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Recovering a time-homogeneous stock price process from perpetual option prices

It is well known how to determine the price of perpetual American options if the underlying stock price is a time-homogeneous diffusion. In the present paper we consider the inverse problem, that is, given prices of perpetual American options for different strikes, we show how to construct a time-homogeneous stock price model which reproduces the given option prices.


Introduction
In the classical Black-Scholes model, there is a one-to-one correspondence between the price of an option and the volatility of the underlying stock. If the volatility σ is assumed to be given, for example by estimation from historical data, then the arbitrage free option price can be calculated using the Black-Scholes formula. Conversely, if an option price is given, then the implied volatility can be obtained as the unique σ that would produce this option price if inserted in the Black-Scholes formula. It has been well documented that if the implied volatility is inferred from real market data for option prices with the same maturity date but with different strike prices, then typically a non-constant implied volatility is obtained. Since the implied volatility often resembles a smile if plotted against the strike price, this phenomenon is referred to as the smile effect. The smile effect is one indication that the Black-Scholes assumption of normally distributed log-returns is too simplistic.
A wealth of different stock price models has been proposed in order to overcome the shortcomings of the standard Black-Scholes model, of which the most popular are jump models and stochastic volatility models. Given a model, option prices can be determined as risk neutral expectations. However, models are typically governed by a small number of parameters and only in exceptional circumstances can they be calibrated to perfectly fit the full range of options data.
Instead, there is a growing literature which tries to reverse the procedure and uses option prices to make inferences about the underlying price process. At one extreme models exist which take a price surface as the initial value of a Markov process on a space of functions. In this way the Heath-Jarrow-Morton [6] interest rate models can be made to perfectly fit an initial term structure. Such ideas inspired Dupire [5] to introduce the local volatility model which calibrates perfectly to an initial volatility surface. For a local volatility model, Dupire derived the PDE P T (T, K) + rKP K (T, K) = 1 2 σ 2 (T, K)K 2 P KK (T, K) where P (T, K) is the European put option price, T is time to maturity and K is the strike price. Solving for the (unknown) local volatility σ(T, K) gives a formula for the time inhomogeneous local volatility in terms of derivatives of the observed European put option prices. The local volatility model gives the unique martingale diffusion which is consistent with observed call prices. (Alternative, non-diffusion models also exist, see for example, Madan and Yor [9].) The recent literature (eg Schweizer and Wissel [13]) has included attempts to extend the theory to allow for a stochastic local volatility surface. However, it relies on the knowledge of a double continuum of option prices, which are smooth. In contrast, Hobson [7] builds models which are consistent with a continuum of strikes, but at a single maturity, in which case there is no uniqueness.
In the current article we present a method to recover a time-homogeneous local volatility function from perpetual American option prices. More precisely, we assume that perpetual put option prices are observed for all different values of the strike price, and we derive a time-homogeneous stock price process for which theoretical option prices coincide with the observed ones.
No-arbitrage enforces some fundamental convexity and monotonicity conditions on the put prices, and if these fail then no model can support the observed prices. If the observed put prices are smooth then we can use the theory of differential equations to determine a diffusion process for which the theoretical perpetual put prices agree with the observed prices, and our key contribution in this case is to give an expression for the diffusion coefficient of the underlying model in terms of the put prices. (It turns out that this expression uniquely determines the volatility co-efficient at price levels below the current stock price, but the volatility function is undetermined above the current stock price level, except through a single integral condition.) The key idea is to construct a dual function to the perpetual put price, and then the diffusion co-efficient can be found easily by taking derivatives of this dual.
The second contribution of this paper is to give time-homogeneous models which are consistent with a given set of perpetual put prices, even when those put prices are not twice differentiable or not strictly convex (in the continuation region where it is not optimal to exercise immediately). Again the key is the dual function, coupled with a change of scale and a timechange. We give a construction of a time-homogeneous process consistent with put prices, which we assume to satisfy the no-arbitrage conditions, but otherwise has no regularity properties.
One should perhaps note that in reality put prices are only given in the market for a discrete set of strike prices. Therefore, as a first step one needs to interpolate between the strikes. If a stock price is modeled as the solution to a stochastic differential equation with a continuous volatility function, then the perpetual put price exhibits certain regularity properties with respect to the strike price. Therefore, if one aims at recovering a continuous volatility, then one has to use an interpolation method that produces option prices exhibiting this regularity. On the other hand, if a linear spline method is used, then a continuous volatility cannot be recovered. This is one of the motivations for searching for price processes which are consistent with a general perpetual put price function (which is convex but may be neither strictly convex, nor smooth).
Whilst preparing this manuscript we came across a preprint by Alfonsi and Jourdain [2]. One of the aims of [2], as in this article, is to construct a time-homogeneous process which is consistent with observed put prices. However the method is different, and considerably less direct. Alfonsi and Jourdain [2] build on their previous work [1] to construct a parallel model such that the put price function in the original model (expressed as a function of strike) becomes a call price function expressed as a function of the initial value of the stock. They then solve the perpetual pricing problem for this parallel model, and subject to solving a differential equation for the optimal exercise boundaries in this model, give an analytic formula for the volatility coefficient. In contrast, the approach in this paper is much simpler, and unlike the method of Alfonsi and Jourdain extends to the irregular case.

The Forward Problem
Assume that the stock price process X is modeled under the pricing measure as the solution to the stochastic differential equation Here the interest rate r is a positive constant, the level-dependent volatility σ : (0, ∞) → (0, ∞) is a given continuous function, and W is a standard Brownian motion. We assume that the stock pays no dividends, and we let zero be an absorbing barrier for X. If the current stock price is x 0 , then the price of a perpetual put option with strike price K > 0 iŝ where the supremum is taken over random times τ that are stopping times with respect to the filtration generated by W . From the boundedness, monotonicity and convexity of the payoff we have: (ii)P is non-decreasing and convex on [0, ∞).
Intimately connected with the solution of the optimal stopping problem (1) is the ordinary differential equation This equation has two linearly independent positive solutions which are uniquely determined (up to multiplication with positive constants) if one requires one of them to be increasing and the other decreasing, see for example Borodin and Salminen [4, p18]. We denote the increasing solution bŷ ψ and the decreasing one byφ. In the current setting,ψ andφ are given bŷ for some arbitrary constant c. For simplicity, and without loss of generality, we choose c so thatφ(x 0 ) = 1. It is well-known that with H z = inf{t ≥ 0 : (This result is easy to check by considering e −rtφ (X t ) and e −rtψ (X t ) which, since they involve solutions to (3), are local martingales.) Given the assumed time-homogeneity of the process X, it is natural to consider stopping times in (1) that are hitting times. Definẽ where the last equality follows from (5). By differentiating (4) we find Then, for K ≤K we havẽ whereas for K >K we haveP (K) = (K − x 0 ). Clearly,P (K) ≥P (K), and of course, as we show below, there is equality.
Proof. As noted above, clearlyP ≥P since the supremum over all stopping times is at least as large as the supremum over first hitting times. For the reverse implication, suppose first that K ≤K. In that casê ϕ(z) ≥ (K − z) + /P (K) by (7). Further, e −rtφ (X t ) is a non-negative local martingale and hence a supermartingale. Thus, for any stopping time τ we have Finally, let K >K. It follows from the first part thatP (K) =K − x 0 , so Proposition 2.1 implies thatP (K) = K − x 0 =P (K), which finishes the proof.
Example. If σ is constant, i.e. if X is a geometric Brownian motion, then where β = 2r/σ 2 . Consequently, the put option price is given bŷ Straightforward differentiation shows that the supremum is attained for is given by (2).
Proof. Since σ is continuous and positive, the functionφ(x) given by (4) is twice continuously differentiable with a strictly positive second derivative. Therefore, for each fixed K there exists a unique z = z(K) ≤ x 0 for which the supremum in Lemma 2.2 is attained, i.e.
Geometrically, z = z(K) is the unique value which makes the (negative) slope of the line through (K, 0) and (z,φ(z)) as large as possible. Algebraically this means that z = z(K) and K satisfy z if K <K. Therefore, if K ≥K, then (i)P is continuously differentiable on (0, ∞), and twice continuously differentiable on (0, ∞) \ {K}.
(ii)P is strictly increasing on (0, ∞) with a strictly positive second derivative on (0,K).
We end this section by showing thatφ can be recovered directly from the put option pricesP (K), at least on the domain (0, x 0 ]. To do this, define the function ϕ : (0, x 0 ] → (0, ∞) by whereP is given by (8).
(ii) f and g are self-dual in the sense that if for z ≤ z 0 we define , (iii) Similarly, assume that g : (0, ∞) → [0, ∞) is a non-negative, nondecreasing convex function with (k−z 0 ) + ≤ g(k) ≤ k and g(k) = k−x 0 for k ≥ k * , and define for z ≤ z 0 . Then g can be recovered by Proof. See Appendix A.1.

The Inverse Problem: The Regular Case
Now consider the inverse problem. Let P (K) be observed perpetual put prices for all nonnegative values of the strike K. The idea is that sincê ϕ satisfies the Black-Scholes equation (3), Corollary 2.6 provides a way to recover the volatility σ(x) for x ∈ (0, x 0 ] from perpetual put prices. In this section we provide the details in the case when the observed put prices are sufficiently regular. We assume that the observed put option price P : [0, ∞) → [0, ∞) satisfies the following conditions, compare Propositions 2.1, 2.3 and 2.4 above.
Motivated by Corollary 2.6, define the function ϕ : (0, Proposition 3.2. The function P can be recovered from ϕ by Proof. This is a consequence of part (iii) of Lemma 2.5.
Proof. For each z ≤ x 0 there exists a unique K = K(z) ∈ (z, K * ] for which the supremum in (14) is attained. Geometrically, K is the unique value which minimises the slope of the line through (z, 0) and (K, P (K)). Clearly, Reasoning as in the proof of Proposition 2.4 one finds that K(z) is contin- and ϕ ′ (x 0 ) = −1/(K * − x 0 ). Differentiating (16) with respect to z gives where the second equality follows by differentiating (15). It follows that ϕ ′′ (z) is continuous and positive, which finishes the proof.
Next, define σ(x) 2 for x < x 0 so that ϕ is a solution to the corresponding Black-Scholes equation, i.e.
Moreover, define σ(x) 2 for x ≥ x 0 to be the constant Now, given this volatility function σ(·), we are in the situation of Section 2, and can thus defineφ to be the decreasing fundamental solution to the corresponding Black-Scholes equation scaled so thatφ(x 0 ) = 1. Moreover, letP (K) be the corresponding perpetual put option price as given by (8). Proof. First recall that any solution to (3) can be written as a linear combination ofψ andφ. Consequently, it follows from the definition of σ that for x ∈ (0, x 0 ] and some constants C 1 and C 2 . The boundary condition ϕ(x 0 ) = 1 yields Moreover, from (19) we have so the requirementφ(x 0 ) = 1 gives the explicit expression for the decreasing fundamental solution. Consequently, Since ϕ ′ (x 0 ) < 0, (21) gives that C 1 = 1 and C 2 = 0, so ϕ =φ on (0, x 0 ]. Proposition 3.2 then yieldŝ which finishes the proof.
Remark. The inverse problem does not have a unique solution. Indeed, σ can be defined arbitrarily for x > x 0 as long as the condition is satisfied, which guarantees that ϕ is C 1 at the pointx 0 . We next show how to calculate the volatility that solves the inverse problem directly from the observed option prices P (K). To do that, note that for each fixed z, the supremum in (14) is attained at some and compare (16) and (17). Since ϕ satisfies the Black-Scholes equation, we get Consequently, to solve the inverse problem one first determines z by and then for this z one determines σ(z) from (25).

The inverse problem: the irregular case
Again, suppose we are given perpetual put prices P (K) and an interest rate r > 0. Our goal is to construct a time-homogeneous process which is consistent with the given prices. Unlike in the regular case discussed in Section 3, we now impose no regularity assumptions on the function P , beyond the necessary conditions stated in Propositions 2.1 and 2.3. For a discussion of the necessity of the condition in Proposition 2.3, see Section 9.1.
(ii) P is non-decreasing and convex.
and such that (e −rt X t ) t≥0 is a local martingale.
Remark. Although we wish to work in the standard framework with rightcontinuous processes, in some circumstances we have to allow for an immediate jump. We do this by making the process right-continuous, except perhaps at t = 0. At t = 0 we allow a jump subject to the martingale condition To exclude a completely degenerate case, henceforth we assume P (x 0 ) > 0. If P (x 0 ) = 0 then necessarily, to preclude arbitrage, P (K) = (K − x 0 ) + and X t = x 0 e rt is consistent with the prices P (K). In this case τ ≡ 0 is an optimal stopping time for every K.
Given P (K) satisfying Hypothesis 4.1 and such that P (x 0 ) > 0, define ϕ by For some values of x, the supremum in (26) may be infinite since P may vanish on a non-empty interval (0, K]. We define and in the case where x > 0 we see that ϕ(x) = ∞ for x < x.
In fact x > 0 if and only if K > 0, and then it is easy to see that these two quantities are equal.
We extend the definition of ϕ to (x 0 , ∞) in any way which is consistent with the convexity, monotonicity and non-negativity properties and such that lim x→∞ ϕ(x) = 0. It is convenient to use ϕ(x) = (x/x 0 ) ϕ ′ (x 0 −)x 0 , for then ϕ ′ (x) is continuous at x 0 , and ϕ is twice continuously differentiable and positive on (x 0 , ∞).
Given ϕ, define s : , with s(x 0 ) = 0. Then s is a concave, increasing function, which is continuous on (x, ∞). (It will turn out that s is the scale function, which explains the choice of label.) The function s has a well-defined inverse g : ). Then f is decreasing and convex, with f (0) = ϕ(x 0 ) = 1.
Example. For geometric Brownian motion, provided β = 1 we have Remark. Recall that a scale function is only determined up to a linear transformation. The choice s(x 0 ) = 0 is arbitrary but extremely convenient, as it allows us to start the process Z, defined below, at zero. The choice Multiplying s by a constant has the effect of modifying the construction defined in the next section, but only by the introduction of a constant factor into the time-changes. It is easy to check that this leaves the final model X t unchanged.
Our goal is to construct a time-homogeneous process which is consistent with observed put prices, and such that e −rt X t is a (local) martingale. In the regular case we have seen how to construct a diffusion with these properties. Now we have to allow for more general processes, perhaps processes which jump over intervals, or perhaps processes which have 'sticky' points. One very powerful construction method for time-homogeneous, martingale diffusions is via a time-change of Brownian motion, and it is this approach which we exploit.
and let A be the right-continuous inverse to Γ, i.e.
Note that Γ is a non-decreasing process, so that A is well defined, and A t is an F B -stopping time for each time t. Set Z 0 = 0 and for t > 0 set Z t = B At and F t = F B At . Note that Z is right-continuous except perhaps at t = 0. The process Z t is a time-changed Brownian motion adapted to the filtration F = (F t ) t≥0 and subject to mild non-degeneracy conditions on ν, (see Lemma 5.1 below), the processes Z t and Z 2 t − A t are local martingales. Further, if ν(dy) = dy/γ(y) 2 then Γ u = u 0 γ(B r ) −2 dr and A t = t 0 γ(Z s ) 2 ds, so that Z t is a weak solution to dZ t = γ(Z t )dW t , and Z is a diffusion in natural scale. The measure ν is called the speed measure of Z, although, as pointed out by Rogers and Williams, ν is large when Z moves slowly.
The measure ν may have atoms, and it may have intervals on which it places no mass. If there is an atom atẑ then dΓ u /du > 0 whenever B u =ẑ, and then the time-changed process is 'sticky' there. Conversely, if ν places no mass in (α, β) then Γ is constant on any time-periods that B spends in this interval, and the inverse time-change A has a jump. In particular, Z t spends no time in this interval. If ν({z}) = ∞ then Γ u = ∞ for any u greater than the first hitting time H B z by B of levelz. Then A ∞ ≤ H B z so that if Z hitsz, thenz is absorbing for Z. The other possibility is that Z tends to this level without reaching it in finite time.
The cases where z ν = 0 or z ν = 0 correspond to the degenerate case X t = x 0 e rt mentioned in the previous section, and we exclude them. The following lemma provides a guide to sufficient conditions for a time-change of Brownian motion to be a local martingale, and therefore provides insight into the constructions of local martingales via time-change that we develop in the next section.
Lemma 5.1. Suppose that either z ν < ∞ or ν charges (a, ∞) for each a, and further that either z ν > −∞ or ν charges (−∞, a) for each a. Then Z t = B At is a local martingale.

Constructing the model
We now show how to choose the measure ν which gives the process we want. Define ν via ν(dy) = 1 2r and let ν(dy) = ∞ for y < s(0) in the case when s(0) > −∞. Similarly, in the case when s(∞) < ∞ we set ν(dy) = ∞ for y > s(∞). Where g ′ is absolutely continuous it follows that ν has a density with respect to Lebesgue measure, but more generally (28) can be interpreted in a distributional sense. Now, for this ν we can use the construction of the previous section to give a process Z t . If we then set X t = g(Z t ), then, subject to the hypotheses of Lemma 5.1, Z t = s(X t ) is a local martingale, so that s is a scale function for X. The process X is our candidate process for which the associated put prices are given by P .
Example. For geometric Brownian motion, In the case β = 1 this simplifies to Recall that Γ u = R L z u ν(dz) and let ξ be the first explosion time of Γ. Note that by construction Γ is continuous for t < ξ, and left continuous at t = ξ. Since ν is infinite outside the interval [s(0), s(∞)] we also have the expression ξ = inf{u : . The inverse scale function g is concave on (s(0), s(∞)), but may have a jump (from a finite to an infinite value) at s(∞). In that case we take it left-continuous at s(∞) so that we may haveḡ := lim z↑∞ g(s(z)) is finite.
For 0 < u < ξ, define M u = e −rΓu g(B u ) and N u = e −rΓu f (B u ). Sketch of proof: Suppose that ϕ is twice continuously differentiable with a positive second derivative. Then g is twice continuously differentiable. For u < ξ, applying Itô's formula to M u = e −rΓu g(B u ) gives But, by definition, dΓ u /du = g ′′ (B u )/(2rg(B u )), so M is a local martingale as required.
A similar argument can be provided for the process N . For the general case, see the Appendix.
Recall that A is the right-continuous inverse to Γ and define the timechanged processesM t = M At andÑ t = N At . Note that these processes are adapted to F and that, at least for t ≤ Γ ξ , we have Γ At = t,M t = e −rt g(Z t ) = e −rt X t andÑ t = e −rt f (Z t ) = e −rt ϕ(X t ).
Similarly, if s(∞) < ∞ then we may have ξ = H B s(∞) . Then either Γ ξ = ∞, whenceM is defined for all t, or Γ ξ < ∞. In the latter case, if ξ = H B s(∞) < ∞ and Γ ξ < ∞ thenM Γ ξ = M ξ = e −rΓ ξḡ . We setM t =M Γ ξ for all t > Γ ξ , and it follows that for t > Γ ξ , X t := e rtM t = e r(t−Γ ξ )ḡ . Thus, for t > Γ ξ , X grows deterministically. An example of this situation is given in Example 8.4 below. (In fact, the case whereḡ < ∞, which depends on the behaviour of the scale function s to the right of x 0 , can always be avoided by suitable choice of the extension to ϕ.) We want to show howM and X inherit properties from M . The key idea below is that, loosely speaking, a time-change of a martingale is again a martingale. Of course, to make this statement precise we need strong control on the time-change. (Without such control the resulting process can have arbitrary drift. Indeed, as Monroe [10] has shown, any semi-martingale can be constructed from Brownian motion via a time-change.) We have the following result the proof of which is given in the Appendix.
Corollary 6.2. The process (e −rt X t ) t≥0 is a local martingale.
We can perform a similar analysis on N andÑ and use similar ideas to ensure thatÑ is defined on R + . The proof thatÑ is a local martingale mirrors that of Corollary 6.2. Corollary 6.3. The process (e −rt ϕ(X t )) t≥0 is a local martingale.

Determining the put prices for the candidate process
Recall the definitions of s, g and ν via s ′ (x) = ϕ(x) − xϕ ′ (x), g ≡ s −1 and ν(dy) = g ′′ (y)/(2rg(y))dy. Suppose that Z is constructed from ν and a Brownian motion using the time-change Γ, and construct the candidate price process via X t = g(Z t ). By Corollary 6.2, the discounted price e −rt X t is a (local) martingale. To complete the proof of Theorem 4.2 we need to show that for the candidate process X t the function is such thatP (K) ≡ P (K) for all K ≥ 0. Unlike the regular case the process X that we have constructed may have jumps. For this reason, for x < x 0 we modify the definition of the first hitting time so that H x = inf{u > 0 : X u ≤ x}.
Theorem 7.1. The perpetual put prices for X are given by P .
Proof: Fix x ∈ (x, x 0 ). Suppose first that x is such that Γ is strictly increasing whenever the Brownian motion B takes the value s(x). Then X Hx = x. More generally the same is true whenever ν((s(x) − δ, s(x)]) > 0 for every δ > 0. By Corollary 6.3 we have that (e −rt ϕ(X t )) t≤Hx is a local martingale, and ϕ is bounded on [x, ∞) so it follows that e −r(t∧Hx) ϕ(X t∧Hx ) is a bounded martingale and ϕ( Otherwise, fix x − (x) = inf{w < x : ν((s(w), s(x)]) = 0} and x + (x) = sup{w > x : ν([s(x), s(w)) = 0}. It must be the case that ϕ is linear on (x − (x), x + (x)) and bounded on [x − (x), ∞) and It follows thatP (K) ≥ sup (Clearly, if x < x then (K − x)/ϕ(x) = 0, so the supremum cannot be attained for such an x.) To prove the reverse inequality, we first claim that the left derivative D − ϕ of the convex function ϕ satisfies To prove (30), first note that choosing Conversely, note that for each δ > 0 there exists a non-empty interval (x 0 − ǫ, x 0 ) on which We next claim that for each fixed K ≤ K * we have for all x. Clearly, this holds for x ≥ K and for x ≤ x 0 . Similarly, if x 0 < x < K, then it follows from (30) and the convexity of ϕ that It follows from (31) and Corollary 6.3 that for any stopping rule τ we have HenceP (K) ≤ P (K) for K ≤ K * , and in view of (29),P (K) = P (K). For K > K * it follows fromP (K * ) = P (K * ) = K * − x 0 , the convexity ofP and Hypothesis 4.1 thatP (K) = K − x 0 = P (K), which finishes the proof.

Examples
The following examples illustrate the construction of the previous sections. The list of examples is not intended to be exhaustive, but rather indicative of the types of behaviour that can arise. In each example we assume x 0 = 1.

Kinks in P
If the first derivative of P is not continuous then we find that ϕ is linear over an interval (α, β) say. Then s ′ is constant on this interval and g is linear over the interval (s(α), s(β)). It follows that ν does not charge this interval, so that Γ u is constant whenever B u ∈ (s(α), s(β)), and A t has a jump. Then Z t jumps over the interval (s(α), s(β)), and X t spends no time in (α, β).
Example 8.2. Suppose P (K) satisfying Hypothesis 4.1 is given by Then P is continuous, but P ′ has a jump at K = 27/32. Using (26) we find that Over the region I = [27/64, 9/16] ϕ is given by linear interpolation. The corresponding scale function is linear on I, and in the construction of Z, ν assigns no mass to s(I). The process X is a generalised diffusion with diffusion coefficient given by σ(x) = √ 2r for x ≤ 27/64, σ(x) = √ r for x ≥ 9/16 (strictly speaking there is some freedom in the choice of σ for x ≥ x 0 ≡ 1, but the constant value √ r is a natural choice) and σ(x) = ∞ for x ∈ I.

Linear parts to P
In this case the derivative of ϕ(x) is discontinuous at a point γ say. Then s ′ is also discontinuous at this point, and g ′ is discontinuous at s(γ). It follows that ν has a point mass at s(γ), and that Γ u includes a multiple of the local time at s(γ).

Example 8.3. Suppose P (K) satisfying Hypothesis 4.1 is given by
Then P is convex, but is linear on the interval [3/4, 1]. Then where we have chosen to extend the definition of ϕ to (1, ∞) in the natural way. Then s(x) = 3 − 2 ln 2 − 3/2x for x < 1/2 and s(x) = 2 ln x otherwise. It follows that g is everywhere convex but has a discontinuous first derivative at z = −2 ln 2, and that the corresponding measure ν has a positive density with respect to Lebesgue measure and an atom of size r −1 /12 at −2 ln 2.
In the terminology of stochastic processes the process Z is 'sticky' at this point -for a discussion of sticky Brownian motion see Amit [3] or, for the one-sided case Warren [14].
If P is piecewise linear (for example if P is obtained by linear interpolation from a finite number of options) then ϕ is piecewise linear, s is piecewise linear, g is piecewise linear, and ν consists of a series of atoms. As a consequence the process Z t is a continuous-time Markov process on a countable state-space (at least whilst Z t < s(x 0 ) ≡ 0), in which transitions are to nearest neighbours only. Holding times in states are exponential and the jump probabilities are such that Z t is a martingale.
In turn this means that X t is a continuous-time Markov process on a countable set of points (at least whilst X t < x 0 ).
This is consistent with a situation in which only two perpetual American put options trade, with strikes 1 and 3/2 and prices 1/3 and 2/3, in which case we may assume that we have extrapolated from the traded prices to a put pricing function P (K) which is consistent with the traded prices. Consider the process X t with state space {0} ∪ {1/2} ∪ [2, ∞) and such that • at t = 0+, X jumps to 1/2 or 2 with probabilities 2/3 and 1/3, respectively; • if ever X t 0 ≥ 2 then thereafter X t = X t 0 e r(t−t 0 ) ; • zero is an absorbing state for X; • if ever X reaches 1/2 then it stays there for an exponential length of time, rate 3r, and jumps to 2 with probability 1/3 and zero with probability 2/3.
Note that for the continuous time Markov process X t , then conditional on X t = 1/2, Also, for this process so we recover the put price function given at the start of the example.
In this case lim x↓0 ϕ(x) < ∞. It follows that s(0) > −∞ and the resulting diffusion X t can hit zero in finite time. In this case we insist that 0 is an absorbing endpoint for the diffusion.
The following example covers the case of mixed linear and smooth parts to P (K), and shows an example where reflection, local times and jumps all form part of the construction. Example 8.6. Suppose P (K) satisfies Note that P ′ has a jump at x = 1/2. We have ϕ(x) = 4 − 4x for x < 1/2 and ϕ(x) = 1/x for 1/2 ≤ x ≤ 1. We assume this formula applies on [1, ∞) also.
The process is not absorbed at 1/2 as this would conflict with the idea that the discounted price process is a local martingale, or equivalently dX t = rX t dt + dM t , for a local martingale M t . However, adding a reflection alone would again destroy the martingale property of e −rt X t . What the general construction above does is that a local time reflection and a compensating downward jump are added. This jump takes the process to zero where it is absorbed.
Alternatively, the process can be formalised as follows.
Then by Skorokhod's Lemma, B t +I t is a reflected Brownian motion (reflected at the level −(ln 2/ √ 2r)) and e √ 2r(Bt+It) ≥ 1/2. Let N λ be a Poisson Process rate λ, independent of B, and let T λ be the first event time. Then the compensated Poisson Process (N λ t − λt) t≥0 and the compensated Poisson Process stopped at the first jump (N λ t∧T λ − λ(t ∧ T λ )) t≥0 are martingales. The time change (N λ It∧T λ − λ(I t ∧ T λ )) t≥0 , is also a martingale.
Take λ = √ 2r and define X via X 0 = 1 and Note that at the first jump time of the time-changed Poisson process, X jumps from 1/2 to zero. By construction (e −rt X t ) t≥0 is a martingale.
8.5 P is zero on an interval: P (K) = 0 for K ≤ K Now we find that ϕ(x) = ∞ for x ≤ x where x = K. Depending on whether the right derivative P ′ (K+) is zero or positive, we may have ϕ(x+) is infinite or finite. In the former case we have that X t does not reach x in finite time.
In the latter case X t does hit x in finite time.
It follows that for x ≥ 1/2, η(x) 2 ≡ (xσ(x)) 2 = 2rx 2 . As before we have is consistent with the observed put prices, but we need to describe what happens when X hits 1/2. The probability that the process ever goes below 1/2 is zero, else the put with strike K = 1/2 would have positive value. The process cannot be absorbed at 1/2 as this would conflict with the idea that the discounted price process is a martingale, or equivalently dX t = rX t dt + dM t , for a local martingale M t . Instead, the time-change Γ includes a multiple of the local time at s(1/2) = −2 ln 2.
In fact g ′′ /g is constant for y > s(1/2) and equal to zero for y < s(1/2). The process Γ u is a linear combination of O + u and L −2 ln 2 u , where O + u is the amount of time spent by the Brownian motion above s(1/2) before time u. It is easy to check using Itô's formula that e −rΓu g(B u ) is a martingale in this case. The process Z t = B At is 'sticky' at s(1/2) (this time in the sense of a one-sided sticky Brownian motion, see Warren [14]) and this property is inherited by X = s(Z).
In this case ϕ ′ (x) is constant on an interval (x, x 0 ). This case is just like Section 8.2.

No options exercised immediately
In Hypothesis 4.1, in addition to (i) and (ii) which are enforcible by noarbitrage considerations, we also assumed (iii) that there exists a finite strike K * such that for all strikes K ≥ K * the put option is exercised immediately. Since K * < ∞ is equivalent to ϕ ′ (x 0 ) < 0, it is apparent from the expression in (6), that provided σ is finite for some x > x 0 , or equivalently ν gives mass to some z > 0, then this property will hold. However, it is interesting to consider what happens when this fails. Suppose P (K) > K − x 0 for all K and lim K P (K) − (K − x 0 ) = 0. Then ϕ ′ (x 0 ) = 0, but ϕ is strictly decreasing on (x, x 0 ). The measure ν places no mass on (0, ∞), the process Z t spends no time on (0, ∞) and X t never takes values above x 0 . In particular X t is reflected (downwards) at x 0 . The resulting model is consistent with observed option prices, but not with the assumption that the discounted price process is a martingale. However, by allowing non-zero dividend rates we can find a model for which the exdividend price process is a martingale, and for which the model prices are given by P (K). See Section 9.2 below. Now suppose lim K P (K) − (K − x 0 ) = δ > 0. If P (x 0 ) = x 0 then P (K) ≡ K, and we have an extreme example which falls into this setting. For P as specified above we have that ϕ(x) = 1 on (x 0 − δ, ∞). The measure ν places no mass on (−δ, ∞) and s(x) = x − x 0 on this region. Except for time 0, the process Z t spends no time in (−δ, ∞) and X t jumps instantly to x 0 − δ, and thereafter spends no time above this point.
Note that if x 0 is not specified, then this case can be reduced to the previous case by assuming x 0 = K − lim K P (K). This obviates the need for a jump at t = 0.

Time homogeneous processes with non-constant interest rates and non-zero dividend processes
In the main body of the paper we have assumed that the interest rate r is a given positive constant, that dividend rates are zero, and that σ is a function to be determined. However, the same ideas can be used to find other timehomogeneous models consistent with observed perpetual put prices, whereby the volatility function is given, and either a state-dependent dividend rate, or a state-dependent interest rate is inferred. Suppose X has dynamics dX t = (r(X t ) − q(X t ))X t dt + X t σ(X t ) dB t . Given put prices P (K) as before, define ϕ via ϕ(z) = inf K:K≥z (K−z)/P (K). Then the relationship between ϕ and the characteristics of the price process X are such that ϕ solves Lϕ = 0, where L is given by Note that we now allow for any of σ, q or r to depend on x. To date we have assumed that r is constant and q is zero and solved for σ, but alternatively we can assume σ(x) is a given function, and r is a positive constant, and solve for q, or assume that q and σ are given, and solve for r. For example, if r and σ are given constants then the (proportional) dividend rate process is given by If q is negative this should be thought of as a convenience yield. By allowing for dividend processes which are singular with respect to calender time, and which are instead related to the local time of X at level x 0 , it is possible to construct candidate price processes which spend no time above x 0 . For example, ifL is the local time at 1 of X, and if then X t reflects at 1, and ifφ(x) = (E 1 [e −rHx ]) −1 for x < 1, thenφ ′ (1−) = 0. This gives an example of a model consistent with the class of option prices described in Section 9.1.

Recovering the model from perpetual calls
The perpetual American call price function C : [0, ∞) → [0, x 0 ] must be non-increasing and convex as a function of the strike K, and must satisfy the no-arbitrage bounds ( If there are no dividends (and if e −rt X t is a martingale) then the perpetual call prices are given by the trivial function C(K) = x 0 .
So suppose instead that the (proportional) dividend rate q is positive. Letψ be the increasing positive solution to normalised so thatψ(x 0 ) = 1. Then, for z > x, E x [e −rHz ] =ψ(x)/ψ(z), and call prices in a model where dX t = (r − q)X t dt + X t σ(X t ) dB t are given bŷ Note that since q > 0 we have γ > 1. Note also thatφ(x) = x β − . The corresponding call prices are given bŷ The example discusses the forward problem, but the discussion of the inverse problem is similar to that in the put case. Given perpetual call prices C(K), for x > x 0 we can define ψ via ψ(x) = inf K:K≤x (x − K)/C(K), and then construct a triple σ(x), q(x), r(x) so that By combining information from put and call prices it is possible to determine a model which simultaneously matches both puts and calls.

A.1 Duality
Proof of Lemma 2.5. It is clear that g is non-negative and non-decreasing since f is positive and non-increasing. The lower bound on g follows from choosing z = z 0 ∧ k in (13), and the upper bound follows since f is nonincreasing. To show that g is convex, first note that g(k) is minus the reciprocal of the slope of the tangent of the function f which passes through the point (k, 0). For two given points k 1 and k 2 with k 1 < k 2 , let l 1 (z) and l 2 (z) be the corresponding tangent lines. Let k = λk 1 + (1 − λ)k 2 for some λ ∈ (0, 1), and let l(z) be the line through the point (0, k) and the intersection point of l 1 and l 2 . If the intersection point is denoted (z, l(z)), then the convexity of f guarantees that which proves that g is convex.
To prove the self-duality, let z ≤ z 0 . By the definition of g we have that For the reverse inequality, let z ≤ z 0 and let l be a tangent line to f through the point (z, f (z)) (such a tangent is not necessarily unique if f has a kink at z). Assume that the point where l intersects the z-axis is given by (k ′ , 0). Then g(k ′ ) = (k ′ − z)/f (z), so which finishes the proof of (ii). The proof of (iii) can be performed along the same lines.

A.2 Time changes of local martingales
Proposition A.1. Suppose (γ u ) u≥0 is a martingale with respect to the filtration G = (G u ) u≥0 , and A t is an increasing process such that A t is a stopping time with respect to G for each t. Defineγ t = γ At , andG t = G At . In general (γ t ) t≥0 is not a martingale. However, if γ is a bounded martingale thenγ is a bounded martingale.
Proof. Given a Brownian motion B, for b > 0, let H B b be the first hitting time of level b.
is not a martingale.
However, if γ is bounded then E[γ t |G s ] = E[γ At |G As ] = γ As =γ s by optional sampling.
Suppose now that we are in the setting of Section 5 where Z t is constructed from the Brownian motion B. In particular, Γ u is an increasing additive functional of B, and A is the right-continuous inverse to Γ.
Proof of Lemma 5.1. Intuitively a time change of Brownian motion is a local martingale, but if the additive functional Γ is constant when B is in [a, ∞) then the resulting process spends no time above a and reflects there. To maintain the local martingale property we need either that the timechanged process never gets to a, or that there are arbitrarily large values at which Γ is strictly increasing.
If [z ν , z ν ] is a bounded interval, then A ∞ ≤ H B z ν ∧ H B zν and (Z t ) 0≤t<∞ = (B At ) 0≤At<A∞ is a bounded martingale by Proposition A.1. Now suppose (z ν , z ν ) = R and suppose that for each a, ν assigns mass to every set (a, ∞) and (−∞, a).
Finally, since L y u is a bounded continuous function with compact support for each fixed u, we conclude that Γ u =Γ u .
Proof of Corollary 6.2. Recall that in our setting Γ defined via (27) grows without bound and is continuous, at least until B hits s(0) or s(∞). Thus, if ξ denotes the first explosion time of Γ, then the inverse function A is defined for every t and A t = ξ for t ≥ Γ ξ . Then, using the extension of the definition ofM beyond Γ ξ as necessary, we havẽ Recall that ϕ is extended to (x 0 , ∞) in such a way that lim x↑∞ ϕ(x) = 0. Therefore either s(∞) < ∞ and ν assigns infinite mass to all points z > s(∞) = z ν , or s(∞) = ∞ and there exists a sequence a n ↑ ∞ such that ν assigns mass to any neighbourhood of a n .
where the second line is redundant if s(0) = −∞. Then A Tn = H B an ∧ ξ is such thatM Tn t :=M t∧Tn = M At∧ξ∧H B an ≤ g(a n ) and T n is a reducing sequence forM . Now suppose s(∞) < ∞ andḡ = ∞. Choose a n ↑ s(∞) such that ν assigns mass to any neighbourhood of a n . Then, on H B s(∞) < H B s(0) we have by the argument after Lemma 6.1 that Γ H B an ↑ ∞ almost surely, and the argument proceeds as before with T n given by (34) a reducing sequence.