Differentiable pricing and automatic differentiation

Definition A computational graph of a program is a directed acyclic graph whose nodes are the inputs, the intermediate values and the outputs of the computation, with an edge from $v_i$ to $v_j$ when $v_j$ is computed by an elementary operation that uses $v_i$. Each edge carries the local partial derivative $\partial v_j/\partial v_i$ of that operation, evaluated at the computed values. If every operation is differentiable at those values, the chain rule says that the derivative of an output with respect to an input is the sum, over all directed paths from the input to the output, of the product of the local derivatives along each path. For one simulated terminal value of a stock without dividends in the Black-Scholes model under a pricing measure $\mathbb{Q}$, fix a realization $z$ of the standard normal variable $W^{\mathbb{Q}}_T/\sqrt{T}$ and compute

with inputs $S_0$, $r$ and $\sigma$, and with $K$, $T$ and $z$ held fixed. The rate reaches the discounted payoff $Y$ along two paths, one through the exponent $a$ and one through the discount factor $D$.

Definition Automatic differentiation (AD) evaluates derivatives of a function defined by a program by applying the chain rule to the program’s elementary operations, each of which has a known local derivative. It uses no step size. At an input where every operation is differentiable at its computed arguments and every branch taken stays the same for all nearby inputs, it returns the derivative of the function that the program computes, up to floating-point rounding. In forward mode a seed vector $\dot\theta$ of input tangents is pushed through the graph in the order of computation,

and one sweep returns $\mathcal{J}\dot\theta$, the directional derivative of every output, where $\mathcal{J}$ is the Jacobian of the outputs with respect to the inputs. A sweep costs a small constant multiple of one evaluation, and a full Jacobian with $n$ inputs takes $n$ forward sweeps. AD differentiates the computation, not the price it approximates. Time discretization, the Monte Carlo sample, exercise decisions and fitted parameters are all part of what is differentiated. If a model parameter is calibrated to market quotes, a sensitivity to a quote needs the dependence of the fit on the quote, obtained either by differentiating through the calibration or by the implicit function theorem applied to its optimality conditions, when these are continuously differentiable and their Jacobian with respect to the parameters is invertible. Holding the fitted parameter fixed answers a different question.

Definition For a scalar output $y$, the adjoint of a node $v_i$ is $\bar v_i = \partial y/\partial v_i$, the rate at which $y$ changes when $v_i$ changes and the change is passed through every operation that uses $v_i$. Seeding $\bar y = 1$ and visiting the nodes in reverse order of computation,

where the sum runs over all edges leaving $v_i$. The adjoints of the inputs form the gradient of $y$. One reverse sweep costs a small constant multiple of one evaluation whatever the number of inputs, so it suits a price with many inputs, while forward mode suits few inputs and many outputs. Reverse mode needs the intermediate values of the forward computation, which it stores or recomputes, and checkpointing stores some of them and recomputes the rest to save memory. A node used in several places must add the contributions of all its edges, and overwriting one with another drops part of the chain rule. Iterative solvers and branches need their own treatment, for example by differentiating the converged equations instead of every iteration.

Definition Let $Y(\theta) = Y(\theta, \omega)$ be a discounted payoff computed from a parameter $\theta$ and random inputs $\omega$, here the normal draw. The pathwise derivative at $\theta$ is $\partial Y(\theta, \omega)/\partial\theta$ with $\omega$ held fixed, wherever it exists. For the call path with $\sigma > 0$, $T > 0$ and $S_T \ne K$,

measured in currency per currency unit of spot, currency per unit of decimal volatility and currency per unit of decimal rate. On paths with $S_T < K$ all three are zero, and at $S_T = K$ the payoff has a kink and no derivative. The pathwise estimator of a price sensitivity $V'(\theta)$ is the average of the path derivatives over independent draws. It estimates the right quantity only when the derivative and the expectation can be exchanged, the subject of the next card. Central differences computed with the same draws check a path derivative, while fresh draws for each bumped input would mix a change of path into the difference.

Definition Let $V(\theta) = \mathbb{E}^{\mathbb{Q}}[Y(\theta)]$ for $\theta$ in an open interval, with each $Y(\theta)$ integrable, and fix $\theta_0$ in it. Suppose that (i) with probability one, $\theta \mapsto Y(\theta)$ is differentiable at $\theta_0$, and (ii) there are $\varepsilon > 0$ and an integrable random variable $M$ such that, for each $\theta$ with $\lvert \theta - \theta_0 \rvert < \varepsilon$, the bound $\lvert Y(\theta) - Y(\theta_0) \rvert \le M \lvert \theta - \theta_0 \rvert$ holds with probability one. Then $V$ is differentiable at $\theta_0$ and

The proof applies dominated convergence to the difference quotients along any sequence $\theta_n \to \theta_0$. For the call delta, $x \mapsto (x u - K)^{+}$ is Lipschitz with constant $u$, so condition (ii) holds with $M = D u$, and $\mathbb{E}^{\mathbb{Q}}[M] = e^{-rT}\, \mathbb{E}^{\mathbb{Q}}[e^{a}] = 1$. Condition (i) holds because $S_T$ has a continuous law when $\sigma > 0$ and $T > 0$, so the kink $S_T = K$ has probability zero. Hence

with

For vega and rho similar bounds hold for $\theta$ in a small closed interval, because the exponential of a normal variable times a polynomial in it is integrable, and the averages of the path derivatives are $S_0 \varphi(d_1)\sqrt{T}$ and $K T e^{-rT}\Phi(d_2)$. In other models a kink of probability zero is not enough by itself, and the integrable bound is what carries the argument. If $\sigma = 0$, the terminal price $S_0 e^{rT}$ is deterministic, and the price $(S_0 - K e^{-rT})^{+}$ has no derivative at $S_0 = K e^{-rT}$, where the kink has probability one.

Definition A cash-or-nothing digital call pays $1$ currency unit at $T$ if $S_T > K$. With $z$ fixed, its discounted path payoff $Y = D\, \mathbf{1}_{\{S_0 u > K\}}$ is piecewise constant in $S_0$, so $\partial Y/\partial S_0 = 0$ on every path with $S_T \ne K$, that is with probability one, and AD through the indicator returns a zero delta on every path. The price $V = e^{-rT}\Phi(d_2)$ has

so the interchange fails. Condition (ii) of the previous card is what breaks. A bump $h$ of spot changes the payoff by $D$ on the event that $S_T$ crosses $K$, whose probability is of order $\lvert h \rvert$, so the difference quotient equals $D/\lvert h \rvert$ on that event, and no integrable $M$ bounds these quotients for all small $h$. The change in price comes from probability mass that moves across the strike, which the path derivative does not see. Methods that handle it under their own assumptions include the likelihood ratio method, which differentiates the lognormal density instead of the payoff and in this model gives

conditioning on part of the randomness to get a smooth conditional expectation, smoothing the indicator at the cost of a bias, and analytic boundary terms. The same failure affects the second derivatives of a call. The pathwise delta $D u\, \mathbf{1}_{\{S_T > K\}}$ jumps at the strike, so differentiating it again path by path returns zero on every path, while the gamma $\varphi(d_1)/(S_0 \sigma \sqrt{T})$ is positive. A valid first-order pathwise derivative does not make the second-order one valid.