Kevin Dowd 27 August 2018
Why do we use the Black ’76 model (Black, 1976) model to price the put options in No-Negative Equity Guarantees (NNEGs) instead of the better-known vanilla Black-Scholes (BS) model (Black and Scholes, 1973; Merton, 1973)?
The reason we don’t use BS is because the BS model assumes that the underlying variable does not earn any yield. In the case of the put options that are used to value NNEGs, the underlying variable is a house, and it makes to think of a house as an asset that bears a continuous yield in the form of a rental rate. For example, if the house is worth £100k and has a rental rate of 2% a year, then the house generates a rental benefit of £2k a year. This rental benefit is the use-benefit of living in the house or the rental income we might get by renting the house out. It should be obvious that this rental rate must be positive: you don’t charge someone a negative rent to stay in your house.
So we need a model that allows for an underlying with a continuous rental benefit, and there are a number we could choose from – we could use the Black ’76 model, or we could tweak the Garman-Kohlhagen foreign currency option model (Garman and Kohlhagen, 1983) or use an appropriate special case of the Margrabe option, the option to exchange one risky asset for another (Margrabe, 1978). All these models are near-relatives of BS and are mathematically equivalent when applied to options on any asset with a continuous yield. So we may as well use the most straightforward model for our purpose and the most straightforward is Black ’76.
The Black ’76 formula for the price of a European put option on a forward contract on a commodity bearing a continuous yield q is
(1) put = exp(-rT) K N(-d2) – exp(-rT) F N(-d1)
where exp(.) is the mathematical exponential function, r is the risk-free rate, T is the maturity of the option (e.g., 10 years), K is the strike price of the option, F is the (current) price of the forward contract, and d1 and d2 are given by:
(2a) d1 = [ln(F/K) + σ^2 t/2]/(σ√t)
(2b) d2 = d1 – σ√t
where ln is the natural logarithm and σ is the volatility of the forward house price.
The term “European” means that the option can only be exercised at the end of the maturity period, and a forward contract on a commodity is a contract in which the price of the commodity (in this case, a house) is agreed now but settled (i.e., paid for) when the house is handed over at some future time T.
The price of the forward contract is
(3) F = S exp((r-q)T)
where S is the spot or current house price and the discount factor r-q is known as the forward rate.
If we wished to, we could use (3) to substitute out F from the put price formula and replace it with the spot price S. Our put price formula is then
(4) put = exp(-rT) K N(-d2) – exp(-rT) S exp(rT) exp(-qT) N(-d1)
= exp(-rT) K N(-d2) – S exp(-qT) N(-d1)
The term S exp(-qT) is known as the deferment house price, the price we would agree and pay now for possession at future time T. (Equivalently, the deferment house price is the present value of the forward price, where the present value is obtained by discounting at the risk-free rate.) The model specified by (4) is mathematically equivalent to that specified by (1)-(3) and we can use either.
Personally, I like the version given by (4) because it highlights the intuition underlying the deferment house price and the key to understanding any model is to grasp its intuition.
Speaking of which, a q > 0 implies that the deferment house price must be less than the current house price. Reasonable people might argue over a suitable value for q, but I would suggest that q should be around 2.5% and definitely not less than 2%, and that a q <= 0 makes no sense i.e. is impossible.
It is then astonishing that standard practice in the UK equity release sector is to work with a forward rate r-q that is taken to be equal to the expected house price inflation rate. So if we take the expected house price inflation rate as 4.25%, as one major firm recently did, then take r as 1.5%, which seems reasonable, we get an implied q rate equal to 1.5% minus 4.25%, which is equal to minus 2.75%.
It also turns out that the assumed q rate makes a big difference to the value of the NNEG. For example, in my Adam Smith Institute report Asleep at the Wheel, I gave the example of a baseline case in which an assumed value of q=2% led to an estimated NNEG value that was 52% of the amount loaned, but the same case with q=minus 2.75, the q rate used by the firm I mentioned, led to a NNEG that was only 3% of the amount loaned. That is some difference and it’s all down to a different assumption about the q rate.
UK equity release firms appear to get very low NNEG valuations because they are using impossibly (not implausibly!) low q rates. But the NNEG is a cost that the equity release firms must bear, so they under-valuing their costs i.e. they are carrying hidden losses.
So the Black ’76 (or equivalent) models make sense in the NNEG context in that they take account of the house price rental, but does this mean that we must use such models?
No.
Firms could use other models such as Monte Carlo or stochastic simulation models. Properly used, such models might allow for a NNEG valuation that takes account of features that Black ’76 and related models do not take account of, such as stochastic volatility or stochastic interest rates. But there are good and bad Monte Carlo models, and if such a model generated results that are different from Black ’76 then the Monte Carlo model and the difference in results should be explained.
What one cannot do is throw stones at Black ’76 (e.g., because one does not like the high NNEG valuations it produces) to justify using some Monte Carlo or any other approach that ‘just’ happens to produce NNEG valuations that are a fraction of those produced by Black ’76.
References
Black, F. (1976) “The Pricing of Commodity Contracts.” Journal of Financial Economics 3: 167-179.
Black, F. and M. Scholes (1973) “The Pricing of Options and Corporate Liabilities.” Journal of Political Economy 81: 637–654.
Dowd, K. (2018) Asleep at the Wheel: The Prudential Regulation Authority and the Equity Release Sector. London: Adam Smith Institute.
Garman, M. K. and S. W. Kohlhagen (1983) “Foreign Currency Option Values.” Journal of International Money and Finance 2(3): 231–237.
Margrabe, W. (1978) “The Value of an Option to Exchange One Asset for Another.” Journal of Finance 33(1): 177-186.
Merton, R. C. (1973) “Theory of Rational Option Pricing.” The Bell Journal of Economics and Management Science 4(1): 141:183.
Prudential Regulation Authority (2018) “Consultation Paper CP13/18 Solvency II: Equity Release Mortgages.” London: Prudential Regulation Authority, July.