Model uncertainty and robust valuation, financial engineering for quants lesson

A one-year call on a stock at 100 with strike 105 can be priced to many decimals once its volatility is fixed, but suppose the volatility is only known to lie between 18 and 28 percent. What range of values does that doubt justify, and what changes when the doubt concerns the model itself rather than one of its inputs? This lesson separates parameter uncertainty from model ambiguity and from numerical error, derives an exact volatility envelope from the sign of vega, bounds first-order changes over a norm ball with the dual norm, and computes robust valuation bounds over families of pricing measures.

Key terms

Parameter uncertainty

Definition Let a model family assign to each parameter vector $\theta$ in a set $\Theta \subseteq \mathbb{R}^d$ the value $V(\theta)$ of a fixed claim, computed under a stated pricing convention that fixes the pricing measure of each member, the numeraire, the payoff and every input outside $\theta$ . Parameter uncertainty is the situation in which the family is taken as given but $\theta$ is only known to lie in an uncertainty set $U \subseteq \Theta$ . It is distinct from numerical error, which is the difference between a computed number and the exact $V(\theta)$ at one fixed $\theta$ . Refining a grid, adding Monte Carlo paths or tightening a root finder reduces numerical error and leaves parameter uncertainty untouched. The set $U$ carries a probability statement only if a stated statistical procedure produced it.

In plain words The pricing formula is trusted, but one of its inputs is not known exactly. The question becomes how far the answer moves as that input ranges over the values still considered possible, and no amount of extra computing precision can settle it.

Example In the Black-Scholes model with the constructed inputs $S = 100$ , $K = 105$ , $\tau = 1$ , $r = 0.02$ and $q = 0$ , the call value $C(\sigma)$ is a function of the volatility alone. At $\sigma = 0.23$ the formula gives $C \approx 7.9011$ , and a Monte Carlo estimate of the same number would have a standard error that falls like $1/\sqrt{n}$ in the number $n$ of paths. Neither fact says whether $0.23$ is the right volatility. The set $U = [0.18, 0.28]$ expresses that doubt, and across it the value runs from about $5.9082$ to about $9.8954$ , as the cards below show.

Model ambiguity

Definition Model ambiguity is uncertainty about the model family or the probability law itself, rather than about inputs inside one family. It is described by a set $\mathcal{Q}$ of candidate pricing measures, or of models that each produce one, for example a constant-volatility model next to a stochastic volatility model calibrated to the same quotes. Each member must satisfy the restrictions it is meant to represent, typically equivalence to $\mathbb{P}$ , the martingale property of the traded prices discounted by the chosen numeraire, and agreement with the quoted prices that are taken as given. A candidate that breaks one of these restrictions is inadmissible for that market. Parameter uncertainty is the special case $\mathcal{Q} = \{\mathbb{Q}_\theta : \theta \in U\}$ of one parametric family. When the restrictions tie several parameters together, combining the separate extreme values of each parameter can leave the admissible set.

In plain words Here the doubt is about which probability law, or which kind of dynamics, should price the claim. Every candidate must still reproduce what the market already shows, and market data narrow the candidates without always pinning one down.

Example In a constructed one-period market, a stock at $100$ pays $120$ , $100$ or $80$ , and cash grows by the factor $1/0.98$ , so the discount factor is $D = 0.98$ . The market is free of arbitrage because $0.98 \cdot 80 < 100 < 0.98 \cdot 120$ . A measure $\mathbb{Q} = (q_u, q_m, q_d)$ with positive entries prices the stock correctly exactly when $0.98(120 q_u + 100 q_m + 80 q_d) = 100$ , which together with $q_u + q_m + q_d = 1$ means

\begin{aligned} q_u - q_d &= \frac{100/0.98 - 100}{20} \\[0.4em] &\approx 0.10204 . \end{aligned}

The admissible measures form a one-parameter family with $0.10204 < q_u < 0.55102$ , $q_d = q_u - 0.10204$ and $q_m = 1 - q_u - q_d$ . A call with strike $100$ pays $20$ in the top state only, so its value $0.98 \cdot 20 \cdot q_u = 19.6\,q_u$ ranges over the open interval $(2.0, 10.8)$ , the interval of arbitrage-free call prices in the sense of Market completeness and nonunique prices. A quoted call price of $6$ would leave the single measure with $q_u = 6/19.6 \approx 0.30612$ . Taking $q_u = 0.5$ and $q_d = 0.01$ , each inside its own range, violates the stock restriction, so that pair is not admissible.

Valuation envelope

Definition For a value function $V$ on an uncertainty set $U$ , let

\begin{aligned} \underline{V}_U &= \inf_{\theta \in U} V(\theta), \\[0.4em] \overline{V}_U &= \sup_{\theta \in U} V(\theta). \end{aligned}

The valuation envelope is the interval $[\underline{V}_U, \overline{V}_U]$ . If $U$ is compact and $V$ is continuous on it, both ends are attained, and if $U$ is also connected, for example an interval or a convex set, every value between them is attained, so the envelope is exactly the image $V(U)$ . The envelope is monotone in the set. If $U \subseteq U'$ , then

\begin{gathered} \underline{V}_{U'} \le \underline{V}_U, \\[0.4em] \overline{V}_{U'} \ge \overline{V}_U, \end{gathered}

because optimizing over more candidates cannot give a worse optimum. The envelope is conditional on $U$ and on the family. It is a confidence interval only when $U$ comes from a confidence procedure. Then, if the family is correct and the procedure covers the true $\theta$ with probability at least $1 - a$ , the envelope covers $V(\theta)$ with probability at least $1 - a$ , since $\theta \in U$ forces $V(\theta)$ into the envelope. Nor is the envelope automatically an interval of arbitrage-free prices, which arises when the family is exactly the set of pricing measures consistent with the traded prices, as in the model ambiguity example.

In plain words The envelope answers one question, which values this model produces as the uncertain inputs range over this set. A larger set can only widen it, and how believable its ends are depends entirely on how the set was chosen.

Example For the call, the set $[0.18, 0.28]$ gives the envelope from about $5.9082$ to about $9.8954$ , a width of about $3.9873$ , and the smaller set $[0.20, 0.26]$ gives the envelope from about $6.7048$ to about $9.0979$ , inside the first one as the nesting property requires. The next card shows why the endpoints of each set suffice. Both sets were chosen by hand, so neither envelope carries a coverage probability, and neither covers models outside the constant-volatility family.

Monotone parameter bound

Definition Let $V$ be continuous and nondecreasing on a closed interval $[a, b]$ . Then $V(a) \le V(\theta) \le V(b)$ for every $\theta \in [a, b]$ , so the envelope is $[V(a), V(b)]$ with both ends attained, and for a nonincreasing $V$ the ends swap. When $V$ is differentiable on $(a, b)$ , the condition $V' \ge 0$ there is sufficient, by the mean value theorem. In the Black-Scholes model with $S, K, \tau > 0$ , constant $r$ and $q$ , and $\sigma > 0$ , the value of a European call is

\begin{aligned} C &= S e^{-q\tau}\Phi(d_1) \\[0.4em] &\quad - K e^{-r\tau}\Phi(d_2), \end{aligned}

with

\begin{aligned} d_1 &= \frac{1}{\sigma\sqrt{\tau}}\Bigl[\ln\frac{S}{K} \\[0.4em] &\qquad + \bigl(r - q + \tfrac12\sigma^2\bigr)\tau\Bigr], \\[0.4em] d_2 &= d_1 - \sigma\sqrt{\tau}. \end{aligned}

Since $d_1^2 - d_2^2 = (d_1 - d_2)(d_1 + d_2) = 2\ln(S/K) + 2(r - q)\tau$ , the ratio $\varphi(d_2)/\varphi(d_1) = e^{(d_1^2 - d_2^2)/2}$ equals $(S/K)\,e^{(r - q)\tau}$ , which is the identity

S e^{-q\tau}\varphi(d_1) = K e^{-r\tau}\varphi(d_2),

and differentiating the call value gives

\begin{aligned} &\frac{\partial C}{\partial \sigma} \\[0.4em] &= S e^{-q\tau}\varphi(d_1)\,\frac{\partial d_1}{\partial \sigma} \\[0.4em] &\quad - K e^{-r\tau}\varphi(d_2)\,\frac{\partial d_2}{\partial \sigma} \\[0.4em] &= S e^{-q\tau}\varphi(d_1)\,\frac{\partial (d_1 - d_2)}{\partial \sigma} \\[0.4em] &= S e^{-q\tau}\varphi(d_1)\sqrt{\tau} > 0 . \end{aligned}

So the call value is strictly increasing in $\sigma$ , and over any interval $[\sigma_{\text{lo}}, \sigma_{\text{hi}}]$ with $\sigma_{\text{lo}} > 0$ its envelope within the constant-volatility family is exactly $[C(\sigma_{\text{lo}}), C(\sigma_{\text{hi}})]$ . The endpoint rule comes from the sign of the derivative, not from a general property of bounded sets, and a value whose derivative changes sign inside $U$ can have an interior extremum. In several parameters, a value that is nondecreasing in each coordinate on a box $\prod_i [a_i, b_i]$ attains its envelope at the two opposite corners $a$ and $b$ , but if a restriction ties the parameters together, those corners may lie outside the admissible set.

In plain words If raising the volatility always raises the price, the cheapest price over a volatility range sits at its lowest volatility and the dearest at its highest, and nothing in between needs to be checked. The sign of vega is what justifies this shortcut, so it has to be proved for each payoff.

Example With $S = 100$ , $K = 105$ , $\tau = 1$ , $r = 0.02$ and $q = 0$ , the volatility $0.23$ gives $d_1 \approx -0.0102$ and $d_2 \approx -0.2402$ , and both sides of the density identity equal about $39.8922$ , which is the vega in currency units per unit of decimal volatility. The endpoints give $C(0.18) \approx 5.9082$ , from $d_1 \approx -0.0699$ and $d_2 \approx -0.2499$ , and $C(0.28) \approx 9.8954$ , from $d_1 \approx 0.0372$ and $d_2 \approx -0.2428$ . Positive vega makes $[5.9082, 9.8954]$ the exact envelope of the constant-volatility family over $[0.18, 0.28]$ , up to the rounding of the printed digits. It says nothing about models outside that family.

Dual norm bound

Definition Let $\lVert \cdot \rVert$ be a norm on $\mathbb{R}^d$ . Its dual norm is

\lVert g \rVert_{*} = \sup_{\lVert u \rVert \le 1} g^{\top}u .

For every $g \in \mathbb{R}^d$ and $\varepsilon > 0$ ,

\sup_{\lVert \Delta\theta \rVert \le \varepsilon} g^{\top}\Delta\theta = \varepsilon\,\lVert g \rVert_{*},

because $\Delta\theta = \varepsilon u$ maps the unit ball onto the ball of radius $\varepsilon$ and $g^{\top}(\varepsilon u) = \varepsilon\, g^{\top}u$ . The infimum is $-\varepsilon\lVert g \rVert_*$ , since the ball is symmetric. If $V$ is differentiable at $\theta_0$ , the first-order change $\nabla V(\theta_0)^{\top}\Delta\theta$ therefore ranges over $[-\varepsilon\lVert \nabla V(\theta_0) \rVert_*, \varepsilon\lVert \nabla V(\theta_0) \rVert_*]$ on the ball. Familiar pairs are the Euclidean norm, which is its own dual, the maximum norm, whose dual is $\lVert g \rVert_1 = \sum_i \lvert g_i \rvert$ , and the ellipsoidal norm $\sqrt{u^{\top}A^{-1}u}$ for a symmetric positive definite matrix $A$ , whose dual is $\sqrt{g^{\top}A g}$ . This is a statement about the linear term only. The full change is

\begin{aligned} &V(\theta_0 + \Delta\theta) - V(\theta_0) \\[0.4em] &= \nabla V(\theta_0)^{\top}\Delta\theta + R(\Delta\theta), \end{aligned}

and the bound becomes a guarantee only once the remainder $R$ is controlled. One way is the mean value theorem. If $V$ is differentiable on an open set containing the ball, then, using $g^{\top}u \le \lVert g \rVert_*\lVert u \rVert$ along the segment from $\theta_0$ to $\theta_0 + \Delta\theta$ ,

\begin{aligned} &\lvert V(\theta_0 + \Delta\theta) - V(\theta_0) \rvert \\[0.4em] &\le \varepsilon \sup_{\lVert \theta - \theta_0 \rVert \le \varepsilon} \lVert \nabla V(\theta) \rVert_* \end{aligned}

for every $\Delta\theta$ with $\lVert \Delta\theta \rVert \le \varepsilon$ .

In plain words To first order a value moves along its gradient, and the worst direction inside an uncertainty ball depends on the shape of the ball as much as on the gradient. The dual norm packs that worst case into one number, but only for the linear part of the change. It describes local sensitivity, not a guarantee over a large set or over other models.

Example For the call at $\sigma = 0.23$ , the one-dimensional ball $\lvert \Delta\sigma \rvert \le 0.05$ gives the first-order change $0.05 \cdot 39.8922 \approx 1.9946$ and the interval from about $5.9065$ to about $9.8957$ . The exact envelope from about $5.9082$ to about $9.8954$ differs at the two ends by about $0.0017$ and $0.0003$ , the contribution of the terms of second and higher order. Here the linear interval happens to contain the exact one, which is not true in general. Now let the rate move too, with rho $\partial C/\partial r \approx 41.6930$ in currency units per unit of decimal rate. The box $\lvert \Delta\sigma \rvert \le 0.02$ , $\lvert \Delta r \rvert \le 0.005$ is the unit ball of the weighted maximum norm $\max(\lvert u_1 \rvert/0.02, \lvert u_2 \rvert/0.005)$ , whose dual is $0.02\lvert g_1 \rvert + 0.005\lvert g_2 \rvert$ , so the first-order bound is

\begin{aligned} &0.02 \cdot 39.8922 + 0.005 \cdot 41.6930 \\[0.4em] &\approx 0.7978 + 0.2085 \approx 1.0063 . \end{aligned}

The ellipse $(\Delta\sigma/0.02)^2 + (\Delta r/0.005)^2 \le 1$ leaves out the corners of the box, and its dual norm gives the smaller bound $\sqrt{0.7978^2 + 0.2085^2} \approx 0.8246$ .

Robust valuation bounds

Definition Let $\mathcal{Q}$ be a nonempty set of admissible pricing measures, let $D$ be the discount factor of the stated numeraire convention, for example $1/B_T$ with the money-market account or a deterministic $D(0, T)$ , and let $X$ be a payoff with $\mathbb{E}^{\mathbb{Q}}[\lvert D X \rvert] < \infty$ for every $\mathbb{Q} \in \mathcal{Q}$ . The robust lower value and the robust upper value are

\begin{aligned} \underline{V} &= \inf_{\mathbb{Q} \in \mathcal{Q}} \mathbb{E}^{\mathbb{Q}}[D X], \\[0.4em] \overline{V} &= \sup_{\mathbb{Q} \in \mathcal{Q}} \mathbb{E}^{\mathbb{Q}}[D X]. \end{aligned}

They need not be attained, as in an open family, and they can be infinite when the expectations are unbounded over the family. Like any envelope, they widen when the family grows. For a fixed payoff the map $\mathbb{Q} \mapsto \mathbb{E}^{\mathbb{Q}}[D X]$ is affine under mixtures, since for $\lambda \in [0, 1]$

\begin{aligned} &\mathbb{E}^{\lambda\mathbb{Q}_1 + (1 - \lambda)\mathbb{Q}_2}[D X] \\[0.4em] &= \lambda\,\mathbb{E}^{\mathbb{Q}_1}[D X] \\[0.4em] &\quad + (1 - \lambda)\,\mathbb{E}^{\mathbb{Q}_2}[D X], \end{aligned}

so adding mixtures of members to the family does not change the bounds. When the family is indexed by a parameter, the shape of the objective in that parameter comes from how the law depends on it, and the extremes have to be found from that objective, by monotonicity, by convexity or by a search over the whole set. With $D = 1$ and a loss in place of $X$ , the upper value has the supremum form of the dual representation of a coherent risk measure.

In plain words Each admissible law gives one price, and the robust bounds report the lowest and the highest of these prices. They are only as meaningful as the list of laws allowed, and each law on the list must still respect the payoff and the market prices that motivated it.

Example A constructed claim pays $10$ in one state and $0$ in the other, the discount factor is $0.98$ , and the pricing probability $p$ of the paying state may lie anywhere in $[0.4, 0.6]$ . The value $0.98 \cdot 10 \cdot p = 9.8\,p$ is affine in $p$ , so the bounds sit at the endpoints, $\underline{V} = 3.92$ and $\overline{V} = 5.88$ . If other traded claims already fixed $p = 0.5$ , every other member of the family would fail that calibration, and the bounds would collapse to $4.9$ . For a contrast, let two independent events $A$ and $B$ each have probability $p$ , and let a claim pay $1$ on $A \cap B^{c}$ . Its expected payoff is $p(1 - p)$ , with derivative $1 - 2p$ and second derivative $-2$ , so over $[0.4, 0.6]$ the maximum $0.25$ sits inside the interval at $p = 0.5$ , while both endpoints give $0.24$ . The expectation is still linear in the joint law, but the probabilities of the joint law, such as $p(1 - p)$ for $A \cap B^{c}$ , are quadratic in $p$ , and this changed family, not the affine claim above, produces the interior maximum.

Check yourself

The volatility set for the call is widened from $[0.18, 0.28]$ to $[0.15, 0.30]$ , with the other inputs unchanged. Find the new envelope and explain why its lower end cannot lie above the old one.

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Vega is positive for every $\sigma > 0$ , so the envelope is again given by the endpoints. At $\sigma = 0.15$ the inputs $d_1 \approx -0.1169$ and $d_2 \approx -0.2669$ give $C \approx 4.7166$ , and at $\sigma = 0.30$ the inputs $d_1 \approx 0.0540$ and $d_2 \approx -0.2460$ give $C \approx 10.6925$ . The new envelope from about $4.7166$ to about $10.6925$ contains the old one from $5.9082$ to $9.8954$ . This is the nesting property. The new set contains the old one, and an infimum over more candidates cannot be larger. The wider envelope reflects a larger stated set, not new information about the market.

A cash-or-nothing call pays $100$ currency units at maturity if $S_T > 105$ , with the same inputs, so its Black-Scholes value is $V(\sigma) = 100\, e^{-r\tau}\Phi(d_2)$ . A colleague evaluates it at $\sigma = 0.18$ and $\sigma = 0.28$ and reports the interval between the two results as its envelope over $[0.18, 0.28]$ . Is that right?

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The endpoint values are $V(0.18) \approx 39.3368$ and $V(0.28) \approx 39.6070$ . The endpoint rule needs monotonicity, so compute the derivative. With $q = 0$ ,

\begin{aligned} &\frac{\partial d_2}{\partial \sigma} \\[0.4em] &= -\frac{\ln(S/K) + r\tau}{\sigma^2\sqrt{\tau}} - \frac{\sqrt{\tau}}{2} \\[0.4em] &= -\frac{d_1}{\sigma}, \end{aligned}

so $\partial V/\partial\sigma = -100\, e^{-r\tau}\varphi(d_2)\, d_1/\sigma$ , whose sign is opposite to that of $d_1$ . Here $\ln(100/105) + 0.02 \approx -0.02879 < 0$ , and $d_1 = 0$ exactly when $\sigma^2 = -2(\ln(S/K) + r\tau)/\tau \approx 0.05758$ , that is, at $\sigma^* \approx 0.2400$ . Below $\sigma^*$ the value rises and above it the value falls, so the maximum is interior. At $\sigma^*$ the input is $d_2 = -\sigma^* \approx -0.2400$ , and $V(\sigma^*) = 100\, e^{-0.02}\Phi(-0.2400) \approx 39.7158$ . The minimum is the smaller endpoint value, $39.3368$ at $\sigma = 0.18$ . The envelope runs from about $39.3368$ to about $39.7158$ , and the colleague’s upper end is too low by about $0.1087$ .

Let both the volatility and the rate of the call be uncertain, with $\lvert \Delta\sigma \rvert \le 0.02$ and $\lvert \Delta r \rvert \le 0.005$ around $\sigma = 0.23$ and $r = 0.02$ . The first-order dual norm bound is about $1.0063$ . Is the interval $7.9011 \pm 1.0063$ guaranteed to contain the call value at every point of the box? Find the exact envelope and a bound that is guaranteed.

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The first-order interval runs from about $6.8948$ to about $8.9074$ . Vega is positive, and rho, $K\tau e^{-r\tau}\Phi(d_2)$ , is positive too, so the call value is increasing in each coordinate and the exact envelope sits at the two opposite corners of the box. Repricing gives $C(0.21, 0.015) \approx 6.8972$ and $C(0.25, 0.025) \approx 8.9089$ . The upper corner lies about $0.0015$ above the first-order interval, so the interval is not a guarantee. It describes the linear term only, and the remainder here adds to the change. A guaranteed bound comes from the mean value theorem. A numerical search over the box puts the largest value of $0.02\,\partial C/\partial\sigma + 0.005\,\partial C/\partial r$ at about $1.0097$ , so every value on the box lies in $7.9011 \pm 1.0097$ , from about $6.8914$ to about $8.9108$ , which contains the exact envelope. In this example the monotone corner argument gives the exact answer, and the dual norm calculations give a local approximation and a looser guarantee.

Model uncertainty and robust valuation

Key terms

Parameter uncertainty

Model ambiguity

Valuation envelope

Monotone parameter bound

Dual norm bound

Robust valuation bounds

Check yourself

How this lesson connects

Builds on

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