Option Pricing with Transaction Costs

C. Barnett & G. Salkin & D.J.Hill Centre for Quantitative Finance Imperial College London SW7 2AZ c.barnett@ic.ac.uk

March 26, 2001

Abstract We price an option when there are transaction costs. A discrete time formula is adapted for a continuous limit.

Contents

1 Introduction 3

2 The Market and the Market Participants 4 2.1 The Participants . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 The Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

3 The Mathematical Structure 6

4 No arbitrage and Non-triviality 7 4.1 Short with cash delivery . . . . . . . . . . . . . . . . . . . . . 7 4.2 Short with asset delivery . . . . . . . . . . . . . . . . . . . . . 9 5 A Single Period Pricing Formula 10

5.1 Redemption Costs . . . . . . . . . . . . . . . . . . . . . . . . . 11 5.2 Adjustments for Multiple Time Periods . . . . . . . . . . . . . 12 5.3 The Pricing Formula . . . . . . . . . . . . . . . . . . . . . . . 14 5.4 Risk Neutral Probabilities ? . . . . . . . . . . . . . . . . . . . 15

6 Multiple Time Periods 16 6.1 The n-period formula . . . . . . . . . . . . . . . . . . . . . . . 16 6.2 Martingale Properties . . . . . . . . . . . . . . . . . . . . . . . 20

7 A Limiting Case 22 7.1 A Central Limit Result . . . . . . . . . . . . . . . . . . . . . . 26

8 Concluding Remarks 36

1 Introduction

This paper is concerned with pricing a European call option and falls into two parts. The first part deals with discrete time and processes, the second with continuous time and processes. In the first part we find an option pricing formula holds when there are proportional transaction costs. Although this problem has been examined before, for example in [4], and is discussed in [7], our case is more general and novel in that we specify the market, it’s participants and how portfolios must be managed. The last of these may appear curious. Our portfolios are liquefied and re-balanced at the end of every time period. The effect of this is to impose the maximum amount of transaction costs. This case is polar to that examined in [4] and if one were concerned with applications to the real world could be passed over without a second thought^{1}. Our view is broader than this. From the theoretical point of view the maximum cost case is as of much interest as the minimum cost case. Moreover, we always had in mind an extension to continuous trading. The maximum cost case lends itself to this in a natural way. The reader who is putting together continuous trading and maximum transaction costs and coming up with an infinity or two is, of course correct! We make our transaction costs a function of the length of the (discrete) time interval, so they are asymptotically zero. We do not propose a mechanism for the generation of transaction costs, they are simply imposed. However, we believe that the assumption that transaction costs are function of the time that assets must be held deserves further investigation^{2}.

In the discrete setting the formula for the option price is a natural and elegant generalisation of that obtained by (CRR), Cox, Ross and Rubinstein, [5]. This involves the emergence of a new interest rate process and a new probability measure which deserves to be called “risk-neutral”. The limit is achieved via the Central Limit Theorem, as one might expect. In taking the limit we prove a central limit result for a class of distributions which we have come to call ‘generalised binomial’. These are the sum of independent Bernoulli variables for which the probability of success changes at each trial. Although this case is covered by Lyapounov’s result we cannot find a reference in the literature and so have included a proof. The more difficult calculation involves the limiting values for the means and variances and some detail is unavoidable. As we pass to the limit this modified interest rate persists and may be thought of as the “average relative spread^{3}” of the prices of the stock over time.

1An error of judgment?

2It wont be all of the story....

Our underlying will be a Cox, Ross and Rubinstein binomial stock where the buying price and selling price are (possibly) different in each time period. We also have a risk-less bond following the usual price process. A portfolio of stocks and bonds is managed to hedge the obligations arising from an option written on the underlying. Of course the mathematics can deal with assets other than options but we felt that keeping close to the CRR format improves the exposition. What makes our case unique is the delivery of the option and the management of the portfolio. We consider two cases; the option is cash settled at some intermediate value between the final buy and sell prices for the stock, or, the asset itself is delivered. As for the management of the portfolio, at the end of each time period the portfolio is completely liquefied before rebalancing. We wanted to model some of the “market microstructure” and for this we found the work of ‘O’ Hara [2] very useful. Our attempts at getting a microstructure forced us to think about the market participants and how they could behave. The constraints we were led to impose produced a system in which behaviour is simple but rather wooden, but perhaps this is inevitable.

2 The Market and the Market Participants

2.1 The Participants

We found that the participants in our ‘market’ and their relationships had to be specified for our arguments to work. We wanted a market with no arbitrage so that a replicating portfolio would determine the price of the option. Also, we wanted a difference between buying and selling prices. If all participants are equivalent, then when John sells to Jane he receives less than she pays. Where does the difference go? This lead us to represent the market in stock by a stock ‘agent’. One can only buy and sell stock through this agent. If you consider this agent as a proactive participant in the market then when John sells and Jane buys the agent makes a risk-free profit. Once again, this wont do. After passing through several stages of emasculation the poor stock agent was retired and replaced with what we came to regard as a slot machine. Feeding in the correct number of bonds leads to the issue of a stock certificate which can be redeemed for bonds by the same procedure^{4}. In any event, we had to settle for a market in which every stock transaction led to the consumption of some wealth. So our market consists of an impersonal stock agent and investors. We also posit the existence of an agent called ‘option writer’. The reason for identifying this last participant was to see if the existence of a bid-offer spread on the underlying would lead to a bid-offer spread for the option. This turned out to be so, but we had to make some assumptions about the option writer and of her relationship with investors. The effect of these assumptions is to make the option writer rather mechanical too. All option transactions go through the option writer. The option writer trades sequentially, meaning that when an option is written the risk averse writer immediately takes a hedging position before passing on to the next transaction. Options may be redeemed, that is, sold back to the option writer. When an option is redeemed the hedging position is resolved so there is no residual risk. Because trading is sequential, this prevents the writer from declaring a two sided price and immediately reselling a redeemed option for an arbitrage profit. The option writer avoids bankruptcy.

3If S has buy price β(S) and sell price γ(S) the relative spread is \frac{β(S)−γ(S)}{S}

2.2 The Market

We assume that our market is non-trivial and arbitrage free. This means that trading in the stock must be risky when compared to trading in the bond, which is a sure investment. We have two views of short selling; borrowing the asset and returning it at a later date, or, borrowing the asset and returning to the owner its cash value at a later date. Note that the first of these is equivalent to returning the buying price for any state of the world, while the second is returning the selling price of the asset for any state of the world. We toyed with the idea of short selling being the return of a cash value intermediate between the buy and sell price, but simplicity urged itself on us. To distinguish the two cases of short selling we shall use the terms “short with cash delivery” and “short with asset delivery”. These two kinds of short selling will not co-exist, short selling of one kind only is allowed throughout the life of the Option. If we have short selling with asset delivery the option will be settled by asset delivery. We note that this is equivalent to cash delivery at the final buying price. If we have short selling with cash delivery, the option is settled at a price intermediate between the final buy and sell prices. Aside from the bid-offer spread there are no taxes or margin requirements and our asset does not pay dividends. The interest rate is constant, the same for borrowing and lending and there are no limits on borrowing. We let r be one plus the single period rate.

4Perhaps the slot machine is steam powered and the spread buys coal for the boiler?

3 The Mathematical Structure

The mathematics we adopt anticipates some of our requirements later on, so may seem a little idiosyncratic at first. For our sample space; let 2 = {0, 1} then Ω = 2^{N} where N is a positive integer, the number of time periods within which the stock evolves. We denote elements of Ω by ω = (ω_{i}). So upward stock movements have been encoded as a 1 while downward movements are encoded as a 0. For each fixed n = 1, …, N, and each fixed binary sequence of length n there is a subset of Ω consisting of all w ∈ Ω whose first n terms agree with this binary sequence. For each fixed n, the collection of all such subsets form a partition of Ω which generates a σ-field, F_{n}. When n = 0 (the subset is Ω and) the σ-field is {Ω, ∅}. We assume that there is a probability measure, P, defined on F_{N}, such that

P (ω) = ∏_{i=1}^{N} p^{ω_{i}}(1 − p)^{1−ω_{i}} = p^{∑_{i} ω_{i}} (1 − p)^{N−∑_{i} ω_{i}}.

This is sufficient to specify the probabilities of all events. The process

S_{n}(ω) = S_{0} u^{∑_{1}^{n} ω_{i}} d^{n−∑_{1}^{n} ω_{i}}

for n = 0, 1, 2 … N, and where S_{0} > 0, is modified to give the buying price of our stock, β_{n}(S_{n}(ω)), and the selling price of our stock, σ_{n}(S_{n}(ω)). We will suppose that

β_{n}(S_{n}(ω) ≥ S_{n}(ω) ≥ σ_{n}(S_{n}(ω)) > 0,

the inequalities being strict where necessary. We note that some of the mathematics works regardless of this assumption. We assume further that β_{n} and σ_{n} are increasing functions of S_{n}(ω). The risk-less bond shall have price process

B_{n}(ω) = r^{n}.

We shall denote our option by C and its buy and sell prices by β_{n}(C_{n}(ω)) and σ_{n}(C_{n}(ω)) respectively. Our notation suggests that the time n buy and sell prices for the stock and the option are given by the same functions. This is not intended.

4 No arbitrage and Non-triviality

4.1 Short with cash delivery

Recall that in the CRR model it is argued that u > r > d, otherwise there would exist profitable risk-less arbitrage opportunities involving only the stock and risk-less borrowing and lending. Something like this occurs in our context.

1. Suppose an investor is short with cash delivery one item of stock over a single time period, (t, t + 1], having invested the proceeds in the riskless bond. At the end of the period she pays for the stock at “what it is worth”. So the liabilities amount to σ_{t+1}(uS_{t}) or σ_{t+1}(dS_{t}) while the asset is rσ_{t}(S_{t}). If σ_{t}(S_{t})r − σ_{t+1}(uS_{t}) ≥ 0 then there is arbitrage. So we assume

r < \frac{σ_{t+1}(uS_{t})}{σ_{t}(S_{t})}.

2. Now suppose that an investor borrows β_{t}(S_{t}) from the bank and buys stock. At the end of the period the investor sells stock at the new price. The liability is rβ_{t}(S_{t}) while the asset is either σ_{t+1}(uS_{t}) or σ_{t+1}(dS_{t}). There will be arbitrage if σ_{t+1}(dS_{t}) − β_{t}(S_{t})r ≥ 0 so we assume

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} < r.

So far then we have

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} < r < \frac{σ_{t+1}(uS_{t})}{σ_{t}(S_{t})}.  (1)

3. Consider the following situation. An investor has the choice of putting β_{t}(S_{t}) in the bank, and receiving β_{t}(S_{t})r after one period, or buying stock for β_{t}(S_{t}) and receiving at least σ_{t+1}(S_{t}d) or at the most σ_{t+1}(S_{t}u) after one period. If β_{t}(S_{t})r > σ_{t+1}(S_{t}u) then the investor would never buy the stock preferring the certainty of the bond and its’ better return. Our assumption that the market is non-trivial means that we must assume

\frac{σ_{t+1}(S_{t}u)}{β_{t}(S_{t})} > r.

On the other hand if β_{t}(S_{t})r ≤ σ_{t+1}(S_{t}d) then the investor would always buy stock rather than the bond. Again our requirement that the market will be non-trivial contradicts this. Combining this with our previous observation we see that we must assume

\frac{σ_{t+1}(S_{t}d)}{β_{t}(S_{t})} < r < \frac{σ_{t+1}(S_{t}u)}{β_{t}(S_{t})}.  (2)

This last pair^{5} of inequalities is tighter than the previous.

4. Consider now an investor who needs to borrow an amount, α, say. This investor has the choice of borrowing from the bank, and owing αr at the end of the time period, or short selling \frac{α}{σ_{t}(S_{t})} of stock (i.e. initially receiving α) and owing at most \frac{α}{σ_{t}(S_{t})} · σ_{t+1}(uS_{t}). If

\frac{α}{σ_{t}(S_{t})} · σ_{t+1}(uS_{t}) ≤ αr

then no one would borrow from the bank preferring to short stock because it is cheaper. To avoid avoid a trivial market, we assume the reverse inequality. This leads to the right side of 1. Alternatively, using the same kind of argument, we can see that setting the interest rate so that

r < \frac{σ_{t+1}(dS_{t})}{σ_{t}(S_{t})}

means that no one would ever short sell the stock as a means of borrowing because borrowing from the bank is cheaper. We could avoid this by assuming

\frac{σ_{t+1}(dS_{t})}{σ_{t}(S_{t})} < r

but we do not use this condition. Perhaps banks should be the cheapest form of borrowing? We remain agnostic.

5Trio?

Our preoccupations have led us to

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} ≤ \frac{σ_{t+1}(dS_{t})}{σ_{t}(S_{t})} < r < \frac{σ_{t+1}(uS_{t})}{β_{t}(S_{t})} ≤ \frac{σ_{t+1}(uS_{t})}{σ_{t}(S_{t})}.  (3)

4.2 Short with asset delivery

As we remarked before, asset delivery is equivalent to cash delivery at the buying price. Paralleling the arguments of the previous subsection,

1. If we short the stock and invest then the asset is rσ_{t}(S_{t}) while the liability is at most β_{t+1}(uS_{t}). If the former exceeds the latter we have arbitrage so

r < \frac{β_{t+1}(uS_{t})}{σ_{t}(S_{t})}.

2. If we borrow β_{t}(S_{t}) and buy stock the liability is rβ_{t}(S_{t}) while the asset is at least σ_{t+1}(dS_{t}). If the latter exceeds the former we have arbitrage. So,

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} < r.

3. If investing is always better than buying stock and selling it the market is trivial. On the other hand if buying stock is always better than investing the market is also trivial. Avoiding these states gives us,

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} < r < \frac{σ_{t+1}(uS_{t})}{β_{t}(S_{t})}.

The left side has already turned up in 2.

4. If an investor needs to borrow α she can short the stock or go to the bank. If she shorts the stock the liability after one time period is at most \frac{α}{σ_{t}(S_{t})}β_{t+1}(uS_{t}). If this is less than αr the bank is redundant. This gives us the right side of 1. If \frac{α}{σ_{t}(S_{t})}β_{t+1}(dS_{t}) is greater than αr then shorting would never be used as a means of borrowing. As before, we can get round this by having

\frac{β_{t+1}(dS_{t})}{σ_{t}(S_{t})} < r.

Putting it all together gives

\frac{σ_{t+1}(dS_{t})}{β_{t}(S_{t})} ≤ \frac{β_{t+1}(dS_{t})}{σ_{t}(S_{t})} < r < \frac{σ_{t+1}(uS_{t})}{β_{t}(S_{t})} ≤ \frac{β_{t+1}(uS_{t})}{σ_{t}(S_{t})}.  (4)

Comparing these inequalities with those of the previous subsection we see that we can satisfy both sets by requiring

\frac{β_{t+1}(dS_{t})}{σ_{t}(S_{t})} < r < \frac{σ_{t+1}(uS_{t})}{β_{t}(S_{t})}.

5 A Single Period Pricing Formula

We consider a single period, from t = 0 to the end of the period at t = 1 when the call option matures. We let β_{0}(C_{0}) be the current price for an investor to buy the call. We let C_{u} be the payoff of the call at the end of the period if the stock price goes up. Similarly we let C_{d} be the payoff of the call at the end of the period if the stock price goes down. The payoff of the option is determined by taking the underlying value to be somewhere intermediate between its’ sell and buy price. This is achieved by setting

C = (λβ_{1}(S_{1}) + (1 − λ)σ_{1}(S_{1}) − K)^{+}

where 0 ≤ λ ≤ 1^{6} and S_{1} is uS_{0} or dS_{0} and C is C_{u} or C_{d} accordingly. We note that if λ = 1 then this is equivalent to asset delivery. Following the treatment in [?], we form a portfolio consisting of Δ stocks and B bonds. We choose Δ and B so that the selling price of the portfolio will be equal to the (cash) value of the option whatever the final outcome.

Lemma 1 There is a portfolio which can be formed at time 0 and which replicates the values of the option at time 1. The portfolio amount are

Δ = \frac{C_{u} − C_{d}}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0})},  (5)
B = \frac{σ_{1}(uS_{0})C_{d} − σ_{1}(dS_{0})C_{u}}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0}))r} .  (6)

Moreover the value of the option at time 0 is

β_{0}(C_{0}) = Δβ_{0}(S_{0}) + B.

6We could make the value of λ path dependent, but we won’t

Proof 1 The values for Δ and B are obtained in the usual way, one just has to remember that it is the selling price of the portfolio that must replicate the values of the option. To show that the buying price for the option must equal that of the replicating portfolio we have:

1. Suppose that β_{0}(C_{0}) > Δβ_{0}(S_{0}) + B. The option writer sells a call and buys the portfolio, a sure profit ensues. So β_{0}(C_{0}) ≤ Δβ_{0}(S_{0}) + B.

2. Suppose that β_{0}(C_{0}) < Δβ_{0}(S_{0})+B. Then the option writer cannot buy a portfolio which meets their obligations at expiry. If the sum β_{0}(C_{0}) could be deployed to this end then there would be arbitrage. The option writer must hedge this risk, so

β_{0}(C_{0}) ≥ Δβ_{0}(S_{0}) + B,

which proves what we wanted. In this context, there is another consequence of assuming

β_{0}(C_{0}) < Δβ_{0}(S_{0}) + B.

Investors can go to the option writer and buy \frac{1}{Δ} calls and −\frac{B}{Δ} bonds. The cost of this portfolio is strictly less than β_{0}(S_{0}). However, at expiry the option component has a cash value, so the value of this portfolio at time 1 is

\frac{Δσ_{1}(S_{1}) + rB}{Δ} − r\frac{B}{Δ} = σ_{1}(S_{1}).

So one may obtain the same return as from the stock with a portfolio which is cheaper. This makes the market in stock trivial, which violates one of our requirements.

We now consider if a written option has a natural selling price.

5.1 Redemption Costs

We have assumed at the outset that our option writer is ‘risk averse’. We elaborate this aspect of the option writer by supposing that transactions occur sequentially and each transaction is closed by the option writer adopting a position which reduces the outstanding risk. Options which have already been written may be sold back to the option writer. We do not allow investors to trade options issued by the option writer among themselves, when an option is sold by an investor it is sold back to, or redeemed by, the option writer. What should the selling price for the option be when an investor returns it to the option writer? First of all, we must specify a protocol for redemption, otherwise the situation is ambiguous. We stipulate that at time t, the new prices for S are announced, the existing portfolio is liquefied and re-balanced and only then can the option be redeemed. For the single period case, there is no existing portfolio. The option writer will have purchased a portfolio of Δ_{0} items of stock and B_{0} items of bond with funds provided by the investor. The option writer must pay to end the obligations implied by the call that has been sold. Of course the investor wants as much as possible, and because the option writer deals with transactions sequentially; offsetting the risk of each transaction before moving to the next, the option writer can find at most Δσ_{0}(S_{0}) + B_{0} in cash for the investor. Clearly the option writer will not declare a redemption price greater than the sum mentioned because this would involve borrowing and the possibility of eventual bankruptcy. So this provides an upper bound for the redemption price. On the other hand, any investor holding an option would not offer it for sale, to anyone, for less than Δσ_{0}(S_{0}) + B^{7} because they can always obtain this amount by shorting the portfolio, with their liabilities being covered by the option at the next time period. This provides a lower bound for the price at which an investor would divest themselves of the option. This sets the redemption price. It is Δσ_{0}(S_{0}) + B.

5.2 Adjustments for Multiple Time Periods

We have considered what portfolio of stock and bond is neccesary to hedge the option over a single time period. Of course what we want to consider are multiple time periods. Here the problem changes. Our target is no longer the cash value of the option at expiry, bar of course at the penultimate time step. Let us consider the situation at the penultimate time step. Using the analysis above, we know that what ever the state of the underlying S, we can form a portfolio of stock and bond which will replicate the cash value of the payoff of the option and that the buying price of this portfolio must agree with the buying price of the option. Moreover, the redemption cost of the option at this time is exactly the selling value of this portfolio that we hold over this final time period. So our target must be to arrive at the penultimate time step with a portfolio whose selling^{8}price is equal to the buying price of the option. So, imagine that at time t we need to arrive (from time t − 1) with a portfolio whose selling value is exactly β_{t}(C_{t}), the buying price of the option, and that this portfolio can be re-balanced and managed thereafter to meet the obligations arising from the option. This portfolio consists of Δ_{t−1} stocks and B_{t−1} bonds. It has a buying price of Δ_{t−1}β_{t−1}(S_{t−1}) + B_{t−1} at time t − 1. If β_{t−1}(C_{t−1}) is strictly greater than the buying price of the portfolio then the option writer can write a call at time t − 1 and buy this portfolio. Her obligations are met and a sure profit ensues. So,

7We assume investors are rational and prefer more wealth to less.

Δ_{t−1}β_{t−1}(S_{t−1}) + B_{t−1} ≥ β_{t−1}(C_{t−1}).

If β_{t−1}(C_{t−1}) < Δ_{t−1}β_{t−1}(S_{t−1}) + B_{t−1} and with this sum the option writer purchases a portfolio of stock and bond whose selling value at time t is β_{t}(C_{t}) or greater, then we have arbitrage. Since we preclude this, any such portfolio has a selling value, V (t), strictly less than β_{t}(C_{t}). If it is possible to form a portfolio with V (t) which can subsequently be managed to meet the obligations of the option then, once again, the option writer has an arbitrage opportunity in writing a call and buying this portfolio. We conclude that

Δ_{t−1}β_{t−1}(S_{t−1}) + B_{t−1} ≤ β_{t−1}(C_{t−1}).

To summarise; the buying cost of the time t − 1 portfolio must agree with the buying cost of the option at time t − 1. Looking one step back in time, our problem is to form a portfolio at time t − 2 whose selling value at time t − 1 is the buying price of the option at time t − 1, because then a portfolio can be managed thereafter to meet the obligations arising from the option. But this problem is identical to the one just solved. In this way we find our way back to time 0 where a time zero price emerges. The arguments above demonstrate the following,

Lemma 2 For times t = 1, …, N, the buying price of the option at time t − 1 must be equal to the buying price of the portfolio whose selling value at time t is the buying price of the option. For time t = N we identify the buying price with the cash value of the option.

8Recall that in our market one is forced to sell all stock at the end of every time period before re-balancing the portfolio.

5.3 The Pricing Formula

From the discussion above it is clear that we need first of all to solve the following single period problem: What portfolio shall we hold at time 0 whose selling price at time 1 is equal to the buying price of the option? All we have to do is modify the relevant terms in equations 5 and 6. So what we have is

Δ = \frac{β_{1}(C_{u}) − β_{1}(C_{d})}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0})},  (7)

and

B = \frac{σ_{1}(uS_{0})β_{1}(C_{d}) − σ_{1}(dS_{0})β_{1}(C_{u})}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0}))r}.  (8)

The buying price of our option is Δβ_{0}(S_{0}) + B, we can write this in a different way

Δβ_{0}(S_{0}) + B = (\frac{β_{1}(C_{u}) − β_{1}(C_{d})}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0})})β_{0}(S_{0}) + \frac{σ_{1}(uS_{0})β_{1}(C_{d}) − σ_{1}(dS_{0})β_{1}(C_{u})}{(σ_{1}(uS_{0}) − σ_{1}(dS_{0}))r}
= \frac{1}{r}(\frac{β_{0}(S_{0})r − σ_{1}(dS_{0})}{σ_{1}(uS_{0}) − σ_{1}(dS_{0})} β_{1}(C_{u}) + \frac{σ_{1}(uS_{0}) − rβ_{0}(S_{0})}{σ_{1}(uS_{0}) − σ_{1}(dS_{0})} β_{1}(C_{d}))

Writing

\frac{β_{0}(S_{0})r − σ_{1}(dS_{0})}{σ_{1}(uS_{0}) − σ_{1}(dS_{0})} = p_{0}

and

\frac{σ_{1}(uS_{0}) − rβ_{0}(S_{0})}{σ_{1}(S_{0}) − σ_{1}(dS_{0})} = 1 − p_{0}.

we rewrite our formula as

β_{0}(C) = \frac{1}{r}(p_{0}β_{1}(C_{u}) + (1 − p_{0})β_{1}(C_{d})).

In view of the relations

\frac{σ_{1}(dS_{0})}{β_{0}(S_{0})} < r < \frac{σ_{1}(uS_{0})}{β_{0}(S_{0})}.

we see that 1 ≥ p_{0} ≥ 0 and p_{0} is a probability measure for this single period case.

In the event that our functions β_{0} and σ_{1} amount to a simple scaling of S by constants;

β_{0}(S_{0}) = β_{0}S_{0} and σ_{1}(S_{1}) = σ_{1}S_{1}

the formula for the buying price of the option takes on an interesting form. First of all the equation for p_{0} simplifies to

p_{0} ≡ \frac{(\frac{β_{0}}{σ_{1}} )r − d}{u − d}, and so 1 − p_{0} ≡ \frac{u − (\frac{β_{0}}{σ_{1}} )r}{u − d}  (9)

which is very similar to the CRR case. We can write r′ = \frac{β_{0}}{σ_{1}} r and see that an effect of the introduction of transaction costs in this setting is to re-calibrate the risk free rate because our options buying price can now be written

β(C) = \frac{β_{0}}{σ_{1}} \frac{1}{r′} \{p_{0}β_{1}(C_{u}) + (1 − p_{0})β_{1}(C_{d})\}.

Which states that the price of the option is the product of the (CRR) price an option (without transaction costs) with an adjusted risk-free rate and the ratio of the buy and sell scalings. Of course if β_{0} = β_{1} = σ_{1} = 1, i.e. no transaction costs, then we recover the usual CRR case. The similarities do not end there. Once again the probability that the stock moves up does not come into the equation, with the consequence that even if different investors have different views on the probabilities of up and down movements of the stock they will still agree on the price.

5.4 Risk Neutral Probabilities ?

Let q be the probability that the stock moves up in the time period starting at 0 and ending with 1. An investor buying the stock at time 0 and selling it at time 1 would have an expected return of

\frac{σ_{1}(uS_{0})q + σ_{1}(dS_{0})(1 − q) − β_{0}(S_{0})}{β_{0}(S_{0})}

If this return were to be the return from the risk free asset, this return is r − 1,^{9} then it is easy to see that q = p_{0}. So, under this new probability, p_{0}, the stock has an expected return which is the same as that for the bond. If we look at our formula for the buying price of the option in the case that the buy and sell prices are linear scalings of the underlying, S, we see that it can be described as the expectation of its payoff under the measure p_{0} discounted by the adjusted risk-free rate and multiplied by the ratio of the buy (for time 0) and sell (for time 1) scalings.

9Recall that r is one plus the risk-free rate

6 Multiple Time Periods

6.1 The n-period formula

We are going to consider only a ‘linear case’. In each time period our buy and sell prices shall be simple linear scalings of the underlying S. Our options payoff is replicated by our portfolio and as we have seen there are arguments to confirm that the buying price of the portfolio at time zero should coincide with the options buying price. In fact we can deploy those arguments at any time, t, 0 < t < T, and state of the stock, where the options buying price and the buying price of the portfolio get out of line. So the buying price of the option throughout it’s life agrees with the buying price of the replicating portfolio (recall that in 2 we identified the value of the option at expiry with its ‘buying price’). Recall how we set up the probability space governing our model as a collection of binary sequences. In each of the time periods, from 0 to 1, from 1 to 2, etc, the differing buy and sell prices give rise to a new probability;

p_{t−1} = \frac{\frac{β_{t−1}}{σ_{t}} r − d}{u − d}

and an ‘adjusted risk-free rate’;

r_{t−1} = \frac{β_{t−1}}{σ_{t}} r

for the time period, t − 1 to t, for t = 1, 2, … N − 1.

We define, for each ω = (ω_{i}) ∈ Ω

Q(ω) = ∏_{j=0}^{n−1} p_{j}^{ω_{j}}[p_{j}^{⊥}]^{1−ω_{j}}  (10)

where

p_{j}^{⊥} = (1 − p_{j}).  (11)

and we set

s(ω) = ∑_{j=0}^{N−1} ω_{j}.  (12)

The set function defined on F by the function Q is easily seen to be a probability measure. We have

Theorem 1 The time zero price of our call option is given by

β_{0}(C_{0}) = \frac{∏_{r=0}^{N−1} β_{r}}{(∏_{t=0}^{N−1} σ_{t+1})} \frac{E^{Q}(β_{N}(C_{N}))}{r̂_{N}}.

where r̂_{N} = ∏_{t=1}^{N} r_{t−1}.

Proof 2 The n-period formula is established by mathematical induction. We assume a formula for the n-period case (the induction hypothesis) and then extend it to the (n + 1)-period case by reducing the problem to an application of the one period case under the induction hypothesis. As with the original CRR paper an investigation of the two period case suggests the general form. We include this here for completeness. So consider the two period case; after one time step the buying price of our option has moved from its initial value to one of two values, β_{1}(C_{u}) or β_{1}(C_{d}). These are, of course, the buying prices of the portfolio whose selling price in the next time period is equal to the payoff of the option. Using the one period analysis we know that we can express each of these as, respectively,

β_{1}(C_{u}) = \frac{β_{1}}{σ_{2}} \frac{1}{r_{1}} \{p_{1}β_{2}(C_{uu}) + (1 − p_{1})β_{2}(C_{ud})\}

and

β_{1}(C_{d}) = \frac{β_{1}}{σ_{2}} \frac{1}{r_{1}} \{p_{1}β_{2}(C_{du}) + (1 − p_{1})β_{2}(C_{dd})\}.

Here r_{1} = \frac{β_{1}}{σ_{2}} r. On the other hand we can express β_{0}(C) in terms of β_{1}(C_{u}) and β_{1}(C_{d}). It will be exactly the buying price, at time 0, of the portfolio whose selling price, at time 1, is β_{1}(C) in any state of the stock. We computed exactly this previously;

\frac{β_{0}}{σ_{1}} \frac{1}{r_{0}} \{p_{0}β_{1}(C_{u}) + (1 − p_{0})β_{1}(C_{d})\}.  (13)

We now substitute for the various states of β_{1}(C) and rearrange things to get

β_{0}(C_{0}) = \frac{β_{0}β_{1}}{σ_{1}σ_{2}r_{0}r_{1}} ∑_{j=0}^{2} \{ ∑_{s(ω)=j} ∏_{k=0}^{2−1} p_{k}^{ω_{k}}[p_{k}^{⊥}]^{1−ω_{k}} \} β_{2}(C_{u^{j}d^{2−j}}).  (14)

It might be helpful to remark that the term ∑_{s(ω)=j} is a sum over just one term if j = 0,(there is just one ω that is all zero’s), two terms if j = 1 (there are two ω’s with a zero and a one), and just one term again if j = 2. We remark also that the term

∑_{s(ω)=j} ∏_{k=0}^{2−1} p_{k}^{ω_{k}}[p_{k}^{⊥}]^{1−ω_{k}}

is simply ∑_{s(ω)=j} p(ω). We now proceed by induction. First of all we will assume that the n-period formula for the buy price of our call is

β(C_{0}) = \frac{∏_{r=0}^{n−1} β_{r}}{(∏_{t=0}^{n−1} σ_{t+1})(∏_{v=0}^{n−1} r_{v})} ∑_{j=0}^{n}( ∑_{s(ω)=j} Q(ω))β_{n}(C_{u^{j}d^{n−j}})  (15)

and we consider the n + 1- period problem. After the first time period has elapsed we have n periods to go and we may indicate the buying price of our option by β_{1}(C_{u}) and β_{1}(C_{d}). Using our hypothesis and advancing the suffices as appropriate to reflect that one time period has elapsed we get

β_{1}(C_{u}) = \frac{∏_{r=1}^{n} β_{r}}{(∏_{t=1}^{n} σ_{t+1})(∏_{v=1}^{n} r_{v})} ∑_{j=1}^{n+1}( ∑_{s(ω)=j} Q^{1}(ω))β_{n+1}(C_{u^{j}d^{(n+1)−j}}) (16)

here Q^{1}(ω) is ∏_{j=1}^{n} p_{j}^{ω_{j}}[p_{j}^{⊥}]^{1−ω_{j}}. We get a similar form for the other term

β_{1}(C_{d}) = \frac{∏_{r=1}^{n} β_{r}}{(∏_{t=1}^{n} σ_{t+1})(∏_{v=1}^{n} r_{v})} ∑_{j=0}^{n}( ∑_{s(ω)=j} Q^{1}(ω))β_{n+1}(C_{u^{j}d^{(n+1)−j}}) (17)

Equation 12 tells us how to put these together. We consider the terms separately.

\frac{β_{0}}{σ_{1}r_{0}} p_{0}β_{1}(C_{u}) = \frac{∏_{r=0}^{n} β_{r}}{(∏_{t=0}^{n} σ_{t+1})(∏_{v=0}^{n} r_{v})} ∑_{j=1}^{n+1}( ∑_{s(ω)=j} p_{0}Q^{1}(ω))β_{n+1}(C_{u^{j}d^{(n+1)−j}})(18)

Now, as p^{1} varies, p_{0}p^{1}(ω) varies over all values, p(ω), where ω has a 1 in the first place. Keep this in mind as we consider the second term.

\frac{β_{0}}{σ_{1}r_{0}} β_{1}(C_{d}) = \frac{∏_{r=0}^{n} β_{r}}{(∏_{t=0}^{n} σ_{t+1})(∏_{v=0}^{n} r_{v})} ∑_{j=0}^{n}( ∑_{s(ω)=j}(p_{0}^{⊥})Q^{1}(ω))β_{n+1}(C_{u^{j}d^{(n+1)−j}}).(19)

We note that as Q^{1}(ω) varies then p_{0}^{⊥}Q^{1}(ω) varies over all values Q(ω) where ω has a zero in the first place. Now equations 17 and 18 each contain an ‘extreme case’ , in 17 there is the case j = n + 1, which corresponds to the underlying S moving up and never down while equation 18 contains the case j = 0, when S only moves down. For each of the terms that remain in one of these equations there is a corresponding term in the other which has the same number of up and down moves of S - these divide equally into those that start with an up move and those that start with a down move. Putting all of this together we see that we have

β_{0}(C_{0}) = \frac{∏_{r=0}^{n} β_{r}}{(∏_{t=0}^{n} σ_{t+1})(∏_{v=0}^{n} r_{v})} ∑_{j=0}^{n+1}( ∑_{s(ω)=j}(Q(ω))β_{n+1}(C_{u^{j}d^{(n+1)−j}}). (20)

Noting that we have already proved the result for the one and two period case (which was all that we employed in the proof), we have proved our formula. We further observe that for an option maturing after N time periods, we can describe the time zero price of C as

β_{0}(C_{0}) = \frac{∏_{r=0}^{N−1} β_{r}}{(∏_{t=0}^{N−1} σ_{t+1})(∏_{v=0}^{N−1} r_{v})}E^{Q}(β_{N}(C_{N}))

Now by defining a new discount process by r̂_{t} = ∏_{j=0}^{t−1} r_{j} the formula simplifies further to

β_{0}(C_{0}) = \frac{∏_{r=0}^{N−1} β_{r}}{(∏_{t=0}^{N−1} σ_{t+1})} \frac{E^{Q}(β_{N}(C_{N}))}{r̂_{N}}.  (21)

This states that our options price is the product of two terms; one the product of the ratios of buy to sell scalings for, respectively, the start and end of the time periods and the second is the ‘risk neutral’ expectation of the payoff discounted by the modified risk-free rate. Of course, if there are no bid offer spreads then we recover the CRR case.

6.2 Martingale Properties

We restrict ourselves to the linear case that buy and sell prices are simple scalings of the underlying dependent only on time but not on the path of the underlying process ^{10}. It is well known that in the CRR case the stock price discounted by the risk free rate is a martingale under the risk neutral measure. Something like this persists in our situation. For each time period, t − 1 to t, we have identified a measure p_{t−1} and an ‘adjusted risk free rate, r_{t−1}. One can verify that the result in section 5.4 generalises to

\frac{σ_{t}(uS_{t−1})p_{t−1} + σ_{t}(dS_{t−1})(1 − p_{t−1}) − β_{t−1}(S_{t−1})}{β_{t−1}(S_{t−1})} = r − 1.  (22)

In the case we are considering this means that

p_{t−1} = \frac{\frac{β_{t−1}}{σ_{t}} r − d}{u − d}  (23)

and so is independent of the state of S at time t − 1. As in the previous section, we define a measure, Q, on Ω by taking the probability of any path, ω = (ω_{t}), to be

Q(ω) = ∏_{t=0}^{t=N−1} p_{t}^{ω_{t}}(1 − p_{t})^{1−ω_{t}}.

This determines the (new) probability completely. We recall that we have a new numeraire ‘process’

r̂_{t} = ∏_{j=0}^{t−1} r_{j}

the point of this becomes plain with the following

Theorem 2 Under the measure Q the underlying S discounted by the numeraire process r̂_{t} is a martingale.

Proof 3 Let <, > denote the inner product in L^{2}(Ω, F_{T} , Q). The conditional expectation onto L^{2}(Ω, F_{t}, Q), which we denote by M_{t}, is characterised by the relation

< M_{t}(X), I_{E} >=< X, I_{E} >

10Work in progress indicates that much of what we have to say here does carry over to the path dependent case.

for every X ∈ L^{2}(Ω, F_{T} , Q) and every E ∈ F_{t}. In fact it is enough for E to be a node of the binary tree for S. Small explanation; recall that an atom of F_{t} consists of all those ω’s which agree with some fixed binary string of length t in their first t places. A node is a collection of atoms, what all of these atoms have in common is that their ω’s have the same number of 1’s in their first t places. The effect of this is that S has the same (single numerical) value on each of these atoms that make up the node. Pictorially they correspond exactly to the branch points (nodes) of the tree for S. Moreover, because the conditional expectations form an increasing family of projection operators, then for our purposes it is enough to take X = \frac{S_{t+1}}{r̂_{t+1}} . Now since E is to be a node of F_{t} then S takes a single numerical value on E which we might as well denote by S_{t}. The set E consists of two disjoint F_{t+1} nodes, which we will denote by E^{+}, E^{−}. We then have

S_{t+1} = uS_{t}I_{E^{+}} + dS_{t}I_{E^{−}}

and, using equation 22

< M_{t}(S_{t+1}), I_{E} > = ∫_{Ω} M_{t}(S_{t+1})I_{E}dQ = ∫_{Ω} S_{t+1}I_{E}dQ
= S_{t}uQ(E^{+}) + S_{t}dQ(E^{−}) = S_{t}(p_{t}u + (1 − p_{t})d)
= S_{t} \frac{β_{t}}{σ_{t+1}} r

(24)

We see that

M_{t}(S_{t+1}) = S_{t}r_{t}.

From this it follows immediately that the process (\frac{S_{t}}{r̂_{t}}) is a martingale under the measure Q.

Some things follow immediately from this. We remind ourselves that we have two discount processes, a selling price process and a buying price process in addition to the underlying process.

Corollary 1 Under the measure Q we have the following:

• The selling price process discounted by the original discount process is a submartingale.

• The buying price process discounted by the original discount process is a submartingale.

Proof 4 It follows immediately from Equation 22 that

M_{t}(σ_{t+1}(S_{t+1})) = β_{t}(S_{t})r

and because we have assumed that buying prices dominate the corresponding selling prices the conclusions follow.

We remark here that Equation 20 suggests that the theorem and the corollary remain true in a more general context.

7 A Limiting Case

In this section we investigate a limiting case similar to that described in the Cox, Ross and Rubinstein paper. As usual, we divide the time interval, [0, T], up into N parts of length, Δt = \frac{T}{N}. In each time period, the upward and downward movements of the stock are given by,

u = 1 + μΔt + σ\sqrt{Δt}, d = 1 + μΔt − σ\sqrt{Δt}

and a risk free investment of a unit of bond over an interval of length Δt will grow to 1 + rΔt. We pause to remark here that we will continue to use u and d when clarity needs it. Our buy and sell prices will involve two strictly positive continuous real valued functions, β(t) and γ(t), defined on [0, T]. We assume that our transaction costs in any time period before time T are a function of the length of the time period. Specifically, where before we assumed that, for example, the buy price over the time period [t, t + Δt) was β_{t}(S_{t}) we now specify that it is (1 + β(t)Δt)S_{t}. In a similar fashion we specify that what was σ_{t}(S_{t}) is now to be (1 − γ(t)Δt)S_{t}^{11}. We have renamed the sell price to avoid confusion with σ, the volatility of the underlying S. Now at time T we specify that the buy and sell prices will be given by (1 + β(T))S_{T} and (1 − γ(T))S_{T} respectively. As before, we are taking our buy and sell prices to be simple scalings of the underlying S. So, for 1 ≤ i ≤ N, on the interval [(i − 1)Δt, iΔt] we will have

11It is clear that we will have to assume that β(t) and γ(t) satisfy certain conditions to keep prices positive, avoid daft situations and so on

r_{(i−1)Δt} = \frac{(1 + β((i − 1)Δt)Δt)}{(1 − γ(iΔt)Δt)}(1 + rΔt)

and

p_{(i−1)Δt} = \frac{\frac{(1+β((i−1)Δt)Δt)}{(1−γ(iΔt)Δt)}(1 + rΔt) − (1 + μΔt − σ\sqrt{Δt})}{2σ\sqrt{Δt}}.

We won’t need the specific forms for p and r until much later, for now we will continue to use the notations, p_{i−1} and r_{i−1}. We recall now our option pricing formula from Theorem 1;

β_{0}(C_{0}) = \frac{∏_{r=0}^{N−1} β_{r}}{(∏_{t=0}^{N−1} σ_{t+1})} \frac{E^{Q}(β_{N}(C_{N}))}{r̂_{N}}.

We will be substituting in this equation: β_{r} = (1 + β(rΔt)Δt) and σ_{t+1} = (1 + σ((t + 1)Δt)Δt) and eventually taking a limit as N ⟶ ∞. To facilitate this we prove a Lemma.

Lemma 3

lim_{N⟶∞} ∏_{r=0}^{N−1}(1 + β(rΔt)Δt) = e^{∫_{0}^{T} β(s)ds}.

Proof 5

log( ∏_{r=0}^{N−1}(1 + β(rΔt)Δt)) = ∑_{r=0}^{N−1} log(1 + β(rΔt)Δt)

= ∑_{r=0}^{N−1} ∑_{j=1}^{∞} \frac{(β(rΔt)Δt)^{j}}{j} (−1)^{j+1}

= ∑_{j=1}^{∞} ∑_{r=0}^{N−1} \frac{(β(rΔt)Δt)^{j}}{j} (−1)^{j+1}

= ∑_{r=0}^{N−1} β(rΔt)Δt + ∑_{j=2}^{∞} ∑_{r=0}^{N−1} \frac{(β(rΔt)Δt)^{j}}{j} (−1)^{j+1}.

When Δt is strictly less than 1, the last term on the right above satisfies

| ∑_{j=2}^{∞} ∑_{r=0}^{N−1} \frac{(β(rΔt)Δt)^{j}}{j} (−1)^{j+1}| ≤ ∑_{j=2}^{∞} N(Δt)^{j} = \frac{N(Δt)^{2}}{1 − Δt} .

Taking the exponential of both sides of this relationship we have

∏_{r=0}^{N−1}(1 + β(rΔt)Δt) = exp( ∑_{r=0}^{N−1} β(rΔt)Δt) · exp(E).

Where E is a bounded term that tends to 0 as N ⟶ ∞. It now follows that the limit of the left side exists and

lim_{N⟶∞} ∏_{r=0}^{N−1}(1 + β(rΔt)Δt) = e^{∫_{0}^{T} β(s)ds}.

Of course the same kind of argument will work for the other term. So we state immediately

Corollary 2 With γ(s) as above we have,

lim_{N⟶∞} ∏_{t=0}^{N−1}(1 − γ((t + 1)Δt)Δt)) = e^{− ∫_{0}^{T} γ(s)ds}.

And so

lim_{N⟶∞} \frac{∏_{r=0}^{N−1}(1 + β(rΔt)Δt)}{∏_{t=0}^{N−1}(1 − γ((t + 1)Δt)Δt)} = e^{∫_{0}^{T} (β(s)+γ(s))ds}.

We note in passing that the function β(s) + σ(s) might be interpreted in terms of the spread for the prices of S. We turn now to consider the term \frac{E^{Q}(β_{N} (C_{N}))}{r̂_{N}}. At this point it is worth reminding ourselves that β_{N}(C_{N}) is simply a shorthand for the payoff of the option. Now, without substituting the values we chose for u and d,

E^{Q}(β_{N}(C_{N})) = ∑_{j=0}^{N}( ∑_{s(ω)=j} Q(ω))β_{N}(C_{u^{j}d^{N−j}})

here Q(ω) = ∏_{j=0}^{N−1} p_{j}^{ω_{j}}[p_{j}^{⊥}]^{1−ω_{j}} and the value of the option at expiry, β_{N}(C_{N}), is

β_{N}(C_{N}) = ((λ(1 + β(T)) + (1 − λ)(1 − γ(T))S(T) − K)^{+}
= ((1 + λβ(T) − (1 − λ)γ(T))S(T) − K)^{+}.

Of course we can write this in terms of u and d if we wish.

The reader will recall the Binomial Option Pricing Formula from the CRR paper;

C = Sϕ(a, N, p′) − Kr̂^{N}ϕ(a, N, p)

we remind ourselves that K is the strike for the option, a the least number of upward moves of the stock that ensures the option finishes in the money, p the risk-neutral probability and p′ = \frac{u}{r}p. Can we recover a similar form for our option pricing formula? The answer is yes. Let b be the least positive integer such that

b > \frac{log(\frac{K}{S_{0}d^{N} (1+(λβ(T)−(1−λ)γ(T)))})}{log(\frac{u}{d})}

For j ≥ b the option is “in the money”, that is

C_{u^{j}d^{N−j}} = (1 + λβ(T) − (1 − λ)γ(T))S_{0}u^{j}d^{n−j} − K.

So, following the treatment in CRR, we can rewrite, our formula for \frac{E^{Q}(β_{N} (C_{N}))}{r̂_{N}}.

E^{Q}(β_{N}(C_{N})) = ∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))((1 + λβ(T) − (1 − λ)γ(T))S_{0}u^{j}d^{N−j} − K)

= ∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))(1 + λβ(T) − (1 − λ)γ(T))S_{0}u^{j}d^{N−j} − K ∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))

= ∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))(1 + λβ(T) − (1 − λ)γ(T))S_{0}u^{j}d^{N−j} − KΨ(b, N, Q).

Here we have written

Ψ(b, N, Q) = ∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))

and we call Ψ a generalised complementary binomial distribution. It is the probability of at least b success in N independent trials where the probability of success in the i − th trial is p_{i−1}. Taking our line of argument from CRR we write \frac{E^{Q}(β_{N} (C_{N}))}{r̂_{N}} as

∑_{j=b}^{N} ∑_{s(ω)=j}(Q(ω))(1 + λβ(T) − (1 − λ)γ(T))S_{0} \frac{u^{j}d^{N−j}}{r̂_{N}} − \frac{K}{r̂_{N}} Ψ(b, N, Q)

which we may write as

(1 + λβ(T) − (1 − λ)γ(T))S_{0}Ψ(b, N, Q′) − \frac{K}{r̂_{N}} Ψ(b, N, Q).

Here Ψ(b, N, Q′) is a generalised complementary binomial distribution, it is the probability of at least b successes in N independent trials where the probability of success at the i − th trial is given by

pi−1^{′} = \frac{u}{r_{i−1}} p_{i−1}, and 1 − pi−1^{′} = \frac{d}{r_{i−1}} (1 − p_{i−1}).

and Q′ is the product measure derived from these probabilities. In order to take a limit as N ⟶ ∞ we need to understand the limiting distribution for the generalised (complementary) binomial.

7.1 A Central Limit Result

As before, we will use u and d rather than their specific forms when clarity need it. Some notation: For i = 1, 2, 3, …, N, we define i(ω) = ω_{i}. So i is an F_{i} random variable which takes the value 1 with probability p_{i−1} and the value 0 with probability 1 − p_{i−1}. For i ≠ j the random variables i(ω) and j(ω) are independent. We can express S in terms of i.

S_{n}(ω) = S_{0} ∏_{i=1}^{n}(1 + μΔt + (2i(ω) − 1)σ\sqrt{Δt})

and then

log(\frac{S_{n}}{S_{0}} ) = ∑_{i=1}^{n} log(1 + μΔt + (2i(ω) − 1)σ\sqrt{Δt})
= ∑_{i=1}^{n}(log uI_{{i(ω)=1}} + log dI_{{i(ω)=0}})
= ∑_{i=1}^{n} log(\frac{u}{d})I_{\{i(ω)=1\}} + n log d
= log(\frac{u}{d}) ∑_{i=1}^{n} i(ω) + n log d.

So that

E^{Q}(log(\frac{S_{N}}{S_{0}} )) = (∑_{i=1}^{N} p_{i−1}) log(\frac{u}{d}) + N log d.

We will write

X_{Ni} = log uI_{{i(ω)=1}} + log dI_{{i(ω)=0}}

We note that for i = 1, 2, …, N the random variables X_{Ni} are an independent family and that ∑_{i=1}^{N} X_{Ni} = log(\frac{S_{N}}{S_{0}} ). It is easy to show that

E^{Q}(X_{Ni}) = p_{i−1} log(\frac{u}{d}) + log d, and Var(X_{Ni}) = (1 − p_{i−1})p_{i−1}(log(\frac{u}{d}))^{2}.

Of course what we are doing here is to set things up so we can apply a version of the Central Limit Theorem. We have in mind using Lyapounov’s Condition to show that the appropriate terms converge in distribution to a standard normal distribution. This is explained very clearly in Section 27, pp 312 in particular, of [1]. We adopt the notation used there. We want to find a limiting distribution for our generalised complementary binomial distribution, Ψ(b, N, Q). First of all, write Ψ^{⊥}(b, N, Q) = 1 − Ψ(b, N, Q). We observe that

Ψ^{⊥}(b, N, Q) = Q{∑_{i=1}^{N} i(ω) ≤ b − 1}.

This can be rewritten, (we add and subtract \frac{N log d}{log( \frac{u}{d})} initially and later multiply top and bottom by log(\frac{u}{d})).

Q\{\frac{∑_{i=1}^{N} i(ω) log(\frac{u}{d}) + N log d − (∑_{i=1}^{N} p_{i−1} log(\frac{u}{d}) + N log d)}{log(\frac{u}{d})\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}} ≤ \frac{b − 1 − (∑_{i=1}^{N} p_{i−1})}{\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}\}

and we can write this as

Q\{\frac{∑_{i=1}^{N} XNi^{′}}{\sqrt{V ar(∑_{i=1}^{N} XNi^{′})}} ≤ \frac{b − 1 − (∑_{i=1}^{N} p_{i−1})}{\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}\}.

Here we have written XNi^{′} = X_{Ni} − E^{Q}(X_{Ni}). The random variables XNi^{′} are independent and so the variance of their sum is the sum of their variances, but we won’t need this fact just yet. We remark that the measure Q depends upon N too but, as with many convergence results, it is best read “probability of..”.

Lyapounov’s condition states that if there is a real number δ > 0 such that for every N and i, |XNi^{′}|^{2+δ} are integrable and

lim_{N} ∑_{k=1}^{N} \frac{1}{(\sqrt{∑_{i=1}^{N} V ar(XNi^{′})})^{2+δ}} E^{Q}(|XNk^{′}|^{2+δ}) = 0

then \frac{∑_{i=1}^{N} X′_{Ni}}{\sqrt{∑_{i=1}^{N} V ar(X′_{Ni})}} will converge in distribution to a standard normal ran-

dom variable. Our first task is to verify that Lyapounov’s condition holds. First of all

XNi^{′} = (1 − p_{i−1}) log(\frac{u}{d})I_{\{i(ω)=1\}} − p_{i−1} log(\frac{u}{d})I_{\{i(ω)=0\}}

so, for δ = 1

E^{Q}(|XNi^{′}|^{3}) = (log(\frac{u}{d}))^{3}p_{i−1}(1 − p_{i−1})((1 − p_{i−1})^{2} + p_{i−1}^{2})
≤ 2(log(\frac{u}{d}))^{3} < ∞.

We already know what ∑_{i=1}^{N} V ar(XNi^{′} is, so, putting the parts together we see that Lyapounov’s condition amounts to considering the behaviour of

\frac{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})(p_{i−1}^{2} + (1 − p_{i−1})^{2})}{(∑_{i=1}^{N} p_{i−1}(1 − p_{i−1}))^{\frac{3}{2}}}

as N ⟶ ∞. Notice that we can write this as

∑_{i=1}^{N} \frac{p_{i−1}(1 − p_{i−1})}{∑_{j=1}^{N} p_{j−1}(1 − p_{j−1})} \frac{(1 − p_{i−1})^{2} + p_{i−1}^{2}}{\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}.

Focusing our attention on the second fraction, for each N we know that there is an index, call it imax, such that the value of (1 − p_{i−1})^{2} + p_{i−1}^{2} is greatest. Similarly, there is an index for which p_{i−1}(1 − p_{i−1}) is least, call this imin. The term we have to consider is dominated by

∑_{i=1}^{N} \frac{p_{i−1}(1 − p_{i−1})}{∑_{j=1}^{N} p_{j−1}(1 − p_{j−1})} \frac{(1 − p_{imax})^{2} + p_{imax}^{2}}{\sqrt{N}\sqrt{p_{imin}(1 − p_{imin})}},

but this is exactly

\frac{(1 − p_{imax})^{2} + p_{imax}^{2}}{\sqrt{N}\sqrt{p_{imin}(1 − p_{imin})}}.

Now, whatever index we are considering, it is easy to show that as N ⟶ ∞ we have p_{i−1} ⟶ p, where p is the usual risk neutral measure given by

p = \frac{(1 + rΔt) − (1 + μΔt − σ\sqrt{Δt})}{(1 + μΔt + σ\sqrt{Δt}) − (1 + μΔt − σΔt)}

= \frac{(r − μ)Δt + σ\sqrt{Δt}}{2σ\sqrt{Δt}}
= \frac{1}{2} + \frac{1}{2}(\frac{r − μ}{σ})\sqrt{Δt}.

This in turn tends to \frac{1}{2} as N ⟶ ∞ showing that p_{i−1} has a strictly positive limit. So the quotient

\frac{(1 − p_{imax})^{2} + p_{imax}^{2}}{\sqrt{N}\sqrt{p_{imin}(1 − p_{imin})}}

tends to the same limit as the quotient

\frac{(1 − p)^{2} + p^{2}}{\sqrt{N}\sqrt{p(1 − p)}}

which is of order \frac{1}{\sqrt{N}} for large N and therefore converges to zero. In order to complete our result we need to find the limiting value of

\frac{(b − 1 − (∑_{i=1}^{N} p_{i−1}))}{\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}.  (25)

Recalling that b is the least positive integer strictly larger than

\frac{log(\frac{K}{S_{0}d^{N} (1+(λβ(T)−(1−λ)γ(T)))})}{log(\frac{u}{d})},

we can write

b − 1 = \frac{log(\frac{K}{S_{0}d^{N} (1+(λβ(T)−(1−λ)γ(T)))})}{log(\frac{u}{d})} + ϵ,

where ϵ ∈ [0, 1). Replacing b − 1 in 25 leads us to consider the term

\frac{\frac{log(_{\frac{K}{S_{0}d^{N} (1+(λβ(T)−(1−λ)γ(T)))}} )}{log( \frac{u}{d})} + ϵ − (∑_{i=1}^{N} p_{i−1})}{\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}

which rearranges to

\frac{log(\frac{K}{S_{0}(1+(λβ(T)−(1−λ)γ(T)))}) + ϵ log(\frac{u}{d}) − (∑_{i=1}^{N} p_{i−1}) log(\frac{u}{d}) − N log d}{log(\frac{u}{d})\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})}}.

Of the terms in the numerator we need only think about

(∑_{i=1}^{N} p_{i−1}) log(\frac{u}{d}) + N log d = E^{Q}(∑_{i=1}^{N} X_{Ni}),

because ϵ log(\frac{u}{d}) ⟶ 0 while the remainder is constant. The denominator may be estimated,

log(\frac{u}{d})\sqrt{N}\sqrt{p_{imin}(1 − p_{imin})} ≤ log(\frac{u}{d})\sqrt{∑_{i=1}^{N} p_{i−1}(1 − p_{i−1})} ≤ log(\frac{u}{d})\sqrt{N}\sqrt{p_{imax}(1 − p_{imax})}.

As we know that any of the p_{i−1}’s converge to p, the usual (risk neutral) measure in the absence of transaction costs, but does it follows that the leftmost and right-most terms are converging to log(\frac{u}{d})\sqrt{N}\sqrt{p(1 − p)} as N gets large? It is the presence of the \sqrt{N} term that is the problem of course, two sequences may get close as N gets large, but multiply them by \sqrt{N} and their difference might not get small. So, we must check. In fact it is enough to consider the term log(\frac{u}{d})\sqrt{N}. Here we need to use the specific form for u and d; we have

u = log(1 + μΔt + σ\sqrt{Δt})d = log(1 + μΔt − σ\sqrt{Δt}).

For sufficiently large values of N, the terms |μΔt+σ\sqrt{Δt}| and |μΔt−σ\sqrt{Δt}| will be strictly less than 1. Using the power series expansion for log,

log(1 + μΔt + σ\sqrt{Δt}) = (μΔt + σ\sqrt{Δt}) − ∑_{k=2}^{∞}(−1)^{k+1} \frac{(μΔt + σ\sqrt{Δt})^{k}}{k} ,

while

log(1 + μΔt − σ\sqrt{Δt}) = (μΔt − σ\sqrt{Δt}) − ∑_{k=2}^{∞}(−1)^{k+1} \frac{(μΔt − σ\sqrt{Δt})^{k}}{k} .

So we can write

|\sqrt{N} log(\frac{u}{d}) − 2σ\sqrt{T} | ≤ ∑_{k=2}^{∞} \sqrt{N} \frac{|μΔt − σ\sqrt{Δt}|^{k} + |μΔt + σ\sqrt{Δt}|^{k}}{k}

≤ ∑_{k=2}^{∞} \sqrt{N} \frac{2(|μ| + |σ|)^{k}\sqrt{Δt}^{k}}{k}

≤ 2(|μ| + |σ|)\sqrt{T} ∑_{k=1}^{∞}(\frac{(|μ| + |σ|)\sqrt{T}}{\sqrt{N}})^{k}.

We can make the sum of the geometric progression as small as we please by taking N sufficiently large. So this shows that in fact

\sqrt{N} log(\frac{u}{d}) ⟶ 2σ\sqrt{T}

as N ⟶ ∞. Since we know that p_{i−1} tends to p and this in turn tends to \frac{1}{2} we conclude that^{12}

log(\frac{u}{d})\sqrt{N}\sqrt{p(1 − p)} ⟶ σ\sqrt{T} .

So via a sandwich theorem, the denominator of our expression is determined. We turn to consider the undetermined part of the numerator. This is the term

(∑_{i=1}^{N} p_{i−1}) log(\frac{u}{d})) + N log d = ∑_{i=1}^{N} p_{i−1} log u + (1 − p_{i−1}) log d.

We have calculate and estimate each term for each i, 1 ≤ i ≤ N. In order to make things readable we are going to contract some of the notation, specifically, let

β((i − 1)Δt) = β, γ(iΔt) = γ.

Then

p_{i−1} = \frac{(1 + βΔt)(1 + rΔt) − (1 + μΔt − σ\sqrt{Δt})(1 − γΔt)}{2σ\sqrt{Δt}(1 − γΔt)}
= \frac{(r − μ + β + γ)Δt + (βr + γμ)Δt^{2} + σγΔt^{\frac{3}{2}} + σ\sqrt{Δt}}{2σ\sqrt{Δt}(1 − γΔt)}
= \frac{σ\sqrt{Δt} + o^{+}(Δt)}{2σ\sqrt{Δt}(1 − γΔt)}.

The expression for 1 − p_{i−1} is similar

1 − p_{i−1} = \frac{σ\sqrt{Δt} − o^{+}(Δt) − 2σγΔt^{\frac{3}{2}}}{2σ\sqrt{Δt}(1 − γΔt)} .

12One might argue that we’ve reduced it to the CRR case, but our parametrisations of u, d and r are slightly different. However it amounts to the same thing in the end

Now for sufficiently large N,

p_{i−1} log u = \frac{σ\sqrt{Δt} + o^{+}(Δt)}{2σ\sqrt{Δt}(1 − γΔt)} ∑_{i=1}^{∞}(−1)^{k+1}k^{−1}(μΔt + σ\sqrt{Δt})^{k}
= \frac{σ\sqrt{Δt} + o^{+}(Δt)}{2σ\sqrt{Δt}(1 − γΔt)}(μΔt + σ\sqrt{Δt}) − \frac{σ\sqrt{Δt} + o^{+}(Δt)}{2σ\sqrt{Δt}(1 − γΔt)}(μΔt + σ\sqrt{Δt})^{2} \frac{1}{2}
+ \frac{σ\sqrt{Δt} + o^{+}(Δt)}{2σ\sqrt{Δt}(1 − γΔt)} ∑_{k=3}^{∞}(−1)^{k+1} \frac{(μΔt + σ\sqrt{Δt})^{k}}{k}

= \frac{σ + o^{+}(\sqrt{Δt})}{2σ(1 − γΔt)}(μ\sqrt{Δt} + σ)\sqrt{Δt} − \frac{σ + o^{+}(\sqrt{Δt})}{2σ(1 − γΔt)}(μ\sqrt{Δt} + σ)^{2}Δt\frac{1}{2}
+ \frac{σ + o^{+}(\sqrt{Δt})}{2σ(1 − γΔt)} ∑_{k=3}^{∞}(−1)^{k+1}Δt^{\frac{k}{2}} \frac{(μ\sqrt{Δt} + σ)^{k}}{k} .

Here o^{+}(\sqrt{Δt}) = \frac{o^{+}(Δt)}{\sqrt{Δt}}. One can do the same job for the other term

(1 − p_{i−1}) log d = \frac{σ − o^{+}(\sqrt{Δt}) − 2σγΔt}{2σ(1 − γΔt)}(μ\sqrt{Δt} − σ)\sqrt{Δt}
− \frac{σ − o^{+}(\sqrt{Δt}) − 2σγΔt}{2σ(1 − γΔt)}(μ\sqrt{Δt} − σ)^{2}Δt\frac{1}{2}
+ \frac{σ − o^{+}(\sqrt{Δt}) − 2σγΔt}{2σ(1 − γΔt)} ∑_{k=3}^{∞}(−1)^{k+1}Δt^{\frac{k}{2}} \frac{(μ\sqrt{Δt} − σ)^{k}}{k} .

Now we form p_{i−1} log u + (1 − p_{i−1}) log d. First, if one adds the leading terms the result breaks into two terms

\frac{μΔt + o^{+}(Δt)}{(1 − γΔt)} + \frac{μ\sqrt{Δt}o^{+}(Δt) + σγΔt^{\frac{3}{2}} − σγμΔt^{2}}{σ(1 − γΔt)} .

We observe that the numerator for the first term is of order Δt while that of the second is of order Δt^{\frac{3}{2}} . Now we must consider the contribution from the second (quadratic) terms. Adding the second terms involving (μ\sqrt{Δt} ± σ)^{2} gives,

\frac{σ^{2}Δt + μ^{2}Δt^{2} + μΔt o^{+}(Δt) + 2μσγΔt^{2} − σ^{2}γΔt^{\frac{3}{2}} − γμ^{2}Δt^{\frac{5}{2}}}{2(1 − γΔt)} .

which can be written

\frac{σ^{2}Δt + o(Δt^{\frac{3}{2}} )}{2(1 − γΔt)}.

Here o(Δt^{\frac{3}{2}} ) means the obvious thing, note that it is a function of the index i. Of course these terms are subtracted from what we obtained from the first (linear) terms. The remaining terms (of our sum from 1 to N) are

p_{i−1} ∑_{k=3}^{∞}(−1)^{k+1}Δt^{\frac{k}{2}} \frac{(μ\sqrt{Δt} + σ)^{k}}{k} + (1 − p_{i−1}) ∑_{k=3}^{∞}(−1)^{k+1}Δt^{\frac{k}{2}} \frac{(μ\sqrt{Δt} − σ)^{k}}{k} .

We have to sum from 1 to N and then let N ⟶ ∞. We estimate the series terms first. When N is large enough,

| ∑_{i=1}^{N} p_{i−1} ∑_{k=3}^{∞}(−1)^{k+1}Δt^{\frac{k}{2}} \frac{(μ\sqrt{Δt} + σ)^{k}}{k} | ≤ N ∑_{k=3}^{∞} Δt^{\frac{k}{2}} \frac{(|μ| + |σ|)^{k}}{k} ,

and

∑_{k=3}^{∞} Δt^{\frac{k}{2}} \frac{(|μ| + |σ|)^{k}}{k} = N(|μ| + |σ|)^{2}Δt ∑_{k=3}^{∞} Δt^{\frac{k−2}{2}} \frac{(|μ| + |σ|)^{k−2}}{k}
= (|μ| + |σ|)^{2}T ∑_{k=1}^{∞} Δt^{\frac{k}{2}} \frac{(|μ| + |σ|)^{k}}{k + 2}

This contribution tends to 0 as N ⟶ ∞. The same is true for the other series term. Recall that the quadratics terms amounted to

\frac{σ^{2}Δt + o(Δt^{\frac{3}{2}} )}{2(1 − γΔt)}.  (26)

Let’s take the part with o(Δt^{\frac{3}{2}} ) first. Let M be a bound for γ. For large enough N,

| ∑_{i−1}^{N} \frac{o(Δt^{\frac{3}{2}} )}{2(1 − γΔt)}| ≤ | ∑_{i=1}^{N} \frac{o(Δt^{\frac{3}{2}} )}{2}| + | ∑_{i=1}^{N} \frac{o(Δt^{\frac{3}{2}} )γΔt}{2(1 − γΔt)}|

≤ NΔt\frac{o(\sqrt{Δt})}{2} + Nmax\{o(Δt^{\frac{3}{2}} )\}MΔt.

The term max\{o(Δt^{\frac{3}{2}} )\}^{13} is of order Δt^{\frac{3}{2}} too, so the contribution from this term tends to 0 as N gets large. Using the same trick with the reciprocal of (1 − γΔt) as above, we see that the other term from equation 26 contributes \frac{σ^{2}T}{2} . Of course, this is subtracted from the overall total. Now we consider the contribution from the leading terms. Recall that these broke into two parts;

13The maximum is taken over i

\frac{μΔt + o^{+}(Δt)}{(1 − γΔt)} + \frac{μ\sqrt{Δt}o^{+}(Δt) + σγΔt^{\frac{3}{2}} − σγμΔt^{2}}{σ(1 − γΔt)} .

We consider the terms of order Δt^{\frac{3}{2}} first. For large enough N,

|\frac{μ\sqrt{Δt}o^{+}(Δt) + σγΔt^{\frac{3}{2}} − σγμΔt^{2}}{σ(1 − γΔt)} | ≤ 2Δt\frac{\{|μo^{+}(\sqrt{Δt})| + |σγΔt^{\frac{1}{2}} | + |σγμΔt|\}}{σ}

so that

| ∑_{i=1}^{N} \frac{μ\sqrt{Δt}o^{+}(Δt) + σγΔt^{\frac{3}{2}} − σγμΔt^{2}}{σ(1 − γΔt)} | ≤ \frac{2NΔt}{σ}max\{|μo^{+}(\sqrt{Δt})| + |σγΔt^{\frac{1}{2}} | + |σγμΔt|\}
= \frac{2T}{σ} o(\sqrt{Δt}).

Here o(\sqrt{Δt}) has the obvious meaning and this shows that the contribution from these terms vanishes as N ⟶ ∞. We are left with

\frac{μΔt + o^{+}(Δt)}{(1 − γΔt)}.

We rewrite this as

\frac{μΔt + o^{+}(Δt)}{(1 − γΔt)} = μΔt + o^{+}(Δt) + \frac{\{μΔt + o^{+}(Δt)\}γΔt}{(1 − γΔt)}.

The contribution of the second term on the right may be estimated, for large enough N,

∑_{i=1}^{N} \frac{\{μΔt + o^{+}(Δt)\}γΔt}{(1 − γΔt)} ≤ 2\{NγμΔt^{2} + NΔt max\{γo^{+}(Δt)\}\}
= 2T γμΔt + 2NΔt max{γo^{+}(Δt)}.

So this contribution vanishes as N gets large. Finally we consider

∑_{i=1}^{N} μΔt + ∑_{i=1}^{N} o^{+}(Δt).

The first bit is μT. An examination of the parts of o^{+}(Δt) coupled with arguments we have already rehearsed above shows that only the term

(r − μ + β + γ)Δt

makes any contribution. Writing in the dependence upon i, we must consider

∑_{i=1}^{N}(r − μ + β((i − 1)Δt) + γ(iΔt))Δt

letting N ⟶ ∞ this is, of course, just

(r − μ)T + ∫_{0}^{T} (β(s) + γ(s))ds.

Putting all the parts together we see that as N gets large the limiting value of the term in equation 25 is

\frac{log(\frac{K}{S_{0}(1+(λβ(T)−(1−λ)γ(T)))}) − (r − \frac{σ^{2}}{2} )T − ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}}.

What this amounts to is that

Ψ^{⊥}(b, N, Q) ⟶ N(\frac{log(\frac{K}{S_{0}(1+(λβ(T)−(1−λ)γ(T)))}) − (r − \frac{σ^{2}}{2} )T − ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}}).

Of course what we are interested in is Ψ(a, N, Q), so, using the symmetry of the standard normal distribution,

Ψ(b, N, Q) ⟶ N(\frac{log(\frac{S_{0}(1+(λβ(T)−(1−λ)γ(T)))}{K}) + (r − \frac{σ^{2}}{2} )T + ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}}).

By adapting the estimates used above, the reader can verify that the other generalised complementary binomial distribution converges too and that the limit is the expected modification to the one above^{14}

Ψ(b, N, Q′) ⟶ N(\frac{log(\frac{S_{0}(1+(λβ(T)−(1−λ)γ(T)))}{K}) + (r + \frac{σ^{2}}{2} )T + ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}}).

14It’s a long furrow to plough!

We can now express the price of our option in a familiar form. Write S0^{′} = S_{0}(1 + (λβ(T) − (1 − λ)γ(T))). Using Lemma 3 and Corollary 2 and the fact that (1 + x)^{\frac{1}{x}} has limit e at x = 0 we can express our price as

β_{0}(C_{0}) = e^{∫_{0}^{T} (β(s)+γ(s))ds}\{S0^{′}N(\frac{log(\frac{S′_{0}}{K}) + (r + \frac{σ^{2}}{2} )T + ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}})

− \frac{K}{e^{rT +∫_{0}^{T} (β(s)+γ(s))ds}} N(\frac{log(\frac{S′_{0}}{K}) + (r − \frac{σ^{2}}{2} )T + ∫_{0}^{T} (β(s) + γ(s))ds}{σ\sqrt{T}})\}.

(28)

8 Concluding Remarks

Once again, our option price is the product of two terms. The first is a function of the relative spread of the prices for S and the second is a familiar option pricing formula albeit with a modified initial stock value and interest rate. Seen in this way, (leaving aside the first term ) the effect of transaction costs on the price of the option is to re-calibrate both the initial stock value and the interest rate. If we take (a commonly used) special case, β(s) = k, γ(s) = k, and λ = \frac{1}{2}, with k of suitable magnitude, then, bearing in mind how we defined the spread at expiry, our formula takes the form

C_{0} = e^{(r+2k)T} \{S_{0}(1 + k)N(\frac{log(\frac{S_{0}(1+k)}{K}) + (r + 2k + \frac{σ^{2}}{2} )T}{σ\sqrt{T}})
− \frac{K}{e^{(r+2k)T}} N(\frac{log(\frac{S_{0}(1+k)}{K}) + (r + 2k − \frac{σ^{2}}{2} )T}{σ\sqrt{T}})\},

and the re-calibration is quite clear. With this example “the spread is constant”, however, the formula allows us to investigate what happens if there is some deterministic change in the spread over [0, T]. For example, if there were a narrowing of the spread, effected by a linear function of s ∈ [0, T], say, β(s) = γ(s) = 2k(1 − \frac{s}{T}), then a quick calculation shows that the riskfree rate, r, is modified by the term 2kT. In general, there will be a t ∈ [0, T] such that

(β(t) + γ(t))T = ∫_{0}^{T} (β(s) + γ(s))ds

but the location of the point t in [0, T] (and hence its’ effect) will depend upon the nature of β and γ.

Our work could be extended in several ways. One could look at more general forms for the buy and sell prices, non-linear functions of S for example. An obvious next step would be to make the buy and sell prices stochastic. The formula above suggests that this could amount to having a stochastic modification to the interest rate. A link with option pricing under stochastic interest rates would be pleasing, but this is all speculation.

References

[1] P.Billingsley. Probability and Measure. J. Wiley 1979.

[2] Maureen ‘O’ Hara. Market Microstructure Theory.

[3] F.Black and M Scholes 1973. The Pricing of Options and Corporate Liabilities. Journal of Political Economy, Vol 81, p637-654

[4] P.Boyle and T Vorst 1992. Option Replication in Discrete Time with Transaction Costs. Journal of Finance, Vol 47, p271-293

[5] J.Cox, S.Ross and Rubinstein 1979. Option Pricing : A Simplified Approach. Journal of Financial Economics, Vol 7, p229-263

[6] R.C. Merton 1973. Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, Vol 4, p141-183

[7] Marek Muselia, Marek Rutowski Martingale Methods in Financial Modelling. Applications of Mathematics; 36, Springer 1998