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The arithmetic stock model with constant volatility has additive price dynamics $d{S}_t=\mu dt+\sigma d{W}_t$, with constant $\mu \in \mathbb{R}$ and $\sigma >0$. Its absolute volatility is constant, and the model can take negative stock values. This differs from a geometric Brownian motion, whose noise is proportional to its current price.

With a bank account satisfying $d{B}_t=r{B}_{t}dt$, assume a finite-horizon equivalent risk-neutral measure exists. The market price of risk is $(\mu -r{S}_t)/\sigma$, and the pricing dynamics become $d{S}_t=r{S}_{t}dt+\sigma d{W}_t^Q$. For $\tau =T-t$, the conditional Gaussian law hasThe variance is ${\sigma }^2(e^{2r\tau }-1)/(2r)$ when $r\ne 0$ and ${\sigma }^2\tau$ when $r=0$. The zero-rate case is obtained by the continuous limit, not by division by zero.

Let $A_t=\exp (t^{-1}{\int }_0^t\log S_{u}du)$ for a positive geometric Brownian motion in the non-dividend-paying Black-Scholes model, and put $\tau =T-t$. Under the risk-neutral measure, the future log-average isThe final integral is a centred normal variable independent of the past, with variance ${\tau }^3/3$. Applying the moment-generating function of a normal distribution proves the displayed identity for every real $p$. Multiplication by $e^{-r\tau }$ prices the payoff $A_T^p$. The historical average must remain a state variable; terminal stock price alone does not determine this payoff.

For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up probability is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm \sigma \sqrt{h}$; expanding $q_h$ gives mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[ref]

The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price drift loses the $\rho$ term, as required for the discounted price to be a martingale. Convergence of claim prices also needs the payoff functional to be continuous at almost every limiting path, together with Uniform integrability when payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.

A share paying a continuous dividend rate ${\delta }_t$ produces dollar gains $d{S}_t+{\delta }_{t}dt$. If its price follows a geometric Brownian motion with drift ${\mu }_{\mathrm{price}}$ and its dividend yield ${\delta }_t/S_t$ is constant, its drift in an investment-consumption problem is their sum. The self-financing portfolio equation uses this total return, even if the agent consumes the dividends rather than reinvesting them. Omitting the dividend yield changes both the optimal stock fraction and the equilibrium interest rate.

A stochastic differential equation with drift and diffusion coefficients affine in its state. In the constant-coefficient additive-noise case $d{X}_t=(a{X}_t+b)dt+\sigma d{W}_t$, multiplying by $e^{-at}$ givesConsequently its conditional distribution is normal, with mean $e^{a\tau }x+b(e^{a\tau }-1)/a$ and variance ${\sigma }^2(e^{2a\tau }-1)/(2a)$ when $a\ne 0$. For $a=0$, these become $x+b\tau$ and ${\sigma }^2\tau$. Additive noise permits negative states; it differs from geometric Brownian motion.

In the basic structural model, the firm's asset value $V$ follows geometric Brownian motion with constant volatility and a single debt obligation of face value $F$ matures at $T$. Default at maturity occurs when $V_T<F$. Equity is a call on firm assets; debt is the risk-free face claim minus a put on those assets, since $\min (V_T,F)=F-(F-V_T{)}^{+}$. This links debt valuation to asset value, volatility and capital structure. The basic model does not permit default before maturity; richer structural models add a default barrier or more detailed obligations.

Applying Itô's formula to $\log S$ gives the geometric Brownian motionThe Gaussian exponential moment $\mathbb{E}{e}^{a{W}_t}=e^{a^{2}t/2}$ then yieldsThis expectation is under the physical measure and therefore uses drift $\mu$.

The stochastic differential computed above isThus this positive no-dividend traded claim behaves as a new geometric Brownian motion asset withThe transformed drift need not be positive, even though the three original market parameters were assumed positive. Under the risk-neutral measure it becomes $r$, as it must for a non-dividend traded asset.

Apply Itô's formula to $\log X$ for positive initial value $x_0$. The drift correction is $-{\sigma }^2/2$, soThis is geometric Brownian motion. Under the physical probability measure, $W_T/\sqrt{T}$ has the standard normal distribution. For positive volatility,The drift here is $\mu$, because this is a physical bankruptcy probability. The risk-neutral measure will instead give drift $r$ for pricing. If $\sigma =0$, the value is deterministic and the default probability is the indicator of $x_0{e}^{\mu T}<m$; the usual positive-volatility convention is used in the remaining formulas.

For geometric Brownian motion, $\log S_u=\log S_0+(\mu -{\sigma }^2/2)u+\sigma W_u$. Averaging in time givesThus $\log A_t$ is normal with mean $\log S_0+(\mu -{\sigma }^2/2)t/2$ and variance ${\sigma }^{2}t/3$. The moment-generating function of a normal distribution gives

Cubing the supplied geometric Brownian motion solution gives $S_t^3=s^3\exp ((3\mu -3{\sigma }^2/2)t+3\sigma W_3(t))$. The moment-generating function of a normal distribution yields $\mathbb{E}[e^{a{W}_3(t)}]=e^{a^{2}t/2}$. Therefore

Apply Itô's formula to the logarithm of the positive geometric Brownian motion. The quadratic-variation correction givesIntegrating yieldsThe increment is a Gaussian random variable with variance $T-t$, independent of current information. Writing it as $W_{\tau }$ means a fresh future Brownian motion ${\tilde{W}}_v=W_{t+v}-W_t$, evaluated at $\tau$; it need not equal the original process at time $T-t$.

While the up-and-out digital put is alive, its pricing domain is $0<S<B$, $t<T$. Its Black-Scholes equation, terminal financial payoff, and absorbing boundary condition areFor completeness, the natural limiting condition is ${\lim }_{S\downarrow 0}H(S,t)=e^{-r(T-t)}$, with $0\le H(S,t)\le e^{-r(T-t)}$. A positive geometric Brownian motion cannot hit zero in finite time, and from a very small starting value the survival and below-strike terminal events have probability tending to one. If the barrier has already been hit, its value is zero regardless of the current stock price.

For $K>B$, the terminal payoff tends to one from inside at $S=B$, while an already hit barrier pays zero. Thus there is a terminal-boundary corner discontinuity; the terminal condition is imposed only for $S<B$, not as a requirement of continuous extension at $(B,T)$.

For a positive starting value, Itô's formula applied to $\log S_u$ gives $d\log S_u=(\mu -{\sigma }^2/2)du+\sigma d{W}_u$. Hence the geometric Brownian motion isThe future Brownian increment $W_T-W_t$ is independent of ${\mathcal{F}}_t$ and has normal distribution $N(0,\tau )$. If ${\tilde{W}}_v=W_{t+v}-W_t$, then $\tilde{W}$ is a fresh Brownian motion and the increment is exactly ${\tilde{W}}_{\tau }$. This specifies the meaning of the paper's abbreviated $W_{\tau }$; the original process value $W_{T-t}$ is not generally that future increment. The stochastic exponential solution also verifies positivity for $S>0$.

On a filtered probability space carrying a standard Brownian motion $W$ with independent increments relative to the filtration, a strictly positive security price follows geometric Brownian motion with constant drift $\mu$ and volatility $\sigma$ if it solves the stochastic differential equationHere $W_t-W_s$ is independent of ${\mathcal{F}}_s$ and has the normal distribution $N(0,t-s)$ for $s<t$. The solution isIndeed applying the Itô formula to this exponential gives the specified drift and volatility. Consequently the price is positive and continuous, andThus disjoint log-return increments are independent and stationary. The parameter $\mu$ is the instantaneous expected relative return under the physical probability measure, not the mean log-return. Under the pricing risk-neutral measure it becomes the risk-free rate for a non-dividend-paying stock.

One alternative is a stochastic volatility model, such as the Heston model:The parameters satisfy $\kappa ,\theta ,\xi >0$ and $|\rho |\le 1$, and $v_0\ge 0$. The square-root Itô diffusion has a nonnegative solution; $2\kappa \theta \ge {\xi }^2$ is a standard sufficient condition for the positive initial variance not to hit zero. Mean reversion of $v_t$ allows persistent high- and low-volatility episodes, and negative $\rho$ can produce the empirical association between falling prices and rising volatility. Mixtures of conditional return distributions produce richer tails and option implied volatility shapes than constant-volatility geometric Brownian motion.

For data showing volatility clustering and option smiles, this is often a more useful fit. It is not universally superior: more parameters create estimation and calibration uncertainty, the model is more expensive to use, and its continuous price paths still exclude price jumps. With two independent noise directions and only a stock and a bank account, the market is generally incomplete, so a volatility risk premium or an extra traded instrument is needed for unique contingent claim pricing. The alternative improves flexibility at the cost of calibration and hedging complexity.

Assume a frictionless market with no taxes or transaction costs, unrestricted short selling and borrowing/lending at the same constant rate $r$, continuous trading, and absence of arbitrage. The non-dividend-paying stock has geometric Brownian motion dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma >0$, and the bank account satisfies $dB=rBdt$. A contingent claim with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal financial payoff kinks can be handled by the smooth value for earlier times. Trading strategies are adapted, self-financing portfolios, and admissible trading strategies so that doubling strategies are excluded.

The Itô formula givesChoose $\Delta =f_S$ shares and put the remaining value $f-S{f}_S$ in the bank account. Its self-financing gain isMatching the contingent claim's gain to this replicating gain cancels the random increment and equates the drifts:This is the Black-Scholes equation. The argument uses the self-financing gain $\Delta dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the bank account. Conversely, a smooth solution with suitable growth and boundary conditions defines a delta hedge whose value replicates the contingent claim; law of one price identifies this value with the contingent claim price. Here $S$ is the current stock price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the contingent claim price. The physical expected return $\mu$ cancels because exposure to price risk has been hedged.

For strike $K>0$ and maturity $T$, a European call option and a European put option have boundary conditionsThe Black-Scholes equation is solved backward from these conditions. At zero stock price, the zero state is absorbing in geometric Brownian motion, soAt the other boundary, with $\tau =T-t>0$,These conditions and a suitable growth bound select the financial solution and provide boundary data for numerical truncations. In the constant-coefficient model, under the risk-neutral measure,For $d_1=[\log (S/K)+(r+{\sigma }^2/2)\tau ]/(\sigma \sqrt{\tau })$ and $d_2=d_1-\sigma \sqrt{\tau }$, the exercise event is $Z>-d_2$. Completing the square in $e^{\sigma \sqrt{\tau }Z}$ times the normal density gives the truncated first moment and hencewhere $\Phi$ is the standard normal cumulative distribution function. These are the Black-Scholes formula values and satisfy the specified terminal and boundary conditions.

Put-call parity is the relationA call minus a put has terminal financial payoff $S_T-K$. A portfolio of one share minus a zero-coupon bond paying $K$ has the same financial payoff, so absence of arbitrage and law of one price equate their current values. The formula here assumes the same strike and maturity and no dividends; known dividends modify the stock-side value.

Separate $L$ into a diffusion generator and the constant killing term $-1/2$. The appropriate stochastic differential equation and its explicit geometric Brownian motion solution areThe second-order term in the Itô formula cancels the drift of $\log |X|$ when $x\ne 0$; if $x=0$, the solution stays zero.

Fix a terminal time $t$. Apply the Itô formula to $e^{-s/2}u(t-s,X_s)$ for $0\le s\le t$. Its drift isAfter localization this is a local martingale, and it is bounded because $u$ is bounded. It is therefore a true martingale. Taking expectations at its endpoints gives the Feynman-Kac formula

For $t>0$ and $x\ne 0$, the change of variable $y=x{e}^z$ in the normal density produces the killed geometric Brownian heat kernel with respect to Lebesgue measure:For $xy>0$, an equivalent form isIn particular,The apparent missing $1/y$ in this last expression is absorbed into the completed square, together with the killing factor; it is not a transcription error. The total mass is $e^{-t/2}$, as expected for killing at rate $1/2$. At $x=0$ the fundamental kernel is instead the measure $e^{-t/2}{\delta }_0(dy)$; it has no Radon-Nikodym derivative with respect to Lebesgue measure. Thus $u(t,0)=e^{-t/2}g(0)$. At $t=0$ the kernel is ${\delta }_x$, and continuity of bounded $g$ gives the initial condition. This also specifies the fundamental solution at the degenerate point omitted by a density-only formula.

The power-wealth investment with running utility objective is homogeneous of degree $\gamma$ in wealth: scaling both wealth and positions by $a>0$ scales the reward by $a^{\gamma }$. Accordingly set $J(w,t)=A(t)w^{\gamma }$. ThenFor completeness the remaining equation is $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, whereThis confirms $J_{ww}<0$. The constant optimal risky fraction also makes wealth a positive geometric Brownian motion when started positive.

Financially the relative risk aversion coefficient is $1-\gamma$. Increasing $\gamma$ reduces risk aversion, raises the risky fraction, and decreases the bond fraction: its derivative is $-(\mu -r)/[{\sigma }^2(1-\gamma {)}^2]<0$. If the risky fraction is below one, some wealth is lent; at one, none is invested in bonds; above one, $B^{\ast }<0$ represents borrowing to leverage the risky position. As $\gamma \uparrow 1$ the risky fraction becomes unbounded, consistent with the absence of a finite unconstrained optimum for linear utility and positive excess drift. As $\gamma \downarrow 0$ the limiting fraction is $(\mu -r)/{\sigma }^2$, the logarithmic-utility allocation obtained by the appropriately rescaled power-utility limit; the literal objective at $\gamma =0$ would be constant and would not select a unique strategy.