7.4 Econometric and Mathematical Based Models
7.4.1 SHILLER-WEISS LOGNORMAL MODEL
One of the very first models on pricing real-estate derivatives was proposed by Shiller and Weiss (1999) in their paper on home equity insurance. Their model for the observed real-estate log returnsln(Ht)is a standard AR(1) process
ln(Ht)=c+ρln(Ht−1)+εt (7.29) This model is clearly in the opposite direction of a random walk, implying that price changes tend to continue through time leading to price inertia. The unconditional mean of the annual log price change is 1−ρc . The model can be rewritten equivalently in moving-average form
ln(Ht)− c
1−ρ =εt+ρεt−1+ρ2εt−2+. . . (7.30) Using this model, the conditional returns and volatilities can be determined analytically.
Because the errors in (7.30) are serially uncorrelated the variance ofln(Pt) is the sum of variances of the terms on the right side. Simple algebra gives
Et[ln(Ht+n)] −Et−1[ln(Ht+n)] = 1−ρn 1−ρ εt
Hence, the innovation proportionality toεtis larger the higher then. Therefore, the Shiller-Weiss model indicates an advantage of hedging real-estate risk with long-term contracts.
To price real-estate futures and options Shiller and Weiss (1999) use the model (7.29) and realize that the terminal distribution of the log prices is Gaussian. This assumption was considered sufficient and relates to options pricing literature before the Black-Scholes model was published in 1973.
Denoting byHthe value of the real-estate index, byμthe expected change in the log real-estate price index between today and T periods from today and by σ2 the variance of the change in the log real-estate price index between today andTperiods from today, standard econometric calculations lead to
var(ln(Ht))= ρ2 (1−ρ2)
Note that since the model (7.29) assumes autocorrelation of consecutive returns the variance of logarithmic returns at different horizons will have to be calculated individually.
For practical implementation it is important to calculate the mean and vari- ance at timetof the future valueHTof the real-estate index. Taking advantage of the lognormally model the following propositions are useful here.
Proposition 7.1.
Et(lnHT−lnHt)=(T−t)c+(T−t−1)ρc+ρ1−ρT−t
1−ρ lnHt (7.31) wherelnHt=lnHt−lnHt−1.
Proposition 7.2.
vart(lnHT−lnHt)= σε2 1−ρ2
⎡
⎣(T−t)−ρ21−ρ2(T−t)
1−ρ2 +2
1≤i<j≤T−t
(ρj−i−ρj+i)
⎤
⎦
(7.32) Shiller and Weiss (1999) price only European derivatives on residential real- estate. Thus, we can assume without loss of generality that all quantities in the model refer to a given maturityT. In essence this means that the total returnR= HHTt is lognormally distributed with meanμand varianceσ2. The following known results are useful for option pricing.
Proposition 7.3. If U and V are two random variables and U=aV, with a being a real number different from zero then6fU(u)= 1afV(u/a).
Proposition 7.4. If ln(R)∼N(μ,σ2)then the cumulative distribution func- tion of random variable R is FR(x)=N ln(x)−μσ
. Moreover, ∞
K
ufR(u)du=eμ+σ
2 2 N
μ−ln(K)
σ +σ
and
K
0
fR(u)du=N
ln(K)−μ σ
In addition, if Ht is a constant, HT =HtR is lognormally distributed with parametersμ+lnHtand varianceσ2.
For pricing a put option the calculations go as follows, whereTis the time to maturity from todaytandHT =RHt,
Put(Ht,K,T,μ,σ,r)=e−r(T−t)E[max(K−HT, 0)] =e−rT K
0 (K−HT)f(HT)dHT
=e−r(T−t)
K
0
f(HT)dHT− K
0
HTf(HT)dHT
=e−r(T−t)
KN
ln(K)−μ−ln(Ht) σ
−eμ+ln(Ht)+σ2/2N
ln(K/Ht)−μ
σ −σ
=e−r(T−t)
KN
ln(K/Ht)−μ σ
−Hteμ+σ2/2N
ln(K/Ht)−μ
σ −σ
The Shiller-Weiss formula for a European put option on the real-estate index with strike priceXand maturityTis
Put(Ht,K,T,μ,σ,r)=Ke−r(T−t)N
ln(K/Ht)−μ σ
−Hteμ+σ
2
2−r(T−t)N
ln(K/Ht)−μ
σ −σ
(7.33) Thus, what Shiller and Weiss do is replace the risk-free rate of returnrwith the expected real-world returnμand the implied volatility with the estimated value from the AR(1) model. Their model falls onto the Black-Scholes model only whenμ=rT−σBS2 T/2 andσ2=σBS2 T. It also falls onto the McDonald
6fdenotes generically here the probability density function of the corresponding variable.
and Siegel (1984) pricing formula for options whose underlying asset earns a below-equilibrium rate of return if their equilibrium rate of return equals the risk-free rate.
One can easily derive under the Shiller-Weiss model the price for a European call option with strike priceKand maturityT. This is
Call(Ht,K,T,μ,σ,r)=Hteμ+σ
2 2−rTN
ln(Ht/K)+μ
σ +σ
−Ke−rTN
ln(Ht/K)+μ σ
(7.34) In addition, the formula for the real-estate forwards is the same as the call with strikeK =0, so
Fwd(Ht,T,μ,σ,r)=Hteμ+σ
2
2 (7.35)
In addition, the Shiller-Weiss formulae for forwards, call and puts verify the put-call parity.
This model has the advantage that it starts from a realistic econometrical behaviour of a real-estate index, capturing the serial correlations that are well documented in real-estate literature. One criticism that has been brought to light by some authors is that the pricing of real-estate derivatives is not done under a risk-neutral measure. In order for someone to apply this model they need to estimate the investor-required rate of return driven by the parameterμ. This shortcoming was solved later on with a more advanced derivatives pricing framework.
Example 7.2. Assume that an investor is looking to trade forwards and Euro- pean put and call options with various maturities on the IPD UK Monthly Index using the Shiller-Weiss model.
The trading date is 2 March 2015 and there are five annual maturities starting with March 2015 up to March 2019. The forward contracts are standard over- the-counter forwards contracts with linear payoff.7The risk-free rate is 0.5% and the estimation of the model parameters from the historical time-series of the IPD UK monthly index gives c=0.00070688, ρ=0.9028;lnHt=0.0129;σε2= 0.000022053. Applying the formulae(7.31) and (7.32)allows the investor to calculate the forward prices using formula (7.35) and the European call and put options prices with formulae(7.34)and(7.33), respectively. The results for K=Ht, that is at-the-money options, are described in Table 7.4.
Example 7.3. Consider now the example of a house owner in UK who would like to hedge the value of her house against a possible market crash. Her house is
7This should not be confused with the EUREX futures prices.
Table 7.4. Calculation of forwards, European call and put options prices on IPD UK Monthly index using Shiller-Weiss model.
Maturity μ σ2 Forward Call Put
March 2015 0.0123 2.2053e-05 1179.61 14.43 0.0072
March 2016 0.1049 0.0080 1299.25 131.29 5.5991
March 2017 0.1435 0.0286 1364.27 193.48 17.7239
March 2018 0.1662 0.0543 1413.65 234.77 28.2100
March 2019 0.1843 0.0816 1459.22 265.77 35.5700
valued at 210,000 GBP in February 2016 and she is looking to use either the Halifax House Price Index or the Nationwide House Price Index for buying put options up to five years ahead. First, the historical time series monthly and seasonally adjusted for the two main real-estate residential UK indices are used to calibrate the Shiller-Weiss model parameters. Then, put option prices are calcu- lated and hedge ratios applied to see how much it would cost the house owner to get the protection via the real-estate put options on those two indices. A heuristic analysis reveals that for the Halifax Index the hedging ratio for a property with value X is given by1000/3given that Xt≈Ht× 10003 so she would need to buy roughly 333 put options written on the Halifax index in order to be hedged in full for her house of 210,000 GBP. Likewise, for Nationwide the hedging ratio is roughly1000/2so 500 options need to be bought. The risk-free rate is considered to be 0.5%, which is equal to the Bank of England Base rate for February 2016.
The results in Table 7.5 show some interesting findings. The price of puts first increases with maturity and then decreases. This is a feature of the autoregressive model specification. Secondly, the hedging costs for the house owner are annual so the monthly hedging costs are quite manageable. Furthermore, the crash costs, calculated from the put options with a strike price equal to 70% of the current value of the real-estate index are effectively zero, for both indices. Thus, the Shiller- Weiss model implies that a property crash in the UK is unlikely in the next five years. The hedging costs seem smaller for the Nationwide index. There are significant differences between the two major UK residential indices as explained in Chapter 2 and therefore there is a degree of model risk involved. In addition, these calculations are also dependable on the goodness-of-fit of the AR(1) model to the Halifax and Nationwide log-return indices.