Black model question

[Josu]

Hi guys.... first of all, apologizes if my english is not very good, I'm not a native speaker so it's not easy for me to express ideas but I'll try...

well, I've been studying quant finances for the last 3 months, on my own, with just the classic Hull book. I've been reading about the Black model (or Black 76) which basically it's said to be an improvement of the classic B-Scholes one , as far as I'm concerned, for valuing options on assets which cannot be modelled using the usual lognormal model. I've got a couple of doubts about it.

1. could anybody tell me one academic example of such an asset? (let's forget that empirically, none of them fit lognormal model ) and in that case, why could we assume that future price of the asset would (theoretically) fit the model?

2. this is related with las question on 1. Taking into acocunt relationship between future price and underlying asset, if we assume log returns of forward prices are normal, wouldn't we be assuming too that asset prices log returns are normal too?? (which would contradict what we say that asset price is not lognormal)

thanks very much for the help!!!
And sorry if questions are bit weird, but I'm quite new on this subject...

 

[bob]

Black's formula, as you point out, isn't so much a different model from BS. It's the same model expressed in terms of a zero-coupon bond as the unit of value (called the "numeraire") instead of the money-market account.

To illustrate the difference, consider the simplest of contracts to hedge: a forward contract: You agree to deliver 1 unit of an asset whose spot price is S at some future time T for an agreed-upon amount K. K is chosen so that the contract has zero net value at inception. For the sake of simplicity, suppose S produces no cash flows between now and T.

In the money-market numeraire, the static hedge goes like this: Purchase the asset today for S, and finance the purchase by selling S shares of the money-market asset. When T rolls around, you deliver the asset in exchange for K, using this amount to close your money-market short. So the fair strike K is equal to S shares of the money market asset at any and all future times T. This is another way of expressing the martingale requirement in the risk-neutral measure.

How you actually calculate K is another question; without other asset information, you cannot actually assign a numerical value to K at time zero. In Black-Scholes, you typically infer an interest rate to time T from bond prices and use this to calculate what K actually is.

Consider an alternate way of hedging the contract: Instead of funding by shorting the money-market asset, do it by shorting a zero-coupon bond with maturity T. In this scheme, once again you deliver the asset at T and close the short by paying K.

If the price of the zero-coupon bond today is P_T, then you can explicitly calculate the fair strike K as S / P_T. This is the forward price of the asset, and as you can see is the price of the asset denominated not in terms of the money-market asset, but as a number of units of the zero-coupon bond maturing at T. The measure induced by this asset is called the T-forward measure, and is the measure in which Black's formula operates.

Why is this useful? Consider an interest rate caplet, which is essentially a call option on an interest rate: In this option, we observe at time t an interest rate to tenor (T - t), where T > t. (Usually this is 3-month LIBOR, so that for example we have t = 6 months and T = 9 months.) At T, the contract pays some multiple of (r - r_K)+, where r is the rate observed and r_K is the strike rate.

How would we value this? As seen at time zero, the value of the payoff is the risk-neutral expectation in the money-market numeraire:
V = E_mm [P_T *(T - t) * (r - r_K)+]
...scaled by the deal notional.

This is not at all easy since the discount factor P_T and the forward rate r are related. The resulting payoff function is convex and not fun to deal with.

But if we denominate all assets in terms of the zero-coupon bond maturing at T, we get a much nicer result:
V / P_T = E_Tfwd [(T - t) * (r - r_K)+]

By choosing the risk-free zero-coupon bond as our numeraire, its value ceases to be random: it is 1 at all times. The payoff must of course also be denominated in terms of this numeraire, but doing so actually simplifies the payoff in this case.

This is why Black's formula is used to value caplets and floorlets, as well as swaptions, which choose a different numeraire to achieve the same simplification of the payoff function.

[[Edit to add:]]
I should point out additionally that, typically of finance, there is at least one significant chunk of fine print I have left out here, relating to the "convexity adjustment" discussion that has been proceeding in another thread. I will simply state without explanation, unless it's asked for, that caps/floors and swaptions--as well as the instruments on which they are options: FRA's and swaps--are amenable to this change-of-numeraire technique without requiring any adjustment to the payoff or the forward rate used.

 

[Josu]

Thanks Bob!! Very clarifying... understood now!