Sep 5, 2026
There in no question the theory developed by (Black and Scholes 1973) and (Merton 1973) for valuing derivative instruments has had a profound effect in the financial world over the past half century. Scholes and Merton won The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel for their contributions. Their eponymous partial differential equation is required study in every mathematical finance program.
Few traders could write it down, much less derive it, but they fully understand how delta is the position they need for their hedge and gamma is what they need to buy to adjust their hedge. They know the definition of vega as the derivative of the value of the position with respect to the multifarious word volatility.
There are other greeks defined as derivatives with respect to model parameters, but trader hedging is delta hedging. That is what shows up in their trade blotter and determines their P&L. Risk managers seem to take comfort in using higher order derivatives produced by models to empirically verify Taylor’s1 formula after market close.
B-S/M used the mathematical theory of Itô processes to show a position in a bond and a stock could perfectly hedge an option over an infinitesimal period of time. As a consequence, the portfolio is risk-free and must grow at the risk-free rate2 if there are no arbitrage opportunities. Earlier models assumed you needed to estimate the risk premium of the uncertainty in stock returns.
Black, Scholes, and Merton were well aware of the shortcomings of their model. Volatility is not constant. Peter Carr (rightly) insisted the word “volatility” should always be used with an adjective. The implied volatility is the constant to be plugged into B-S/M pricing formula that recovers the option price. Traders use implied volatility as a proxy for price just as they use yield as a proxy for the price of a bond: a constant that is not constant to plug into a mutually agreed upon formula.
The question the B-S/M theory has no answer to is how to provide a price given the strike and expiration of an option.
Suppose you are a trader and your boss tells you to make a market for a three month at-the-money call option on stock XYZ. Further suppose the stock is very liquid but has no traded options. Should we use historical realized volatility? If there are other stocks in the same sector that have traded options how do we incorporate that information?
This is not unlike the issue of finding the price of a bond that is thinly traded. How do you incorporate the limited data you have to estimate its price?
What follows is a simple and rigorous mathematical theory that provides a framework for tackling this fundamental problem.
There is a theory for valuing, hedging, and measuring the risk of any collection of instruments that generalizes the (Black and Scholes 1973) and (Merton 1973) models. (Ross 1978) showed
If there are no arbitrage opportunities in a market, then there must exist a (not generally unique) positive linear operator that can be used to value all marketed assets.
He used this to establish the Modigliani-Miller capital structure theorem and a resolution to the closed-end mutual fund paradox, among other results.
Let us stress again the power of these results. To begin with, expectational mechanisms to describe how we anticipate the index, X_t, will behave are irrelevant. Furthermore, there is no need for X_t to follow a diffusion process to apply the usual option analysis or, for that matter, for the stream c to follow any specific process. All that is necessary is the assumption that current asset values do not permit arbitrage. In particular, then, we do not even have to assume that the current index price is in equilibrium to correctly appraise the project.
Expectation mechanisms are irrelevant? No need for prices to follow a diffusion process? No need for equilibrium? Experts in the academic field seem to have trouble realizing they can cast off their cognitive anchor in the unnecessary accoutrements of the classical Black-Scholes and Merton theory.
If your career involves publishing papers, you may need to deal with a referee pointing out an error that might delay publication. If you are a quant responsible for turning mathematics into software that will be used to run a business, there can be more serious issues. The first time a trader shows up at your desk and says, “The P&L seems a little off.” you better have a good explanation. If you don’t, the second visit might include the head of the trading floor and angry questions like, “You told me this position was hedged! Were you lying to me then or are you lying to me now?”3
There is no need to assume stocks involve a “real world” measure that gets immediately thrown out for a “risk neutral” measure. Ross showed how to get the same results as B-S/M and how to generalize those from a bond, stock, and option to any collection of instruments. He seemed to realize the difficulty he would face trying to communicating that.
I believe that most of these are well known (at least at some level of consciousness), but the failure to understand either their universality or their applicability is endemic.
B-S/M answers the question of how much to trade but their answer to when to trade is “continuously” and claim the hedge is riskless. This note presents a model that does not give an obviously wrong answer and exhorts readers to improve upon it.
We reify Ross’s stream of cash flows c to a stochastic process (C_t) that is 0 except at a finite number of times. This leads to a simple and mathematically rigorous model to enable progress on the fundamental issues of when and how much to trade, and how to manage the associated risk.
Stephen (Ross 1978) showed there is no need for probability theory, Itô processes, or partial differential equations. Positive linear operators give rise to positive measures that can be divided by their total mass resulting in a “risk-neutral probability measure.” They are positive measures having mass 1, but are not the probability of anything. There is no need for a so-called real world measure that gets immediately thrown out. His model works for any collection of instruments, not just the bond, stock, and option B-S/M considered.
Every arbitrage-free model of prices (X_t) and cash flows (C_t) has the form \tag{1} X_t D_t = X_0 M_t - \sum_{s\le t} C_s D_s where (D_t) are positive (valuation) measures and (M_t) is a vector-valued martingale measure indexed by market instruments.
Exercise. Show \tag{2} X_t D_t = (X_u D_u + \sum_{t < s\le u} C_s D_s)|\mathcal{A}_t.
Hint: Replace X_u D_u by (1) and cancel terms in the sum. Use the martingale measure condition M_u|\mathcal{A}_t = M_t, u \ge t where the vertical bar indicates restriction of a measure.
If there are no cash flows then (2) can be interpreted as appropriately discounted prices are a martingale. This was pointed out by Paul (Samuelson 1965) prior to Black, Scholes and Merton.
If X_u D_u goes to 0 as u goes to infinity then (2) can be interpreted as price is the present value of appropriately discounted future cash flows. This was pointed out in Securities Analysis by (Graham and Dodd 1934).
For example, the B-S/M model of a bond and a stock having prices X_t = (R_t,S_t) and no dividends is D_t = e^{-\rho t}P and M_t = (1, e^{\sigma B_t - \sigma^2/2})P where P is Wiener measure on \Omega = C[0,\infty) and (B_t) is standard Brownian motion, B_t(\omega) = \omega(t).
Exercise. Show R_t = e^{\rho t} and S_t = S_0e^{\rho t + \sigma B_t - \sigma^2 t/2} where we assume initial prices {X_0 = (1, S_0)}.
B-S/M considered a dynamic hedge in the bond and stock that replicates a European option payoff. If the option pays \nu(S_T) at expiration T we need to find a trading strategy with value \nu(S_T) at T having no intermediate cash flows. The value of the option is the cost of setting up the initial hedge which they show to be E[\nu(S_T)D_T]. See the B-S/M appendix for the derivation.
Ross does not use probability measures. In the Ross model D_T is just a positive measure. Multiplying a measure by a function, \nu(S_T)D_T, is also a measure and he shows the option value is \nu(S_T)D_T(\Omega).
Instead of conditional expectation we use the simpler notion of restriction of measure.
Exercise. Show Y = E[X|\mathcal{A}] if and only if {Y(P|\mathcal{A}) = (XP)|\mathcal{A}}.
Hint: The definition of conditional expectation Y = E[X|\mathcal{A}] where \mathcal{A} is an algebra of sets on \Omega is Y is \mathcal{A}-measurable and {E[Y1_A] = E[X1_A]} for all A\in\mathcal{A}.
The value, or mark-to-market, of a hedge at any time is the amount that could potentially be obtained by liquidating the position at current market prices. If the value of the option is known as a function of hedging instrument prices then the delta hedge is the mathematical derivative of option value with respect to underlying prices. This gives an approximate hedge over a short period of time.
Every model specifies a sample space \Omega of possible outcomes. B-S/M use the space of continuous functions on the interval [0, \infty). Their notion of information available at time u is that given a stock trajectory (S_t)_{t\ge0} you know S_t exactly for t < u and nothing about the value of S_t if t \ge u.
Partial information about a sample space is specified by a partition: a collection of pairwise disjoint subsets having union equal to the sample space. No information is the singleton partition \{\Omega\}. Complete information is the partition of singletons {\{\{\omega\}\mid\omega\in\Omega\}}. Partial information is knowing which atom in the partition \omega\in\Omega belongs to.
If X\colon\Omega\to\boldsymbol{{R}} is a function and \mathcal{A} is a partition of \Omega then X is \mathcal{A}-measurable if and only if it is constant on each element of the partition.
Exercise. Show X\colon\mathcal{A}\to\boldsymbol{{R}} is a well-defined function on the atoms of \mathcal{A}.
Hint: Show if \omega and \omega' belong to the same atom of the partition then X(\omega) = X(\omega').
Let I be the set of market instruments and T be the set of trading times. We use \mathcal{A}_t for the partition of \Omega representing information available at time t\in T. We assume \mathcal{A}_u is a refinement of \mathcal{A}_t when u > t. This means every A\in\mathcal{A}_t is the union of atoms in \mathcal{A}_u contained in A, u > t. Information increases over time.
Every instrument has prices and cash flows. We make the usual idealized assumption that any amount can be bought or sold at the given price and the unequivocal fact cash flows accrue in proportion to the amount of the instrument held.
Prices and cash flows are functions X_t,C_t\colon\mathcal{A}_t\to\boldsymbol{{R}}^I with values indexed by the set of market instruments. We assume prices are bounded and that cash flows are 0 except at a finite number of times, just like in the real world.
A trading strategy is a sequence of stopping times \tau_0 < \tau_1 < \cdots \tau_n and trading sizes \Gamma_j\colon\mathcal{A}_{\tau_j}\colon\boldsymbol{{R}}^I.
A stopping time is a function \tau\colon\Omega\to T where \tau = t is known at time t, i.e., {\{\omega\mid\tau(\omega) = t\}} is a union of atoms of \mathcal{A}_t for all t\in T. The partition corresponding to a stopping time is {\mathcal{A}_\tau = \cup\{\{\tau = t\}\mid t\in T\}}.
Exercise. Show \mathcal{A}_\tau is a partition of \Omega for any stopping time \tau.
Trading sizes accumulate to positions \Delta_t = \sum_{\tau_j < t} \Gamma_j. Note the strict inequality. It takes some time for a trade executed at t to settle. To simplify formulas we write {\Delta_t = \sum_{s < t} \Gamma_s} where \Gamma_s = \Gamma_j when s = \tau_j and is zero otherwise.
The trading account accrues cash flows proportional to the current position and debit the amount paid for trades just executed at the current market price. The amount showing up in the trading account at time t is {A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t}.
The value, or mark-to-market, is the amount that could be putatively obtained from liquidating the existing positions and trades just done at current market prices: {V_t = (\Delta + \Gamma_t)\cdot X_t}. No actual trading is involved with calculating the value. Figuring out what prices to use for thinly traded instruments is problematic.
The profit and loss over a period is the difference in values.
Trading strategies create synthetic market instruments. Using the definitions of V_t and A_t \tag{3} V_t D_t = (V_u D_u + \sum_{t < s\le u} A_s D_s)|\mathcal{A}_t. Compare this to equation (2) and note how prices (X_t) and cash flows (C_t) correspond to values (V_t) and amounts (A_t). This is the skeleton key to valuing derivatives.
Exercise. Prove equation (3) from equation (2) using the definitions of V_t and A_t.
Hint: Start with V_t D_t = (\Delta_t + \Gamma_t)\cdot X_t D_t then equation (2) with u - t sufficiently small so X_t D_t = (X_u D_u + C_u D_u)|\mathcal{A}_t. Replace the \Delta_t\cdot C_t term with A_t + \Gamma_t\cdot X_t and use finite induction.
Arbitrage is a trading strategy that makes money on the first trade and never loses money until it is closed out (position is zero). Mathematically, A_{\tau_0} > 0, A_t \ge0 for t > \tau_0, and \sum_{j=0}^n \Gamma_j = 0. Note this definition does not require probability.
If (D_t) are positive measures on \mathcal{A}_t, t\in T, then V_{\tau_0} D_{\tau_0} = (\sum_{j = 1}^n A_{\tau_j} D_{\tau_j}) \mid \mathcal{A}_{\tau_0}
No need for Hahn-Banach.
Every positive random variable with finite expected value and log-variance can be written F = fe^{sX - \kappa(s)} where f = E[F], s^2 = \operatorname{Var}(\log F), and \kappa(s) = \log E[e^{sX}] is the cumulant of the mean 0 and variance 1 random variable X. We call f the forward and s the vol.
The forward value of a put option having strike k is \begin{aligned} v &= E[\max\{k - F,0\}] \\ &= E[(k - F)1(F \le k)] \\ &= k P(F\le k) - E[F 1(F\le k)] \\ &= k P(F\le k) - f [(F/f) 1(F\le k)] \\ &= k P(F\le k) - f P_s(F\le k) \\ \end{aligned} where P_s = e^{sX - \kappa(s)}P.
Exercise. Show P_s is a probability measure.
Since F\le k if and only if X \le (\log k/f + \kappa(s))/s E[\max\{k - F,0\}] = k\Psi(x,0) - f\Psi(x,s) where x = x(k, f, s) = (\log k/f + \kappa(s))/s is the moneyness and \Psi(x,s) = P_s(X\le x).
Option delta is the derivative of value with respect to forward price \partial_f v = E[-1(F\le k)\partial_f F] = -E[1(F\le k)e^{sX - \kappa(s)}] = -\Psi(x,s).
Option gamma is the second derivative of value with respect to forward price \partial_f^2 v = -\partial_x\Psi(x,s)\partial_f x = \partial_x\Psi(x,s)/fs.
Option vega is the derivative of value with respect to vol \partial_s v = E[-1(F\le k)\partial_s F] = -E[1(F\le k)F(X - \kappa'(s))] Since \partial_s \Phi(x,s) = \partial_s E[e^{sX - \kappa(s)} 1(X \le x)] = E[e^{sX - \kappa(s)}(X - \kappa'(s)) 1(X \le x)] we have \partial_s v = f\partial_s\Phi(x,s).
The Fischer (Black 1976) model uses s = \sigma\sqrt{t} and X standard normal. Recall {E[\exp(N)] = \exp(E[N] + \operatorname{Var}(N)/2)} if N is normal.
Exercise. Show the cumulant of a standard normal is s^2/2.
Recall E[\exp(N) f(M)] = E[\exp(N)] E[f(M + \operatorname{Cov}(N,M)] if N and M are jointly normal.
Exercise. If X is standard normal show \Psi(x, s) = \Psi(x - s, 0).
Hint: Use \Psi(x, s) = E[e^{sX - s^2/2} 1(X \le x)].
Exercise. Show the Black forward put value is {v = k N(-d_-) - f N(-d_+)} where N(x) = \Psi(x, 0) and {d_\pm = (\log f/k \pm \sigma^2 t/2)/\sigma\sqrt{t}}.
Exercise. Show the Black delta is -N(-d_+).
Exercise. Show the Black gamma is N'(d_+)/f\sigma\sqrt{t}.
Exercise. Show the Black vega is \partial_\sigma v = f N'(d_+)\sqrt{t}.
The B-S/M model uses {f = S_0e^{\rho t}} where S_0 is the initial stock price, {s = \sigma\sqrt{t}} and the option value is discounted by {e^{-\rho t}}.
Exercise. Derive the classical B-S/M value and greeks.
Risk-free vs risk-neutral. Funding rate for strategies.
Commodity contracts – storage costs
Continuous dividend rate. Hah!
The original Black and Scholes paper had some mathematical mistakes and unnecessary assumptions. Robert Merton learned about Itô processes as a student at Caltech and graciously waited to publish his rigorous derivation, back in the day when gentlemen scholars respected priority.
We give a schematic review of Merton’s mathematically correct derivation of their eponymous partial differential equation to compare with Ross’s theory.
Every stochastic process (X_t)_{t\ge0} is determined by its joint cumulative distribution function F_{t_0,\ldots,t_n}(x_0,\ldots,x_n) = P(X_{t_0} \le x_0,\ldots, X_{t_n}\le t_n). Given samples from a stochastic process
A process is stationary if (X_t) and X_{t + s} - X_s have the same joint cumulative distribution. The mathematical import is that you can start sampling at any time and get the same statistics. The reality is that this is rare in financial time series.
A process has independent increments if given t_0 < t_1 < \cdots t_n then X_{t_0}, X_{t_1} - X_{t_0}, \ldots, X_{t_n} - X_{t_{n-1}} are independent.
He also showed if a stochastic process is stationary with independent increments it is completely determined by its distribution at a single time and its distribution is infinitely divisible
A random variable X is infinitely divisible if for any n\in\boldsymbol{{N}} X has the same law as X_1 + \cdots + X_n where (X_i) are independent have have the same law.
Kolmogorov showed if for any n\in\boldsymbol{{N}} X = X_1 + \cdots + X_n where X_i have finite variance, have the same distribution, and are independent then \log E[e^{sX}] = \gamma s + \int_{-\infty}^\infty K_x(s)\,dG(s) where K_x(s) = (e^{sx} - 1 - sx)/s^2 = \sum_{k=2}^\infty s^{n-2}x^n/n!.
They assume the bond price satisfies dR_t/R_t = \rho\,dt and stock price satisfies the stochastic differential equation dS_t/S_t = \mu\,dt + \sigma\,dB_t where \rho, \mu, and \sigma are constant and (B_t) is standard Brownian motion.
Mathematically rigorous proofs require the SDE be expressed as an Itô integral.
but we will get by with the Itô calculus (dt)^2 = 0, dt\,dB_t = dB_t\,dt = 0, and (dB_t)^2 = dt.
Suppose 0 = t_0 < t_1 < \cdots t_n = t. Define {\Delta t_j = t_{j + 1} - t_j} and {\Delta t = \max_{0\le j < n} \Delta t_j}.
Exercise. Show \sum_{0\le j < n} \Delta t_j = t.
Hint: It is a telescoping sum.
Exercise. [(dt)^2 = 0] Show \sum_{0\le j < n} (\Delta t_j)^2 goes to zero as \Delta t goes to zero.
Hint: Start with \sum_{0\le j < n} (\Delta t_j)^2 \le \Delta t \sum_{0\le j < n} \Delta t_j
Exercise. [dt\,dB_t = 0] Show E[\sum_{0\le j < n} \Delta t_j\,\Delta B_j] = 0 and \operatorname{Var}(\sum_{0\le j < n} \Delta t_j\,\Delta B_j) goes to zero as \Delta t goes to zero.
Hint: Use \operatorname{Var}(\Delta B_j) = \Delta t_j and (\Delta B_j) are independent.
Exercise. [(dB_t)^2 = dt] Show E[\sum_{0\le j < n} (\Delta B_j)^2] = t and \operatorname{Var}(\sum_{0\le j < n} (\Delta B_j)^2) goes to zero as \Delta t goes to zero.
Hint: You will need to compute \operatorname{Var}(\Delta B_t^2) = E[\Delta B_t^4] - E[\Delta B_t^2]^2. Use E[e^{sX}] = e^{s^2/2} if X is standard normal and read off the moments using E[e^{sX}] = \sum_{n=0}^\infty s^n E[X^n]/n!.
We need the facts that if X and Y are Itô processes then so are aX for a\in\boldsymbol{{R}}, X + Y, and XY. Recall {d(XY) = (X + dX)(Y + dX) - XY = X\,dY + Y\,dX + dX\,dY}.
Itô showed that Y_t = f(t,X_t) is an Itô diffusion that satisfies dY_t = f_t(t, X_t) + f_x(t, X_t) + \frac{1}{2} f_{xx}(t, X_t) if the partial derivatives of f satisfy certain technical conditions. This can be generalized to vector valued processes X = (X_i). dY_t = f_t(t, X_t) + \sum_i f_{x_i}(t, X_t) + \sum_{i,j} \frac{1}{2} f_{x_ix_j}(t, X_t)
Merton considered strategies where the position held in each instrument depended on the time where M_t = m(t, R_t, S_t) is the amount held in the bond and N_t = n(t, R_t, S_t) is the amount held in the stock at time t. The value of the position at t is V_t = M_t R_t + N_t S_t.
Dropping the t subscript and using the Itô calculus \begin{aligned} dV &= (M\,dR + R\,dM + dM\,dR) + (N\,dS + S\,dN + dN\,dS \\ &= dM(R + dR) + dN(S + dS) + M\,dR + N\,dS \\ \end{aligned} The first two terms in the last equation are the cost of adjusting the bond and stock hedge at time t + dt. A trading strategy is self-financing if this is 0.
Note M\,dR = (V - NS)\rho\,dt since M = (V - NS)/R. Using the Itô formula for V_t = v(r, R_t, S_t) we have \begin{aligned} dV_t &= (V_t - N_t S_t)\rho\,dt + N_t\,dS_t \\ &= V_t\,dt + v_r\,dR + v_s\,dS + \frac{1}{2}v_{ss} (dS_t)^2 \\ &= (v_t + \rho R_t v_r + \frac{1}{2}\sigma^2 S_t^2 v_{ss})\,dt + v_s\,dS_t. \\ \end{aligned} Equating the dS_t terms gives the stock hedge N_t = v_s and equating the dt terms gives (v - s v_s)\rho = v_t + \rho r v_r + \frac{1}{2}\sigma^2 s^2 v_{ss}.