V(r, H, k) = A(r, k) - J(r, H, k) J (r, H, k) = D(r, H, k) + C(r, H, k)
to be perfectly competitive and frictionless. There are no capital market imperfections of the kind considered in previous chapters. This perfectly competitive economy is one where it is not possible for traders in financial assets to arbitrage by taking profits on risky securities. Under these circumstances risk adjustments can be made to the price of risky assets such that their rate of return equals the risk-free rate of interest. In the absence of arbitrage all traders can adopt long or short positions in securities to obtain the risk-free rate of return. These risk adjustments are important because we can now value a security (mortgage) by discounting its expected cash flows at the risk-free rate of interest, equivalent to assuming that both borrowers and lenders are risk-neutral.
The general discussion of the option theoretic approach to valuing the risky mortgage noted the role of r and H in determining value. These are the state variables and it is the stochastic behaviour of interest rates and house prices that underpins the option-like characteristics of the risky mortgage. For example, interest rate volatility will determine the value of the option to prepay. It is assumed that the value of the contingent security (the mortgage) has no effect on the fundamental determinants of asset prices (e.g. house price). This analytical framework also means that the personal characteristics and preferences of individuals have no impact upon the value of the risky mortgage.
How then can we represent the stochastic behaviour of the state variables, which underpin mortgage valuation? Equation (9.4) represents the expected behaviour of interest rates. Interest rate changes, dr, are expected to occur at a rate mr(r, H, t) where r is the spot rate of interest, H is the level of house prices and t is a point of time t. The argument st(r, H, t) is the instantaneously adjusted standard deviation. The term dzr is a Wiener process, which ensures that interest rate changes proceed in a random independently distributed unbiased manner, that is they follow Brownian motion. Equation (9.5) shows the same process for the stochastic determination of house prices, dH. The disturbance terms of interest rate and house price changes dzr, dzH may also be correlated through p(r, H, t).
In a perfectly competitive no arbitrage economy continuously traded assets have an expected rate of return that equals the risk-free rate of interest. This is not a problem when we treat property as a continuously traded asset. Finding a risk adjustment for interest rates is a problem dr = mr(r, H, t)dt + sr(r, H, t)dzr dH = mH(r, H, t)dt + oh(i, H, t)dzu
because a positive term structure with interest rates expected to rise can lead to a risk premium added to the current rate. In this case the no arbitrage modelling becomes extremely complex. One solution is to assume that the spot rate of interest contains all of the information implicit in the term structure. This is known as the Local Expectations Hypothesis (LEH).1 This is the assumption adopted here.
The analysis has now established how the state variables behave stochastically through time, and noted that in a perfectly competitive no arbitrage economy risk-neutrality can be assumed. The general framework for evaluating the value of a mortgage can now be considered. That is, the expected value of the risky mortgage can be determined. This in turn will offer important insights into the default and prepayment decisions of households and the relationship between them.
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