Halts in Trading I

Ivona Milic, C.Barnett & G.R. Salkin Centre for Quantitative Finance Imperial College London SW7 2AZ e-mail: c.barnett@ic.ac.uk

May 29, 2002

Abstract We consider the problem of hedging a contingent claim

when there is a period during which no trading is allowed.

Key Words: Hedging, Superhedging, Interrupted Trading

JEL classification: G10, G12, G13

Mathematics Subject Classification (1991): 60H30, 62F03, 62P05, 90A09

1 Introduction

The motivation for this work arose from the problem of pricing and hedging a European Call option when there is a period during the life of the option where no trading can occur. We can use only those strategies whose portfolio amounts are fixed from some deterministic time T_{1} and up to some other deterministic time T_{2}. We consider multiple halting periods later in the paper. So, these strategies can only be continued during the halting period(s) in a certain way. We will consider two cases; first, that all wealth has to put into the bond during the halting period(s), second, that the portfolio can be re-balanced as we please at the onset of a halting period. Our results show that the times at which the halting periods ensue does not figure in the final price of a claim. Rather it is the aggregate length of halts which is important. We analyse two models, the discrete binomial stock and the continuous Black-Scholes stock. From the practical point of view, the obvious example of a halt in trading occurs when markets close but price sensitive information continues to accumulate.

2 A single halting period

Initially we work with the multiplicative binomial stock of Cox, Ross and Rubinstein. There are two assets, the (risk-less) bond, B_{t} = r^{t}, where r is one plus the risk-free interest rate, and S, the stock. We assume it pays no dividends. We shall assume that we are to be the writer of a claim, X, we have in mind a European Call Option on S. First of all, we consider the case when we “know” that there will be a halt in trading between the times T_{1} and T_{2} and we have to decide how we are to hedge our obligations. We make the following observations. Since we are working in a binomial world, the value of our claim X is a bounded random variable. Accordingly, there are claims attainable despite the halt in trading which dominate the claim X. So the set of attainable claims (usually, attainable will mean ”attainable despite the halt in trading”) which dominate X is non-empty. The price of any attainable claim, Y, will be given by the usual risk-neutral pricing formula, \frac{E^{P} (Y)}{r^{T}}, where P is the risk-neutral measure. Certainly then,

X_{T}^{+} = inf\{\frac{E^{P} (Y)}{r^{T}} : Y is attainable, Y ≥ X\}

is finite. If X were to be sold for a value strictly greater that X_{T}^{+} there would be an arbitrage opportunity. Clearly, the existence of an attainable claim whose payoff is exactly X_{T}^{+} is of interest. The writer of X can buy this claim and meet their obligations. In a complete market, without halts in trading, the price of X is the time zero price of the self-financing portfolio which replicates X. In our market, with a halt in trading, we take the view that the price of X should be the the least of the prices of those attainable claims which dominate X, if this exists. What we obtain in the sequel is the existence, structure and price of a claim attainable under a halt in trading which has this minimal cost. We write X_{ij}, H_{ij}, .. (and sometimes X(i, j), H(i, j), ..) to denote the value of the claim when the stock has executed i upward moves and j downwardmoves. Although we will return to this “path independent” case relatively quickly we consider initially a more general class of claims. We can meet our obligations at time T if and only if we have sufficient wealth to purchase the replicating portfolio for X at time T_{2}. A remark about the information structure at times T_{1} and T_{2} will help at this point. The underlying sample space, Ω, is (some encoding of) the set of all paths of S. The σ-field at time t, F_{t}, is generated by the partition, {E_{i}^{t}}, of Ω where each E_{i}^{t} is a set of paths which agree with each other up to the t-th place but are thereafter arbitrary. There are 2^{t} sets in this partition. A random variable which is F_{t} measurable is completely specified by it’s values on the atoms E_{i}^{t}. Each atom, E_{i}^{T_{1}} splits into 2^{T_{2}−T_{1}} atoms of F_{T_{2}} by time T_{2}. In principle, the replicating portfolio for X at time T_{2} may require a different position in S and B for each atom of F_{T_{2}} at time T_{2}. This displays the nature of our problem nicely; we can adopt only one position on E_{i}^{T_{1}} but it must serve up to 2^{T_{2}−T_{1}} positions at time T_{2}.

2.1 No stock component over [T_{1}, T_{2})

Let us write the atoms of F_{T_{2}} as E_{ij}^{T_{2}}, 1 ≤ i ≤ 2^{T_{1}} and 1 ≤ j ≤ 2^{T_{2}−T_{1}} to indicate that this atom arose from the atom E_{i}^{T_{1}} at time T_{1}. Let X_{T_{2}}(i, j) be the value of the replicating portfolio for X in the market without a halt on the atom E_{ij}^{T_{2}}. This defines an F_{T_{2}} random variable X_{T_{2}}. Now since all wealth must reside in the bond over [T_{1}, T_{2}] then on the atom E_{i}^{T_{1}} at time T_{1} one must hold at least

\frac{max\{X_{T_{2}}(i, j) : 1 ≤ j ≤ 2^{T_{2}−T_{1}}\}}{r^{T_{2}−T_{1}}}

in the bond. This prescription defines an F_{T_{1}} measurable random variable, H_{T_{1}}, which at time T_{2}ismax{X_{T_{2}}(i, j) : 1 ≤ j ≤ 2^{T_{2}−T_{1}}} on the atom E_{ij}^{T_{2}}, we will call it H_{T_{2}}.

Lemma 1 Let Y be a claim attainable under a halt in trading. Suppose that Y ≥ X. If we write Y_{t} to denote the value of the replicating portfolio for Y at time t (ditto X), then we have Y_{T_{1}} ≥ H_{T_{1}} and Y_{T_{2}} ≥ H_{T_{2}}.

Proof 1 The conditional expectation onto F_{T_{2}} preserves positivity and so

Y_{T_{2}} = B_{T_{2}}M_{T_{2}}(\frac{Y}{B_{T}} ) ≥ B_{T_{2}}M_{T_{2}}(\frac{X}{B_{T}} ) = X_{T_{2}}.

Letting E_{ij}^{T_{2}} be an atom of F_{T_{2}} arising from the atom E_{i}^{T_{1}} of F_{T_{1}} , the value of Y_{T_{2}} on each of the atoms E_{ij}^{T_{2}}, for i fixed, is the same. This value must exceed those of X_{T_{2}} on each of these atoms. So Y_{T_{2}} must exceed the value of H_{T 2} on each of these atoms. Discounting by r^{T_{2}−T_{1}} we arrive at the value of Y_{T_{1}} and H_{T_{1}} on the atom E_{i}^{T_{1}}. This shows that Y_{T_{1}} ≥ H_{T_{1}}.

We now know how to conduct our hedging up to time T_{2}. But we are not finished, we still do not know what to do between T_{2} and T. There is another difficulty: For every state at time T_{2} arising from an atom E_{i}^{T_{1}} the difference of H_{T_{2}}(i, j) − X_{T_{2}}(i, j) is non-negative. For specific claims it could be strictly positive in some states. So, if we are to sell our claim X for the price of the hedging portfolio for H_{T_{1}} then we have effectively created an arbitrage opportunity because at time T_{2}, after our obligations are completely covered by purchase of the replicating portfolio, we have left over a random cash amount (possibly) with strictly positive expectation. What are we to do with this? The answer is that we give it back to the buyer of the claim, after all, they paid for it. We call this term the Rebate. It is defined on the atom E_{ij}^{T_{2}} by

R_{T_{2}} = H_{T_{2}}(i, j) − X_{T_{2}}(i, j).

We note that the value of the Rebate depends upon which atom of F_{T_{1}} the path passed through at time T_{1} and which atom of F_{T_{2}} it hits at time T_{2}. Both X and R are (in general) not attainable under the halt in trading, but their sum is. One can, equivalently, return to the buyer of X the random amount, R_{T} (ω) = r^{T −T_{2}}R_{T_{2}}(ω), at time T. Our portfolio management strategy is now clear; over [0, T_{1}] we manage a hedging portfolio for the random variable H_{T_{1}}. At time T_{1} we re-balance the portfolio putting all the wealth into the bond - because we must. At time T_{2} we rebalance the portfolio by purchasing the replicating portfolio for the claim X and investing the Rebate, R, into the bond. Over [T_{2}, T] we manage the replicating portfolio so that it achieves X and we leave the Rebate alone in the bond. So at time T our portfolio achieves the random variable, H_{T} = X_{T} + R_{T}. It is clear that H_{T} is a claim which is attainable under the halt in trading and H_{T} ≥ X. What is more, we have

Theorem 1 The price of H is

\frac{E^{P} (H)}{r^{T}} = X_{T}^{+} = inf\{\frac{E^{P} (Y)}{r^{T}} : Y is attainable under the halt in trading Y ≥ X\}

Proof 2 Suppose that Y is attainable under the halt in trading and that Y ≥ X. Then we know that Y_{T_{2}} ≥ H_{T_{2}}. Now the process, (\frac{M_{t}(H)}{r^{t}}) is a martingale under P and so

E^{P} (\frac{H}{r^{T}} ) = E^{P} (M_{T_{2}}(\frac{H}{r^{T}} )) = E^{P} (\frac{H_{T_{2}}}{r^{T_{2}}} ) ≤ E^{P} (\frac{Y_{T_{2}}}{r^{T_{2}}} ) = E^{P} (\frac{Y}{r^{T}} ).

H is not unique? We could adopted any admissible trading strategy at time T_{2} with the random initial endowment R_{T_{2}} and manage it (in a self-financing way) until T. Taking a “commercial view” momentarily, we see that under these circumstances the writer can offer the claim X plus the Rebate R_{T_{2}} managed in any admissible manner over [T_{2}, T] . All of these claims have the same price, but their payoffs will differ. So, we will price the case in which the Rebate has been invested in the bond over [T_{2}, T] . This takes care of the other cases too.

2.2 Deriving the Price of H

TWe will refer to H as the “claim plus rebate” at any time in [0, T], it is attainable in the market with the halt in trading. Recall that we have denoted the atoms of F_{T_{1}} by E_{i}^{T_{1}}, here 1 ≤ i ≤ 2^{T_{1}}. We take a further liberty by associating the atom E_{1}^{T_{1}} with the least value of S_{T_{1}}, that is S_{0}d^{T_{1}}, and E_{r}^{T_{1}} for \binom{T_{1}}{1} ≥ r ≥ 2 with S_{0}ud^{T_{1}−1}, and so on. In this way clusters of F_{T_{1}} atoms are associated with the actual values that S adopts at time T_{1}, that is with the nodes of the binary tree for S. The following result summarises what we know so far.

Theorem 2 The value of the claim plus rebate, H, at time t = 0 is:

\frac{1}{r^{T_{1}}} E(H_{T_{1}}) = \frac{1}{r^{T_{1}}} ∑_{i=1}^{2^{T_{1}}} H_{T_{1}}(E_{i}^{T_{1}})P (E_{i}^{T_{1}})

= \frac{1}{r^{T_{1}}} ∑_{l=0}^{2^{T_{1}}}p^{l}(1 − p)^{T_{1}−l}H_{T_{1}}(E_{i}^{T_{1}})

here p is the risk-neutral measure and we have written H_{T_{1}}(E_{i}^{T_{1}}) for the value that H_{T_{1}} takes on that atom.

In the event that H is not path dependent the last formula above simplifies. The value of H will depend only on the value of S_{T_{1}} and so it will be constant on the clusters of atoms that correspond to the nodes of the binary tree for S. So, our formula becomes

\frac{1}{r^{T_{1}}} E(H_{T_{1}}) = \frac{1}{r^{T_{1}}} ∑_{l=0}^{T_{1}} \binom{T_{1}}{l}p^{l}(1 − p)^{(T_{1}−l)}H_{T_{1}}(l, T_{1} − l) (1).

2.3 Monotone Claims

We consider the case when the claim X is a monotone function of S (only) at time T. We call these monotone claims. We have

Lemma 2 If claim X is a monotone claim and t ≤ T then X_{t} is a monotone function of S at time t. Put another way, if X is monotone at T then it is monotone at every earlier time.

Proof 1 The proof is a backwards induction on T − t.

If the claim is monotone then max{X_{T_{2}}(k, s) : i ≤ k, j ≤ s and s + k = T_{2}} is easily identified. If the claim X is monotone increasing then

max{X_{T_{2}}(k, s) : i ≤ k, j ≤ s and s + k = T_{2}} = X_{T_{2}}((i + (T_{2} − T_{1}), j)

while if it is monotone decreasing

max{X_{T_{2}}(k, s) : i ≤ k, j ≤ s and s + k = T_{2}} = X_{T_{2}}(i, (j + (T_{2} − T_{1})).

So if the claim is increasing, respectively decreasing,

H_{T_{1}}(i, j) = \frac{1}{r^{L}} X_{T_{2}}((i + L), j), respectively H_{T_{1}}(i, j) = \frac{1}{r^{L}} X_{T_{2}}(i, (j + L))

where L=T_{2} − T_{1} is the length of the halt. Let us rewrite equation (1) at the end of the last section. For a monotone increasing claim we have

\frac{1}{r^{T_{1}}} E(H_{T_{1}}) = \frac{1}{r^{T_{1}}} ∑_{l=0}^{T_{1}} \binom{T_{1}}{l}p^{l}(1 − p)^{(T_{1}−l)}(\frac{1}{r^{L}} X_{T_{2}}((l + L), T_{1} − l)  (1)

For a monotone decreasing claim our expression becomes

\frac{1}{r^{T_{1}}} E(H_{T_{1}}) = \frac{1}{r^{T_{1}}} ∑_{l=0}^{T_{1}} \binom{T_{1}}{l}p^{l}(1 − p)^{(T_{1}−l)}(\frac{1}{r^{L}} X_{T_{2}}(l, (T_{1} − l + L)).

We do the increasing case and we use X_{ij} rather than X_{T_{2}}(i, j) for the values of things. Also, we write p^{⊥} = 1 − p.

Theorem 1 An attainable claim of least price which dominates X exists. The claim H described above is one such claim and its price is

H_{0} = X_{T}^{+} = \frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X_{(L+s)((T −L)−s)}.

Proof 1 For each l, with 0 ≤ l ≤ T_{1}, we have

X_{(l+L)(T_{1}−l)} = \frac{1}{r^{T −T_{2}}} ∑_{k=0}^{T −T_{2}} \binom{T −T_{2}}{k}p^{k}(p^{⊥})^{T −T_{2}−k}X_{(l+L+k)(T_{1}−l+(T −T_{2})−k)}.

So, our expression for H_{0} becomes the following;

\frac{1}{r^{T_{1}}} ∑_{l=0}^{T_{1}} \binom{T_{1}}{l}p^{l}(p^{⊥})^{(T_{1}−l)} \frac{1}{r^{L}} \frac{1}{r^{T −T_{2}}} ∑_{k=0}^{T −T_{2}} \binom{T −T_{2}}{k}p^{k}(p^{⊥})^{T −T_{2}−k}X_{(l+L+k)(T_{1}−l+(T −T_{2})−k)}.

This expression ‘simplifies’ to

\frac{1}{r^{T}} ∑_{l=0}^{T_{1}} \binom{T_{1}}{l} ∑_{k=0}^{T −T_{2}}\binom{T − T_{2}}{k}p^{k+l}(p^{⊥})^{T −T_{2}−k+T_{1}−l}X_{(l+L+k)(T_{1}−l+(T −T_{2})−k)},

writing this in terms of L, the length of the halt in trading we get

\frac{1}{r^{T}} ∑_{l=0}^{T_{1}} ∑_{k=0}^{T −T_{2}}\binom{T_{1}}{l}\binom{T − T_{2}}{k}p^{k+l}(p^{⊥})^{T −L−(k+l)}X_{(L+(l+k))((T −L)−(k+l))}.

Using

∑_{r=0}^{s} \binom{T_{1}}{s − r}\binom{T − T_{2}}{r} = \binom{T − T_{2} + T_{1}}{s} = \binom{T − L}{s}.

The result now follows immediately.

We can get a corresponding formula for a monotone decreasing claim. We spare the reader the details which are in [2].

H_{0} = X_{T}^{+} = \frac{1}{r^{T}} ∑_{s=0}^{T −L}\binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X_{(s)(T −s)}

3 Multiple Halts

To make the exposition clear we will first discuss the case of two halts. Assume that the first halting period starts at time T_{1} and finishes at time T_{2}, and the second halting period starts at time T_{3} and finishes at time T_{4}, where 0 ≤ T_{1} ≤ T_{2} ≤ T_{3} ≤ T_{4} ≤ T. Consider first, the last halting period from T_{3} to T_{4}. A moments thought will convince the reader that this is exactly the situation analysed in the previous section. So, at time T_{3} we know the value of the portfolio we need to hold, we may express it as “the discounted value of the claim plus the rebate at time T_{4} conditional on the value of S at time T_{3}”. We can write this as H_{T_{3}} = e^{−r(T_{4}−T_{3})}H_{T_{4}}, where H_{T_{4}} is exactly X_{T_{4}} + R_{T_{4}} the value of the claim X and the rebate R at time T_{4}. The random variable, H_{T_{3}}, is the target of our self-financing trading strategy over [T_{2}, T_{3}]. The value of this self-financing trading strategy at time T_{2}, we will call it H_{T_{2}}, determines the target for our portfolio at time T_{1}. Of course, because there is a halt over [T_{1}, T_{2}], all we can do is ensure that we have sufficient wealth to be able to purchase H_{T_{2}} at time T_{2}. So H_{T_{2}} becomes our “claim” and there will be an associated rebate, R_{T_{2}}, whose value depends upon the value of S at time T_{1}. The random variable e^{−r(T_{2}−T_{1})}(H_{T_{2}} + R_{T_{2}}) is our target over [0, T_{1}]. We know that a self-financing portfolio exists which will achieve this random variable at time T_{1}. So, we know that there is a trading strategy which will eventually achieve

X_{T} + e^{r(T −T_{2})}R_{T_{2}} + e^{r(T −T_{4})}R_{T_{4}}.

Using earlier results we can show that this strategy is least cost. Moving to the general case. Let m be a positive integer. We suppose that there will be halts in trading between times T_{2i−1} and T_{2i}, we let L_{i} = T_{2i} − T_{2i−1}, 1 ≤ i ≤ m. Also let X be a monotone claim. Then we have

Theorem 2 There is an attainable claim which dominates X and has least cost. If X is an increasing claim then the claim H described above is one such and its price is

H_{0} = X_{T}^{+} = \frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X_{(L+s)((T −L)−s)}.

While if X is a decreasing claim, there is an attainable claim of least cost dominating X. The price of any one of these is

H_{0} = X_{T}^{+} = \frac{1}{r^{T}} ∑_{s=0}^{T −L}\binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X_{(s)(T −s)}.

Where L = ∑_{i=1}^{m} L_{i}.

So when we have multiple halts it is equivalent to the case of a single halt whose length is the aggregate of the lengths of the halts.

Proof 2 By induction on m the number of halts.

3.1 Pricing a European Call Option

Let C be a European Call Option on S. Clearly C is a monotone increasing function of S. We suppose that there are a finite number of halts in trading with aggregate length L over [0, T] and we will refer to the sum of the rebates paid at time T as simply the rebate. Accordingly we have the following time zero price for C,

H_{0} = \frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}C_{(L+s)((T −L)−s)}.

Recalling [1] we can write our price as

H_{0} = S_{0} \frac{u^{L}}{r^{L}} Φ(a, T − L, p′) − Kr^{−T} Φ(a; T − L; p)
= S_{0} \frac{u^{L}}{r^{L}} Φ(a, T − L, p′) − \frac{K}{r^{L}} \frac{1}{r^{T −L}} Φ(a; T − L; p)

here Φ denotes the complementary binomial distribution function and a is the smallest non-

negative integer greater than \frac{log(_{\frac{K}{S_{0}d^{T}}} )}{log( \frac{u}{d})} − L, ie. greater than \frac{log(_{\frac{K}{S_{0}d^{T −L}u^{L}}})}{log( \frac{u}{d})}. So our Call plus Rebate looks like a normal call option with expiry now at time T − L, the initial stock value equal to S_{0} \frac{u^{L}}{r^{L}} and the strike modified to \frac{K}{r^{L}}. Moreover, the the integer a is the ‘appropriate’ one. Some observations • When the length of the halt, L, is one, there is no halt but we do not recover the

CRR formula. A moments thought will convince the reader that this is appropriate. However, see the next section.

• The price depends only on the length of the halts, L, and not on the times when they commence.

• The price for the option now depends upon u, the ‘best’ return for the stock. And so it should! However, this is in contrast to the Black-Scholes world.

• For a Call the price C_{0} is less than H_{0} (because H_{T} = C_{T} + R_{T}). However, we have not considered how much larger it is.

Let P be the plain vanilla put option. An argument entirely similar to that above shows that

H_{0} = Kr^{−T} Φ[0; T − L − a; p] − S_{0} \frac{d^{L}}{r^{L}} Φ[0; T − L − a; p′]

where p = \frac{r−d}{u−d} and p′ = (\frac{u}{r})p and a is the smallest non-negative integer greater than

\frac{log(\frac{Su^{T −L}d^{L}}{K})}{log(\frac{u}{d})}.

The interpretation of this price in terms of a put option with modified expiry, strike and initial stock value is left to the reader.

4 Non-zero Stock component over [T_{1}, T_{2}]

In this section we assume one is allowed to hold whatever (fixed) portfolio amounts one desires during the halt. Once again, we want a least-cost attainable claim which dominates X. We can meet our obligations if and only if we have sufficient wealth to purchase the replicating portfolio at time T_{2}. We restrict our attention to claims which are not path dependent so that at any time, t, the value of the claim is a function of the values of S, i.e., the nodes of the binary tree for S. Let N_{i} be a node of the tree for S at time T_{1}. At time T_{2}, as S varies over the nodes arising from N_{i}, our approximating portfolio will be a linear function of S. Observe also that the slope of the line that represents the portfolio value (plot portfolio value against S) is simply the stock holding while it’s intercept with the vertical axis is the bond holding. Moreover, there is a value, call it S′, which lies between its least and greatest values of S on the time T_{2} nodes which is exactly its value on the time T_{1} node from which those time T_{2} nodes arose. The portfolio value at S′, therefore, is just the portfolio value at time T_{1} plus B_{T_{2}} − B_{T_{1}}. Therefore, if for the range of values of S we are considering one line lies above another then the portfolio that this represents has greater value than the other at time T_{1}. So the geometry of these lines will help us choose the least-cost portfolio. Rather than try to solve the problem in full generality we restrict the class of claims that we consider. First some notation; adapting our earlier notation, let us denote the time T_{1} nodes of the tree for S by (i, j), with i + j = T_{1}, and those time T_{2} nodes originating from (i, j) by (i + k, j + l), here k and l are non-negative integers and k + l = T_{2} − T_{1}. We write the value of X, S, etc, on these nodes as X(node), etc, also write L = T_{2} − T_{1}, the length of the halt. Write

\frac{ΔX}{ΔS} = \frac{X(i + L, j) − X(i, j + L)}{S(i + L, j) − S(i, L + j)}.

Definition 1 A claim, X, is said to be monotone convex at time T_{2} if it is either an increasing function of S_{T_{2}} or it is a decreasing function of S_{T_{2}} and for each L ≥ 1 and k+l = L satisfies,

X(i, j + L) + \frac{ΔX}{ΔS} (S(i + k, j + l) − S(i, L + j)) ≥ X(i + k, j + l).

Our X is only defined for a discrete set of values of S. We will extend X to a continuum of values of S by linear interpolation. In view of our earlier remarks we believe that the following is clear and omit a proof: Our extended X is a convex function of S, that is, if S ≤ S‘ then for α ∈ [0, 1],

X((1 − α)S + αS′) ≤ (1 − α)X(S) + αX(S′).

Given the discussion above we have ‘proved’ the following.

Claim 1 Let X be monotone convex. The linear portfolio which holds

\frac{ΔX}{ΔS}

items of stock and

X(i, L + j) − \frac{ΔX}{ΔS} S(i, L + j)

of bond on the nodes (i+k, j +l) at time T_{2} is the linear portfolio which dominates the claim on these nodes and has least value at time T_{1}.

Notice that this portfolio is simply the one determined by the line through (S(i, L + j), X(i, L + j)) and (S(i + L, j), X(i + L, j)) in the (S, X) plane. The way forward is now clear. For each node at time T_{1}, (i, j), we form the portfolio consisting of \frac{ΔX}{ΔS} items of stock and (X(i, L + j) − \frac{ΔX}{ΔS} S(i, L + j))r^{−L} of bond. The value of this holding defines the F_{T_{1}} random variable that we must hedge over [0, T_{1}] . At time T_{2} we re-balance the portfolio into the the hedging portfolio for X and the Rebate for the client, this, once again, being the residual amount left over after the hedge for X has been achieved. As before the management of the Rebate until time T can follow any possible strategy over [T_{2}, T]. For the purposes of pricing, one can assume it left in the bond. This results in a claim attainable under this halt in trading. From the arguments set out above it is clear that this trading strategy has price X_{T}^{+}.

4.1 The Price of the Claim plus Rebate

We can turn now to the specific form for the price of the Claim plus the Rebate. Once again we use the notation H for the hedging portfolio We know that at time T_{1} at the node (i, j) we want to hedge an amount of wealth,

\frac{ΔX}{ΔS} S(i, j) + \frac{X(i, L + j) − \frac{ΔX}{ΔS} S(i, L + j)}{r^{L}}.

Which is

H(i, j) = \frac{(u^{L} − r^{L})X(i, j + L) + (r^{L} − d^{L})X(i + L, j)}{r^{L}(u^{L} − d^{L})}.

Now, the value we need to calculate is

\frac{1}{r^{T_{1}}} ∑_{k=0}^{T_{1}} \binom{T_{1}}{k}p^{k}(p^{⊥})^{T_{1}−k}H(k, T_{1} − k).

Substituting for H(k, T_{1} − k) gives two terms (in X). For the first of these,

\frac{X(k, T_{1} − k + L)}{r^{L}} = \frac{1}{r^{T −T_{1}}} ∑_{l=0}^{T −T_{2}} \binom{T − T_{2}}{l}p^{l}(p^{⊥})^{T −T_{2}−l}X(k + l, T − (k + l)).

Leaving out the term \frac{(u^{L}−r^{L})}{(u^{L}−d^{L})}, the contribution from this term to H_{0} is

\frac{1}{r^{T_{1}}} ∑_{k=0}^{T_{1}} \binom{T_{1}}{k}p^{k}(p^{⊥})^{T_{1}−k} \frac{1}{r^{T −T_{1}}} ∑_{l=0}^{T −T_{2}} \binom{T − T_{2}}{l}p^{l}(p^{⊥})^{T −T_{2}−l}X(k + l, T − (k + l))

which we can rewrite as

\frac{1}{r^{T}} ∑_{k=0}^{T_{1}} ∑_{l=0}^{T −T_{2}} \binom{T_{1}}{k}\binom{T − T_{2}}{l}p^{l+k}(p^{⊥})^{T −L−(l+k)}X(k + l, T − (k + l)).

This is familiar territory, we sum ‘across the diagonals’. We get

\frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(s, T − s).

Putting back the factor we left out gives the following:

\frac{(u^{L} − r^{L})}{(u^{L} − d^{L})} \frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(s, T − s).

For the other term in the equation for H(k, T_{1} − k) we get

\frac{(r^{L} − d^{L})}{(u^{L} − d^{L})} \frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(L + s, T − L − s).

So the price of our Claim plus Rebate is the sum of the two preceeding terms above. What strikes one immediately is that we have seen the terms in this formula before in the previous section. If we write

Q_{L} = \frac{(r^{L} − d^{L})}{(u^{L} − d^{L})} and 1 − Q_{L} = \frac{(u^{L} − r^{L})}{(u^{L} − d^{L})}

then Q_{L} can be thought of as a probability. This probability is the “risk-neutral” probability for a stock that makes an upward move of u^{L} or a downward move of d^{L} in a single time period. In this (hypothetical) situation the risk-free rate for a single time period is r^{L}. The term that Q_{L} multiplies in the price for the claim plus rebate is the price that the claim X would have if it were monotone increasing, there is a halt in trading of length L and all wealth must reside in the bond during the halt. Similarly, 1 − Q_{L} multiplies the price that the claim X would have if it were monotone decreasing and during the halt all wealth must reside in the bond. Now our claim is monotone and so one of the “would have” statements is actually true while the other is not. Obviously, one can interpret the price as arising from a claim which is, with probability Q_{L}, “as if” X were monotone increasing and all wealth resides in the bond during the halt, and with probability 1 − Q_{L}, “as if” X were monotone decreasing and all wealth resides in the bond during the halt.

We will write X(i, j) as X(S_{0}, i, j) to highlight that the value of X will change as S_{0} does. Moreover, if we assume that our claims are akin to Calls and Puts in that,

X(S_{0}, s, T − s) = X(S_{0}d^{L}, s, T − L − s),

and

X(S_{0}, s + L, T − L − s) = X(S_{0}u^{L}, s, T − L − s),

then we can interpret our price as follows. Consider the term

\frac{1}{r^{T}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(s, T − s).

We can write this as

\frac{1}{r^{L}} \frac{1}{r^{T −L}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(S_{0}d^{L}, s, T − L − s)

turning it into a claim where the initial stock value is S_{0}d^{L} and with expiry T − L we write this as

\frac{1}{r^{L}} \frac{1}{r^{T −L}} ∑_{s=0}^{T −L} \binom{T − L}{s}p^{s}(p^{⊥})^{T −L−s}X(S_{0}d^{L}, s, T − L).

Writing X_{0}(S_{0}d^{L}, T − L) for the price of this claim and X(S_{0}d^{L}, T − L) for the payoff, the actual claim, r_{L} = r^{L}, then we can express the price of our Claim plus Rebate as

\frac{1}{r_{L}} (Q_{L}X_{0}(S_{0}u^{L}, T − L) + (1 − Q_{L})X_{0}(S_{0}d^{L}, T − L)).

This reduces to the usual CRR formula when L = 1 Indeed this can be thought of as a single period pricing formula. The single period interest rate is r_{L}. The price is the discounted, (risk-neutral) Q_{L}-expectation of a claim on an asset, S^{1}, which moves from S_{0} to S_{0}u^{L} with probability Q_{L} or from S_{0} to S_{0}d^{L} with probability 1 − Q_{L} and whose payoff is the claim X(S_{0}u^{L}, T −L) if S^{1} moves to S_{0}u^{L}, or the claim X(S_{0}d^{L}, T −L) if S^{1} moves to S_{0}d^{L}. Seen in this way, the halt is compressed into a single time period with the stock movements and interest rate suitably adjusted. Equivalently, we can think of this situation as a modification of the usual CRR structure. We have T − L + 1 time periods. In all of these time periods bar one the stock moves in the usual way, the risk-neutral measure is the usual p and the interest rate is the usual r. In the other, exceptional time period, the up movement is given by u^{L}, the downward move by d^{L} and the interest rate by r^{L} while Q_{L} is the risk neutral measure. Irrespective of when the exceptional time period occurs the formula for the price will be that given above. Roughly speaking, the exceptional terms commute themselves out from the others. So, a halt in trading amounts to looking at a modified claim over a time period shrunk by the length of the halt into which a single exceptional time period is interpolated. When X is a European Call, C, on S we can use the analysis of the previous section. The Call is a convex monotone increasing function of S. We spare the reader the details, they are in [2], but following the treatment in [1] gives the price of Call as

Q_{L}\{S_{0} \frac{u^{L}}{r^{L}} Φ(a, T − L, p′) − Kr^{−T} Φ(a; T − L; p)\}

+ (1 − Q_{L})\{S_{0} \frac{d^{L}}{r^{L}} Φ(b, T − L, p′) − Kr^{−T} Φ(b; T − L; p)\}.

Here Φ is the complementary binomial distribution function, p′ = (\frac{u}{r})p,

(p^{⊥}′) = (\frac{d}{r})(p^{⊥}) and a is least positive integer greater than \frac{log(_{\frac{K}{S_{0}d^{T −L}u^{L}}})}{log( \frac{u}{d})} and b is the least

positive integer greater than \frac{log(_{\frac{K}{S_{0}d^{T}}} )}{log( \frac{u}{d})}. The formula for a European Put option is as one would expect.

5 Multiple Halts

When we have more than one halt we accumulate rebate terms after each halt and as before we invest these into the bond until time T. Our trading strategy is best described by working back from the end of the penultimate halt. At this point our single period analysis tells us for each state of the stock what wealth we require to form the dominating portfolio of least cost at the onset of the last halt. This wealth is a monotone convex function of S because X is. This wealth becomes the target of a least cost dominating portfolio which we form at the onset of the penultimate halt. One then moves back in time. This process is iterated back to time zero. The argument which shows that this gives us an attainable claim that dominates X and is of least cost is very similar to that given in section 3 and will be omitted. We set up notation for a statement of the result. Let there be m ≥ 1 halts in [0, T] occurring over intervals [T_{2i−1}, T_{2i}], 1 ≤ m. Let the length of the last halt be L_{1}, the length of the penultimate halt, L_{2}, ...., the length of the first halt, L_{m}. After the i − th halt there will be a rebate amount whose time T value we denote by simply R_{i}. Let B_{m} denote the set of binary sequences in R^{m} and δ̂ = (δ_{i}) a typical element of B_{m}. Set L̂_{m} = (L_{1}, L_{2}, …, L_{m}) and s(m) = ∑_{i=1}^{m} L_{i}. for 1 ≤ m, set

\frac{r^{L_{i}} − d^{L_{i}}}{u^{L_{1}} − d^{L_{1}}} = Q_{i}

and write Q_{i}^{⊥} = 1 − Q_{i}. With these notations we have

Theorem 3 The time zero price of X + ∑_{i=1}^{m} R_{i} is given by

\frac{1}{rT} ∑_{δ̂∈Bm} ∏_{i=1}^{m} Q_{i}^{δi} (Q_{i}^{⊥})^{1−δi} ∑_{s=0}^{T −s(m)} \binom{T −s(m)}{s}p^{s}(p^{⊥})^{T −s(m)−s}X(s+L̂_{m}·δ̂,T −s−L̂_{m}·δ̂).

Proof 3 By induction on the number, m, of halts.

So, once again, the effect of the halts is independent of the start times of the halting periods but does depend upon their aggregate length. The payoff of this claim is X + ∑_{i=1}^{m} R_{i}, where R_{i} is the time T value of the i − th rebate. Consequently, we can see that the order of the halts does affect the price and hedging of the claim.

6 Continuous Time

6.1 A Single Halting Period

Some of what we have done does not carry over to continuous time. When S is a log-normal asset of constant volatility and rate of return it is sure that over any time interval of positive length, [t, t + δ], the difference, (S_{t+δ} − S_{t})(ω) can be arbitrarily large. Consequently, if all wealth must reside in the bond for the halting period, [T_{1}, T_{2}], and we wish to form any super-hedge for time T_{2} for a claim which is monotone increasing in S, then we may require an unbounded amount of wealth. However, the situation is different when we allow a mixture of stock and bond into our hedging portfolio, the stock component can track the values of S, but we still have to deal with the possibility that our claim can become unbounded as the values of S increase. Before we go any further we give a statement of our problem.

Our model is of a ‘Black-Scholes’ stock, S, which, under the risk-neutral measure, P, has dynamics given by

S_{t} = S_{0} + ∫_{0}^{t} rS_{s}ds + ∫_{0}^{t} σS_{s}dW_{s}

where r > 0 is the risk-free rate and σ is the constant volatility. The process, (W_{t}), is a standard Brownian Motion under P. The time interval in which we work is [0, T], T > 0, and there are times, T_{1} and T_{2} and there is a complete halt in trading over the interval (T_{1}, T_{2}), so that one can take a position at time T_{1} but it cannot be changed until time T_{2}. Under this restriction the market is incomplete in as much as not all claims are attainable. Despite this, we shall take the following point of view; whatever the value of S_{T_{1}}, it is ‘very unlikely’ that S_{T_{2}} will be far different from S_{T_{1}}. So, we shall accept a certain probability of shortfall in our hedging. More precisely; for each positive real number, A, there is K ∈ [0, 1], such that

P {ω : (W_{T_{2}} − W_{T_{1}})(ω) ∈ [−A, +A]} = 1 − K.

Writing L = T_{2} − T_{1} and {ω : (W_{T_{2}} − W_{T_{1}})(ω) ∈ [−A, +A]} = E, this is equivalent to

P \{ω : S_{T_{2}} ∈ [S_{T_{1}} exp^{−σA+(r− \frac{σ^{2}}{2} )L}, S_{T_{1}} exp^{σA+(r− \frac{σ^{2}}{2} )L}]\} = 1 − K

Because W_{T_{2}} − W_{T_{1}} is N(0, \sqrt{T_{2} − T_{1}}) every K ∈ [0, 1] determines a unique A via the prescription in the first of the equations above, so we will speak in terms of the probability, ”K”. Our hedging strategy shall be, what ever the value of S_{T_{1}}, to find the least cost (linear) super-hedge for our claim on the set E. This super-hedge buys the replicating portfolio on E and rebates the excess to the client, while on Ω \ E we arrange ‘best possible’ under-hedge. We shall takes as ‘best possible’ the quantile hedge described in the section 2 of [5], which maximises the probability of dominating X starting from an initial endowment strictly less than that required for replicating X over the interval [T_{2}, T].

We will restrict ourselves to claims which are monotone convex functions of S_{T_{2}} because we can apply some of the insights from the discrete case considered above. We note immediately that for each value of S_{T_{1}} the position adopted at time T_{1} is a linear function of S_{T_{2}}. Consequently, for monotone and convex claims, there is a short-fall on the set of probability K. Despite this, the price we shall pass on to the client will be that of forming the super-hedge on the set E.

For a claim, X, which is a monotone convex function of S we can apply the arguments developed in the discrete case; at time T_{2} we must hold \frac{ΔX}{ΔS} items of stock, and X − \frac{ΔX}{ΔS} S items of bond. Of course here the meaning of these terms has changed and it is clear that some contraction of notation will improve the exposition; write

exp^{σA+(r− \frac{σ^{2}}{2} )L} = exp^{+} = e^{+} and exp^{−σA+(r− \frac{σ^{2}}{2} )L} = exp^{−} = e^{−}.

The expressions for the stock and bond holdings become

\frac{ΔX}{ΔS} = \frac{X(S_{T_{1}} exp^{+}) − X(S_{T_{1}} exp^{−})}{S_{T_{1}} exp^{+} −S_{T_{1}} exp^{−}}

while

X − \frac{ΔX}{ΔS} S = X(S_{T_{1}} exp^{−}) − \frac{ΔX}{ΔS} S_{T_{1}} exp^{−} .

The wealth we are required to hold at time T_{1} as a function of S_{T_{1}} is, symbolically,

H_{T_{1}}(S_{T_{1}}) = \frac{ΔX}{ΔS} S_{T_{1}} + exp^{−rL}(X − \frac{ΔX}{ΔS} S_{T_{1}})

one can verify that this is

H_{T_{1}}(S_{T_{1}}) = \frac{(exp^{+} − exp^{rL})X_{T_{2}}(S_{T_{1}} exp^{−}) + (exp^{rL} − exp^{−})X_{T_{2}}(S_{T_{1}} exp^{+})}{exp^{rL}(exp^{+} − exp^{−})}.

the price of H_{T_{1}} will be the price of our ‘claim plus the rebate’. Writing E_{P} for the expectation with respect to the risk-neutral measure P, we know that

H_{0}(S_{0}) = e^{−rT_{1}}E_{P} (H_{T_{1}}(S_{T_{1}}).

Now the random variable X = σW_{T_{1}} + (r − \frac{σ^{2}}{2} )T_{1} is normal with mean (r − \frac{σ^{2}}{2} )T_{1} and variance σ^{2}T_{1}. Let f_{X} be the density for X. Then we can rewrite our price as

H_{0}(S_{0}) = e^{−rT_{1}} ∫_{R} H_{T_{1}}(S_{0}e^{x})f_{X}(x)dx.

Here

H_{T_{1}}(S_{0}e^{x}) = \frac{(exp^{+} − exp^{rL})X_{T_{2}}(S_{0}e^{x} exp^{−}) + (exp^{rL} − exp^{−})X_{T_{2}}(S_{0}e^{x} exp^{+})}{exp^{rL}(exp^{+} − exp^{−})}.

This integrand obviously breaks into two parts and we will deal with the first term only, the calculation is similar for the second. Taking in to account the discount factors the first term of our integral is

e^{−rT_{2}} \frac{(e^{+} − e^{rL})}{(e^{+} − e^{−})} ∫_{R} X_{T_{2}}(S_{0}e^{x}e^{−})f_{X}(x)dx.

Now we know also that for a given value of S_{T_{2}},

X_{T_{2}}(S_{T_{2}}) = e^{(T −T_{2})}E_{P} (X_{T} (S_{T_{2}}e^{Y}))

where Y = σ(W_{T} − W_{T_{2}}) + (r − \frac{σ^{2}}{2} )(T − T_{2}) is a normal random variable with mean (r − \frac{σ^{2}}{2} )(T − T_{2}) and variance σ^{2}(T − T_{2}) which is independent of the random variable X. So the first term of our integral is

e^{−rT} \frac{(e^{+} − e^{rL})}{(e^{+} − e^{−})} ∫_{R} ∫_{R} X_{T} (S_{0}e^{x+y}e^{−})f_{Y} (y)f_{X}(x)dydx.

Because of independence the joint density of X and Y is just the product of the densities of X and Y so this becomes

e^{−rT} \frac{(e+−erL)}{(e+−e−)} ∫_{R} ∫_{R} X_{T} (S_{0}e^{x+y}e^{−})f_{X,Y} (x,y)dydx=e^{−rT} \frac{(e+−erL)}{(e+−e−)} E_{P} (X_{T} (S_{0}e^{−}e^{X+Y} )).

The random variable S = X +Y is normal with mean (r− \frac{σ^{2}}{2} )(T −L) and variance σ^{2}(T −L). So we can express the first term of our integral as

e^{−rT} \frac{(e^{+} − e^{rL})}{(e^{+} − e^{−})} ∫_{R} X_{T} (S_{0}e^{s}e^{−})f_{S}(s)ds = e^{−rT} \frac{(e^{+} − e^{rL})}{(e^{+} − e^{−})}E_{P} (X_{T} (S_{0}e^{−}e^{X+Y})).

The other term is

e^{−rT} \frac{(e^{rL} − e^{−})}{(e^{+} − e^{−})} ∫_{R} X_{T} (S_{0}e^{s}e^{+})f_{S}(s)ds = e^{−rT} \frac{(e^{rL} − e^{+})}{(e^{+} − e^{−})}E_{P} (X_{T} (S_{0}e^{+}e^{X+Y})).

We look at special the case of a European Call Option. Leaving aside some of the exponential terms for now; a routine calculation shows that

exp^{−rT} ∫_{R}(S_{0} exp^{s} e^{−}−K)^{+}f_{S}(s)ds = exp^{−rL} S_{0}e^{−} Φ(^{\frac{log( \frac{S_{0}e^{−}}{K} )+(r+ \frac{1}{2} σ^{2})(T −L)}{σ\sqrt{(T −L)}}} )
− Ke^{−rT} Φ(^{\frac{log( \frac{S_{0}e^{−}}{K} )+(r− \frac{1}{2} σ^{2})(T −L)}{σ\sqrt{(T −L)}}} )

Of course we shall write,

d_{±}^{∓} = \frac{log(\frac{S_{0}e^{∓}}{K}) + (r ± \frac{1}{2}σ^{2})(T − L)}{σ\sqrt{(T − L)}}

so that this contracts to

exp^{−rT} ∫_{R}(S_{0} exp^{s} e^{−} − K)^{+}f_{S}(s)ds = exp^{−rL} S_{0}e^{−}Φ(d_{+}^{−}) − K exp^{−rT} Φ(d_{−}^{−}).

The exponential terms that we left out earlier can now be inserted. We note that under the assumption that σA − \frac{σ^{2}}{2} L > 0 then e^{+} > e^{rL} > e^{−} and the quotients involving these terms can be thought of as a probability. Write

Q_{L,A} = \frac{(e^{+} − e^{rL})}{(e^{+} − e^{−})} then 1 − Q_{L,A} = \frac{(e^{rL} − e^{−})}{(e^{+} − e^{−})}

and the price for our Call is looking very similar to that we discovered for the discrete case. It is

H_{0}(S_{0}) = Q_{L,A}(exp^{−rL} S_{0}e^{−}Φ(d_{+}^{−}) − K exp^{−rT} Φ(d_{−}^{−}))
+ (1 − Q_{L,A})(exp^{−rL} S_{0}e^{+}Φ(d_{+}^{+}) − K exp^{−rT} Φ(d_{−}^{+})).

We might interpret this as a price as reflecting the structure of the claim: consider a claim which with probability Q_{L,A} is a call on a stock, S^{−} struck at K with S_{0}^{−} = S_{0}e^{−}, volatility σ, risk-free rate r and time to maturity T − L, and with probability 1 − Q_{L,A} a call on a stock, S^{+} with S_{0}^{+} = S_{0}e^{+} and all other features in common with S^{−}. The value of this, discounted at the risk-free rate by the length of the halt, is the value of our claim.

6.2 Multiple Halting Periods

For monotone and convex claims we can arrive at a formula for multiple halts. Take the case of two halting periods. The first one starts at time T_{1} and finishes at time T_{2}, where L_{2} = T_{2} − T_{1}, and the rebate associated with this halting period is R_{2}. The second one starts at time T_{3} and finishes at time T_{4}, where L_{1} = T_{4} − T_{3}, and the rebate associated with this halting period is R_{1}. We are counting the halts ”backwards”, i.e. the last one is of length L_{1}, etc., because the general case is achieved by induction on the number of halts. Our aim is to explain the proof for two halts and to leave to the reader the details of the general case. So, turning our attention to the last halting period; we take the view here that for the stock values at time T_{4} to lie outside of a range determined by the positive real number A_{1}, in exactly the fashion described above in the case of one halt, is ”very unlikely”. Accordingly there is an F_{T_{4}} event of probability 1 − K_{1}, where 0 < K_{1} < 1, on which the hedger will be perfectly hedged and on the complement of this event will be under-hedged. Some notation will ease the exposition; for i = 1, 2 let

exp^{σA_{i}+(r− \frac{σ^{2}}{2} )L_{i}} = e(i)^{+} and exp^{−σA_{i}+(r− \frac{σ^{2}}{2} )L_{i}} = e(i)^{−},

while

Q_{i} = \frac{e(i)^{+} − exp^{−rL_{i}}}{e(i)^{+} − e(i)^{−}} and Q_{i}^{⊥} = 1 − Q_{i}.

From our analysis of the single halting period case, we know that the portfolio we must hold at time T_{3}, the start of the last halt, is for each value of S_{T_{3}},

H_{T_{3}}(S_{T_{3}}) = e^{−rL_{1}}(Q_{1}X_{T_{4}}(S_{T_{3}}e(1)^{−}) + Q_{1}^{⊥}X_{T_{4}}(S_{T_{3}}e(1)^{+})).

There is a time T_{2} portfolio, P (S_{T_{2}}), say, which can be managed to perfectly replicate H_{T_{3}}(S_{T_{3}}). However our hedger thinks it unlikely that the stock values will lie outside of the range determined by the positive real number, A_{2}, which defines an F_{T_{2}} event of probability K_{2}. Accordingly the portfolio adopted at time T_{1} is

H_{T_{1}}(S_{T_{1}}) = e^{−rL_{2}}(Q_{2}P_{T_{2}}(S_{T_{1}}e(2)^{−}) + Q_{2}^{⊥}P_{T_{2}}(S_{T_{1}}e(2)^{+})).

The price of this portfolio is

H_{0}(S_{0}) = e^{−rT_{1}}E_{P} (H_{T_{1}}(S_{T_{1}})).

We will return to this shortly. Looking back at our expression for H_{T_{3}}(S_{T_{3}}). We observe that that

X_{T_{4}}(S_{T_{3}}e(1)^{−}) = e^{−r(T −T_{4})}E_{P} (X_{T} (S_{T_{3}}e(1)^{−}e^{Z})

where Z = σ(W_{T} − W_{T_{4}}) + (r − \frac{σ^{2}}{2} )(T − T_{4}) is an N((r − \frac{σ^{2}}{2} )(T − T_{4}), σ^{2}(T − T_{4})) random variable. We can write this as

X_{T_{4}}(S_{T_{3}}e(1)^{−}) = e^{−r(T −T_{4})} ∫_{R} X_{T} (S_{T_{3}}e(1)^{−}e^{z})P_{Z}(z)dz

where P_{Z}(z) is the appropriate normal density function. There is a corresponding formula for the terms involving e(1)^{+}. Because the number of terms we have to consider is set to double and double again we will trace the ”evolution” of just one term in what follows. The term X_{T_{4}}(S_{T_{3}}e(1)^{−}) contributes the following to H_{T_{3}}(S_{T_{3}}).

e^{−rL_{1}}Q_{1}e^{−r(T −T_{4})} ∫_{R} X_{T} (S_{T_{3}}e(1)^{−}e^{z})P_{Z}(z)dz

Now the value of the portfolio P_{T_{2}}(S_{T_{2}}) is

e^{−r(T_{3}−T_{2})}E_{P} (H_{T_{3}}(S_{T_{2}}e^{Y}))

where Y = σ(W_{T_{3}} −W_{T_{2}})+(r − \frac{σ^{2}}{2} )(T_{3} −T_{2}) is an N((r − \frac{σ^{2}}{2} )(T_{3} −T_{2}), σ^{2}(T_{3} −T_{2})) random variable. The expectation can be written as

E_{P} (H_{T_{3}}(S_{T_{2}}e^{Y}) = ∫_{R} H_{T_{3}}(S_{T_{2}}e^{y})P_{Y} dy

and we note that Y and Z are independent. Our single term that we are following will make a contribution of

e^{−r(T_{3}−T_{2})e^{−rL_{1}} Q_{1}e^{−r(T −T_{4})} ∫_{R}(∫_{R} X_{T} (S_{T_{2}} e^{y}e(1)^{−}e^{z})P_{Z}(z)dz)P_{Y} (y)dy}

tidying up this is

e^{−r(T −T_{2})}Q_{1} ∫_{R} ∫_{R} X_{T} (S_{T_{2}}e^{y}e(1)^{−}e^{z})P_{Z}(z)P_{Y} (y)dzdy.

The drift of this calculation is now clear. The portfolio, P_{T_{2}}(S_{T_{2}}), is the ‘target’ for a portfolio to be adopted at time T_{1}, the start of the first halt. However the hedgers view is such that there is a positive real number A_{2} associated with an F_{T_{2}} event of probability 1 − K_{2} on which the hedge will be perfect and outside of which there will be a shortfall. We know that the hedge is

H_{T_{1}}(S_{T_{1}}) = e^{−r(T_{2}−T_{1})}(Q_{2}P_{T_{2}}(S_{T_{1}}e(2)^{−}) + (Q_{2}^{⊥}P_{T_{2}}(S_{T_{1}}e(2)^{+})

and our term above will make a contribution to this, again we single out one term of this contribution;

e^{−r(T −T_{1})}Q_{1}Q_{2} ∫_{R} ∫_{R} X_{T} (S_{T_{1}}e(2)^{−}e^{y}e(1)^{−}e^{z})P_{Z}(z)P_{Y} (y)dzdy.

Using the risk-neutral valuation formula for H_{T_{1}} over [0, T_{1}] and noting that S_{T_{1}} = S_{0}e^{X} where X = σ(W_{T_{1}} − W_{0}) + (r − \frac{σ^{2}}{2} )T_{1} is an N((r − \frac{σ^{2}}{2} )T_{1}, σ^{2}T_{1}) random variable which is independent of both Y and Z. We can rewrite our contributing term above as

e^{−rT} Q_{1}Q_{2} ∫_{R} ∫_{R} ∫_{R} X_{T} (S_{0}e^{x}e(2)^{−}e^{y}e(1)^{−}e^{z})P_{Z}(z)P_{Y} (y)P_{X}(x)dzdydx.

Writing S = X + Y + Z this may be described as

e^{−rT} Q_{1}Q_{2} ∫_{R} X_{T} (S_{0}e^{s}e(2)^{−}e(1)^{−})P_{S}(s)ds

where S is an N((r − \frac{1}{2}σ^{2})(T − L), σ(T − L)) random variable and L is the aggregate length of the halts. Let B_{2} denote the binary vectors in R^{2} and δ = (δ_{1}, δ_{2}) a typical element of B_{2}. If we write

N(S_{0}, δ) = ∫_{R} X_{T} (S_{0}e^{s}(∏_{i=1}^{2}(e(i)^{−})^{δ_{i}}(e(i)^{+})^{1−δ_{i}}))P_{S}(s)ds

then we can express our price as

H_{0}(S_{0}) = e^{−rT} ∑_{δ∈B_{2}} ∏_{i=1}^{2} Q_{i}^{δ_{i}}(Q^{⊥})^{1−δ_{i}}N(S_{0}, δ).

This is amenable to generalisation to m halting periods, L_{i}, in which there are m choices of probabilities, K_{i}, and positive real numbers, A_{i}, chosen so that σA_{i} − (r − \frac{σ^{2}}{2} )L_{i} > 0. A proof by induction on m the number of halts (see also [2]) shows that

H_{0}(S_{0}) = e^{−rT} ∑_{δ∈Bm} ∏_{i=1}^{m} Q_{i}^{δ_{i}} (Q_{i}^{⊥})^{1−δ_{i}} N(S_{0},δ)
= e^{−rT} ∑_{δ∈Bm} ∏_{i=1}^{m} Q_{i}^{δ_{i}} (Q_{i}^{⊥})^{1−δ_{i}} ∫ X(S_{0}e^{s}(∏_{i=1}^{m}(e(i)^{−})^{δ_{i}} (e(i)^{+})^{1−δ_{i}}))P_{S}(s)ds.

Here S is an N((r − \frac{1}{2}σ^{2})(T − L), σ(T − L)) random variable and L = ∑_{i=1}^{m} L_{i}. As one can imagine, when X is a European Call option we can identify each of the terms N(S_{0}, δ) with a difference of factors involving the standard normal distribution function. These can in turn be interpreted as the prices of call options whose time to expiry and initial price have been modified, much as in the single halt case. Returning to the case of a single halting period and a claim X, we consider what, if anything can be said about the limiting case, A ⟶ ∞. Although the values of both S and X can become infinite as S does it may still be true that \frac{ΔX}{ΔS} remains finite and will be independent of the value of S at the start of the halt . For example, with a European Call Option this ratio tends to 1 as A ⟶ ∞. This seems to be telling us to hold one item of stock for the period of the halt. This will come as no great revelation to practitioners. It is supported by the fact that the limit of the value of the pricing formula for a European Call is S_{0}, the price of one item of stock at time zero.

We have been unable to identify work directly related to ours in the literature, although the paper, [5], [6], may be connected. In a paper which is in preparation, [3], we investigate this and we consider random halting times.

References

[1] Cox, J. C., Ross, S. A. ,Rubinstein, M. Option Pricing: a simplified approach. J. Finan. Econom. 7, (1979), pp 229-263.

[2] Milic I.: PhD Thesis, Imperial College, 2002.

[3] Barnett, C. Milic, I, Salkin, G: Halts in Trading II, in preparation.

[4] Barnett, C. Liu, W. P., Salkin, G. R. A minimum variance result under Market Conditions.

[5] Follmer and Leukart, Quantile Hedging.

[6] Schweizer, M. Optimal Hedging in Discrete Time. Math. Op. Res. 1995 20 1-32.