Valuing Seats in Futures Markets with Random Jumps in Cumulative Trading

Owning a seat is what enables locals to trade in futures markets. Seats are usually assigned by the Clearinghouse but can be successively traded among locals by paying officially quoted prices. This paper develops a pricing model for seats in futures markets. The existing literature on the argument is extended by introducing the new hypothesis that the number of contract traded since the seat’s ownership acquisition up to current time t, what we’ll refer to as "cumulative trading", is a quantity following a mixed diffusion-jump stochastic process instead of a pure diffusion one. The basic structure of the model is one interpreting the seat’s price as the value of an European down and out call option written on the gross profits accrued through futures trading allowed by the seat’s ownership, having an exercise price equal to the current value of futures trading-related operating costs. The paper shows that the introduction of random jumps in cumulative trading has a deterministic impact on the gross profits’ and operating costs’ dynamics, translating in a seat’s price change that can randomly occur either as a price increase or decrease. The Clearinghouse in a futures market, among other functions, has the one of assigning seats to locals in order to qualify them as market makers. Each local can own one or more seats enabling him/her to operate different contracts with different degrees of freedom. Seats can be freely traded among locals as a regular asset. Such a kind of trades are not so frequent even if it must be recognized that a seats’ secondary market exists, although it is quite thin in terms of transactions. The existence of a market implies one market price, at least, of the traded asset. There are models, like Bhasin & Brown (1994) and Paris (2000), dealing with the problem of seats’ pricing in futures markets. Both these models point out the fact that the current seat’s value must reflect, somehow, the specific local’s features (e.g. trading ability, information, etc.), implying that the price of a seat changes according to its owner’s specific qualities. The Paris’ paper explicitly recognizes this fact by modeling the seat’s value as the price of a European down and out call option written on the gross profits accrued to the seat’s owner through the seat-related futures trading. The exercise price of such an option is represented by the operating costs incurred as a consequence of futures trading. A further contribution of this work is to show how the seats’ pricing mechanism affects the futures market’s efficiency, liquidity and competitiveness. Gross profits and operating costs ultimately depend, among other variables, on the number of contract traded since the seat’s aquisition up to current time; what will be referred to as "cumulative trading". The stochastic dynamics of such a variable is crucial in computing the seat’s price. Past results were derived under the assumption of cumulative trading following a pure diffusion process. This paper introduces the realistic hypothesis of cumulative trading being a diffusion process characterized by random jumps, related, for example, to sudden informational flows, generating significant increases in futures trading activity. What we investigate is the impact of jumps in cumulative trading on the seat’s market value. Random jumps in cumulative trading affect the drift and not the dispersion terms of the dynamics characterizing gross profits and operating costs. It means that locals price their seats accounting for the possibility of jumps in cumulative trading. Numerical simulations show that the relationship between the seat’s prices under the jump and no-jump assumptions is not stable. It changes depending on the underlying variable which is allowed to vary. Substantially different results are produced by those variables influencing the initial values of gross profits and operating costs with respect to those affecting the variations of the aforementioned quantities. The paper is organized as follows: section 1 presents the general theoretical framework by summarysing the seat’s pricing model developed in Paris (2000). In that model the cumulative trading’s dynamics are dealt with as a pure diffusion process. Section 2 introduces the assumption of random jumps in cumulative trading and derives the new partial differential equation whose solution gives the seat’s value. In section 3 the partial differential equations, under both the assumptions concerning the cumulative trading’s process, are numerically solved by applying Monte Carlo methods. Section 4 is devoted to some concluding remarks. 1. Sketching the basic pricing model. Let us consider a market for futures contracts operated by J locals owning, for sake of simplicity, just one seat each. We denote with J j n j t j t j t ∈ ∀ = ~ ~ φ π the random value of the gross profits accrued to the jth local through the seat-related futures trading by time t and with J j n b a c j t j t ∈ ∀ + = ~ ~ the random value of the operating costs incurred by the jth local by time t, as a consequence of the seat-related futures trading. For a given time horizon1, T, the payoff generated by the seat ownership, under different scenarios, can be easily computed and the consequent action entertained by the seat’s owner can be specified: • if j T j T c ~ ~ ≥ π , then the seat’s owner pays off the costs out of the gross profits and retains the difference ( ) j T j T c~ ~ − π , if positive; • if j T j T c ~ ~ < π , then the seat’s owner returns the seat to the Clearinghouse2 and fails. A special situation occurs when at an arbitrary time t T < the following inequality is satisfied: ( ) ( ) w c E E j T j t j T j t − ≤ ~ ~ π , where w is the collateral posted by the local to guarantee his/her ability to meet his/her obligations. Having denoted with E the expectation operator, the last 1 It must be recognized that the time horizon is absolutely arbitrary in the model. We can interpret T as the time when the local considers either the opportunity to remain in the market or to abandon it. Nothing tells us that T has to be the same for all the locals operating in the same market. However, we assume a unique T across locals, based on the hypothesis that all the market makers converge toward a common expiration to evaluate their position. 2 With respect to our study the Clearinghouse is the futures market organization supervising the seats’ assignment to market makers and the seats’ trading among market makers. inequality implies that the expected gross profits generated by the seat-related futures trading are so small that operating costs will be possibly not covered even using collateral. This fact certainly put the seat’s owner at a competitive disadvantage with respect to other locals, if known by the market. Collateral will be more expensive and the local has an incentive to get out of the market. In other words ( ) w c E F j T j t j t − = ~ can be interpreted as a time-dependent lower bound of expected gross profits below which the seat’s economic value vanishes. Conclusively, it can be said that the payoff generated at time T by the ownership of a seat is: ( ) j T j T j T c S ~ ~ , 0 max − = π (1.1) subject to the condition ( ) T t F E j t j T j t ∈ ∀ > π~ . This is the same payoff of a European down and out call option written on the gross profits generated by the seat-related futures trading with operating costs as time-varying exercise price. The pricing of such an option depends on the behaviour of the stochastic variables characterizing the payoff (1.1). We assume the gross profits’ value satisfying the following equation: j t j t j t n ~ ~ ~ φ π = while the operating costs are given by3: j t j t n b a c ~ ~ + = . In particular, j t φ~ denotes the average gross profit rate per contract associated to the jth local4 and j t n ~ indicates the number of contracts traded by the jth local by time t, as a consequence of the seat’s ownership; this is what we’ll refer to as "cumulative trading". Both these variables are assumed to follow Ito processes specified as follows: d r dt s dz j t j t j φ φ = + dn f dt g dz j j j n = + where r j j = αφ and f n j j = β are the drifts while s j j = σφ and g n j j = γ are the dispersion terms. dzφ and dzn are two standard brownian motions whose instantaneous correlation coefficient is ρφn . a is the coefficient expressing the fixed cost component, constant across locas and b is the cost rate per contract, the same across locals. 3 Inspite of its simple structure, such a functional form of operating costs reflects quite well the reality of futures markets. Furthermore, it is computationally convenient. Those are the reasons why we don’t make more complicate assumptions (i.e. quantity discounts). 4 It can be interpreted as the average of the marginal profit rates associated to single trades, with respect to all the contracts traded since the beginning of the trading activity through the seat, time 0, up to current time t. By considering both gross profits and operating costs as functions of time and the aforementioned Ito processes, Ito’s lemma can be applied to derive their dynamics. They are given by the following two stochastic differential equations: d e dt i dz i dz j j j n j n π φ φ = + + (1.2) dc h dt k dz j j j n = + (1.3) where e n r f s g j j j j j j j n = + + φ ρφ and h bf j j = are the respective drifts while i n s j j j φ = and i g n j j j = φ are the dispersion terms of (1.2) and k bg j j = is the dispersion term of (1.3). By rewriting (1.2) in a more compact way as follows: d e dt i dz j j j π π = + (1.4) where i i i s g j n j j n π φ φ ρ = + + 2 2 2 and ( ) ( ) ( ) j n j n j i t z i t z i t z π φ φ + = , and applying Ito’s lemma with respect to (1.4) the dynamics of the seat’s value can be derived. It is: dz i S dt i S t S e S dS j j j j j j j j j j j π π ∂π ∂ ∂π ∂ ∂ ∂ ∂π ∂ +     + + = 2 2 2