BSM equation

Question 1

elaborate on the traditional valuation equation

Answer 1

We have that the value of a stock is equal to appreciation in stock price plus dividends payed over the time, then discounted.

S(t) = 1/(1+r) [D(t+h)h + S(t+h)]

This can then be written as

S(t+h) - S(t) + D(t+h) h = S(t)r

Simply: Change in stock + cash payout must equal stock price times the total return of stock

We then divide by h and let h go small

dS(t)/dt - D(t) = rS(t)

Question 2

when we use the differential equation to solve shit, what do we need?

Answer 2

Boundary conditions. Terminal or initial or both. Depends on the thing we’re working iwth.

Question 3

elaborate on the choice between terminal and initial boundary conditions

Answer 3

It depends on what we have.

If we want to price a bond and we know that it will pay 1USD at maturity, we have the termination condition. It makes no sense to talk about initial conditions, because this is what we are trying to solve for.

Question 4

what is the black and scholes problem

Answer 4

A market maker selling a call that buy shares to offset the risk

Question 5

what is the point of this chapter?

Answer 5

To understand the movements

Question 6

build the valuation function in differential form

Answer 6

We start by the classical case: The total return on some asset, when discounted appropriately, equals the price. for some stock, this is:

S(t) = (D(t+h)h + S(t+h))/(1+r_h)

this can be written as:

S(t) (1+r_h) = D(t+h)h + S(t+h)

S(t) + S(t)r_h = D(t+h)h + S(t+h)

S(t+h) - S(t) + D(t+h)h = S(t)r_h

Divide by h, and let h grow small

dS(t)/dt + D(t) = rS(t)

Here we have a differential equation that explains how the fucntion S, which is the stock price, move as a function of time.

Question 7

the valuation function is given as:

dS(t)/dt + D(t) = rS(t)

what is the purpose of this?

Answer 7

We can solve it if we have boundary conditions. The boundary conditions provide key elements on the characteristics of the stock.

For instance, if we know with certainty that the stock is worth X at expiration, and it doesnt pay dividends, then the equation is reduced to:

dS(t)/dt = rS(t)

Solving it:

dS(t) = rS(t) dt

1/S(t) dS(t) = r dt

integrating

ln(S(t)) = r t + C

S(t) = Ce^(rt)

With the bounadry condition of certainty, we get:

S(T) = Ce^(r(T-T)) = X

C = X

S(t) = Xe^(r(T-t))

I got the signs mixed.