Category: derivatives

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Libor flat

I mentioned David Land’s bemused question to the Equity Release working party yesterday. If the working party hasn’t yet fixed the right method of calculating the forward, isn’t that a pretty major source of possible error?

No coherent answer emerged, but Land raised an interesting point. If we can’t lower the value of the no negative equity guarantee by putting in an optimistic growth forecast, perhaps we can tweak the funding rate instead? He drops a hint when he suggests that there’s a large range of possible funding rates that you could think about, and that ‘The PRA thinks that you could possibly fund a house at Libor flat, which seems remarkably difficult’.

Nice try, but there is a problem with that idea too.

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Time decay

Most people with even the slightest familiarity with option pricing will know of time decay, or the tendency of the option value to decrease through time independent of any interest rate effect. The chart above shows the price of call option through 40 years, struck at 100 with the underlying price also constantly at 100, and with interest rate set to zero to remove the effect of interest carry. I.e. the only change to the model is the time to expiry. The effect of time decay or theta is apparent.

Now I have discussed in several posts, such as here, how a real option can be replicated by means of a synthetic option – a series of linear positions in the underlying market adjusted frequently to match the delta or sensitivity of the real option.

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An ingenious objection

Ingenious objections are always interesting, particularly when they are wrong. An actuary has objected to my post here, where I considered a put written at 90 on a price series which goes 95,96,95 etc. The point of the post was to show that the standard formula works perfectly well for such a distribution. The objector objects that if the series really does oscillate between 95 and 96, the price will clearly never reach 90, so the standard pricing formula must be wrong. The true price of the put must be zero.

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Weird distributions #2

The Institute says, in its reply to CP 13/18  (p.10), that

Using the Black-Scholes formula in pricing NNEG will affect the cost of the guarantee, since allowance is not made for the features of mean reversion, momentum and jumps described above. Under geometric Brownian motion the volatility increases with the square root of time while for other models it does not; the value for long term derivatives such as NNEG could materially differ from that assumed under the Black-Scholes model.

This second article on strange distributions discusses the mean reversion claim.

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Weird distributions #1

Turn to any textbook treatment of the Black-Scholes model, and you will find a list of things that the model ‘assumes’. Wikipedia is no exception. These divide into assumptions about the market, such as no arbitrage, ability to short sell etc., which I shall set aside for now, and assumptions about the asset process. Foremost among these are that

  • Future returns are independent of past values, i.e. the process is random
  • Log returns are Gaussian, or normally distributed
  • Volatility is constant
  • Drift is constant

Now it is true that if these conditions are satisfied, then the model will work (I shall discuss an appropriate sense of ‘work’ below). That is, these are sufficient conditions (p implies q). But it is also commonly assumed1 that if they are not satisfied, then the model will not work (not p implies not q), i.e. it is assumed that the conditions are necessary, as well as sufficient.

Nothing could be further from the truth.

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Why the forward price has an interest rate term

Another day, and another challenge to the method we have proposed for the valuation of the no negative equity guarantee. In our presentation here (slide 9, equation 8) we use the term r-q in the calculation of the forward house price at time t, so aren’t we incorporating both a house price growth assumption (r = interest rate) and a net rental rate q? Thus aren’t we – implausibly – assuming that future house price growth will be equal to the interest rate?

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