Forward Futures ~ Future Forwards. Thought of the Day 126.0

Forward contracts and futures are much alike, except that the former are traded over- the-counter and the latter are traded in exchanges. Since the exchanges would like to organize trading such that contract defaults are minimized, an investor who buy a futures in an exchange is requested to deposit funds in a margin account to safeguard against the possibility of default (the futures agreement is not honored at maturity). At the end of each trading day, the futures holder will pay to or receive from the writer the full amount of the change in the futures price from the previous day through the margin account. This process is called marking to market the account. Therefore, the payment required on the maturity date to buy the underlying asset is simply the spot price at that time. However, for a forward contract traded outside the exchanges, no money changes hands initially or during the life-time of the contract. Cash transactions occur only on the maturity date. Such difference in the payment schedules may lead to differences in the prices of a forward contract and a futures on the same underlying asset and date of maturity. This is attributed to the possibility of different interest rates applied on the intermediate payments.

Here, let us argue how the equality of the two prices when the interest rate is constant contours. First, consider one forward contract and one futures which both last for n days. Let F_i and G_i represent the forward price and the futures price at the end of the i^th day. We aim to show F₀ = G₀. Let S_n denote the asset price at maturity. Let the constant interest rate per day be δ. Suppose we initiate the long position of one unit of the futures on day 0. The gain/loss of the futures on the i^th day is (G_i – G_i-1), and this amount grows to the currency value (G_i – G_i-1)e^δ(n-i) at maturity, which is the end of the nth day (n ≥ i). Therefore, the value of this one long futures position at the end of the n^th day is the summation of (G_i – G_i-1)e^δ(n-i), where i runs from 1 to n. The sum can be expressed as,

∑_i=1ⁿ (G_i – G_i-1)e^δ(n-i)

The summation of gain/loss of each day reflects the daily settlement nature of a futures. Instead of holding one unit of futures throughout the whole period, the investor now keeps changing the amount of futures to be held on each day. Suppose he holds α_i units at the end of the (i − 1)^th day, i = 1,2,···,n, α_i to be determined. Since there is no cost incurred when a futures is transacted, the investor’s portfolio value at the end of the n^th day becomes

∑_i=1ⁿ α_i (G_i – G_i-1)e^δ(n-i)

On the other hand, since the holder of one unit of the forward contract initiated on day 0 can purchase the underlying asset which is worth S_n using F₀ currency units at maturity, the value of the long position of one forward at maturity is S_n − F₀. Now, we consider the following two portfolios:

Portfolio A :

long position of a bond with par value F₀ maturing on the n^th day

long position of one unit of forward contract

Portfolio B : long position of a bond with par value G₀ maturing on the n^th day

long position of e^-δ(n-i) units of futures held at the end of the (i − 1)^th day, i = 1, 2, · · · n

At maturity (end of the n^th day), the values of the bond and the forward contract in Portfolio A become F₀ and S_n − F₀, respectively, so that the total value of the portfolio is S_n. For Portfolio B, the bond value is G₀ at maturity. The value of the long position of the futures (number of units of futures held is kept changing on each day) is obtained by setting α_i = e^-δ(n-i). This gives

∑_i=1ⁿ e^-δ(n-i)(G_i – G_i-1)e^δ(n-i) = ∑_i=1ⁿ (G_i – G_i-1) = G_i – G₀

Hence, the total value of Portfolio B at maturity is G₀ + (G_n − G₀) = G_n. Since the futures price must be equal to the asset price S_n at maturity, we have G_n = S_n. The two portfolios have the same value at maturity, while Portfolio A and Portfolio B require an initial investment of F₀e^−δn and G₀e^−δn currency units, respectively. In the absence of arbitrage opportunities, the initial values of the two portfolios must be the same. We then obtain F₀ = G₀, that is, the current forward and futures prices are equal.

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Hyperstructures

In many areas of mathematics there is a need to have methods taking local information and properties to global ones. This is mostly done by gluing techniques using open sets in a topology and associated presheaves. The presheaves form sheaves when local pieces fit together to global ones. This has been generalized to categorical settings based on Grothendieck topologies and sites.

The general problem of going from local to global situations is important also outside of mathematics. Consider collections of objects where we may have information or properties of objects or subcollections, and we want to extract global information.

This is where hyperstructures are very useful. If we are given a collection of objects that we want to investigate, we put a suitable hyperstructure on it. Then we may assign “local” properties at each level and by the generalized Grothendieck topology for hyperstructures we can now glue both within levels and across the levels in order to get global properties. Such an assignment of global properties or states we call a globalizer. 

To illustrate our intuition let us think of a society organized into a hyperstructure. Through levelwise democratic elections leaders are elected and the democratic process will eventually give a “global” leader. In this sense democracy may be thought of as a sociological (or political) globalizer. This applies to decision making as well.

In “frustrated” spin systems in physics one may possibly think of the “frustation” being resolved by creating new levels and a suitable globalizer assigning a global state to the system corresponding to various exotic physical conditions like, for example, a kind of hyperstructured spin glass or magnet. Acting on both classical and quantum fields in physics may be facilitated by putting a hyperstructure on them.

There are also situations where we are given an object or a collection of objects with assignments of properties or states. To achieve a certain goal we need to change, let us say, the state. This may be very difficult and require a lot of resources. The idea is then to put a hyperstructure on the object or collection. By this we create levels of locality that we can glue together by a generalized Grothendieck topology.

It may often be much easier and require less resources to change the state at the lowest level and then use a globalizer to achieve the desired global change. Often it may be important to find a minimal hyperstructure needed to change a global state with minimal resources.

Again, to support our intuition let us think of the democratic society example. To change the global leader directly may be hard, but starting a “political” process at the lower individual levels may not require heavy resources and may propagate through the democratic hyperstructure leading to a change of leader.

Hence, hyperstructures facilitates local to global processes, but also global to local processes. Often these are called bottom up and top down processes. In the global to local or top down process we put a hyperstructure on an object or system in such a way that it is represented by a top level bond in the hyperstructure. This means that to an object or system X we assign a hyperstructure

H = {B0,B1,…,Bn} in such a way that X = bn for some bn ∈ B binding a family {bi1n−1} of Bn−1 bonds, each bi1n−1 binding a family {bi2n−2} of Bn−2 bonds, etc. down to B0 bonds in H. Similarly for a local to global process. To a system, set or collection of objects X, we assign a hyperstructure H such that X = B0. A hyperstructure on a set (space) will create “global” objects, properties and states like what we see in organized societies, organizations, organisms, etc. The hyperstructure is the “glue” or the “law” of the objects. In a way, the globalizer creates a kind of higher order “condensate”. Hyperstructures represent a conceptual tool for translating organizational ideas like for example democracy, political parties, etc. into a mathematical framework where new types of arguments may be carried through.

Financial Forward Rate “Strings” (Didactic 1)

Imagine that Julie wants to invest $1 for two years. She can devise two possible strategies. The first one is to put the money in a one-year bond at an interest rate r₁. At the end of the year, she must take her money and find another one-year bond, with interest rate r_1/2 which is the interest rate in one year on a loan maturing in two years. The final payoff of this strategy is simply (1 + r₁)(1 + r_1/2). The problem is that Julie cannot know for sure what will be the one-period interest rate r_1/2 of next year. Thus, she can only estimate a return by guessing the expectation of r_1/2.

Instead of making two separate investments of one year each, Julie could invest her money today in a bond that pays off in two years with interest rate r₂. The final payoff is then (1 + r₂)². This second strategy is riskless as she knows for sure her return. Now, this strategy can be reinterpreted along the line of the first strategy as follows. It consists in investing for one year at the rate r₁ and for the second year at a forward rate f₂. The forward rate is like the r_1/2 rate, with the essential difference that it is guaranteed : by buying the two-year bond, Julie can “lock in” an interest rate f₂ for the second year.

This simple example illustrates that the set of all possible bonds traded on the market is equivalent to the so-called forward rate curve. The forward rate f(t,x) is thus the interest rate that can be contracted at time t for instantaneously riskless borrowing 1 or lending at time t + x. It is thus a function or curve of the time-to-maturity x², where x plays the role of a “length” variable, that deforms with time t. Its knowledge is completely equivalent to the set of bond prices P(t,x) at time t that expire at time t + x. The shape of the forward rate curve f(t,x) incessantly fluctuates as a function of time t. These fluctuations are due to a combination of factors, including future expectation of the short-term interest rates, liquidity preferences, market segmentation and trading. It is obvious that the forward rate f (t, x+δx) for δx small can not be very different from f (t,x). It is thus tempting to see f(t,x) as a “string” characterized by a kind of tension which prevents too large local deformations that would not be financially acceptable. This superficial analogy is in the follow up of the repetitious intersections between finance and physics, starting with Bachelier who solved the diffusion equation of Brownian motion as a model of stock market price fluctuations five years before Einstein, continuing with the discovery of the relevance of Lévy laws for cotton price fluctuations by Mandelbrot that can be compared with the present interest of such power laws for the description of physical and natural phenomena. The present investigation delves into how to formalize mathematically this analogy between the forward rate curve and a string. We formulate the term structure of interest rates as the solution of a stochastic partial differential equation (SPDE), following the physical analogy of a continuous curve (string) whose shape moves stochastically through time.

The equation of motion of macroscopic physical strings is derived from conservation laws. The fundamental equations of motion of microscopic strings formulated to describe the fundamental particles derive from global symmetry principles and dualities between long-range and short-range descriptions. Are there similar principles that can guide the determination of the equations of motion of the more down-to-earth financial forward rate “strings”?

Suppose that in the middle ages, before Copernicus and Galileo, the Earth really was stationary at the centre of the universe, and only began moving later on. Imagine that during the nineteenth century, when everyone believed classical physics to be true, that it really was true, and quantum phenomena were non-existent. These are not philosophical musings, but an attempt to portray how physics might look if it actually behaved like the financial markets. Indeed, the financial world is such that any insight is almost immediately used to trade for a profit. As the insight spreads among traders, the “universe” changes accordingly. As G. Soros has pointed out, market players are “actors observing their own deeds”. As E. Derman, head of quantitative strategies at Goldman Sachs, puts it, in physics you are playing against God, who does not change his mind very often. In finance, you are playing against Gods creatures, whose feelings are ephemeral, at best unstable, and the news on which they are based keep streaming in. Value clearly derives from human beings, while mass, charge and electromagnetism apparently do not. This has led to suggestions that a fruitful framework to study finance and economy is to use evolutionary models inspired from biology and genetics.

This does not however guide us much for the determination of “fundamental” equa- tions, if any. Here, we propose to use the condition of absence of arbitrage opportunity and show that this leads to strong constraints on the structure of the governing equations. The basic idea is that, if there are arbitrage opportunities (free lunches), they cannot live long or must be quite subtle, otherwise traders would act on them and arbitrage them away. The no-arbitrage condition is an idealization of a self-consistent dynamical state of the market resulting from the incessant actions of the traders (ar- bitragers). It is not the out-of-fashion equilibrium approximation sometimes described but rather embodies a very subtle cooperative organization of the market.

We consider this condition as the fundamental backbone for the theory. The idea to impose this requirement is not new and is in fact the prerequisite of most models developed in the academic finance community. Modigliani and Miller [here and here] have indeed emphasized the critical role played by arbitrage in determining the value of securities. It is sometimes suggested that transaction costs and other market imperfections make irrelevant the no-arbitrage condition. Let us address briefly this question.

Transaction costs in option replication and other hedging activities have been extensively investigated since they (or other market “imperfections”) clearly disturb the risk-neutral argument and set option theory back a few decades. Transaction costs induce, for obvious reasons, dynamic incompleteness, thus preventing valuation as we know it since Black and Scholes. However, the most efficient dynamic hedgers (market makers) incur essentially no transaction costs when owning options. These specialized market makers compete with each other to provide liquidity in option instruments, and maintain inventories in them. They rationally limit their dynamic replication to their residual exposure, not their global exposure. In addition, the fact that they do not hold options until maturity greatly reduces their costs of dynamic hedging. They have an incentive in the acceleration of financial intermediation. Furthermore, as options are rarely replicated until maturity, the expected transaction costs of the short options depend mostly on the dynamics of the order flow in the option markets – not on the direct costs of transacting. For the efficient operators (and those operators only), markets are more dynamically complete than anticipated. This is not true for a second category of traders, those who merely purchase or sell financial instruments that are subjected to dynamic hedging. They, accordingly, neither are equipped for dynamic hedging, nor have the need for it, thanks to the existence of specialized and more efficient market makers. The examination of their transaction costs in the event of their decision to dynamically replicate their options is of no true theoretical contribution. A second important point is that the existence of transaction costs should not be invoked as an excuse for disregarding the no-arbitrage condition, but, rather should be constructively invoked to study its impacts on the models…..

Hedging. Part 3. Futures Contract.

Futures

The futures price F(t₀, t_*, T) is given by

F(t₀, t_*, T) = E[t₀, t_*][p(t_*, T)] = ∫DAe^{-∫t_*Tdx f(t_*, x)} = F(t₀, t_*, T) exp {Ω_F (t₀, t_*, T) —– (1)

where F(t₀, t_*, T) represents the forward price of the same contract F(t₀, t_*, T) = P(t₀, T)/P(t₀, t_*) and

Ω_F (t₀, t_*, T) = – ∫_to^t_* dt ∫_t^t_* dx σ(t, x) ∫_t*^T dx’ D(x, x’; t, T_FR) σ (t, x’) —– (2)

For μ = 0, equation 2 collapses to

Ω_F (t₀, t_*, T) = – ∫_to^t_* dt ∫_t^t_* dx σ(t, x) ∫_t*^T dx’ σ (t, x’) —– (3)

which is equal to the one-factor HJM model. For the one factor HJM model with exponential volatility, equation 3 becomes

Ω_F (t₀, t_*, T) = – σ²/2λ³ (1 – e^{-λ(T – t_*)})(1 – e^{-λ(t_* – t₀)})²

Observe that the propagator modifies the product of the volatility functions with μ serving as an additional model parameter. Prices for call options, put options, caps, and floors proceed along similar lines with an identical modification of the volatility functions. Let us take an example and compute the appropriate hedge parameters for futures contract for a period of one year. The proposition expresses the futures price F(t₀, t_*, T) in terms of the forward price

P(t, T)/P(t, t_*) = e^{-∫_t*T dx f (t, x)}

and the deterministic quantity Ω_F (t₀, t_*, T) found in equation 2. the dynamics of the futures price dF (t, t*, T) is given by

dF (t, t_*, T)/F(t, t_*, T) = dΩ_F (t, t_*, T) – ∫_t*^T dxdf(t, x) —– (4)

⇒

(dF (t, t_*, T) – E[dF (t, t_*, T)])/F(t, t_*, T) = -dt ∫_t*^T dx σ (t, x) A(t, x) —– (5)

squaring both sides to the instantaneous variance of the futures price

Var [dF (t, t_*, T)] = dt F²(t, t_*, T) ∫_t*^T dx ∫_t*^T dx’ σ (t, x) D(x, x’) σ (t, x’) —– (6)

This updates the definition in terms of the futures contract.

Definition: Futures Contract: Let F_i denote the futures price F(t, t_*, T) of a contract expiring at time t_* on a zero-coupon bond maturing at time T_i. The hedged portfolio in terms of the futures contract is given by

∏(t) = P + ∑_i=1^NΔ_iF_i

where F_i represents observed market prices. Defining, for notational simplicity,

L_i = PF_i ∫_t*^T_i dx ∫_t^T dx’σ (t, x) D(x, x’; t, T_FR) σ (t, x’)

M_ij = F_iF_j ∫_t*^T_i dx ∫_t^T_j dx’σ (t, x) D(x, x’; t, T_FR) σ (t, x’)

The hedge parameters and the residual variance when futures contracts are used as underlying instruments have identical expressions to the theorem and the corollary.

Corollary: Hedge Parameters and Residual Variance using Futures Hedge parameters for a futures contract that expires at time t_∗ on a zero coupon bond that matures at time T_i equals

Δ_i = – ∑_j=1^NL_jM_ij^-1

while the variance of the hedged portfolio equals

Var = P² ∫_t^T dx ∫_t^T dx’ σ (t, x) σ (t, x’) D(x, x’; t, T_FR) – ∑_i=1^N∑_j=1^NL_jM_ij^-1

for L_i and M_ij in definition.