SUPERREPLICATION OF OPTIONS ON SEVERAL UNDERLYING ASSETS

ERIK EKSTRÖM, SVANTE JANSON AND JOHAN TYSK

Abstract. We investigate when a hedger who over-estimates the volatility will superreplicate a convex claim on several underlying assets. It is shown that the classical Black and Scholes model is the only model, within a large class, for which over-estimation of the volatility yields the desired superreplication property. This is in contrast to the onedimensional case, in which it is known that over-estimation of the volatility with any model guarantees superreplication of convex claims.

1. Introduction

For options written on one underlying asset it is well-known that convexity of the contract function ensures certain monotonicity properties of the option price with respect to the volatility. For example, if the contract function is convex, then the option price increases with the volatility, see Bergman, Grundy and Wiener (1996), El Karoui, Jeanblanc-Picque and Shreve (1998), Hobson (1998) or Janson and Tysk (2003a). It is also known that a hedger who over-estimates the volatility with any model will superreplicate a given convex claim, see El Karoui et al (1998) or Hobson (1998). Crucial both for the monotonicity result and for the superreplication property is the fact that the price of a convex claim is convex (as a function of the current stock price) at any time before maturity. This fact, however, is one-dimensional in nature; it is easy to find examples of an option with a convex pay-off of two underlying assets which has a non-convex price, see Janson and Tysk (2003a).

Janson and Tysk (2003b) consider second order parabolic differential equations of the form

(1)  \frac{∂G}{∂t} = LG,

where the differential operator

L = ∑_{i,j=1}^{n} a_{ij}(x, t)\frac{∂^{2}}{∂x_{i}∂x_{j}} + ∑_{i=1}^{n} b_{i}(x, t)\frac{∂}{∂x_{i}} + c(x, t)

is elliptic. The authors find a necessary and sufficient local condition on the operator L that guarantees that the unique (satisfying appropriate growth conditions) solution to the equation (1) remains convex at every time t > 0 provided the initial condition is convex. If L satisfies the condition it is said to be locally convexity preserving (LCP). In the present paper we apply the results of Janson and Tysk (2003b) to the problem of monotonicity in the volatility; in particular we study the problem of superreplication of options written on several underlying assets.

Key words and phrases. Parabolic equations; superreplication; convexity; options.

Consider a market consisting of a bank account with price process

B(t) = B(0) exp { ∫_{0}^{t} r(u) du},

where the interest rate r is a deterministic function, and n risky assets, with the price X_{i} of the ith asset satisfying the stochastic differential equation

(2)  dX_{i} = μ_{i}(X, t) dt + ∑_{j=1}^{n} β_{ij}(X, t) dW_{j}.

In this equation the drift μ_{i} is some deterministic function of the current stock prices and time, W is an n-dimensional Brownian motion and the diffusion matrix β = (β_{ij}(x, t))_{i,j=1}^{n} is assumed to be non-singular for all x with positive components. Note that the only source of randomness in the diffusion matrix β is in the dependence on the current stock prices. In finance it is natural to consider stock prices that cannot become negative. Therefore we let 0 be an absorbing barrier, i.e. if X_{i} is 0 at some time, then X_{i} remains 0 forever.

Given a finite time horizon T > 0, there exists a unique probability measure ℙ̃ equivalent with the original measure ℙ such that

dX_{i} = r(t)X_{i} dt + ∑_{j=1}^{n} β_{ij}(X, t) dW̃_{j}

for some ℙ̃-Brownian motion W̃, i.e. B_{t}^{−1}X_{t} is a local martingale under ℙ̃. Let g : ℝ_{+}^{n} → ℝ be continuous and of at most polynomial growth. Standard arbitrage theory yields that the price at time t of the option which at time T_{0} ≤ T pays g(X(T_{0})) is F (X(t), t), where

(3)  F (x, t) = exp { − ∫_{t}^{T_{0}} r(u) du}Ẽ_{x,t}g(X(T_{0})).

Here Ẽ denotes expected value with respect to the measure P̃ and the indices indicate that X_{t} = x. Moreover, this pricing function F solves the Black-Scholes parabolic differential equation

(4)  \frac{∂F}{∂t} + LF = 0,

where

(5)  L = \frac{1}{2} ∑_{i,j=1}^{n} a_{ij}(x, t)\frac{∂^{2}}{∂x_{i}∂x_{j}} + ∑_{i=1}^{n} rx_{i} \frac{∂}{∂x_{i}} − r,

with terminal condition

F (x, T_{0}) = g(x).

In this equation the coefficients a_{ij} = a_{ij}(x, t) are the entries of the n × nmatrix ββ^{∗}. Note that the invertibility of β guarantees parabolicity of the equation (4) (since the direction of the time variable is opposite to the customary one).

We will say that a model for the stock price vector X (or the diffusion matrix β) is convexity preserving if, for any T_{0} ≤ T, the price of an option with a convex pay-off g(X(T_{0})) at T_{0} is convex in X(t) at all times t prior to T_{0}. In Section 2 we introduce the LCP-condition. Then we show that, for n ≥ 2, the only convexity preserving model which is standard, see Definition 2.5, is geometric Brownian motion. In Section 3 we show that if the hedger uses a convexity preserving model, then over-estimation of the diffusion matrix (in the sense of quadratic forms, see below) guarantees superreplication of convex claims.

2. Convexity preserving models

Assume that the diffusion coefficients β_{ij} are linear in x_{i} and independent of x_{l}, l ≠ i. Then, under the measure ℙ̃, the stock price vector X satisfies

(6)  dX_{i} = r(t)X_{i} dt + X_{i} ∑_{j=1}^{n} σ_{ij}(t) dW̃_{j}

for some deterministic functions σ_{ij}. Such a process X is called n-dimensional geometric Brownian motion with time-dependent volatility, or simply geometric Brownian motion. Thus, to show that a process X defined by (2) is geometric Brownian motion one has to show that β_{ij}(x, t) = x_{i}σ_{ij}(t) for all i and j.

Theorem 2.1. Geometric Brownian motion is convexity preserving.

Proof. Let T_{0} ≤ T. It is well-known that if X is geometric Brownian motion as in (6), then

X_{i}(T_{0}) = x_{i} exp \{ ∫_{t}^{T_{0}} (r(u) − \frac{1}{2} ∑_{j=1}^{n} σ_{ij}^{2}(u)) du + ∑_{j=1}^{n} ∫_{t}^{T_{0}} σ_{ij}(u) dW̃_{j}\},

where x_{i} = X_{i}(t). Let g : ℝ_{+}^{n} → ℝ be convex. The price F (X(t), t) at time t of an option which at time T_{0} pays g(X(T_{0})) is given by

F (x, t) = exp { − ∫_{t}^{T_{0}} r(u) du}Ẽ_{x,t}g(X(T_{0})),

which is convex in x since X_{i}(T_{0}) is linear in x_{i} and g is convex. Thus geometric Brownian motion is convexity preserving. □

Next we investigate which models are convexity preserving. It turns out that if one imposes some conditions on the diffusion matrix, then geometric Brownian motion is the only model which is convexity preserving. We first introduce the LCP-condition:

Definition 2.2. Assume that the coefficients of the differential operator L are in C^{3}(ℝ_{+}^{n} × [0, T]). Let x ∈ ℝ_{+}^{n} be an interior point, and let t ∈ [0, T]. Then L is said to be locally convexity preserving (LCP) at (x, t) if

D_{uu}(Lf)(x, t) ≥ 0

whenever u ∈ ℝ^{n} \ {0}, f ∈ C^{∞}(ℝ_{+}^{n}) is convex in a neighborhood of x and D_{uu}f(x) = 0.

If G is a solution to (1), then the infinitesimal change of G during a short time interval Δt is approximately Δt(LG). Thus the LCP-condition is intuitively the right condition to preserve convexity: if, at some instant, convexity is almost lost in some direction u, then the infinitesimal change of G is convex in that direction.

For simplicity we will in this paper work under the following assumption.

Hypothesis 2.3. The diffusion matrix β is in C(\overline{ℝ_{+}^{n}} × [0, T]) ∩ C^{3}(ℝ_{+}^{n} × [0, T]) and is such that, for any vector of non-negative initial values of the stocks, there exists a unique strong solution X to (2) with absorption at 0 of the i:th component X_{i} for all i.

We also assume that the diffusion matrix β is such that, for any smooth terminal value g, the function F defined by (3) has continuous derivatives D_{x}^{k}D_{t}^{m}F, m ∈ {0, 1}, 0 ≤ |k| + 2m ≤ 4 up to time T_{0}.

Remark The assumption that β is C^{3} is unnecessarily strong, see section 3 in Janson and Tysk (2003b). To clarify the presentation, however, we keep this assumption.

Theorem 2.4. Let the diffusion matrix β satisfy Hypothesis 2.3, and let L be the corresponding differential operator as in (5). Now, if β is convexity preserving, then L is LCP at all points (x_{0}, T_{0}) such that x_{0} ∈ ℝ_{+}^{n} is interior and T_{0} ∈ (0, T).

Proof. Suppose that f ∈ C^{∞}(ℝ_{+}^{n}) is convex in a neighborhood of some point x_{0} in the interior of ℝ_{+}^{n}, and suppose that D_{uu}f(x_{0}) = 0 for some direction u ≠ 0. Then there exists a smooth convex function g : ℝ_{+}^{n} → ℝ which equals f in a neighborhood of x_{0}, for details see Lemma 3.2 in Janson and Tysk (2003b). Since β is convexity preserving there exists a solution F (x, t) (defined by (3)), which is convex in x for all t ≤ T_{0}, to equation (4) with terminal condition F (x, T_{0}) = g(x). By assumption, D_{x}^{k}D_{t}^{m}F, m ∈ {0, 1}, 0 ≤ |k| + 2m ≤ 4, exist and are continuous up to time T_{0}. Since D_{uu}F (x_{0}, t) ≥ 0 for all t ≤ T_{0} and D_{uu}F (x_{0}, T_{0}) = 0, we have that D_{t}D_{uu}F (x_{0}, T_{0}) ≤ 0. Therefore, using equation (4),

D_{uu}(Lf)(x_{0}, T_{0}) = D_{uu}(LF)(x_{0}, T_{0}) = −D_{uu}(D_{t}F)(x_{0}, T_{0})
= −D_{t}D_{uu}F (x_{0}, T_{0}) ≥ 0.
□

Remark Janson and Tysk (2003b) show that the LCP-condition is both necessary and sufficient to guarantee that an operator L defined on ℝ^{n} × [0, T] is convexity preserving. To get sufficiency in the present setting we would have to add some conditions on the boundary of ℝ_{+}^{n} in the definition of LCP. In our analysis, however, we only need that LCP is a necessary condition, and we therefore leave the rather technical considerations about the appropriate LCP-condition for boundary points. We now present the class of models under consideration.

Definition 2.5. A model with a diffusion matrix β that satisfies Hypothesis 2.3 is standard if the following conditions are satisfied:

(i) the diffusion coefficient β_{ij} is a function only of x_{i} and t; (ii) the diffusion coefficient β_{ij} = 0 for x_{i} = 0; (iii) for all i = 1, .., n and fixed times t, the volatility

\sqrt{a_{ii}(x_{i}, t)}/x_{i} = \sqrt{β_{i1}^{2}(x_{i}, t) + .. + β_{in}^{2}(x_{i}, t)}/x_{i}

of the i:th asset is not an increasing function of x_{i}, unless it is constant.

Remark Note that (i) does not exclude dependence between the assets. Instead it merely says that the volatility of the ith asset depends only on the value of that asset and time. Note further that condition (ii) allows volatilities tending to infinity for asset values close to zero. Condition (iii) seems to be satisfied for virtually all models for option pricing; in fact many models have larger volatilities at zero than at infinity. We can now state our main theorem.

Theorem 2.6. Let n ≥ 2. Then geometric Brownian motion (with timedependent volatility) is the only standard model that is convexity preserving.

Remark It follows, for example, that the constant elasticity of variance model (which is a standard model) in which the price of the ith asset is given by

dX_{i} = r(t)X_{i} dt + σ_{i}X_{i}^{γ} dW̃_{i}

with 0 < γ < 1, is not convexity preserving. We start with a lemma.

Lemma 2.7. Let β be a C^{2} diffusion matrix such that β_{ij} is a function of x_{i} alone, and such that the property (ii) in Definition 2.5 is satisfied. Assume that there exist constants B_{ij} such that

(7)  ∑_{k=1}^{n} β_{ik}(x_{i})β_{jk}(x_{j}) = B_{ij}x_{i}x_{j}

for all i, j with i ≠ j. Then there exists an index l ∈ {1, ..., n} such that (β_{l1}, ..., β_{ln}) = f(x_{l})v for some constant vector v ∈ ℝ^{n} and some function f.

If, in addition, (7) also holds for i = j, then β is the diffusion matrix of a geometric Brownian motion.

Proof. Let β_{i} = (β_{i1}, ..., β_{in}) and β′′_{i} = (β′′_{i1}, ..., β′′_{in}) for i = 1, ..., n. Here the double primes refer to differentiation with respect to the x_{i}-variables. If β′′_{i} ≡ 0 for some i, then all components of β_{i} are affine functions of x_{i}. From (ii) it follows that the components in fact are linear, and thus the lemma is true in this particular case.

Now, assume that we can choose coordinates x_{1}, ..., x_{n} > 0 such that β′′_{i}(x_{i}) ≠ 0. From (7) we have

(8)  β′′_{i}(x_{i}) · β_{j}(x_{j}) = 0

for i ≠ j. Since β_{1}(x_{1}), ..., β_{n}(x_{n}) are linearly independent (recall that the diffusion matrix is assumed to be non-singular), it follows from (8) that β′′_{1}(x_{1}), ..., β′′_{n}(x_{n}) are linearly independent. Now, still keeping x_{1}, ..., x_{n−1} fixed it follows from (8) that, for any x_{n}, β_{n}(x_{n}) is in the orthogonal (1dimensional) complement of β′′_{1}(x_{1}), ..., β′′_{n−1}(x_{n−1}). Thus β_{n}(x_{n}) = f(x_{n})v for some constant vector v ∈ ℝ^{n}, which finishes the first part of the lemma.

Next, assume in addition that (7) also holds for i = j, i.e. that a_{ii}(x_{i}) = B_{ii}x_{i}^{2} for some positive constants B_{ii}. Fix x_{2}, ..., x_{n} and let v_{i} = \frac{1}{x_{i}} β_{i}(x_{i}) for i = 2, ..., n. Then

\frac{1}{x_{1}} β_{1}(x_{1}) · v_{i} = B_{1i}

for all x_{1}. Since v_{i}, i = 2, ..., n are linearly independent it follows that

\frac{1}{x_{1}} β_{1}(x_{1}) ∈ \{β_{1}(1) + tw; t ∈ ℝ\}

for some vector w ∈ ℝ^{n}. But since the inner product

(\frac{1}{x_{1}} β_{1}(x_{1})) · (\frac{1}{x_{1}} β_{1}(x_{1})) = B_{11}

is constant we find that \frac{1}{x_{1}} β_{1}(x_{1}) is a vector of constant length. By continuity it follows that \frac{1}{x_{1}} β_{1}(x_{1}) is a constant vector. Similarly we deduce that all entries in the ith row of β are linear in x_{i}, and thus the lemma follows. □

Proof of Theorem 2.6. Assume that β is a standard diffusion matrix which is convexity preserving, and let L be the corresponding differential operator appearing in the Black-Scholes equation (4). We suppress the time variable t in the calculations below. First choose two components x_{i} and x_{j}. For fixed s, let f(x_{i}, x_{j}) = \frac{1}{2}(sx_{i} −x_{j})^{2}. Then f is constant along lines x_{j} = sx_{i} +x_{i,0}, where x_{i,0} is a constant. If u = e_{i} +se_{j} then, since L is convexity preserving, it follows from Theorem 2.4 that

0 ≤ 2D_{uu}(Lf)
= (∂_{x_{i}}^{2} + 2s∂_{x_{i}}∂_{x_{j}} + s^{2}∂_{x_{j}}^{2})( ∑_{i,j=1}^{n} a_{ij} \frac{∂^{2}f}{∂x_{i}∂x_{j}} + 2r ∑_{i=1}^{n} x_{i} \frac{∂f}{∂x_{i}} − 2rf)
= (∂_{x_{i}}^{2} + 2s∂_{x_{i}}∂_{x_{j}} + s^{2}∂_{x_{j}}^{2})(s^{2}a_{ii} − 2sa_{ij} + a_{jj} + 2rx_{i}f_{x_{i}} + 2rx_{j}f_{x_{j}} − 2rf)
= −2s^{3}∂_{x_{j}}^{2}a_{ij} + s^{2}(∂_{x_{i}}^{2}a_{ii} − 4∂_{x_{i}}∂_{x_{j}}a_{ij} + ∂_{x_{j}}^{2}a_{jj}) − 2s∂_{x_{i}}^{2}a_{ij}.

Since s is arbitrary we find that

∂_{x_{j}}^{2}a_{ij} = ∂_{x_{i}}^{2}a_{ij} = 0,

so a_{ij} = B_{ij}x_{i}x_{j} + B_{1}x_{i} + B_{2}x_{j} + B_{3}. The condition (ii) guarantees that a_{ij}(x_{i}, x_{j}) vanishes for x_{i} = 0 and for x_{j} = 0. It follows that B_{1} = B_{2} = B_{3} = 0. Hence a_{ij} = B_{ij}x_{i}x_{j} for i ≠ j. It remains to show the same for the diagonal elements a_{ii}. Using the first part of Lemma 2.7 we may assume that say β_{n}(x_{n}) = f(x_{n})v for some constant vector v ∈ ℝ^{n}.

Now, choose a row β_{i}, i ∈ {1, .., n − 1}, in the diffusion matrix, and consider the matrix block

A_{in} = ( \begin{matrix}a_{ii} & a_{in} \\ a_{ni} & a_{nn} \end{matrix} ) = ( \begin{matrix}a_{ii}(x_{i}) & B_{in}x_{i}x_{n} \\ B_{in}x_{i}x_{n} & a_{nn}(x_{n}) \end{matrix} ) .

If A is convexity preserving we must have that A_{in} is convexity preserving for equations in two spatial variables x_{i} and x_{n}. Two different cases can occur. First, assume that B_{in} = 0. Then

(9)  A_{in} = ( \begin{matrix}g^{2}(x_{i}) & 0 \\ 0 & h^{2}(x_{n}) \end{matrix} ) ,

where g and h are some positive functions. From Example 5.3 in Janson and Tysk (2003b) we see that in order to have a convexity preserving operator we must have

g(x_{i})g′′(x_{i}) + h(x_{n})h′′(x_{n}) ≥ 0

for all x_{i} and x_{n}. Hence at least one of g and h is convex. Let this function be g. By (ii) in Definition 2.5 we conclude that g(0) = 0. From (iii) the function g is seen to be linear using the fact that g(x_{i})/x_{i} = \sqrt{a_{ii}(x_{i}, t)}/x_{i} is increasing since g is a convex function vanishing at 0. Then the above inequality has only one term, so h is convex, and thus also linear by the same argument. Therefore a_{ii}(x_{i}) = C_{1}^{2}x_{i}^{2} and a_{nn}(x_{n}) = C_{2}^{2}x_{n}^{2} for some constants C_{1} and C_{2}.

Next, if B_{in} ≠ 0, then

a_{in}(x_{i}, x_{n}) = B_{in}x_{i}x_{n} = β_{i}(x_{i}) · β_{n}(x_{n}) = f(x_{n})(β_{i}(x_{i}) · v)

for some constant vextor v. It follows that β_{n}(x_{n}) = x_{n}u for some constant vector u ∈ ℝ^{n}. Therefore

A_{in} = ( \begin{matrix}g^{2}(x_{i}) + D^{2}x_{i}^{2} & DEx_{i}x_{n} \\ DEx_{i}x_{n} & E^{2}x_{n}^{2} \end{matrix} )

for some non-zero constants D and E and some function g which is strictly positive for x_{i} > 0. From Corollary 5.2 by Janson and Tysk (2003b) it follows that

(10)  D_{uu}\sqrt{b^{2}a_{ii}(x_{i}) − 2abDEx_{i}x_{n} + a^{2}E^{2}x_{n}^{2}} ≥ 0

for all directions u = ae_{i} + be_{n}. Direct calculations show that

D_{uu}\sqrt{b^{2}a_{ii}(x_{i}) − 2abDEx_{i}x_{n} + a^{2}E^{2}x_{n}^{2}}
= a^{2}b^{2}(b^{2}a_{ii}(x_{i}) − 2abDEx_{i}x_{n} + a^{2}E^{2}x_{n}^{2})^{−3/2} ·
((b(D − E)g(x_{i}) − (bDx_{i} − aEx_{n})g′(x_{i}))^{2}
+(b^{2}g^{2}(x_{i}) + (bDx_{i} − aEx_{n})^{2})g(x_{i})g′′(x_{i})).

Here

b^{2}a_{ii} − 2abDEx_{i}x_{n} + a^{2}E^{2}x_{n}^{2} = b^{2}g^{2}(x_{i}) + (bDx_{i} − aEx_{n})^{2} > 0

if b ≠ 0 and x_{i} > 0. We claim that

(11)  g′′(x_{i}) ≥ 0

for all x_{i}. Indeed, for given x_{i} and x_{n}, let K_{1} = Ex_{n}g′(x_{i}) and K_{2} = (D − E)g(x_{i}) − Dx_{i}g′(x_{i}). Then we know that

(K_{1}a + K_{2}b)^{2} + (b^{2}g^{2}(x_{i}) + (bDx_{i} − aEx_{n})^{2})g(x_{i})g′′(x_{i}) ≥ 0

for all a, b ≠ 0. If K_{1} and K_{2} both are 0, then (11) follows immediately. If K_{1} and K_{2} both are non-zero, then a, b ≠ 0 can be chosen so that K_{1}a+K_{2}b = 0, and thus (11) follows. If K_{1} = 0, K_{2} ≠ 0 and g(x_{i})g′′(x_{i}) < 0, then b = 1 and a very big yields a contradiction, so (11) follows. The case K_{1} ≠ 0 = K_{2} is similar. Considering the inequality (11) together with the assumptions (ii) and (iii) of Definition 2.5 we find that a_{ii}(x_{i}) = C^{2}x_{i}^{2} for some constant C. □

Remark The theorem tells us that to be sure to preserve convexity, regardless of the (convex) contract function, geometric Brownian motion should be used to model the asset prices. However, given a particular convex claim, it can of course be the case that the class of models that guarantee convex option prices is bigger. For example, if the contract function g(x_{1}, x_{2}) = g_{1}(x_{1}) + g_{2}(x_{2}), where g_{1} and g_{2} both are convex, then all models with diagonal diffusion matrices satisfying (i) in Definition 2.5 are convexity preserving. The reason is that such a claim is the sum of two one-dimensional claims, both of which have convex prices.

3. Superreplication of convex claims

Assume that an option writer believes that the diffusion matrix is β as in (2), whereas the true stock price vector X̃ evolves according to

(12)  dX̃_{i} = μ̃_{i}(X̃, t) dt + ∑_{j=1}^{n} β̃_{ij}(X̃, t) dW_{j}

for some functions μ̃_{i} and β̃_{ij}. He will then (incorrectly) price an option on the stocks according to (3), where X is a diffusion with diffusion matrix β. Moreover, if he tries to replicate the option with the hedging strategy suggested by his model, then he will form a self-financing portfolio which has initial value F (X(0), 0) and is such that it at each instant t contains \frac{∂F}{∂x_{i}} (X̃(t), t) numbers of shares of the ith asset (and the remaining amount invested in the bank account). In this section we provide conditions under which the terminal value of the hedger’s portfolio exceeds the option pay-off g(X(T_{0})) almost surely.

Given two diffusion matrices β and β̃ we say that β dominates β̃ if A(x, t) = β(x, t)β^{∗}(x, t) ≥ β̃(x, t)β̃^{∗}(x, t) = Ã(x, t) as quadratic forms for all x and t.

Theorem 3.1. Assume that a hedger overestimates (underestimates) the volatility, i.e he uses a diffusion matrix β which dominates (is dominated by) the true diffusion matrix β̃. Moreover, assume that β is convexity preserving. Then the hedger will superreplicate (subreplicate) any convex claim written on X.

Proof. Let g : ℝ_{+}^{n} → ℝ be a convex contract function, and define F as in (3). Since β is convexity preserving, F (x, t) is convex in x for all t ∈ [0, T_{0}]. The value V (t) of the self-financing portfolio with \frac{∂F}{∂x_{i}} (X̃(t), t) shares of the ith asset and initial value V (0) = F (X̃(0), 0) has the dynamics

dV = r(V (t) − ∑_{i=1}^{n} X̃_{i}(t)\frac{∂F}{∂x_{i}} (X̃(t), t)) dt + ∑_{i=1}^{n} \frac{∂F}{∂x_{i}} (X̃(t), t) dX̃_{i}(t).

Consider the process Y (t) := V (t)−F (X̃(t), t). From Ito’s formula it follows that

dY = dV − \frac{∂F}{∂t} (X̃(t), t) dt − ∑_{i=1}^{n} \frac{∂F}{∂x_{i}} (X̃(t), t) dX̃_{i}

−\frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} ã_{ij}(X̃(t), t)\frac{∂^{2}F}{∂x_{i}∂x_{j}} (X̃(t), t) dt

= rV (t) dt − \frac{∂F}{∂t} (X̃(t), t) dt − r ∑_{i=1}^{n} X̃_{i}(t)\frac{∂F}{∂x_{i}} (X̃(t), t) dt

−\frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} ã_{ij}(X̃(t), t)\frac{∂^{2}F}{∂x_{i}∂x_{j}} (X̃(t), t) dt

= r(V (t) − F (X̃(t), t)) dt

+\frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} (a_{ij}(X̃(t), t) − ã_{ij}(X̃(t), t))\frac{∂^{2}F}{∂x_{i}∂x_{j}} (X̃(t), t) dt

= rY (t) dt + \frac{1}{2} ∑_{i=1}^{n} ∑_{j=1}^{n} (a_{ij}(X̃(t), t) − ã_{ij}(X̃(t), t))\frac{∂^{2}F}{∂x_{i}∂x_{j}} (X̃(t), t) dt,

where we have used that F solves the Black-Scholes equation (4) with diffusion matrix β. Now, assume that the hedger overestimates the volatility, i.e. that β(x, t) dominates β̃(x, t) for all x and t. Then, since F is convex, the last double sum is non-negative. Thus Y (0) = 0 and Y (T_{0}) ≥ 0. It follows that the final value V (T_{0}) of the hedger’s portfolio satisfies V (T_{0}) ≥ F (X̃(T_{0}), T_{0}) = g(X̃(T_{0})), so the hedger superreplicates. The case of underestimation of the volatility is similar. □

Theorem 2.6 together with Theorem 3.1 shows that if a hedger wants to be sure to superreplicate a convex claim on several underlying assets, then he should overestimate the true diffusion matrix β̃ with a diffusion matrix β such that β_{ij}(x, t) = x_{i}σ_{ij}(t) for some functions σ_{ij}(t). Note that there is no assumption on the true diffusion matrix to be LCP; what matters is that the hedger should use a model which is LCP. Actually, it is not essential that the true diffusion matrix β̃ is a function of time and the current stock prices. In fact, the theorem is true also in the case when β̃ is some adapted process dominated by β(X̃(t), t) for all t almost surely. This is of course in analogy with the one-dimensional case, see El Karoui, Jeanblanc-Picque and Shreve (1998). Remark By considering an American option as the limit of a sequence of European options (for the one-dimensional case, see El Karoui, Jeanblanc-Picque and Shreve (1998) or Ekström (2003)) it is clear that a model which gives convex European option prices also gives convex American option prices. For American claims, an option writer needs to be sure that the value V of his hedging portfolio at each instant t satisfies V_{t} ≥ g(X̃(t)). It can be shown that this is indeed the case if the option writer overestimates the diffusion matrix with a convexity preserving model.

Remark Janson and Tysk (2003b) show that if A ≥ Ã as quadratic forms, and if either A or Ã is convexity preserving, then the corresponding solutions F and F̃ satisfy the inequality F (x, t) ≥ F̃(x, t) for all x and t. Thus, if one knows that the true diffusion matrix can be bounded both above and below by some diffusion matrices corresponding to geometric Brownian motions, then one also has upper and lower bounds for the price of an option. This can also be seen as a consequence of Theorem 3.1.

References

[1] Bergman, Y.Z., D.B. Grundy and Z. Wiener (1996): General Properties of Option Prices. J. Finance, 51, 1573-1610.

[2] Ekström, E. (2003): Properties of American Option Prices. Submitted for publication.

[3] El Karoui, N., M. Jeanblanc-Picque and S. Shreve (1998): Robustness of the Black and Scholes Formula. Math. Finance, 8, No. 2, 93-126.

[4] Hobson, D. (1998): Volatility Misspecification, Option Pricing and Superreplication via Coupling. Ann. Appl. Probab., 8, No. 1, 193-205.

[5] Janson, S. and J. Tysk (2003a): Volatility time and Properties of Option Prices. To appear in Ann. Appl. Probab.

[6] Janson, S. and J. Tysk (2003b): Preservation of Convexity of Solutions to Parabolic Equations. Submitted for publication.

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