VOLATILITY TIME AND PROPERTIES OF OPTION PRICES

SVANTE JANSON AND JOHAN TYSK

Abstract. We use a notion of stochastic time, here called volatility time, to show convexity of option prices in the underlying asset if the contract function is convex as well as continuity and monotonicity of the option price in the volatility. The volatility time is obtained as the almost surely unique stopping time solution to a random ordinary differential equation related to volatility. This enables us to write price processes, or processes modelled by local martingales, as Brownian motions with respect to volatility time. The results are shown under very weak assumptions and are of independent interest in the study of stochastic differential equations. Options on several underlying assets are also studied and we prove that if the volatility matrix is independent of time, then the option prices decay with time if the contract function is convex. However, the option prices are no longer necessarily convex in the underlying assets and the option prices do not necessarily decay with time, if a time dependent volatility is allowed.

1. Introduction

Consider the spot price S of some asset following the risk neutral process

dS = Sσ(S, t) dB,  (1)

with initial condition S(t) = s, where B is a Brownian motion and σ is called the volatility of S. We here compute the price with respect to some suitable numeraire process, for instance the price of a zero coupon bond, maturing at some future time T, to avoid the drift in the process for S associated with interest rates. We are interested in general properties of prices of simple contingent claims maturing at T. The arbitrage free price of a simple claim with contract function Φ is given by

F (s, t) = E_{s,t}[Φ(S(T)],  (2)

according to [B-S]. In [B-G-W] it is shown that the price F (s, t) is a convex function of s if Φ is a convex function. From the Black–Scholes equation

F_{t} + \frac{1}{2}s^{2}σ^{2}(s, t)F_{ss} = 0,  (3)

corresponding to (2) through the Feynman–Kac stochastic representation formula, it follows that the convexity of F in s is equivalent to F decaying with time when the price of the underlying asset is constant. In [B-G-W]

Date: September 24, 2001; revised April 8, 2002. Key words and phrases. Volatility, stochastic time, contingent claim, convexity, time-

decay.

Mathematics Subject Classification (2000): 91B28, 60H30, 35K15. JEL Classification: G13, C60.

it is also shown that the price F is monotonous in the volatility. However, the arguments presented in [B-G-W] for these results require the volatility to be a differentiable function of the underlying asset price.

In the present paper, see Sections 5 and 7, we generalize the results above to volatilities that are not even continuous in time and only satisfy a local Hölder(1/2) condition in s. We believe that these conditions, especially the lack of continuity assumption in t, are natural in applications as well as mathematically satisfying. To obtain these results, we use the standard fact that a local martingale can be represented as a time-change of a Brownian motion. In our context, this entails defining a notion of stochastic time for risky assets which we refer to as volatility time. It is defined in Section 3 and its aim is to reduce the study of price processes modelled by local martingales in the form (1) to the study of Brownian motion. This stochastic time has also been used by D. Hobson in [Ho] in the same context. However, in [Ho], existence and uniqueness of volatility time, which is needed in these applications, is assumed to hold without further discussing conditions for this. Sections 3 and 4 in the present paper deal with finding very general conditions under which the volatility time exists uniquely. The methods used in these sections are also very general in nature, and we believe they can be of independent interest in the study of stochastic differential equations. From these arguments we are also able to establish the continuity, see Section 6, of the option price in the volatility, a result which we have not seen elsewhere. The question of the continuity of option prices under perturbations of the volatility is of obvious interest in applications. A related reference, where the relation between options prices and volatility is treated, especially in the context of hedging, is [K-J-S].

We also consider properties of option prices in the case of several underlying assets, see Section 8. Let the assets S_{i} have risk neutral processes given by

dS_{i} = S_{i}(t) ∑_{j=1}^{n} σ_{ij}(S(t), t) dB_{j}

for i = 1, …, n, where B_{j} are independent Brownian motions and S(t) = (S_{1}(t), …, S_{n}(t)). The matrix σ with entries σ_{ij} is called the volatility matrix. As above we get rid of the interest rate by using a bond as a numeraire. The pricing function of a contingent claim with the contract Φ(S(T)) is given by

F (s, t) = E_{s,t}[Φ(S(T))].

Alternatively, one has that the pricing function is a solution of the partial differential equation

F_{t} + \frac{1}{2} ∑_{i,j=1}^{n} s_{i}s_{j}F_{s_{i}s_{j}}C_{ij} = 0

with the boundary condition F (s, T) = Φ(s), where C_{ij} = [σσ^{∗}]_{ij}. However, as in the case of one underlying asset, we will only consider the solutions that come from the stochastic representation. Our main result for several underlying assets is that if the volatility matrix is independent of time, and the contract function is convex, then indeed the option prices decay with time. However, many examples show that the case of several underlying assets is essentially different from the case of one underlying asset. For instance, even though the contract function is convex and the solutions decay with time, the option price for a fixed time t, need not be a convex function of the underlying assets, in contrast to the one dimensional case. Furthermore, if a time dependent volatility matrix is allowed, it is easy to find examples where the option prices do not decrease with time. The examples in [B-G-W] on for instance ”bloating” option prices in the case of stochastic volatility have counterparts in our examples on the loss of timedecay for options on several underlying assets.

We have mentioned both the partial differential equation approach in (3) as well as the martingale approach, compare (2), to option pricing; see e.g. [H-S] for an equivalence theorem under very general conditions. Section 2 provides examples of solutions to Black–Scholes equation that clearly do not represent arbitrage free option prices and also do not have the desired convexity properties. This will serve as a motivation for us to, in the rest of the paper, use the martingale approach. However, this does of course not prevent a translation of our result to the language of partial differential equations, compare for instance related results in [G-G-I-S] and [L-P].

Acknowledgement. We take this opportunity to thank Christer Kiselman who gave us lot of assistance concerning questions of convexity and constructed the examples of Section 2.

2. A non-trivial solution to the Black–Scholes equation with zero contract function

It is well-known that there are solutions to the Black–Scholes equation that are not given by the stochastic representation formula (2). In this section we show that if one considers such solutions, one can easily construct solutions that do not have any reasonable economical interpretation and also that fail to have the desired convexity properties. After this section we will therefore only consider solutions given by the Feynman–Kac representation formula. In fact, our Theorem 2 of Section 3 is an alternative formulation of the stochastic representation formula.

Proposition 1. There is a function F (s, t) that solves the Black–Scholes equation (3) with contract function identically equal to zero and which is strictly increasing in time. In other words, in this example F is concave as a function of s.

Proof. Let σ^{2}(s, t) = α(T − t)^{−1} for some constant α so that the Black– Scholes equation becomes

F_{t} + \frac{αs^{2}}{2(T − t)}F_{ss} = 0.

Then a direct computation shows that

F (s, t) = φ(T − t)s^{2}

where φ(y) = −y^{α}, for y positive and zero otherwise, satisfies the given Black–Scholes equation with value 0 on expiration but with F_{t} > 0. It is easily seen that this solution has m continuous derivatives if α > m. A smooth example is given by the same construction but with φ(y) = − exp(−1/y) for positive y and the volatility σ^{2}(s, t) = (T − t)^{−2}. The smooth example can be improved by instead considering the function ϕ(y) = − exp(−| log y|^{γ}) for y ≤ 1/2, smoothly extended to larger y, for some γ > 1. In that case, for t close to T,

σ^{2}(s, t) = γ \frac{| log(T − t)|^{γ−1}}{T − t} .

Hence the rate at which σ^{2}(s, t) blows up in the smooth case and in the C^{m} case only differs by a power of log(T − t). □

We note that the solutions to the Black–Scholes equation described above can be added for instance to a solution with a contract function of a call option. The resulting sum is not convex in s since the price of a call option is bounded by s and the examples given here decay like −s^{2} for s large.

3. Existence and uniqueness of volatility time

We study the local martingale that describes the value of the underlying asset using the stochastic time change by Dambis, Dubins and Schwarz, cf. e.g. [R-Y, Section V.1], which converts the price process to a Brownian motion. We call this stochastic time volatility time, since it is related to the volatility. Its main properties are collected in Theorem 1 below, and as a consequence we have the formula in Theorem 2 yielding an alternative to the Feynman–Kac representation formula.

In option pricing, we are generally only interested in non-negative values, but we state the result in a more general form and consider the non-negative case in the next section. We also change notation, writing α(x, t) = xσ(x, t).

The proof of Theorem 1 is rather technical and occupies the rest of this section, although the idea is simple. We start with some definitions.

Definition 1. A function α(x, t), defined on ℝ × [t_{0}, ∞) for some t_{0} ∈ ℝ, is said to be locally Hölder(1/2) in the x-variable if for every K > 0, there exists a constant C_{K} such that

|α(x, t) − α(y, t)| ≤ C_{K}|x − y|^{1/2} when |x|, |y|, and |t| ≤ K. (4)

Definition 2. Let X_{t} be a stochastic process on [t_{0}, ∞) that satisfies the stochastic differential equation

dX_{t} = α(X_{t}, t) dB̃_{t}, X_{t_{0}} = x_{0},

where B̃ is a Brownian motion and x_{0} ∈ ℝ. Then the quadratic variation τ(t) = ⟨X, X⟩_{t} is called the volatility time of X corresponding to the equation (7). Explicitly, by a standard formula,

τ(t) = ∫_{t_{0}}^{t} α^{2}(X_{s}, s) ds, t ≥ t_{0}.  (5)

Theorem 1. Assume that α(x, t) is measurable on ℝ × [t_{0}, ∞) and locally Hölder(1/2) in x and that for some constant C and all x and t ≥ t_{0},

α(x, t) ≤ C(1 + |x|).  (6)

Let X_{t} be a stochastic process on [t_{0}, ∞) that satisfies the stochastic differential equation

dX_{t} = α(X_{t}, t) dB̃_{t}, X_{t_{0}} = x_{0},  (7)

where B̃ is a Brownian motion and x_{0} ∈ ℝ, and let τ(t) be the corresponding volatility time. Then there exists (possibly on a larger probability space) a Brownian motion B with B_{0} = x_{0} such that

X_{t} = B_{τ(t)}, t ≥ t_{0}.  (8)

This τ satisfies the (random) integral equation

τ(t) = ∫_{t_{0}}^{t} α^{2}(B_{τ(s)}, s) ds, t ≥ t_{0},  (9)

or, equivalently, τ is locally absolutely continuous with τ(t_{0}) = 0 and

τ′(t) = α^{2}(B_{τ(t)}, t), a.e. t ≥ t_{0}.  (10)

Moreover, for each t ≥ t_{0}, τ(t) is a stopping time with respect to the completed filtration (B_{s}) generated by B.

Conversely, given a Brownian motion B with B_{0} = x_{0}, there exists a solution τ(t) to equation (9) such that τ(t) is a stopping time for each t; this solution is almost surely unique and X_{t} = B_{τ(t)} is a solution to (7) for some Brownian motion B̃_{t}.

Our treatment will center around the equation (9), or, equivalently, (10). We say that a stochastic process τ(t) satisfying (9) is a stopping time solution if τ(t) is a stopping time with respect to (B_{s}) for every t ≥ t_{0}.

Theorem 2. Let α be as in Theorem 1. If X_{t} is a solution to (7) and τ(t) is a stopping time solution to (9), then for any T ≥ t_{0}, X_{T} and B_{τ(T)} have the same distribution and thus for any measurable function Φ,

EΦ(X_{T}) = EΦ(B_{τ(T)}),  (11)

in the sense that if one of the expectations exists so does the other and they are equal.

Remark 1. The local Hölder(1/2) condition in (4) is weaker than the more commonly used local Lipschitz condition. Note also that no continuity of α in t is assumed. We also remark that the bound in (6) is used to prevent the solution from exploding.

Remark 2. If α > 0, then τ is strictly increasing and has a continuous increasing inverse A_{t}, t ≥ 0, with A_{0} = t_{0}. Note that A_{t} is adapted and that (10) is equivalent to

A′_{t} = α^{−2}(B_{t}, A_{t}).  (12)

Conversely, every adapted solution to (12) is the inverse of a stopping time solution to (10). Hobson [Ho] uses (12), assuming that α is such that the equation almost surely has a unique solution, without further discussing conditions for this. (Lipschitz continuity in both x and t is enough.) We will instead work with (10) and (9). The main advantage of this approach is that we do not need any continuity of α in t; as a bonus it also allows α to vanish on part of the space. (In other situations, it may be simpler to work with (12); in particular, this is the case when α is time-independent.)

Proof of Theorem 1. We may for notational convenience assume that t_{0} = 0. We first observe that the local Hölder(1/2) condition (4) implies pathwise uniqueness and thus, see [R-Y, Theorem IX.(1.7)], uniqueness in law for the solutions to (7). For a global Hölder(1/2) condition, i.e. if C_{K} does not depend on K, this is stated in [R-Y, Theorem IX.(3.5.ii)]; the general case follows the same proof or by stopping.

Formula (5) for the quadratic variation of the continuous local martingale X_{t} is standard, and so is the fact that B_{t} = X_{τ^{−1}(t)} is a Brownian motion, see for instance [R-Y, Section V.1]. Here τ^{−1} = inf{s : τ(s) > t} is well-defined on [0, τ(∞)) since τ is continuous and increasing. If τ is constant on some intervals, τ^{−1} will have jumps, but then X_{t} is constant on the same intervals so X_{τ^{−1}(t)} is nevertheless continuous. It is possible that τ(∞) < ∞, in which case we have have to enlarge the probability space in order to define B_{t}, but that makes no significant difference. Clearly, (8) holds by the definition of B_{t} and (9) follows by (5) and (8). Furthermore, B_{0} = X_{τ^{−1}(0)} = x_{0}. Since α^{2} is locally bounded, (9) implies that τ(t) is locally Lipschitz, and thus locally absolutely continuous. Thus τ(t) is differentiable almost everywhere and (10) holds.

In order to show the almost sure uniqueness of the solution to (9), we study this equation pathwise as a (non-stochastic) ordinary differential equation with a given Brownian path B. A problem that we now encounter is that even if α is Lipschitz, the Brownian path B is not, and thus the standard theorems on existence and uniqueness of solutions to ordinary differential equations do not apply. Instead, we will construct a solution which is a stopping time, and then use the uniqueness of solutions to (7) to show that the solution is almost surely unique. We prove some lemmas, and begin with a non-stochastic existence result for (9) for the case of bounded α; we write β = α^{2}.

Lemma 1. Let β be a bounded and measurable function on ℝ×[0, ∞) which is continuous in x for each fixed t. If B(t) is a continuous function on [0, ∞), then there exists a solution f to

f(t) = ∫_{0}^{t} β(B(f(s)), s) ds, t ≥ 0.  (13)

The solution can be chosen so that, as B varies over C[0, ∞), f(t) is, for each t ≥ 0, a stopping time with respect to the right-continuous filtration B_{t+}^{0} = ⋂_{u>t} B_{u}^{0}, where B_{u}^{0} is generated by the coordinate maps B ↦ B(s), s ≤ u.

Proof. We begin by defining approximate solutions f_{n}, n = 1, 2, …, by f_{n}(0) = 0 and, inductively for k = 0, 1, …,

f_{n}(t) = f_{n}(k/n) + ∫_{k/n}^{t} β(B(f_{n}(k/n)), s)ds, \frac{k}{n} < t ≤ \frac{k + 1}{n}.  (14)

We can also write this definition as

f_{n}(t) = ∫_{0}^{t} β(B(f_{n}([ns]/n)), s) ds.  (15)

We will show that f(t) = lim sup_{n→∞} f_{n}(t) yields a solution to (13). We first verify that our construction yields stopping times. Write the functions defined above as f_{n}(t, B) and f(t, B) and let t, u ≥ 0. Note first that (14) implies that f_{n}(t, B) is a continuous and thus measurable function of B ∈ C[0, ∞). Hence B ↦ f(t, B) is measurable. Suppose that f(t, B) ≤ u and that B(s) = B_{1}(s) for s ∈ [0, u + ϵ), where ϵ > 0. Then, for some n_{0}, f_{n}(t, B) ≤ u + ϵ for n ≥ n_{0}. It then follows easily from (14) that f_{n}(s, B_{1}) = f_{n}(s, B) for 0 ≤ s ≤ t and n ≥ n_{0} and thus f(t, B_{1}) = f(t, B). It follows by Galmarino’s test, see [R-Y, Exercise I.(4.21)], that f(t, B) + ϵ is a B_{t}^{0}-stopping time. Letting ϵ tend to 0, we find that f(t, B) is a B_{t+}^{0}- stopping time.

To show that f actually is a solution, let M be a fixed number with M ≥ sup β and say that a function g : [0, ∞) → [0, ∞) is a ϕ-approximate solution to (13), where ϕ : [0, ∞) → [0, ∞) is a measurable function with ϕ ≤ M, provided

g(0) = 0  (16)
|g(s) − g(t)| ≤ M|s − t| for all s, t ≥ 0,  (17)
|g′(t) − β(B(g(t)), t)| ≤ ϕ(t) for a.e. t ≥ 0.  (18)

Note that (17) implies that that g is locally absolutely continuous, and thus g′(t) exists almost everywhere and g(t) = ∫_{0}^{t} g′(s) ds. It follows that, given (17), (18) is equivalent to

|g(u) − g(t) − ∫_{t}^{u} β(B(g(s)), s) ds| ≤ ∫_{t}^{u} ϕ(s) ds  (19)

for all t, u with 0 ≤ t ≤ u.

Lemma 2. If f_{1} and f_{2} are ϕ-approximative solutions to (13), then so are f_{1} ∨ f_{2} and f_{1} ∧ f_{2}.

Proof. We consider f = f_{1} ∨ f_{2}; the case f = f_{1} ∧ f_{2} being similar. Clearly (16) and (17) hold for f by the corresponding properties for f_{1} and f_{2}. It follows from (17) that f′(t), f′_{1}(t) and f′_{2}(t) exist almost everywhere. Now, let t be such that these derivatives exist and further that (18) holds for f_{1} and f_{2}. If f_{1}(t) > f_{2}(t), then f(s) = f_{1}(s) for all s in a neighborhood of t, and thus f′(t) = f′_{1}(t) and

|f′(t) − β(B(f(t)), t)| = |f′_{1}(t) − β(B(f_{1}(t)), t)| ≤ ϕ(t).

The case f_{2}(t) > f_{1}(t) is, of course, analogous. Finally, assume f_{1}(t) = f_{2}(t), and let b = β(B(f(t)), f) = β(B(f_{1}(t)), t) = β(B(f_{2}(t)), t). Then for i = 1, 2, we have |f′_{i}(t) − b| ≤ ϕ(t), so if ϵ > 0, there exists a δ > 0 such that

|f_{i}(t + h) − f(t) − hb| ≤ |h|(ϕ(t) + ϵ)

for |h| < δ. In particular, for such h,

|f(t + h) − f(t) − hb| ≤ |h|(ϕ(t) + ϵ)

and since we have assumed that f′(t) exists, we conclude that |f′(t) − b| ≤ ϕ(t) + ϵ. Since ϵ > 0 is arbitrary, we find that |f′(t) − b| ≤ ϕ(t), as required.

□ Lemma 3. Suppose that f_{n} is a ϕ_{n}-approximate solution to (13), n ≥ 1. Then lim sup_{n→∞} f_{n} and lim inf_{n→∞} f_{n} are ϕ-approximative solutions with ϕ = lim sup_{n→∞} ϕ_{n}.

Proof. Consider first the special case when f(t) = lim_{n→∞} f_{n}(t) and ϕ(t) = lim_{n→∞} ϕ_{n}(t) exist for all t ≥ 0. Clearly, (16) and (17) hold for f since they hold for each f_{n}. Moreover, (19) holds with f_{n} and ϕ_{n} for each n and by dominated convergence on both sides (19) holds for f too. (Recall that β and ϕ_{n} are bounded by M, and that x → β(B(x), s) is continuous.) Thus f is a ϕ-approximative solution.

In general, it follows from Lemma 2 that if n, m ≥ 1, then max_{n≤k≤n+m} f_{k} is a sup_{k≥n} ϕ_{k}-approximative solution. Letting first m → ∞ and then n → ∞ and twice applying the special case just proved, we see that sup_{k≥n} f_{k} is a (sup_{k≥n} ϕ_{k})−approximative solution and that lim sup f_{n} is a (lim sup ϕ_{n})- approximative solution. The case of lim inf is similar. □

Conclusion of the proof of Lemma 1. For almost every t, by (15),

f′_{n}(t) − β(B(f_{n}(t)), t) = β(B(f_{n}([nt]/n)), t) − β(B(f_{n}(t)), t).

Moreover, since sup β ≤ M,

0 ≤ f_{n}(t) − f_{n}([nt]/n) ≤ M(t − [nt]/n) ≤ M/n

and f_{n}(t) ≤ Mt. Consequently, if

ϕ_{n}(t) = sup_{0≤y≤x≤Mt,|x−y|≤1/n} |β(B(x), t) − β(B(y), t)|,

then

|f′_{n}(t) − β(B(f_{n}(t)), f)| ≤ ϕ_{n}(t)

and, since (16) and (17) evidently hold, f_{n} is a ϕ_{n}-approximative solution. Moreover, since x ↦ β(B(x), t) is continuous for every t, and thus uniformly continuous on compact sets, ϕ_{n}(t) → 0 as n → ∞. Lemma 3 now shows that lim sup_{n} f_{n} is a 0-approximative solution, i.e. a solution to (13), thus finishing the proof of Lemma 1. □

We now wish to extend Lemma 1 to unbounded α and β. In that case we can no longer guarantee the absence of explosions for all continuous functions B, but we prove that this is true for almost every Brownian path B. We thus have to switch to stochastic arguments including switching to the notation of Theorem 1. We begin with a standard estimate. Note that by a solution to (9) we mean a stochastic process τ such that almost surely (9) holds for all t ≥ t_{0}.

Lemma 4. If B_{t} is a Brownian motion with B_{0} = x_{0} and τ(t), t ≥ 0, is a stopping time solution to (9), where α is measurable on ℝ × [t_{0}, ∞) and satisfies (6), then

Eτ(t) ≤ (1 + x_{0}^{2})e^{2C^{2}(t−t_{0})}, t ≥ t_{0}.

Proof. Let N > 0 be an integer. It follows from (9) that

τ(t) ∧ N = ∫_{t_{0}}^{t} α^{2}(B_{τ(s)}, s)1_{[τ(s)<N]} ds ≤ ∫_{t_{0}}^{t} α^{2}(B_{τ(s)∧N}, s) ds.

Moreover, (6) implies α^{2}(x, s) ≤ C^{2}(1 + |x|)^{2} ≤ 2C^{2}(1 + x^{2}). Hence, using the fact that τ(s) ∧ N is a bounded stopping time,

E(τ(t) ∧ N) ≤ ∫_{t_{0}}^{t} E(2C^{2}(1 + B_{τ(s)∧N}^{2})) ds
= 2C^{2} ∫_{t_{0}}^{t}(1 + EB_{τ(s)∧N}^{2}) ds
= 2C^{2} ∫_{t_{0}}^{t}(1 + x_{0}^{2} + E(τ(s) ∧ N)) ds.

Gronwall’s lemma, see e.g. Appendix §1 in [R-Y], shows that

1 + x_{0}^{2} + E(τ(t) ∧ N) ≤ (1 + x_{0}^{2})e^{2C^{2}(t−t_{0})},

and the result follows by letting N → ∞. □

Remark 3. Lemma 4 can be improved to

Eτ(t)^{p} < ∞

for all t ≥ t_{0} and p < ∞. Indeed, if p ≥ 1, by Hölder’s inequality and the Burkholder–Davis–Gundy inequalities, see for instance [R-Y, §IV.4],

E(τ(t) ∧ N)^{p} ≤ (t − t_{0})^{p−1}E ∫_{t_{0}}^{t} α^{2p}(B_{τ(s)∧N}, s) ds
≤ C_{p}(t − t_{0})^{p−1}E ∫_{t_{0}}^{t}(1 + B_{τ(s)∧N}^{2p}) ds
≤ C′_{p}(t − t_{0})^{p−1} ∫_{t_{0}}^{t}(1 + x_{0}^{2p} + E(τ(s) ∧ N)^{p}) ds

and the result follows by Gronwall’s lemma as above.

Lemma 5. If B_{t} is a Brownian motion with B_{0} = x_{0} and α is measurable on ℝ×[t_{0}, ∞), continuous in x, and satisfies (6), then there exists a stopping time solution τ to (9).

Proof. Assume, as we may, that t_{0} = 0. For N = 1, 2, …, define β_{N}(x, t) = α^{2}(x, t) ∧ N^{2}, and let τ_{N}(t) be the solution to

τ′_{N}(t) = β_{N}(B_{τ_{N} (t)}, t)

constructed in Lemma 1. Define τ(t) = lim inf_{N→∞} τ_{N}(t); this is a (B_{t+}^{0})- stopping time. Moreover, Lemma 4 shows that sup_{N} Eτ_{N}(t) < ∞, so by Fatou’s lemma

Eτ(t) ≤ lim inf_{N→∞} Eτ_{N}(t) < ∞

and τ(t) is almost surely finite.

Next, let t ≥ 0. Consider a point ω in the probability space Ω and assume τ(t) < ∞ at ω. Take A_{1} = A_{1}(ω) > τ(t) and A_{2} = A_{2}(ω) = max_{x≤A_{1}} |B(x)|. For infinitely many N, τ_{N}(t) < A_{1}; fix one such N with N > C(1 + A_{2}). In the proof of Lemma 1, the solution τ_{N} was constructed as lim sup_{n→∞} τ_{Nn}. For sufficiently large n, we thus have τ_{Nn}(t) < A_{1}. For these n, the definition (14) uses, on the interval [0, t], only β_{N}(B(x), s) with x ≤ τ_{Nn}(t) < A_{1} and thus |B(x)| ≤ A_{2} and |α(B(x), s)| ≤ C(1 + A_{2}). It follows that β_{N}(B(x), s) = α^{2}(B(x), s) = β_{M}(B(x), s) for all such x and s and all M ≥ N. Hence τ_{Mn}(s) = τ_{Nn}(s) for s ≤ t and M ≥ N and these n, and thus τ_{M}(s) = τ_{N}(s) for s ≤ t and M ≥ N. Consequently, τ(s) = τ_{N}(s) for s ≤ t, and τ satisfies (9) on [0, t]. Hence τ is a solution to (9). □

Conclusion of the proof of Theorem 1. By assumption, X_{t} is a local martingale with respect to some complete right-continuous filtration (F_{t}). Then τ^{−1}(t) is a right-continuous family of (F_{t})-stopping times, B_{t} is a Brownian motion with respect to F̂_{t} = F_{τ^{−1}(t)} and τ(t) is a (F̂_{s})-stopping time. Let (B_{t}^{0}) be the filtration generated by B_{t}, and B_{t} the completion of B_{t+}^{0}. We have B_{t} ⊆ F̂_{t}, but we have not yet shown that τ(t) is a (B_{t})-stopping time.

Conversely, if B_{t} is a Brownian motion with respect to a complete rightcontinuous filtration (F_{t}), and τ(t) are (F_{t})-stopping times satisfying (9), then X_{t} = B_{τ(t)} defines a local martingale (with respect to the σ−fields F_{τ(t)}, see Proposition V.1.5 in [R-Y]), with the quadratic variation, by (9),

⟨X, X⟩_{t} = τ(t) = ∫_{0}^{t} α^{2}(X_{s}, s) ds.

Hence, if we define B̃ by

B̃_{t} = ∫_{0}^{t} α(X_{s}, s)^{−1}1_{[α(X_{s},s)≠0]} dX_{s} + ∫_{0}^{t} 1_{[α(X_{s},s)=0)]} d\overline{B}_{s}

where \overline{B} is another Brownian motion, independent of everything else, then B̃_{t} is a local martingale with quadratic variation ⟨B̃, B̃⟩_{t} = t, i.e. another Brownian motion by Lévy’s theorem (see for instance [R-Y, Theorem IV.(3.6)]), and (7) holds.

Now, suppose that \overline{τ}(t) is another family of (F_{t})-stopping times satisfying (9). Then \overline{X}_{t} = B\overline{τ} (t) is another solution to (7) (for another B̃_{t}), but as is pointed out in the beginning of the proof, (4) implies uniqueness in law for (7). Hence, the processes X_{t} and \overline{X}_{t} have the same distribution, and (5) implies that

Eτ(t) = ∫_{0}^{t} Eα^{2}(X_{s}, s) ds = E\overline{τ}(t),

for each t. Moreover, τ(t) ∨ \overline{τ}(t) is another solution to (9), by applying Lemma 2 with ϕ = 0 locally; thus the argument just given shows

Eτ(t) = E(τ(t) ∨ \overline{τ}(t)) = E\overline{τ}(t),

which is possible only if τ(t) = \overline{τ}(t) almost surely, since Eτ(t) < ∞ by Lemma 4. In other words, two solutions to (9) that are (F_{t})-stopping times are almost surely identical. Since Lemma 5 provides a solution which is a family of B_{t}-stopping times, and B_{t} ⊆ F_{t}, every solution consists of B_{t}- stopping times.

Similarly, if B is any Brownian motion and τ(t) and \overline{τ}(t) are two B_{t}- stopping time solutions to (9), the same argument shows that τ = \overline{τ} almost surely. This completes the proof of Theorem 1. □

Proof of Theorem 2. By the argument above, B_{τ(t)} is another solution to (7), for some other Brownian motion B̃, and by uniqueness in law as discussed above, X_{t} has the same distribution as B_{τ(t)}. □

Remark 4. We have shown in Theorem 1 that there is only one stopping time solution to (9) (up to almost sure equivalence). We do not know whether there are other solutions that are not stopping times. Note that there is no uniqueness in Lemma 1 under our weak assumptions; a simple example is given by β(x, t) = max(2|x|, 1) and B(t) = \sqrt{t}, when f(t) = min((t − a)_{+}^{2}, (t−a)_{+}) is a solution for every a ≥ 0. Thus there exist Brownian paths where the solution to (9) is non-unique, although it is possible that the set of such paths has probability 0.

4. Existence and uniqueness for volatility time for absorbed processes

In applications to option pricing, we are only interested in non-negative values of the underlying asset. We thus consider a stochastic process X_{t} ≥ 0 that satisfies (7) when X_{t} > 0. We also assume that X_{t_{0}} = x_{0} > 0. We allow the possibility that X_{t} reaches 0 in finite time, and then assume that X_{t} remains 0 forever, i.e. 0 is an absorbing state.

In this setting we need α to be defined on (0, ∞)×[t_{0}, ∞) only; we extend α to ℝ × [t_{0}, ∞) by defining α(x, t) = 0 for x ≤ 0. We still assume that α is measurable and that the bound (6) holds, but we require the Hölder(1/2) condition only locally on (0, ∞) × [t_{0}, ∞), i.e. (4) holds for x, y ∈ [K^{−1}, K] and |t| ≤ K. Note that we allow a discontinuity of α at x = 0. In this context it is customary to use the volatility σ ≥ 0 as in (1) instead of α; they are related by α(x, t) = xσ(x, t). Note that α is locally Hölder(1/2) on (0, ∞) × [t_{0}, ∞) if and only if σ is, and that (6) holds if and only if

σ(s, t) ≤ C(1 + s^{−1}).

Then the following version of the results in Section 3 holds.

Theorem 3. Suppose that α is measurable on (0, ∞)×[t_{0}, ∞) for some real t_{0}, and that (6) holds and also that α is locally Hölder(1/2) on (0, ∞) × [t_{0}, ∞). Let further α(x, t) = 0 for x ≤ 0. Then, for every x_{0} > 0, and a Brownian motion B̃, there exists an almost surely unique stochastic process X_{t} on [t_{0}, ∞) such that (7) holds and 0 is an absorbing state for X_{t}. Moreover, for a Brownian motion B_{t} with B_{0} = x_{0}, there exists an almost surely unique stopping time solution τ(t) to (9). Finally, for any such X_{t} and B, X_{T} and B_{τ(T)} have the same distribution and thus, for any measurable function Φ,

EΦ(X_{T}) = EΦ(B_{τ(T)})

Proof. We may again assume that t_{0} = 0. We first consider the process X_{t}. Define for N ≥ 1/x_{0},

α_{N}(x, t) = α(x ∨ (1/N), t).

Then α_{N} satisfies (6) and (4), and it follows from Theorem 1 that there exists a unique solution X^{(N)} to

dX_{t}^{(N)} = α_{N}(X_{t}^{(N)}, t) dB̃_{t}, X_{0}^{(N)} = x_{0}.  (20)

By stopping X^{(N)} at T_{N} = inf{t : X_{t}^{(N)} = 1/N}, we obtain a solution to (7) and (20) on [0, T_{N}]. The same holds if M > N and we stop X^{(M)} when it reaches 1/N. Since the latter solution may be restarted to give a solution to (20) defined for all t ≥ 0, the uniqueness of X^{(N)} implies that X^{(M)} = X^{(N)} on [0, T_{N}] for M > N. It follows that T_{M} > T_{N} when M > N; i.e. the sequence (T_{N})_{N} of stopping times is strictly increasing. Define T_{∞} = lim_{N→∞} T_{N} = sup_{N} T_{N} and

X_{t} = \begin{cases}lim inf_{N→∞} X_{t}^{(N)} & if t < T_{∞}; \\ 0 & if t ≥ T_{∞}. \end{cases}

Then X_{t} = X_{t}^{(N)} when 0 ≤ t ≤ T_{N}; thus X is a solution to (7) on [0, T_{N}] for each N, and thus on [0, T_{∞}). In particular, X_{t} is a local martingale on [0, T_{∞}). Since a local martingale is a time change of a Brownian motion, it almost surely either converges or oscillates between −∞ and +∞; by construction X_{t} ≥ 0 for t < T_{∞}, so the latter alternative is impossible and lim_{t↑T_{∞}} X_{t} exists almost surely. Moreover, since X_{T_{N}} = X_{T_{N}}^{(N)} = 1/N, this limit is 0, so t ↦ X_{t} is continuous at T_{∞} too, and thus for all t ≥ 0.

Conversely, if X̃_{t} is any solution to (7) with 0 absorbing and X̃_{0} = x_{0} we obtain by stopping at the time T̃_{N} when X̃_{t} reaches 1/N a solution to (20) on [0, T̃_{N}]. As in the first part of the proof it follows that T̃_{N} = T_{N} and that X̃ = X^{(N)} = X on [0, T_{N}]. Hence, X̃ = X on [0, T_{∞}), and since X̃_{T_{∞}} = X_{T_{∞}} = 0 and 0 is an absorbing state for both processes,

X̃ = X.

It follows further that the distribution of (X_{t})_{t} does not depend on the choice of Brownian motion B̃_{t}.

We now consider the volatility time τ. We argue as in the proof of Lemma 5, this time defining

β_{N}(x, t) = \begin{cases}α^{2}(x, t) ∧ N^{2} & if x ≥ 1/N; \\ α^{2}(1/N, t) ∧ N^{2} & if x < 1/N. \end{cases}  (21)

We again let τ_{N}(t) be the stopping time solutions to τ′_{N}(t) = β_{N}(B_{τ_{N} (t)}, t) constructed in Lemma 1, and let \overline{τ}(t) = lim inf_{N→∞} τ_{N}(t); this is by Fatou’s lemma and Lemma 4 an almost surely finite stopping time. Consider a point ω ∈ Ω. Let T_{0} = inf\{t : B_{t} = 0\} and suppose that t ≥ 0 and \overline{τ}(t) < T_{0}. Take A_{1} = A_{1}(ω) with \overline{τ}(t) < A_{1} < T_{0} and let A_{2} = A_{2}(ω) = max_{x≤A_{1}}(|B(x)| + |B(x)|^{−1}). It follows as in the proof of Lemma 5, that \overline{τ}(s) = τ_{N}(s) for all s ≤ t and N ≥ N_{0}(ω) = (1 ∨ C)(1 + A_{2}). Consequently, \overline{τ} satisfies (9) for all t such that \overline{τ}(t) < T_{0}. Note also that \overline{τ}(s) ≤ \overline{τ}(t) when s ≤ t. Now, suppose that \overline{τ}(t) ≥ T_{0} for some finite t, and let t_{0} = inf\{t : \overline{τ}(t) ≥ T_{0}\}. We claim that \overline{τ}(t) ↑ T_{0} as t ↑ t_{0}. Indeed, let \overline{τ}(t_{0}−) = sup_{s<t_{0}} \overline{τ}(s) and assume \overline{τ}(t_{0}−) < T_{0}. Taking A_{1} ∈ (\overline{τ}(t_{0}−), T_{0}) and A_{2} as above we find as before that \overline{τ}(s) = τ_{N}(s) for all s < t_{0} and N ≥ N_{0}. In particular, since τ_{N_{0}} is continuous, τ_{N_{0}}(t_{0}) = \overline{τ}(t_{0}−) < A_{1} and thus τ_{N_{0}} < A_{1} for t ≤ t_{0} + ϵ, for some ϵ > 0. The same argument now shows that τ_{N}(t) = τ_{N_{0}}(t) for all N ≥ N_{0} and t ≤ t_{0} + ϵ, and thus \overline{τ}(t) = τ_{N_{0}}(t) < A_{1} < T_{0} for t ≤ t_{0} + ϵ, which contradicts the definition of t_{0}. Hence \overline{τ}(t_{0}−) = T_{0} as claimed. It follows that

τ(t) = \overline{τ}(t) ∧ T_{0} = \begin{cases}\overline{τ}(t) & if t < t_{0}; \\ T_{0} & if t ≥ t_{0}. \end{cases}

is continuous, and that is satisfies (9) for all t ≤ t_{0}, and thus for all t since α(B_{T_{0}}, t) = 0. Moreover, each τ(t) is a stopping time. This proves the existence of a stopping time solution τ.

Let τ be any stopping time solution to (9). It follows as in the proof of Theorem 1 that X_{t} = B_{τ(t)} is a solution to (7) for some Brownian motion B̃_{t}. We next show that this solution is absorbed at 0.

Consider again an ω ∈ Ω. If B_{τ(t)} < 0 for some t = t_{1}, say, then by continuity, this holds in an open interval t_{1} − ϵ < t < t_{1} + ϵ. In this interval we thus have α(B_{τ(t)}, t) = 0, and hence τ′(t) = 0. In other words, τ(t) and thus B_{τ(t)} are constant on t_{1} − ϵ < t < t_{1} + ϵ. Consequently, for each y < 0, the set E_{y} = {t : B_{τ(t)} = y} is a both closed and open subset of [0, ∞), and since B_{τ(0)} = x_{0} > 0, E_{y} = ∅ for every y < 0. Consequently, B_{τ(t)} ≥ 0 for all t.

Next, almost surely, every interval (T_{0}, T_{0} + ϵ) contains points t where B_{t} < 0. Hence, if τ(t) > T_{0} there exists an s ≤ t such that B_{τ(s)} < 0, which is a contradiction. Consequently, τ(t) ≤ T_{0} almost surely. It follows that if B_{τ(t)} = 0 for some t, then τ(t) = T_{0}. Hence, for u ≥ t we have τ(t) ≤ τ(u) ≤ T_{0}, so τ(u) = T_{0} and B_{τ(u)} = 0. In other words, B_{τ(t)} is absorbed at 0. It now follows, arguing as in the proof of Theorem 1, that if τ_{1} and τ_{2} are two stopping time solutions to (9), then so are τ_{1} ∧ τ_{2} and τ_{1} ∨ τ_{2}, and hence, from the uniqueness assertion in the first part of the proof, τ_{1} = τ_{2} almost surely. The proof is complete. □

5. Time decay of option prices

In this and the following two sections, the general results above on volatility time will be applied to several problems concerning option pricing, using arguments similar to [Ho]. (See also [Ha] for related arguments.) The following simple lemma provides a link between mononotonicity properties of option prices and monotonicity properties of stopping times.

Lemma 6. Suppose that B_{t} is a Brownian motion with B_{0} = x_{0} ∈ ℝ and that τ_{1} and τ_{2} are two stopping times such that τ_{1} ≤ τ_{2} almost surely and Eτ_{2} < ∞. If Φ is any convex function, then

EΦ(B_{τ_{1}}) ≤ EΦ(B_{τ_{2}}),

where the expectations are finite or +∞.

Proof. As is well-known, the stopped Brownian motion B_{τ_{2}∧t} is a square integrable martingale, and

B_{τ_{1}} = E(B_{τ_{2}}|F_{τ_{1}}).

If |Φ(x)| = O(1 + |x|), then EΦ(B_{τ_{1}}) and EΦ(B_{τ_{2}}) are finite and by the conditional version of Jensen’s inequality,

Φ(B_{τ_{1}}) ≤ EΦ(B_{τ_{2}}|F_{τ_{1}})

almost surely and thus

EΦ(B_{τ_{1}}) ≤ EΦ(B_{τ_{2}}).

In general, Φ is the limit of an increasing sequence of convex functions Φ_{n} such that Φ_{n} = Φ on [−n, n] and Φ_{n} is linear on (−∞, −n] and on [n, ∞). The result follows by the preceding case and monotone convergence. □

In our first application we compare option prices at two different times. We begin with a simple monotonicity result for the corresponding volatility times.

Lemma 7. Let α be as in Theorem 1 or 3 and let t_{0} ≤ t_{1} ≤ t_{2}. If B_{t} is a Brownian motion and τ_{1} and τ_{2} are the stopping time solutions to

τ′_{j}(t) = α^{2}(B_{τ_{j}(t)}, t), t ≥ t_{j},

with τ_{j}(t_{j}) = 0, j = 1, 2, then

τ_{1}(t) ≥ τ_{2}(t)

almost surely for every t ≥ t_{2}.

Proof. Define τ(t) = τ_{1}(t) ∧ τ_{2}(t). As in the proof of Lemma 2,

τ′(t) = α^{2}(B_{τ(t)}, t)

for almost every t ≥ t_{2}. Moreover, τ_{1}(t_{2}) ≥ 0 and thus τ(t_{2}) = τ_{2}(t_{2}) = 0. By the uniqueness assertions in Theorems 1 and 3, for any t ≥ t_{2}, τ(t) = τ_{2}(t) almost surely, and thus τ_{1}(t) ≥ τ_{2}(t). □

Remark 5. We conjecture that actually τ_{1}(t) > τ_{2}(t) almost surely except in trivial cases, but we have not been able to show this without stricter requirements on α.

Theorem 4. Let α be as in Theorem 1 or 3 and let t_{0} ≤ t_{1} ≤ t_{2} ≤ T. Let X_{t}^{(1)} and X_{t}^{(2)} be solutions to

dX_{t} = α(X_{t}, t) dB̃_{t},

where B̃ as a Brownian motion, with X_{t_{1}}^{(1)} = x_{0} = X_{t_{2}}^{(2)}. In the setting of Theorem 3, assume further that 0 is an absorbing state for X_{t}^{(j)}, j = 1, 2. Finally, let Φ be a convex function. Then

EΦ(X_{T}^{(1)}) ≥ EΦ(X_{T}^{(2)}).

Before proving this theorem we note that interpreting X as a price process with volatility given by α we have, in view of equation (2) that option prices with convex contract functions decay with time, or in view of equation (3), that the price is convex as a function of the price of underlying asset. This has earlier been proved by [B-G-W] and [Ho] under somewhat different conditions.

Proof. Let τ_{1} and τ_{2} be as in Lemma 7. Thus

τ_{2}(T) ≤ τ_{1}(T)

almost surely. By Lemma 4, Eτ_{1}(T) < ∞, and Lemma 6 yields, together with Theorem 2 or 3,

EΦ(X_{T}^{(2)}) = EΦ(B_{τ_{2}(T)}) ≤ EΦ(B_{τ_{1}(T)}) = EΦ(X_{T}^{(1)}).  □

Remark 6. A basic idea in our approach is to compare different price processes (or, as in Theorem 4, the same price processes started at different times), by generating them from the same Brownian motion, using different time changes as defined by the respective volatility times. This idea, an example of coupling, is also used by Hobson [Ho].

We know that the prices tend to the contract function as time t approaches the time of expiration T and that the prices remain convex under the conditions of Theorem 4. But what happens if we let t tend to minus infinity? We have the following result.

Theorem 5. Let α(x, t) = xσ(x, t) be as in Theorem 3 with t_{0} = −∞ and assume further that σ(x, t) ≥ γ(x)δ(t) for some positive functions γ and δ such that inf_{a≤x≤b} γ(x) > 0 for any 0 < a < b < ∞ and ∫_{−∞}^{0} δ^{2}(t) dt = ∞.

Then, the price of a call option with any strike price K at a fixed time T increases to s, the price of the underlying asset, as time tends to minus infinity.

Proof. Let X_{u}^{(t)}, u ≥ t, denote the solution to dX_{u} = α(X_{u}, u) dB̃_{u} starting at X_{t} = x_{0} = s. Recall that the price of the option at time t is according to (2) given by EΦ(X_{T}^{(t)}) with the contract function Φ(s) = (s − K)_{+}.

Consider a decreasing sequence t_{n} ↓ −∞ of times less than T, and let τ_{n}(t) denote the solution to (9) starting at t_{0} = t_{n}. By Lemma 7, τ_{n}(T) is an increasing sequence; denote its limit by τ(T). Let as before T_{0} = inf{t : B_{t} = 0}. We claim that τ(T) = T_{0} almost surely. Indeed, we know that each τ_{n}(T) ≤ T_{0}, so τ(T) ≤ T_{0}. Suppose that, at some ω ∈ Ω, τ(T) < T_{0}. Then a = inf_{0≤u≤τ(T)} B_{u} > 0 and b = sup_{0≤u≤τ(T)} B_{u} < ∞, and thus, by the assumptions, α(B_{u}, t) ≥ cδ(t) for some c > 0 and all u ≤ τ(T). Then, for each n,

τ(T) ≥ τ_{n}(T) = ∫_{t_{n}}^{T} α^{2}(B_{τ_{n}(u)}, u) du ≥ ∫_{t_{n}}^{T} c^{2}δ(u)^{2} du,

which tends to ∫_{−∞}^{T} c^{2}δ(u)^{2} du = ∞ as n → ∞, a contradiction. Hence τ_{n}(T) → T_{0} almost surely, and thus B_{τ_{n}(T)} → 0.

Since X_{T}^{(t_{n})} and B_{τ_{n}(T)} have the same distribution, by Theorem 3, this shows that X_{T}^{(t_{n})} → 0 in probability as n → ∞. Since the sequence t_{n} is arbitrary, this shows that X_{T}^{(t)} → 0 in probability as t → −∞. It follows by dominated convergence that E(X_{T}^{(t)} ∧ K) → 0. Moreover, X_{u}^{(t)} is a martingale and thus EX_{T}^{(t)} = x_{0}, and consequently

E(X_{T}^{(t)} − K)_{+} = EX_{T}^{(t)} − E(X_{T}^{(t)} ∧ K) → x_{0} = s.  □

Remark 7. Note that this result is not the consequence of the declining value of currency units since we in this article express assets in constant units.

Instead, the proof shows that X_{T}^{(t)} → 0 in probability as t → −∞. In other words, in the long run the asset is with large probability almost worthless (relative to bonds), but with a small probability it has a very large value, making the expectation constant; thus making the strike price irrelevant.

6. Continuity of option prices

In this section we consider continuity of option prices under perturbations of the volatility. We begin with a weak lemma on (pathwise) monotonicity; it will be improved in Lemma 10.

Lemma 8. Suppose that β(x, t) and β̃(x, t) are locally Hölder(1/2) in x the sense of Definition 1 with t_{0} = 0 and suppose further that

β(x, t) + ϵ ≤ β̃(x, t)

for all x, t and some ϵ > 0. If B(t) is a continuous function on [0, ∞), and

τ′(t) = β(B(τ(t)), t), τ̃′(t) = β̃(B(τ̃(t)), t),

then τ(t) < τ̃(t) for all t > 0.

Proof. Suppose that τ(t) = τ̃(t) for some t ≥ 0. Let x = τ(t) and y = B(x). By the local Hölder assumption, for |s − t| ≤ 1 and some finite C,

|τ′(s) − β(y, s)| = |β(B(τ(s)), s) − β(y, s)| ≤ C|B(τ(s)) − y|^{1/2}
= C|B(τ(s)) − B(τ(t))|^{1/2},

which by the continuity of B and τ is less than ϵ/3 if |s − t| is small enough. Similarly,

|τ̃′(s) − β̃(y, s)| < ϵ/3

if |s − t| is small enough. Consequently, for some δ > 0 and |s − t| < δ,

τ̃′(s) − τ′(s) > β̃(y, s) − β(y, s) − 2ϵ/3 ≥ ϵ/3.

Hence τ̃(s) − τ(s) is 0 for s = t, negative for s ∈ (t − δ, t) and positive for s ∈ (t, t + δ). In particular, taking t = 0, τ̃(s) − τ(s) is positive for small positive s. If τ̃(s) ≤ τ(s) for some s > 0, we let t be the infimum of all such s and obtain t > 0 and τ̃(t) = τ(t). As just proved, then τ̃(s) < τ(s) for some s < t, which is a contradiction thus completing the proof. □

Lemma 9. Suppose that α and α_{1}, α_{2}, …, satisfy the conditions of Theorem 1 or 3 uniformly, i.e. with the same constants C and C_{K} in (6) and (4) for all α_{n}. Suppose further that α_{n}(x, t) → α(x, t) as n tends to infinity for all x and t. If B_{t} is a Brownian motion and τ, τ_{1}, τ_{2}, …, are stopping time solutions to

τ′ = α^{2}(B_{τ(t)}, t), τ′_{n} = α_{n}^{2}(B_{τ_{n}(t)}, t)

with τ(t_{0}) = τ_{n}(t_{0}) = 0, then

τ_{n}(t) → τ(t)

almost surely as n tends to infinity for every t ≥ t_{0}.

Proof. Assume again that t_{0} = 0 and let β = α^{2} and β_{n} = α_{n}^{2}. Suppose first that α_{1}, α_{2}, …, are uniformly bounded, say |α_{n}| ≤ C, and satisfy (4) with some constant C_{K} not depending on n. Consider a fixed Brownian path B_{t} and let

B_{t}^{∗} = sup_{0≤s≤t} |B_{s}|.

Let M = C^{2}, and note that τ_{n}(t) ≤ Mt for all n and t. Then, for almost every t ≥ 0,

|τ′_{n}(t) − β(B_{τ_{n}(t)}, t)| = |β_{n}(B_{τ_{n}(t)}, t) − β(B_{τ_{n}(t)}, t)|
≤ ϕ_{n}(t)

where

ϕ_{n}(t) = sup{|β_{n}(x, t) − β(x, t)| : |x| ≤ B_{Mt}^{∗}}
≤ 2C sup{|α_{n}(x, t) − α(x, t)| : |x| ≤ B_{Mt}^{∗}}.

Hence, τ_{n} is a ϕ_{n}-approximative solution to (13).

For fixed t, the functions α_{n}(x, t) − α(x, t) are uniformly continuous on the compact set K = {x : |x| ≤ B_{Mt}^{∗}} and converge pointwise to 0; hence they converge uniformly to 0 on K, and thus ϕ_{n}(t) → 0 as n → ∞. Lemma 3 now shows that lim inf_{n→∞} τ_{n} and lim sup_{n→∞} τ_{n} both are solutions to (13).

Thus lim inf_{n→∞} τ_{n} and lim sup_{n→∞} τ_{n} are two stopping time solutions to (9), so by the uniqueness in Theorem 1

lim inf_{n→∞} τ_{n} = lim sup_{n→∞} τ_{n} = τ

almost surely, which proves the result in this case.

Assume next that the α_{n} satisfy the conditions of Theorem 1, uniformly in n. Let α̃(x, t) = 1+C(1+|x|) and β̃(x, t) = α̃^{2}(x, t). Then α̃ also satisfies the conditions of Theorem 1, so there exists a stopping time solution τ̃ to

τ̃′(t) = α̃^{2}(B_{τ̃(t)}, t).

Moreover, for every n, α̃^{2}(x, t) > 1 + α_{n}^{2}(x, t), and thus τ_{n}(t) ≤ τ̃(t) by Lemma 8.

For 1 ≤ n ≤ ∞ and N ≥ 1 define β_{nN} = α_{n}^{2} ∧ N^{2}, where α_{∞} = α, and let τ_{nN} be the stopping time solution to

τ′_{nN} = β_{nN}(B_{τ_{nN} (t)}, t).

Suppose that t ≥ 0 and that τ̃(t) < ∞. By the proof of Lemma 5 and the uniqueness in Theorem 1, τ_{n} = lim inf_{N→∞} τ_{nN} almost surely for each n ≤ ∞. Moreover, taking A_{1} > τ̃(t) in the proof of Lemma 5 so that A_{1} > τ_{n}(t) for all n, there exists N_{0}(ω) < ∞, independent of n, such that τ_{n}(t) = τ_{nN}(t) for all n ≤ ∞ and N ≥ N_{0}. Since τ_{nN}(t) → τ_{∞N}(t) almost surely for each N by the first part of the proof, we find that τ_{n}(t) → τ_{∞}(t) = τ(t) almost surely, as asserted.

Finally, assume that α and the α_{n} satisfy the conditions of Theorem 3, uniformly in n, and that B_{0} = x_{0} > 0. Consider an ω ∈ Ω and let T_{0} = inf{t : B_{t} = 0}. Then, almost surely, 0 < T_{0} < ∞, and thus sup{|B_{t}| : t ≤ T_{0}} < ∞. By (6), there thus exists a constant C(ω) such that |α_{n}(B_{s}, t)| ≤ C for s ≤ T_{0} and t ≥ 0 and thus |τ′_{n}(t)| ≤ C^{2} for almost every t ≥ 0. Hence, the functions t ↦ τ_{n}(t) are Lipschitz, uniformly in n, and thus lim inf_{n→∞} τ_{n}(t) and lim sup_{n→∞} τ_{n}(t) are Lipschitz and hence continuous.

If t is such that τ(t) < T_{0} and lim sup_{n→∞} τ_{n}(t) < T_{0}, we take A_{1}(ω) with τ(t) < A_{1} < T_{0} and lim sup_{n→∞} τ_{n}(t) < A_{1}, and thus τ_{n}(t) < A_{1} for all large n, say n ≥ n_{0}(ω). Let β_{N} be defined by (21) and let τ_{nN} be the stopping time solution to τ′_{nN}(t) = β_{N}(B_{τ_{nN} (t)}, t). It follows from the proof of Theorem 3, arguing as in the preceding case of the present proof, that τ_{∞}(t) = τ_{∞N}(t) and τ_{n}(t) = τ_{nN}(t) for N ≥ N_{0}(ω) and n ≥ n_{0}(ω). The first part of the proof yields that τ_{nN}(t) → τ_{∞N}(t) for each N, and it follows that lim_{n→∞} τ_{n}(t) = τ(t).

It follows, by first considering rational t, that the three continuous functions lim sup_{n→∞} τ_{n}(t), lim inf_{n→∞} τ_{n}(t) and τ(t) almost surely coincide for all t such that all three functions are stricly less than T_{0}. Since all three functions are continuous, non-decreasing and bounded by T_{0}, it follows that they hit T_{0} simultaneously (if at all) and then remain constant. Hence, almost surely, the three functions coincide for all t, i.e lim_{n→∞} τ_{n}(t) = τ(t). □

Theorem 6. Suppose that α and α_{1}, α_{2}, …, satisfy the conditions of Theorem 1 or 3 uniformly and suppose that

α_{n}(x, t) → α(x, t),

as n → ∞ for all x and t. Let X_{t} and X_{t}^{(n)} be solutions to

dX_{t} = α(X_{t}, t) dB̃_{t}, dX_{t}^{(n)} = α_{n}(X_{t}^{(n)}, t) dB̃_{t}

with X_{t_{0}} = X_{t_{0}}^{(n)} = x_{0}. In the setting of Theorem 3, assume further that 0 is an absorbing state for X_{t} and X_{t}^{(n)}. Let T ≥ t_{0}. Then X_{T}^{(n)} converges in distribution to X_{T} as n tends to infinity. Further, if Φ is a continuous function with |Φ(x)| ≤ C_{1}(1 + |x|)^{k} for some C_{1} and k < ∞, then

EΦ(X_{T}^{(n)}) → EΦ(X_{T}).

Proof. Let τ and τ_{n} be as in Lemma 9. Then τ_{n}(T) → τ(T) almost surely as n tends to infinity, and thus B_{τ_{n}(T)} → B_{τ(T)} almost surely and thus in distribution. Since B_{τ_{n}(T)} and X_{T}^{(n)} as well as B_{τ(T)} and X_{T} agree in distribution, by Theorem 1 or 3, the first assertion follows.

For the second we define, as in the proof of Lemma 9, α̃(x, t) = 1 + C(1 + |x|) and let τ̃ be a stopping time solution to τ̃′(t) = α̃^{2}(B_{τ̃(t)}, t). By Lemma 8, τ_{n}(T) ≤ τ̃(T) and thus

Φ(B_{τ_{n}(T)}) ≤ C_{1}(1 + |B_{τ_{n}(T)}|)^{k} ≤ C_{1}(1 + B_{τ̃(T)}^{∗})^{k}.

Moreover, by the Burkholder–Davis–Gundy inequalities and Remark 3,

E(B_{τ̃(T)}^{∗})^{k} ≤ C_{k}E(τ̃(T)^{k/2}) < ∞.

Hence dominated convergence yields EΦ(B_{τ_{n}(T)}) → EΦ(B_{τ(T)}), which yields the second assertion by Theorem 1 or 3. □

7. Monotonicity in volatility

Using the continuity established above we can now sharpen Lemma 8. This will enable us to derive the monotonicity of option prices in the volatility.

Lemma 10. Suppose that α and α̃ satisfy the conditions of Theorem 1 or 3 and that

|α(x, t)| ≤ |α̃(x, t)|

for all x and t. If B_{t} is a Brownian motion and τ(t) and τ̃(t) are stopping time solutions to

τ′(t) = α^{2}(B_{τ(t)}, t), τ̃′(t) = α̃^{2}(B_{τ̃(t)}, t),

with τ(0) = 0 = τ̃(0), then

τ(t) ≤ τ̃(t),

almost surely for every t.

Proof. We may replace α and α̃ by their absolute values and thus assume that they are both non-negative. Let α̃_{n}(x, t) = α̃(x, t) + 1/n and let τ̃_{n} be the corresponding solution to (9). By Lemma 8, τ_{n}(t) ≥ τ(t) almost surely for each n, and by Lemma 9 τ̃_{n}(t) → τ̃(t) almost surely as n tends to infinity. □

Theorem 7. Suppose that α and α̃ are as in Lemma 10. Let X_{t} and X̃_{t} be solutions to

dX_{t} = α(X_{t}, t) dB̃_{t}, dX̃_{t} = α̃(X̃_{t}, t) dB̃_{t}

with X_{0} = X̃_{0} = x_{0}. If T ≥ t_{0} and Φ is a convex function, then

EΦ(X_{T}) ≤ EΦ(X̃_{T}).

Proof. Immediate by Theorem 1 or 3 and Lemmas 6, 10 and 4. □

8. Several underlying assets

We now consider some properties of option prices in the case of several underlying assets. Let the assets S_{i} have risk neutral processes given by

dS_{i} = S_{i}(t) ∑_{j=1}^{n} σ_{ij}(S(t), t) dB_{j}

for i = 1, …, n, where B_{j} are independent Brownian motions and S(t) = (S_{1}(t), …, S_{n}(t)). The matrix σ with entries σ_{ij} is called the volatility matrix, where we assume that the each of the entries satisfies the assumptions on volatility of the previous sections. As before we get rid of the interest rate by using a bond as a numeraire. The pricing function of a contingent claim with the contract Φ(S(T)) is given by

F (s, t) = E_{s,t}[Φ(S(T))].

Alternatively, one has that the pricing function is a solution of the partial differential equation

F_{t} + \frac{1}{2} ∑_{i,j=1} s_{i}s_{j}F_{s_{i}s_{j}}C_{ij} = 0

with the boundary condition F (s, T) = Φ(s), where C_{ij} = [σσ^{∗}]_{ij}. The following is our main result on several underlying assets.

Theorem 8. If the volatility matrix is independent of time, and the contract function is convex, then the option price F (s, t) given by

F (s, t) = E_{s,t}(Φ(S(T)).

decreases with time.

Proof. Since

F (s, t) = E_{s,t}(Φ(S(T)),

we know that if F (s, T) = Φ(S(T)) ≥ 0, then F (s, t) ≥ 0 for all t ≤ T. We also note that if the contract function is an affine function then F (t, s) = F (T, s) for all t ≤ T. First, we will show that if the contract function is convex then F (s, t) ≥ F (s, T) for all t ≤ T and all s. To show this for some particular s_{0} we compare the solution F with a solution U having a supporting hyperplane at s_{0} as contract function. We then have F (s, T) − U(s, T) ≥ 0, because F is convex, and thus F (s, t) − U(s, t) ≥ 0 for every t ≤ T. Moreover, F (s_{0}, T) = U(s_{0}, T) and U(s_{0}, t) = U(s_{0}, T) for all t ≤ T because U is affine. Hence,

F (s_{0}, t) ≥ U(s_{0}, t) = U(s_{0}, T) = F (s_{0}, T).

Thus we have for arbitrary s and t_{1} ≥ 0 that

F (s, T − t_{1}) ≥ F (s, T).

Now, let us consider both sides of this inequality as contract functions and consider the corresponding solutions at some time T − t_{2} where t_{2} ≥ 0. The corresponding solutions satisfy the same inequality by the argument above. However, the time-independence of the equation yield that these solutions are simply given by translates in time of F (s, t) and we obtain

F (s, T − t_{1} − t_{2}) ≥ F (s, T − t_{2})

which is the desired monotonicity in t. □

Remark on convexity and time-decay of option prices and monotonicity in volatility. Consider a market with two underlying assets S_{1} and S_{2}. Let this market have a diffusion matrix which is independent of time in accordance with the theorem above and with a convex contract function. Then the theorem yields that the option price decays with time. However, if we let the the contract function be that of a call option, with strike price K on one of the assets, say S_{1}, but we let the volatility of S_{1} depend on S_{2} in such a way that the volatility has a strict local maximum for some value s_{2,0} of S_{2}. It is then easy to see that the solution to the pricing equation is not convex near the point (K, s_{2,0}) in the S_{1} − S_{2}-plane. Thus convexity is lost but not time-decay of the prices. Of course, if the price remains a convex function of the asset values we see directly from the differential equation that the prices will decay with time. Another property which is lost with the convexity is the monotonicity in volatility. In the present example we see that if the volatility in S_{2} is made larger then the value of the option at (K, s_{2,0}) decreases.

In the time-dependent case there are no general results corresponding to the theorem above. Instead, different conditions on the contract functions can be combined with various classes of volatility matrices to guarantee time-decay of the solutions. The technique with volatility time is harder to use, since each asset, in general, has its own volatility time. However, if these agree one can show results corresponding to those in the case of one underlying asset. One has that if the volatility matrix σ is a diagonal matrix with

S_{i}σ_{ii} = v(t, S(t))

where v(t, S(t)) satisfies the conditions of α in Theorem 1, then the option price F (s, t) decays with time provided that the contract function Φ is subharmonic. This case is rather special and we leave the proof to the interested reader.

Remark on options as “bloating” assets. One can easily modify the example above by letting the volatility for the second asset be time-dependent such that it is very large at some time and then decreases to a very small value until the time of expiration. Then the option price at (K, s_{2,0}) will increase with time during some interval. This corresponds directly to examples in [B-G-W] of bloating option prices when the volatility is stochastic.

We finally state one observation on independent processes of option prices.

Proposition 2. If the volatility of S_{i}, for every i, only depends on time and S_{i}, then the option prices decay with time if the contract function is a finite sum of products of positive convex functions of one of the underlying assets. Again we have to assume the regularity conditions of the previous theorems.

Proof. This follows directly from the risk neutral valuation formula and Theorem 3. □

References

[B-G-W] Bergman, Y., Grundy, B., Wiener, Z.: General properties of option prices. Jour-

nal of Finance 51 (1996), 1573–1610. [B-S] Black, F., Scholes, M.: The pricing of options and and corporate liabilities. Journal

of Political Economy 81 (1973), 659–683. [G-G-I-S] Giga, Y., Goto, S., Ishii, H., Sato, M.-H.: Comparison principle and convexity

properties for singular degenerate parabolic equations on unbounded domains. Indiana Univ. Math. J. 40 (1991), 443–470.

[Ha] Hajek, B.: Mean stochastic comparison of diffusions. Z. Wahrsch. Verw. Gebiete 68 (1985), 315–329.

[H-S] Heath, D., Schweizer, M.: Martingales versus PDEs in finance: an equivalence

result with examples. J. Appl. Prob. 37 (2000), 947–957.

[Ho] Hobson, D.: Volatility misspecification, option pricing and superreplication via coupling. Ann. Appl. Prob. 8 (1998), 193–205.

[L-P] Liu, D.F., Porru, G.: Convexity and monotonicity of solutions to some second

order parabolic equations. Rend. Sem. Fac. Sci. Univ. Cagliari 64 (1994), 43–53. [K-J-S] El Karoui, N., Jeanblanc-Picqu, M., Shreve, S.: Robustness of the Black-Scholes

formula. Math. Finance 8 (1998), 93–126. [R-Y] Revuz, D., and Yor, M.: Continuous martingales and Brownian motion

(Grundlehren Math. Wiss. 293), 3rd ed., Springer, Berlin, Heidelberg, New York,

Department of Mathematics, Uppsala University, PO Box 480, SE-751 06 Uppsala, Sweden

E-mail address: svante.janson@math.uu.se, johan.tysk@math.uu.se
URL: http://www.math.uu.se/~svante/,  http://www.math.uu.se/~johant/