures. Strict preferences are represented by comparing its real values, and indifference means equal values. This is cardinal utility on lotteries rather than merely an ordinal ordering of deterministic outcomes.
Set . On a finite measurable space, points in the same atom of a sigma-algebra give the same Dirac measure, so is a measurable function and constant on atoms. Every probability measure is a finite mixture of representative Dirac measures with its atom masses as weights. Affineness then proves the expected-utility representation, even when the sigma-algebra is smaller than the power set.
Two real affine functionals on a convex set of lotteries that represent the same strict preferences differ by with . Equal utility values must first be shown to give equal values of the other functional. The induced function on the real interval of utility values preserves mixtures; anchor interpolation, including exterior points represented through an anchor, makes it linear throughout. Constant representations are handled separately and still admit a positive slope.
For three strictly ordered affine-utility values , the middle lottery is indifferent to the mixture of the extreme lotteries with weight on the best one. This weight lies strictly between zero and one. It is the exact mixture-indifference property underlying cardinal utility comparisons.
Mixing two lotteries with the same third lottery and the same positive weight preserves their strict preference order. For affine utility on probability measures, the difference of the two mixed utilities is the original difference multiplied by that weight. At weight zero the strict preference disappears, so positivity is essential.
For an increasing differentiable strictly concave utility function satisfying the Inada conditions, the inverse marginal utility maps each positive shadow price to the unique positive payoff with that marginal utility. For normalized CRRA utility, .
Multiplicative habit utility modifies current consumption utility through its ratio to habit . With , , and , it is . Its homogeneity in has degree , while consumption holding habit fixed has effective relative risk aversion .
After scaling wealth by habit and applying the wealth-variable Legendre dual, the effective consumption shadow price is proportional to . Its power in the utility conjugate keeps the dual equation nonlinear. When , habit is fixed and the dual equation is a linear Euler differential equation with effective relative risk aversion .
When habit evolves by , increasing consumption both costs wealth and increases future habit. Consequently the consumption coefficient in the Hamilton-Jacobi-Bellman equation is . A finite interior optimum requires this effective shadow price to be positive.
An exponentially weighted consumption habit is . Differentiation gives . The state remains positive from positive initial habit and nonnegative consumption.
The Inada conditions specify infinite marginal utility at zero consumption or wealth and zero marginal utility at infinity. For an increasing strictly concave utility function, they make the inverse marginal utility map cover all positive shadow prices, facilitating the state-price budget constraint method.
This smooth utility has and . Its derivative is strictly decreasing, diverges at zero and tends to zero at infinity, so the utility is strictly concave and satisfies the Inada conditions. Yet its curvature vanishes at one. The inverse marginal utility cannot have a finite derivative at , since differentiating there would give zero equal to one. Its dual is differentiable but not twice differentiable at that price.
A constant relative risk aversion utility has . Up to a positive scale and an additive constant, it is for and for . Power utility gives multiplicative homogeneity of an investment value; logarithmic utility gives additive scaling.
For an interior optimal payoff in a complete two-state market under CRRA utility, . If the two positive payoffs differ, taking logarithms gives the displayed coefficient. The multiplier can then be recovered from either state. A positive coefficient requires the price-density and payoff orders to be compatible with decreasing marginal utility.
This utility is linear in the monetary transfer: allocation value minus payment. It supports the incentive comparisons and payment identities used in mechanism design.
Risk neutrality means evaluating uncertain monetary gains by their expected value. For auction payment identities, quasilinear utility gives a bidder's expected utility as its value times its winning probability minus expected payment.
Constant absolute risk aversion utility has . For Gaussian wealth , maximizing is equivalent to maximizing
A holding-dependent cost proportional to the absolute position size. In a one-period gain objective it changes the sample payoff to , giving gain parameter for positive holdings and for negative holdings. The objective is concave but can have a kink at zero.
If a negative differentiable concave utility function satisfies on an open interval, outer secants at four points bound the sample derivative on a smaller compact interval. Negativity makes the endpoint values absolutely integrable. The dominated convergence theorem then gives and continuous derivative when is continuous. This is a justification of differentiation, rather than an assumption of integrability of marginal utility.
A terminal portfolio wealth floor is a prescribed lower bound, possibly a random variable, on terminal portfolio portfolio wealth. In a complete market, its state-price density cost cannot exceed initial portfolio wealth. Nonnegative admissibility replaces a possibly negative floor by its positive part.
For a nonnegative terminal wealth floor whose state-price density cost is strictly below the available budget, pointwise maximization of over gives the displayed payoff. The positive multiplier exhausts the state-price budget constraint. This requires the usual utility and integrability hypotheses ensuring that the multiplier exists and the payoff is replicable.
Finite lower and upper utility bounds together with a positive discount rate bound the absolute time integral by . This justifies expectation and integration in marginal-utility verification. Bounded utility alone on an undiscounted infinite horizon does not prevent both positive and negative parts from having infinite integral.
An incentive fee above a hurdle changes a manager’s utility function into . For constant absolute risk aversion utility, each branch is strictly concave, but the marginal reward jumps upward at the hurdle. Thus the effective terminal-wealth utility is not concave, and concavification of incentive utility can reveal optimal risk-taking lotteries.
Concavification replaces a nonconcave incentive utility function by its least concave majorant. In a complete market with an atomless state-price density, the optimizing payoff may avoid all intervals where the majorant strictly exceeds the original utility, making the relaxed optimum exactly attainable. At a pricing atom corresponding to a linear segment, a lottery between contact points can attain the same value at unchanged cost.
With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts can improve expected utility maximization through a fair lottery paying those two values. The probability of is . In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.
For below hurdle and above it, with , an interior common tangent touches at and . The slope is and . The formula requires ; a smaller hurdle gives a different endpoint-at-zero concavification.
An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.
A fixed loan principal contributes a constant interest outflow . If the objective stops when available portfolio wealth reaches zero, the value function has an absorbing boundary . This differs from requiring the portfolio to finance the debt service forever.
For the wealth-variable Legendre dual , a finite slope at an absorbing zero-wealth boundary makes identically zero for . Matching gives . It does not generally give : killing at ruin permits the dual curvature to jump.
When the market price of risk is constant and nonzero spot volatility gives access to the driving Brownian motion, changes in volatility only rescale the dollar holding needed for fixed Brownian portfolio exposures. With unrestricted holdings and constant interest and discount rates, the CRRA utility value is consequently independent of the factor state. Zero volatility requires separate treatment.
A consumption satisfaction stock smooths past consumption by . Unlike multiplicative habit utility, the running utility function may depend on this stock alone. A unit of present consumption raises future consumption satisfaction while costing one unit of financial portfolio wealth.
In singular stochastic control, an unrestricted consumption rate can approximate instantaneous transfers from portfolio wealth into a consumption satisfaction stock. In the relaxed problem a nondecreasing control records these transfers. A transfer decreases portfolio wealth and increases consumption satisfaction by the same amount. Rate controls need not attain the supremum when the relaxed optimum has a jump.
If consumption contributes to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires . Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.
For CRRA utility of a consumption satisfaction stock, jointly scaling initial portfolio wealth, consumption satisfaction and controls gives homogeneity of degree . The value function therefore depends on one dimensionless ratio after factoring out . The effective consumption coefficient becomes .
A discounted-value transversality condition rules out residual value at infinity in an infinite-horizon Hamilton-Jacobi-Bellman equation verification. Under appropriate integrability, admissibility, and localization assumptions, identifies the economic value among formal differential-equation solutions.
When asset correlation varies with a traded market index, the index is both an asset and a state variable. Writing and optimizing Brownian portfolio exposures accounts for the cross derivative between wealth and . Power homogeneity reduces the Hamilton-Jacobi-Bellman equation to an ordinary differential equation in .
For a complete-market investment-consumption problem with constant relative risk aversion utility, the nonlinear wealth-homogeneity coefficient equation may contain . Writing cancels this gradient square against the one from . In the index-driven correlation model the result is , a linear differential equation; the positive economic solution gives consumption .
Intertemporal hedging demand adjusts the myopic risky position when investment opportunities depend on a stochastic state. Shared noise between wealth and that state contributes a cross derivative to the Hamilton-Jacobi-Bellman equation. In a complete diffusion market the optimal exposures combine the myopic market price of risk term and the value-gradient hedge term.
A high-water mark investment tax charges increments of historical maximum wealth rather than every positive instantaneous return. For the convention charging , the high-water mark tax boundary condition is when the optimal policy raises the maximum. At sufficiently high tax, a wealth-cap investment boundary can make further maximum increases suboptimal.
A wealth-cap investment boundary prevents wealth from exceeding a fixed historical maximum. In the wealth-variable Legendre dual for wealth normalized by that maximum, and vanishing optimal portfolio volatility gives . For power utility and a high-water tax, the smooth tax-paying solution has boundary dual curvature , so it cannot be admissible when .
At an active maximum-raising boundary in high-water mark investment taxation, the finite-variation part of the Itô formula is , so smooth optimality sets this coefficient to zero. If the maximizing policy avoids increasing the maximum, the corresponding inequality is compatible with a wealth-cap investment boundary instead.
For a constant-coefficient single-asset investment-consumption problem with constant relative risk aversion utility, put and . When , the infinite-horizon value is and the optimal controls are and . The case uses logarithmic utility.
For with having the standard normal distribution and an increasing concave utility function , define . Then and Gaussian integration by parts gives . Moreoveris negative semidefinite, so is jointly concave in and .
A utility indifference price for a claim is the cash amount at which optimally buying the claim leaves the investor with the same maximal expected utility as trading without it.
For Gaussian income of mean , standard deviation and tradable correlation , exponential utility makes the maximum acceptable per-period fee equal to the mean minus the correlated-income hedge cost and the residual variance penalty. Compare optimized utilities with and without the contract; a fee above this level makes the contract strictly inferior. At perfect correlation the residual penalty vanishes, leaving only the mean and hedge-cost terms.
Let be a vector space of random variables and supposehas an optimizer for every . If is increasing and concave, then is increasing and concave. For concavity, combine optimizers at two values using the same convex coefficient as the values themselves.
If is concave and , thenis concave in . By Jensen inequality, ; concavity then makes nonincreasing on .
The concave conjugate gives the weak-duality bound . Equality holds when and the payoff exhausts the deflated budget.
For a negative, strictly increasing, concave differentiable utility with finite expected utility at every holding, a gain having both signs makes the objective coercive at both ends. At a finite optimum its derivative is zero. Secant domination for expected utility derivatives gives integrability of ; boundedness of on the remaining compact set gives integrability of the positive marginal utility itself. Its normalization has expectation one and prices the gain at zero.
At a nonzero optimum, the derivative of expected utility of a proportional transaction cost payoff is zero. Normalized marginal utility therefore prices at for a positive optimum or for a negative optimum. At a zero optimum the one-sided derivative inequalities put inside that interval, so density one works. The competing one-sided-payoff alternative prevents the coercive-maximizer argument from being assumed when a net gain has only one sign.
For a positive state-price density pricing every attainable payoff, a feasible positive terminal payoff is optimal at its budget if its marginal utility is for some scalar . The pointwise conjugate inequality gives for every competitor; the marginal-utility identity makes equality hold for . Finite states make integrability straightforward. In a complete market an interior optimum also necessarily has this proportionality, by variations preserving the state-price budget.
For concave increasing utility, the concave supporting-tangent inequality gives . If the proposed consumption exhausts the state-price budget and every competitor has no larger budget, integration proves optimality. The objective must be well defined; a positive discount rate and finite utility bounds are a convenient sufficient condition.
For increasing strictly concave wealth value , the dual is convex. At an interior optimum , , , and . This sign convention is the negative of the concave Legendre dual. It can turn the optimized portfolio term of a Hamilton-Jacobi-Bellman equation into a linear second derivative.
For independent return innovations and a predictable process of holdings , wealth obeysThe investor chooses the holdings to maximize expected utility of terminal wealth.
With a unit cash asset, independent Gaussian price increments of mean and positive variance , and exponential terminal utility , the optimal predictable risky holding is the displayed constant number of shares. The one-step exponential-loss exponent is ; completing its square gives a minimal multiplier . Backward induction proves optimality among adapted strategies and gives value .
An income stream with per-period variance and correlation with a Gaussian traded price increment produces a hedge holding in addition to speculative demand. The residual income variance is . With independent period pairs, the same conditional quadratic minimization applies at each date. A deterministic income fee changes the optimized value but not the hedge holding.
If terminal utility is increasing and concave and each Bellman supremum is attained, backward induction shows that every remaining-horizon value function is increasing and concave. Concavity follows by combining optimal portfolios for two initial wealths with the same convex coefficient.
An equivalent martingale measure is a probability measure equivalent to the physical measure under which discounted asset prices are martingales. In a finite one-period market, its existence is equivalent to no arbitrage.
The probability of a market event under an equivalent martingale measure for a specified numéraire. It can differ from the physical probability. For a random numéraire, impose the martingale condition on asset-price ratios rather than treating the numéraire return as deterministic.
For an integrable adapted discounted price process in finite discrete time, the following are equivalent under a probability measure : is a -martingale; every bounded predictable self-financing strategy has a martingale discounted value; and every such strategy has expected terminal value equal to its initial value. To recover the martingale property from the expectation identity, trade one asset only during one period on an arbitrary event in the preceding sigma-algebra.
Under an equivalent martingale measure , the no-arbitrage value of a replicable payoff is its discounted conditional expectation under .
A martingale deflator is a strictly positive adapted process whose product with each cum-dividend asset gain is a local martingale. After normalization by a numéraire, a martingale deflator determines an equivalent martingale measure.
Let be positive definite. Market completeness restricts each parent atom to at most successors, while positive definiteness forces exactly independent successor price vectors. The positive martingale deflator supplied by the fundamental theorem of asset pricing then has the unique one-step ratio . Its product deflates the prices into martingales.
A local martingale deflator is a strictly positive process such that multiplying each traded asset price by produces a local martingale. It can support no-arbitrage arguments even when the deflated prices are not true martingales.
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Mathematical finance is a field of applied mathematics that focuses on the mathematical modeling and analysis of financial markets and instruments. It integrates concepts from probability theory, statistics, differential equations, and stochastic calculus to understand and manage financial risks and to price financial derivatives. Key areas of mathematical finance include: 1. **Option Pricing**: Developing models to determine the fair value of options and other derivatives. The Black-Scholes model is one of the most famous examples.