7.4 Econometric and Mathematical Based Models
7.4.3 FABOZZI-SHILLER-TUNARU MODEL
In Fabozzi et al. (2009) and Fabozzi et al. (2012), an advanced framework was developed for pricing real-estate derivatives combining the econometric approach with the no-arbitrage approach. The idea is to use continuous-time mean-reverting models for the real-estate indices that allow us to harness all the properties known for these type of stochastic processes, serial correlation in particular, and also to subordinate the pricing of derivatives to the risk-neutral valuation that is the main pillar of derivatives pricing.
The main ingredients are the real-estate price index{It}t≥0, the associated log-price process given forYt=logIt, a long-run trend (LRT) term{t}t≥0
such thattis deterministic given the informationFt at timetand smooth enough forddtt to exist, and an interest rate process{rt}t≥0that is generating a risk-neutral money market account described byB0=1 anddBt=rtBtdt. In addition we permit, if need be, an income/dividends cash-flow process{Dt}t≥0
such thatDtis generated by holdingItover the infinitesimal interval[t,t+dt].
The Fabozzi-Shiller-Tunaru (FST) model first assumes that dYt=
dt
dt −θ(Yt−t)
dt+σdWtP (7.71) under the physical measure P. Modelling on the log-price scale ensures the positivity of the real-estate index at all moments in time.
Denoting by Yt=Yt−t the detrended log-price index, the dynamics evolution described in equation (7.71) is given by the solution to the equation dYt= −θYtdt+σdWtP (7.72) which is just the stochastic differential equation of the well-known OU process.
The solution to this equation is given by YT =Yteθ(t−T)+σ
T
t
eθ(u−T)dWu (7.73)
This means that
YT =T+(Yt−t)eθ(t−T)+σ T
t
eθ(u−T)dWuP (7.74) which implies that conditionally
YT|Yt,∼N(my;t,T;σy;t,T2 ) (7.75) with
my;t,T =T+(Yt−t)eθ(t−T) (7.76)
σy;t,T2 = σ2
2θ[1−e2θ(t−T)] (7.77)
SinceIt=eYt, applying Ito’s lemma to (7.71) gives dIt
It = dt
dt −θ(log(It)−t)+1 2σ2
dt+σdWt (7.78)
Denoting the drift side byα(It,t)=
dt
dt −θ(log(It)−t)+ 12σ2 , under the physical measureP,
dIt=α(It,t)Itdt+σItdWtP (7.79) The income stream, if it is paid, is assumed to follow a diffusion process similar to
dDt=δ(It)It+γ (It)ItdWtP (7.80) Remark that the real-estate price index{It}t≥0and the associated income cash- flow process{Dt}t≥0are subject to the same information filtration represented by the one-dimensional Wiener process{Wt}t≥0.
A risk-neutral valuation requires changing the dynamics equations from physical measurePto the risk-neutral measureQ. This switch is realized via the market price of risk processλ≡λ(It,Dt,t). Then, under the martingale pricing measureQwe would have that
dIt= [α(It,t)−λσIt]dt+σItdWtQ (7.81) dDt= [δ(It)It−λγ (It)It]dt+γ (It)ItdWtQ (7.82) In our frameworkλas a function of the real-estate index, the associated income and time can take a wide variety of functional specifications from constant to linear, affine, quadratic, and so on. For practical purposes we take λconstant,δ(It)=δ= constant and alsoγ (It)=γ= constant. This simplifi- cation will allow us to derive analytical formulae for vanilla derivatives such as futures and swaps.
Under measureQit is easier to follow up the modelling on the log-price space. Thus
dYt= dt
dt −θ(Yt−t)−λσ
dt+σdWQt (7.83) dDt= [δ−λγ]dt+γItdWtQ (7.84) Therefore, under the risk-neutral pricing measure Q and with constant parameters
YT|Yt, ∼N(my;t,T;σy;t,T2 ) (7.85) where
my;t,T =T+(Yt−t)eθ(t−T)− λσ
θ [1−eθ(t−T)]
which it can be rearranged as
my;t,T =T− λσ
θ +(Yt−t+λσ
θ )eθ(t−T) (7.86) One can observe that the effect of changing the probability measure from P toQis equivalent to changing only the mean of the log-price by adjusting the long-run trend fromttot−λσθ . The volatility part remains unchanged.
Pricing real-estate futures
Standard asset pricing theory says that the futures prices on the real-estate indexItwith maturityTare given byFt(T)=EQt (IT). From (7.85) it is clear thatItis log-normal distributed with the indicated parameters (see (7.76)) and therefore we get our first important real-estate derivatives pricing results under the FST model
Ft(T)=exp
my;t,T+ 1 2σy;t,T2
(7.87) While parametersθ,σ and the quantitiestandTare considered under the physical measure, the parameterλneeds to be calibrated from futures market prices with corresponding maturities.
While the formula (7.87) looks great from an analytical point of view there is a hidden problem. In practice, only looking at the real-estate indexIt will not help us identify the risk-neutral pricing measureQ. This is not caused by the modelling framework offered by the FST model. It has to do with the fact that trading the real-estate index in the spot market is not possible. Hence, our real-estate derivatives, including futures, are written on an observable but not tradable indexIt.
In essence the futures contracts complete the market, allowing traders to fix the pricing measure Qon a tradable instrument. The measureQis then discovered by reverse-engineering the values of the market price of riskλfrom the quoted market futures prices. The parameterλcan be considered to be time dependent with the desired maturity so thatλ1is recovered from the real-estate futures with maturityT1, thenλ2from the real-estate futures with maturityT2 and so on. More advanced specifications of the market price of riskλ(It,Dt,t) can be investigated by traders.
Hence, the futures contracts play a fundamental role in real-estate markets.
They allow investors, regulators, auditors and so on to recover the implied forward looking market view on futures values of real-estate. Secondly, real- estate futures allow us to fix the pricing measure for other derivatives such as European options, calls and puts, path-dependent options and so on.
The formula (7.87) for the futures price is applicable to a standard difference contract. However, the EUREX futures contract is designed as a total return payoff type of contract. Therefore, in order to be able to compare our futures model prices correctly, we need to calculate the total return model futures price. This is given by the formula
FFST/Eurext (T)=100×Ft(T)
It (7.88)
whereFt(T)is given in (7.87) whileItis the current level of IPD index.
Example 7.4. We describe here how to price the IPD UK All Property futures curve, that is for all five annual maturities in the first week of March 2015. For simplicity we assume that there is no income stream.11First we need to determine the fundamental term t to be used for model specification under the real- measure in(7.71). Figure 7.1 shows the fitting of a linear modelt=α+βt on the log-linear scale for the IPD index at monthly frequency. Analysts may use different specifications for , including those based on lagged real-estate index levels, macroeconomic variables and interest rates as illustrated in Tunaru (2013).
3 3.5 4 4.5 5 5.5 6 6.5 7 7.5
Dec-86 Aug-87 Apr-88 Dec-88 Aug-89 Apr-90 Dec-90 Aug-91 Apr-92 Dec-92 Aug-93 Apr-94 Dec-94 Aug-95 Apr-96 Dec-96 Aug-97 Apr-98 Dec-98 Aug-99 Apr-00 Dec-00 Aug-01 Apr-02 Dec-02 Aug-03 Apr-04 Dec-04 Aug-05 Apr-06 Dec-06 Aug-07 Apr-08 Dec-08 Aug-09 Apr-10 Dec-10 Aug-11 Apr-12 Dec-12 Aug-13 Apr-14 Dec-14
logIPD Linear trend
Figure 7.1.The fitting of a log-linear trend for the IPD UK All Property index between December 1986 and February 2015, monthly data series.
Source of Data: Bloomberg.
11 The parameter estimation will become more involved and it may require more advanced estima- tion methods such as Kalman filtering.
Table 7.8. Regression fitting results for the IPD UK All Prop- erty index, monthly between December 1986 and February 2015. TheR2=95.38% indicates an excellent fit.
Coefficient Estimate Standard Error t-stat
α 4.7839 0.0162 295.77
β 0.0827 0.0010 83.35
Table 7.9. Calibration of Fabozzi-Shiller-Tunaru model for EUREX IPD All Property futures prices on 2,3,4,5 March 2015.
Maturity Dec 15 Dec 16 Dec 17 Dec 18 Dec 19
Futures 117.9 108.25 105.25 104.5 104.5
λ 4.1102 −0.0560 −0.9034 −1.1657 −1.2663
The linear regression fit on the log scale of the IPD index is excellent as illus- trated in Table 8. The R2=95.38%and also the residual analysis indicate an excellent fit.
The next step consists of estimating parametersθandσof the OU model(7.73).
This can be done by using the exact discretization of the OU process and using again a regression model format, with a monthly frequency. For parameter esti- mation one can use either OLS or maximum likelihood. In our case the values of parameter estimates are virtually identicalθ =0.0554andσ =0.038.
Now we will show how to price Eurex IPD UK All Property Total Return futures prices for December 2015, December 2016, December 2017, December 2018 and December 2019 on four consecutive days, 2,3,4,5 March 2015. The current values for the IPD index for March 2015 are It =1165.113 so that Yt=log(St)=7.0605and the fitted fundamental termt=7.1207.
The observed Eurex IPD futures prices are calibrated with the help of the market price of risk parametersλ. The calibration is done one-to-one for each maturity and it is repeated each day. The results are described in Table 7.9. The market price of risk indicates a positive outlook for 2015, which is not surprising given that the March 2015 contract was due to expire very soon, followed by a downward negative outlook for the next four years. The futures prices are matched exactly for those values ofλwhich will fix the pricing measure for other derivatives on the IPD index such as European call and put options.
Pricing real-estate swaps
Now consider pricing a real-estate swap making payments at timest1<t2< . . .
<tn =T. The pricing is done at time t≤t0 where t0 is the start date of the swap.
For simplicity we consider first pricing the swaplet over the period(tk−1,tk].
This swaplet will exchange attk the total returnItk−Itk−1 possibly plus the income over the same periodtk
tk−1estkrududDsagainst a reference interest rate such as LIBOR, operating over the same period, plus a spreadρ, applied to the value of the index at timetk−1. The same set-up was followed in section 7.3.2.
IfLk is the LIBOR rate paid at tk for the period (tk−1,tk] and, without reduction of generality, denotingIk≡Itkfor allk∈ {0, 1, 2,. . .,n}the swaplet payment is given by
Sk=Ik−Ik−1+ tk
tk−1
estkrududDs−k(Lk+ρ)Ik−1 (7.89) wherek=tk−tk−1It is known thatkLk= p(tk−11,tk) −1, wherep(s,u)is the price of a zero coupon bond at timesand maturityu. This implies that
Sk=Ik− Ik−1 p(tk−1,tk)+
tk
tk−1
estkrududDs−kρIk−1 (7.90) Denoting by DF(t,k)=exp(−tk
t rudu) the discount factor at time t for maturitytkthe risk-neutral valuation of the swaplet attis then given by
k=EQt[p(t,k)Ik] −EQt [p(t,k) Ik−1
p(tk−1,tk)] −kρEQt [p(t,k)Ik−1] (7.91) +EQt[p(t,k)
tk
tk−1
estkrududDs]
Clearly quite accurate calculations can be obtained for a wide variety of model specifications for short ratertand taking into account various day-count conventions for k. Without a great loss of generality, in order to simplify the exposition, we shall assume thatrt≡r= constant andk≡= constant.
Then, the value of thekswaplet is
k=e−r(tk−t)EQt[Ik] −e−r(tk−1−t)EQt [e−r(tk−tk−1) Ik−1
p(tk−1,tk)] (7.92)
−ρe−r(tk−t)EQt[Ik−1] +EQt[ tk
tk−1
e−r(s−t)dDs]
The second term can be simplified toEQt[Ik−1]so that k=e−r(tk−t)EQt [Ik] − e−r(tk−1−t)−ρe−r(tk−t)
EQt[Ik−1] +EQt tk
tk−1
e−r(s−t)dDs (7.93)
Buttimer et al. (1997) assume thatγ ≡0 andδ(It)=δ. ThendDt =δXtdt and the value of thekswaplet at timetis
k=ert
e−rtkEQt [Ik] −
e−rtk−1−ρe−rtk
EtQ[Ik−1] +EQt tk
tk−1
e−rsδIsds
(7.94) The CREIL swap price is the value of the spreadρobtained by requiring that
k=n
k=1
k=0.
This gives the formula of the CREIL swap rate as ρ=
k=n
k=1e−rtk−1EQt [Ik−1]−k=n
k=1e−rtkEQt[Ik]−δk=n
k=1EQt[tk
tk−1e−rsIsds]
k=n
k=1e−rtkEQt[Ik−1]
(7.95) If the real-estate index does not provide any extra income (dividends) and it is only a capital based index, or the swap only covers the capital apprecia- tion/depreciation of the index, all that is required is to takeδ=0,γ =0 in the above and derive the same way that the swap price is
ρ= 1
k=n
k=1e−rtk−1EQt[Ik−1] −k=n
k=1e−rtkEtQ[Ik] k=n
k=1e−rtkEQt [Ik−1] (7.96) One can easily observe that the swap price is a combination of futures prices.12 This can be rewritten as
ρ= 1
k=n
k=1EQt
e−rtk−1[Ik−1] −e−rtkIk k=n
k=1e−rtkEQt [Ik−1] (7.97) As in Bjork and Clapham (2002) thenρwould be equal to zero by applying the martingale condition on the process of a discounted underlying asset so the numerator in (7.97). However, this would only work if one could trade costlessly in the underlying “asset” It. Shiller and Weiss (1999), Otaka and Kawaguchi (2003), Fabozzi et al. (2009) and Fabozzi et al. (2012) made the point that it is notcurrentlypossible to trade in the real-estate market as in equity or foreign exchange markets. For a start it is not possible to short sale real-estate directly and furthermore, the real-estate markets are not fungible,
12Under the assumption of constant risk-free raterfutures prices should also be equal to forward prices.
they present long time transaction costs and trading in the spot real-estate market represented by an index is virtually impossible.