Valuation of no-negative-equity guarantees with a lower reflecting barrier

Abstract

<jats:title>Abstract</jats:title> <jats:p>If the general level of house prices falls a long way, policymakers may introduce new policies which seek to support prices. This paper considers the effect of such interventions on the valuation of no-negative-equity guarantees (NNEG) in equity release mortgages. I model interventions by a reflecting barrier expressed as a fraction of the current level of house prices. Reflection at the barrier is instantaneous, so the no-arbitrage property is preserved, and hence risk-neutral valuation of NNEG is possible. The reflecting barrier can alternatively be justified as a representation of the different economic nature of the underlying housing (and particularly freehold land) assets in NNEG valuations, compared with the underlying equity assets in many other option valuations.</jats:p>

Arbitrage Problems with Reflected Geometric Brownian Motion

“…NNEG valuation plays a crucial role in the design and management of ERMs, and several valuation approaches have been considered (see Buckner and Dowd 2020, for a survey of NNEG valuation). Thomas (2021) recently proposed a new approach, built on the assumption that the government will intervene in the residential real estate market if property prices fall by more than a certain proportion, thereby establishing a de facto lower bound for house prices. Based on this idea, he proposed an RGBM model for house prices, where the lower reflecting boundary b > 0 corresponds to the level at which the government will enter the property market to prop up prices.…”

“…Such a process is obtained by applying Skorokhod's (1961) construction to a vanilla geometric Brownian motion, causing it to reflect off a lower boundary. Veestraeten (2013), Hertrich (2015), and Hertrich and Zimmermann (2017) have used RGBMs to model exchange rates constrained by central bank target zone policies, while Thomas (2021) recently modelled house prices as an RGBM, under the assumption that the government will support the property market if prices fall enough. Veestraeten (2008) claimed that the RGBM model does not offer any arbitrage opportunities.…”

“…He then obtained an expression for the density of the putative equivalent risk-neutral probability measure and used it to derive pricing formulae for European puts and calls. With some modifications to cater for dividends, and following an amendment to the put pricing formula by Hertrich and Veerstraeten (2013), Veestraeten's (2008) option pricing formulae have been used by Veestraeten (2013), Hertrich (2015), Hertrich and Zimmermann (2017) and Thomas (2021).…”

Long-term option pricing with a lower reflecting barrier

“…The failure of standard put-call parity hints that the barrier model allows some sort of arbitrage. Previous papers (Thomas, 2021;Hertrich & Zimmermann, 2017;Hertrich, 2015;Veestraeten, 2008Veestraeten, , 2013 all noted that the assumed instantaneous nature of reflection rules out the obvious arbitrage of buying when the spot price is exactly at the barrier, and then selling a moment later after it rises. However, this argument renders impractical only immediate arbitrage (arbitrage with no interim losses); it does not exclude arbitrage with interim losses, which requires credit to cover losses which may be incurred before an eventual gain.…”

“…For puts with a lower barrier, Hertrich & Veestraeten (2013) stated the result, and Hertrich (2015) and Hertrich & Zimmermann (2017) gave a full derivation. Thomas (2021) suggested an application to the valuation of no-negative-equity guarantees on equity release mortgages. Each of these papers focused mainly either on call options (Veestraeten, 2008(Veestraeten, , 2013 or on put options (Hertrich, 2015;Hertrich & Zimmermann, 2017;Thomas, 2021), but not both; this may explain why the complications highlighted in the present paper have not previously been considered.…”