Martingale deflator

A martingale deflator rescales asset values so that their deflated gains have a stated martingale property. For a price $S$ with cumulative cash dividends $D$, the correctly deflated gain process iswith a strictly positive adapted deflator $Y$. Dividends are deflated at their payment times; multiplying $Y_t$ by the undiscounted total $S_t+D_t$ is generally different. A local martingale deflator requires the local-martingale property, while true expectation pricing requires the corresponding true-martingale and integrability conditions.

For a positive bank account $B$, put $Z_t=Y_t{B}_t/(Y_0{B}_0)$. If $Z$ is a true mean-one martingale, it defines an equivalent probability measure by its terminal density. Deflated-price local martingales then yield an equivalent local martingale measure. Obtaining an equivalent martingale measure, and ordinary conditional-expectation pricing, additionally requires true martingales for the asset-price ratios. Positivity and a local property alone do not justify that promotion.

Work in discounted units with a unit bank account and a square invertible volatility matrix. For asset dynamics $dS=\mathrm{diag}(S)(\mu dt+\sigma dW)$, let $\lambda ={\sigma }^{-1}\mu$ and $Z=\mathcal{E}(-\int {\lambda }^{T}dW)$. The Itô product rule cancels the drift of each $Z{S}^i$, leaving the displayed stochastic integral. When $\mu$ is bounded and $\sigma {\sigma }^T\ge \epsilon I$, $\lambda$ is bounded and the Novikov condition makes $Z$ a true martingale on every finite horizon. If $\sigma$ is also bounded, each $Z{S}^i/S_0^i$ satisfies the Novikov condition as a stochastic exponential with bounded integrand ${\sigma }_i-\lambda$, so it too is a true martingale.

Consider an arbitrage-free complete finite-state market with $n$ traded assets, strictly positive physical probabilities on all successor states, and conditional second-moment matrixpositive definite. Completeness permits at most $n$ successors from each parent atom, while positive definiteness forces at least $n$ independent successor price vectors. Thus there are exactly $n$.

The fundamental theorem of asset pricing supplies strictly positive one-step state prices. If their vector is $q$, their physical probabilities are $p_s$, and the independent successor price vectors are the columns of $C$, then $P_{t-1}=Cq$ and $V_t=C\mathrm{diag}(p)C^{\top }$. ConsequentlyThese unique ratios satisfy $\mathbb{E}[Z_t{P}_t\mid {\mathcal{F}}_{t-1}]=P_{t-1}$. Their successive product is a martingale deflator. The no-arbitrage hypothesis is essential: completeness and an invertible second-moment matrix alone do not make the regression expression positive.^[ref]

A local martingale deflator is a strictly positive process $Y$ such that multiplying each traded asset price by $Y$ produces a local martingale. It can support no-arbitrage arguments even when the deflated prices are not true martingales.

For a self-financing portfolio whose product with a strictly positive local deflator is a local martingale, nonnegative wealth makes that product a supermartingale. Starting from zero forces it to vanish almost surely at each time. Continuous paths then give simultaneous vanishing at every time on one probability-one event.

In a one-factor market with the usual natural Brownian filtration, assume nonzero traded diffusion exposure and a strictly positive local martingale deflator $Y$, normalized by $Y_0=1$. Let $H\ge 0$ be a terminal contingent claim with $\mathbb{E}[Y_{T}H]<\infty$. DefineThe Brownian martingale representation theorem, with localization when needed, represents the true conditional-expectation martingale $M$ as a stochastic integral. Solving the exposure equation against the nonzero traded diffusion constructs a self-financing portfolio with wealth $X\ge 0$ and terminal value $H$. The dollar portfolio from a deflated wealth martingale gives this calculation explicitly.

For any competing nonnegative-wealth self-financing portfolio, the deflated wealth is a nonnegative local martingale, hence a supermartingale. Its initial capital is therefore at least $\mathbb{E}[Y_{T}H]$, and the constructed strategy attains that cost. Without normalization, the cost is $\mathbb{E}[Y_{T}H]/Y_0$. Boundedness of $H$ alone does not replace the required deflated-payoff integrability in an arbitrary model.

In the completed natural Brownian filtration, suppose $d{S}_t/S_t={\mu }_{t}dt+{\sigma }_{t}d{W}_t$ with ${\sigma }_t\ne 0$, and let $d{\zeta }_t=-{\zeta }_t(r_{t}dt+{\kappa }_{t}d{W}_t)$, ${\kappa }_t=({\mu }_t-r_t)/{\sigma }_t$. For an affordable nonnegative terminal claim $X$, represent the conditional-expectation martingale $N_t=\mathbb{E}[{\zeta }_{T}X\mid {\mathcal{F}}_t]$ as $d{N}_t=h_{t}d{W}_t$. Then $w_t=N_t/{\zeta }_t$ and the displayed dollar amount in the stock replicate $X$. The Itô product rule gives $d(\zeta w)=\zeta (\sigma \theta -\kappa w)dW$, proving the formula; share holdings are ${\theta }_t/S_t$.

In a one-factor diffusion market, the market price of risk is the excess drift per unit spot volatility. It is the coefficient in the Brownian motion part of a local martingale deflator, $d{Y}_t=-Y_t(r_{t}dt+{\lambda }_{t}d{W}_t)$.

In a one-Brownian-factor arbitrage-free market, every traded derivative with nonzero diffusion exposure $s$ has the same excess-drift/exposure ratio. Otherwise holdings $s_2$ in the first and $-s_1$ in the second, financed in the bank account, cancel all diffusion risk while leaving a nonzero excess drift. At zero exposure, the correct statement is $m-rV=\theta s=0$, not division by zero. The short rate itself need not be a traded asset, so its physical drift does not alone fix this risk price.

A positive normalized continuous local martingale deflator requires a locally square-integrable Brownian diffusion coefficient. If the forced market price of risk is $-t^{-1/2}$, the coefficient has magnitude $Z_t/\sqrt{t}$. Since $Z_0>0$, continuity makes its squared integral diverge near zero. The bank account can nevertheless exist because $t^{-1/2}$ itself is integrable; finite integrated interest is weaker than square-integrability of the required risk compensation.

A state-price density is a positive random variable or process that converts future payoffs into present values by expectation. In a one-period zero-cost return model $X$, its normalization is commonly $\mathbb{E}\rho =1$ and $\mathbb{E}[\rho X]=0$.

At an interior consumption and share-holding optimum, the Lagrange multiplier on the discrete dividend budget equation is ${\Lambda }_t={\beta }^t{U}^{\prime }(C_t)$. Varying the next holding by a variable measurable at date $t$ gives the displayed Euler equation. Thus the positive normalized process ${\zeta }_t={\Lambda }_t/{\Lambda }_0$ is a state-price density for traded dividend-inclusive gains. In an incomplete market, different investors can have different such densities; equality of their marginal-utility densities is an additional risk-sharing condition, not a consequence of trading one asset alone.

A positive state-price density prices a productive stock through a conditional dividend sum plus a future deflated-price term. The displayed condition removes that terminal term, yielding transversality and fundamental dividend prices. An individual consumption budget instead has terminal term $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$, so its vanishing must be imposed or proved for the particular holdings as well.

Iteration of the pricing Euler equation givesThe displayed fundamental price follows when the future dividend series is integrable and ${\mathbb{E}}_t[{\zeta }_N{S}_N]\to 0$. Without this dividend-price transversality condition, a deflated price martingale component can survive. For individual holdings, the analogous terminal term is $\mathbb{E}[{\zeta }_N{S}_N{\theta }_{N+1}]$. Telescoping the discrete dividend budget equation gives the initial wealth cost of consumption minus this terminal term; its vanishing is needed for budget equality.

Let $a>0$, $\lambda >0$, $\sigma \ge 0$ and $\alpha a\ge {\sigma }^2/2$. For an Ornstein-Uhlenbeck process $d{X}_t=\sigma d{W}_t-\lambda X_{t}dt$ under a reference measure, the processis strictly positive and a supermartingale. The Itô formula identifies its short rate asFor $\tau =T-t$, conditional Gaussian second moments give the unit zero-coupon bond priceThis is $\mathbb{E}[{\zeta }_T\mid {\mathcal{F}}_t]/{\zeta }_t$. The money-market density has diffusion coefficient $\sigma X_t/(a+X_t^2/2)$, of magnitude at most $\sigma /\sqrt{2a}$. The Novikov condition therefore validates the corresponding pricing measure on finite horizons. The reference OU dynamics are not automatically the risk-neutral dynamics.

On a finite state space, the Arrow–Debreu state price is the current price of a payoff equal to one in state $s$ and zero elsewhere. It equals the physical state probability times the state-price density in that state. With a riskless asset of deterministic prices $B_0,B_1$, the Arrow state prices sum to $B_0/B_1$. Normalizing them gives the risk-neutral probabilities, while division by the physical probabilities instead recovers the pricing density.

If a traded payoff matrix $A$ has initial price vector $q=A^{T}p$ with $p\ge 0$, there cannot be a negative-cost portfolio with nonnegative terminal payoff. Nevertheless a nonzero nonnegative attainable payoff supported only in zero-price states can cost zero and give an arbitrage. When every state has positive physical probability, excluding this possibility requires strictly positive Arrow state prices, not merely a nonnegative supporting vector.

A local martingale deflator only forces deflated asset prices to be local martingales. True expectation pricing requires additional martingale and integrability properties. In a Brownian one-factor market, continuity of the coefficients ensures a pathwise finite market-price-of-risk square integral, but not expectation-one of its stochastic exponential. The reciprocal three-dimensional Bessel strict local martingale gives a counterexample to promoting this local conclusion automatically.

A self-financing investor with consumption satisfies $X=H\cdot P$ and $dX=H\cdot dP-cdt$. The Itô product rule and $d[Y,X]=H\cdot d[Y,P]$ give the displayed equation. For a positive local martingale deflator $Y$, the first term is a stochastic integral against the vector of deflated asset-price local martingales, not their current levels.

If wealth and consumption are nonnegative, this local martingale is bounded below by minus the finite initial deflated capital. Adding that capital gives a nonnegative local martingale, hence a supermartingale by the Conditional Fatou lemma. Consequently expected discounted consumption cannot exceed initial deflated wealth. The consumption sign and predictable stochastic integrability are part of the hypotheses.

A state-price density converts terminal wealth and consumption into initial cost. For a nonnegative admissible self-financing portfolio with consumption, the deflated wealth plus cumulative deflated consumption is a supermartingale, giving $\mathbb{E}[{\zeta }_{T}X]+\mathbb{E}{\int }_0^T{\zeta }_t{c}_{t}dt\le w$. In a complete market, an integrable nonnegative terminal claim with full budget equality is replicable.

For $0<R<1$, nonnegative consumption with state-price budget $\mathbb{E}\int Z_t{c}_{t}dt\le x$ satisfies this bound under CRRA utility. Apply Holder inequality on probability-times-time measure to $(Zc{)}^{1-R}$ and $e^{-bt}{Z}^{R-1}$, with conjugate exponents $1/(1-R)$ and $1/R$. If the price integral $D$ is finite, equality forces full budget use and $c_t=(x/D)e^{-bt/R}{Z}_t^{-1/R}$. Financing this process is a separate feasibility issue, so the equality pattern alone does not assert attainability in an incomplete market.

For a bounded payoff vector $P$, the condition that $h\cdot p\le 0\le h\cdot P$ almost surely forces $h=0$ makes $F(h)=\mathbb{E}{e}^{-h\cdot P}+h\cdot p$ coercive. At a minimizer $h_{\ast }$, differentiation gives $\mathbb{E}[P{e}^{-h_{\ast }\cdot P}]=p$. The resulting positive pricing kernel is bounded above and away from zero. A direct coercivity proof uses compactness of the unit sphere and separates directions with negative-payoff probability from directions with nonnegative payoffs.

For price vectors $P_0,P_1$, a one-period martingale deflator is a pair $Y_0>0$, $Y_1>0$ satisfying $\mathbb{E}[Y_1{P}_1]=Y_0{P}_0$ componentwise.