“equation 10.”

There is a paper they wrote in August 1970, I think still not published, where equation (4) is substantially the same as eq. 10. This was just before they met up with Merton.

The genesis of the whole thing is fascinating. Black discovered what we now know as the PDE in 1968 or 1969, but incredibly Black, despite having a PhD in physics, had never studied differential equations. From what I can gather from the historical evidence they eventually solved it by taking Case Sprenkle’s formula, which is substantially the same as the BS formula except with empirically, rather than theoretically derived discount rates, and differentiated it.

In the 1970 paper they write “By working with the formulas developed by Sprenkle *we found a solution* to the differential equation and boundary condition.”

Black thought, on the basis of no-arbitrage arguments, that the appropriate discount rate was risk free. At that point they were ignorant of stochastic calculus. Enter Merton, who introduced them to the work of McKean, which was an introduction to Western readers of the work of Ito. See FN 5 of BS 1973.

The employment of the c.n.d. function is due to Case Sprenkle, probably the first to do so. If you look at Churchill 1963 p.155, referenced on p.644 of BS 1973, you see the function does not appear anywhere, nor AFAICS is it anywhere to be found in the original edition of Churchill 1941 (*Fourier Series and Boundary Value Problems*). In the 1970 paper they write “Sprenkle … shows that equation (8) can be simplified if it is written in terms of the cumulative normal density function N(d)”. Equation (8) is obviously an integration.

Churchill nowhere mentions probability, that I can find, which is naturally associated by the cnd function, but not otherwise. Churchill was mostly writing about heat functions. Note also, as Sprenkle suggests, the c.n.d in those pre-computer days could only be evaluated through tables. Does anyone here remember tables? I imagine anyone who graduated in the 1970s would. I hated them.