Black Scholes

Hi,

What probabilities do d1 and d2 represent in the Black Scholes formulae?

 

If you integrate (St - K) from k to infinity, assuming
St ~ LogNor(Mu = Ln(S0 + [r-s^2/2)*(T-t)], sigma = (s^2)*(T-t)],
using the formula from page 18 of the actuarial tables you will get a solution equal to the Black-Scholes formula.

 

So d2=Pr[St>K]? what about d1?

 

N(d1) = P[S>K] when the payoff of a vanilla European Call Option is a share, an effect of Oxymoron's approach above.

so asset or nothing call has form ct = So*N(d1)

N(d2) = P[S>K] when the payoff of a vanilla European Call Option is cash.

so cash or nothing call has form ct = Qexp(-rT)*N(d2)

Hence German Kohlhagen form for a vanilla European call paying S-K (Share minus cash) has form;-

ct = sN(d1) - kexp(-rT)N(d2)

**d1 and d2 alone aren't probabilities, they are numbers.

 

Phi(d2) = risk-neutral probability that ST > K.

This has proved a really useful result in answering a number of CT8 exams questions.

Phi(d1) = Phi(d2) x Present value of (ST | ST > K) x (1/St)

This doesn't have a nice and neat interpretation as a risk-neutral probability and to date hasn't been of relevance to any CT8 exam questions.

 

Graham Aylott what do u think of my definition of N(d1) above, assuming you are familiar with Oxymoron's approach I.e N(d1) results when you integrate s*f(s) from K to infinity?

 

The whole of the left-hand term in the Black-Scholes formula represents the expected present value of a payment of ST if given ST > K, ie:

EPV[ST | ST> K] = St*Phi(d1).

However, as risk-neutral prob[ST > K] = Phi(d2), it must be the case that:

PV[ST | ST> K] * Phi(d2) = St*Phi(d1)

So:

Phi(d1) = PV[ST | ST> K] * Phi(d2) / St

ie Phi(d1) doesn't have an interpretation as a risk-neutral probability. This is because it also incorporates the (random) growth in St.

As far as the exam is concerned, the important things are to:

(1) be able to derive the pricing formulae for specific derivatives, using the general risk-neutral ("5-step") pricing formula and the formula on p18 in the Tables

(2) remember that Phi(d2) is the risk-neutral prob that ST > K, which can also be useful for valuing derivatives (and 1 - Phi(d2) is the risk-neutral prob of default in the Merton model.)