قیمت گذاری تعادل عمومی از مشتقات CPI

We examine the issue of pricing forward futures and option contracts written on the Consumer Price Index (CPI), the change of which is a measure of inflation affecting the economy. Traditional approaches postulate an exogenous process for the price level and then derive CPI derivatives prices by standard arbitrage arguments. By contrast, we build the general equilibrium of a continuous time monetary economy that is affected by both real and nominal shocks. The price level and thus the inflation rate are found endogenously and solutions for the prices of CPI derivatives are obtained, which are in closed form in a specialized version of the economy.

On June 21, 1985, the Coffee, Sugar, and Cocoa Exchange in New York proposed for trade a new futures contract on the Consumer Price Index (CPI). This happened many years after several economists urged the regulatory authorities to create such a market because of persistent inflation. However, in spite of original enthusiasm, the CPI futures market experienced little activity and was finally closed in 1991. The reason for the failure was attributed to the non-existence of a primary market for inflation to trade against the futures.1 The continuous issue by the US Treasury of TIPS (Treasury Inflation Protected Securities) since 1997 primed a new interest for CPI derivatives, whether forwards/futures or options. TIPS, although they do not provide a perfect protection against inflation for technical reasons, in particular the indexation lag issue and the taxation of revenues, could in effect be used as approximate devices for hedging such CPI derivatives. At least four other arguments can be advanced for the potential interest of such markets. The first one is related to the current conduct of monetary policy. An important variable upon which the Federal Reserve grounds its policy interventions is inflation expectations on the part of economic agents. CPI derivatives will provide undoubtedly a close to perfect estimator of such expectations, in addition to revealing useful information as to the inflation risk premium and real interest rates. Standard techniques, such as the comparison of yields on nominal bonds with yields on indexed bonds, rely on too many assumptions to deliver estimates robust and reliable enough for policy purposes. The creation of futures/forward markets and options markets would enhance the informativeness of the bond and TIPS markets. A second argument in favor of the development of CPI derivatives markets is the high degree of expertise achieved by present traders, as evidenced by the development of derivative assets written on non-financial underlyings such as electricity, weather, CPU capacity or natural catastrophes. Therefore, even though TIPS provide only imperfect hedges against inflation, this should not be a concern from a trading standpoint. The third argument is the recent revival in the strategic asset allocation literature of the study of how uncertain inflation impacts on long-lived investors’ optimal portfolios. In particular, Campbell and Viceira (2001) in discrete time and for infinitely lived individuals, and Brennan and Xia (2002) in continuous time and for finite-horizon investors derive optimal dynamic portfolio strategies when only nominal assets are available for trade so that there exists no riskless security. It is found that the optimal portfolio composition depends crucially on the stochastic behavior of changes in the investment opportunity set, of which inflation is a critical element. Two complementary recent papers by Kothari and Shanken (2004) and Roll (2004) reinforce this point. They show that, as the correlation between stock returns and TIPS returns is negative, according to both calibrated simulation and empirical tests, including TIPS in an already diversified portfolio significantly enhances its mean–variance efficiency. Also, in a more general setting of a monetary production economy where the (stochastic) inflation rate is endogenously derived, Lioui and Poncet (2004) show that the representative agent’s dynamic portfolio strategy involves a hedge against inflation risk and that the nominal and real pricing kernels (stochastic discount factors) upon which expected returns on all assets depend are affected by random inflation. Finally, the fourth argument is empirical in essence. On the one hand, Roll (2004) finds out that the volatility of the US real rate of interest has dramatically increased in 2001–2003 as compared to the period 1999–2000. On the other hand, the Japanese government has decided to auction ¥100 billion ($920 m) of ten-year inflation-linked bonds on March 4, 2004, and plans to issue a further ¥600 billion of the same by the end of March 2005. This will tap demand from, in particular, pension funds, who want to hedge their future liabilities, which will rise with inflation. And this happens in a country that has been fighting against deflation for several years! The preceding arguments make it useful to have a model of CPI derivatives pricing similar to what already exists for stocks, currencies and commodities. The approach that has been followed so far rests on the standard no arbitrage argument. For instance, Bodie (1990) simply applied the standard Black–Scholes formula to price CPI options. Jarrow and Yildirim (2003) recently went a step further and obtained the price of CPI options when both real and nominal term structures are stochastic. The latter have been derived from the prices of nominal bonds and TIPS, making along the way a simplifying assumption to deal with the indexation lag issue. In addition, the dynamics of the price level was given exogenously and assumed to follow a log normal distribution. Obtaining a credible endogenous price level requires a complete model of a monetary economy affected by both monetary and real shocks. Standard equilibrium models aimed at pricing financial assets or derivatives are essentially grounded on Cox et al. (1985a), CIR thereafter. Although theirs is a real economy, their results have been extensively used as if they were derived in a genuine monetary economy. This is of course legitimate only if the inflation rate is the constant zero. The best example is CIR’s (1985b) own model of the term structure. Most monetary extensions of CIR (1985a) simply added an exogenous process for the price level and derived equilibrium values. Other attempts, like Bakshi and Chen (1996) and Basak and Gallemeyer (1999), derived an endogenous price level. However, money is neutral in their respective economies, and their results are therefore identical to the ones that would obtain by simply using CIR (1985a) plus an appropriate exogenous process for the price level. Buraschi and Jiltsov (2004) extend Bakshi and Chen’s (1996) analysis to a production economy with taxes. Introducing depreciation of the firms’ capital that is imperfectly indexed on inflation yields money non-neutrality. However, once this (possibly important) friction is removed, money does not affect the dynamics of physical capital any more in this model and recovers its neutrality. In a recent paper, Lioui and Poncet (2004) extended CIR’s real economy to a monetary economy affected by both real (technological) and monetary shocks. Although their monetary economy is frictionless, money non-neutrality obtains as an inherent feature. Monetary shocks are transmitted to the real sector through changes in real wealth and interest rates. Because money has a positive value at equilibrium, economic agents hold real cash balances from one period to the next. The effective ex post opportunity cost of holding one unit of real balances between two dates is the real interest rate plus the realized rate of depreciation of the purchasing power of money. As of time t, the nominal interest rate represents a deterministic cost from t to t + dt, while the effective cost of holding money from t to t + dt is random if inflation is stochastic. To the extent that either real output or the money supply process, or both as here, are stochastic, so must be the inflation rate. Money thus is a risky asset. The volatility of its (negative) real rate of return affects the investor’s wealth dynamics, and, consequently, all his or her optimal decisions. Since the volatility of inflation is affected by monetary policy, so will be all endogenous variables. We adopt a monetary economy that is a special case of Lioui and Poncet (2004). While they focused on the term structure of interest rates, we focus here on asset pricing and thus derive the equilibrium nominal and real pricing kernels. Our economy is simpler than theirs in that we start with a single (nominal) state variable instead of an arbitrary number (in their most general version) and of a nominal state variable and a real one (in their specialized version). Furthermore, we assume away, in the penultimate section, the nominal state variable and work with lognormal processes in order to obtain closed form solutions. We thus are able to price, in implicit then explicit forms, CPI derivatives, namely forward or futures contracts and European options written on the price level. Incidentally, the methodology developed here could be used to price a wide range of primitive and derivative assets. The remainder of the paper is articulated as follows. Section 2 provides the main assumptions and the structure underlying our monetary economy. The nominal and real pricing kernels are obtained and discussed in Section 3. All equilibrium values, and the properties of CPI derivatives prices, are derived and discussed in Section 4. Closed form solutions for CPI option prices are computed and commented on in a special case. The last section concludes. Most proofs are gathered in a mathematical Appendix.

We have built the general equilibrium of a continuous time monetary economy that is affected by both real and nominal shocks and in which money is inherently not neutral. We have derived implicit formulas for the equilibrium values of the real and nominal pricing kernels, the real and nominal interest rates, the inflation risk premium and the price level. A cash-and-carry type relationship still holds for forward contracts on the CPI but not for futures. The Black–Scholes–Merton type formulas for options cannot be obtained in general. Numerical methods or Monte Carlo simulations (now routinely applied and mastered) are then required and the practitioners are left with the usual tradeoff between realism and tractability. When the economy is specialized so that all processes are lognormal, closed form solutions for the prices of all derivatives – forwards, futures, and options – written on the CPI have been obtained, in particular Black–Scholes–Merton type formulas for European options. However, since the CPI index itself is not traded, the derivations cannot rely on standard arbitrage arguments but necessitate, and justify, the general equilibrium approach adopted here. Since inflation impinges on economic agents’ behavior (in terms of both consumption and portfolio allocation) and welfare, and present TIPS are imperfect hedges against uncertain inflation, the introduction of such CPI derivatives should be Pareto-improving. If the gain might be minimal in countries experiencing low and stable inflation rates, in other countries it could be substantial and worth the effort to design such new instruments.