Chapter 26 - BSM equation

Question 1

Very broadly speaking, what did Black and Scholes assume that made them arrive at the PDE

Answer 1

The assumed that stock prices follow geometric brownian motion. Then they could use Ito’s lemma to relate how the option value is affected by the stock price movement.

This yield a PDE with some trickery.

Question 2

what is the difference between black scholes equation and black scholes formula

Answer 2

The equaiton is the PDE.

The formula is the solved PDE that spits out call value.

Question 3

how does this relate to the greeks

Answer 3

Vt is the theta

Vss is gamma

Vs is delta

Question 4

name the assumptions we have made to build the black scholes equation

Answer 4

1) Stock prices follow geometric brownian motion
2) Constant volatility
3) Underlying asset pays continuous dividends
4) The shit we compute price for pays no dividends, and its price is determined based on the price of the underlying
5) The interest rate is fixed
6) No transaction costs (delta hedge possible)

Question 5

What do we need to determine the correct FORMULA for an option given that the stock prices follow geometric brownian motion?

Answer 5

the prices must follow hte black scholes equation, and satisfy boundary conditions.

however, there are varying kinds of boundary conditions that are used?

Question 6

what is the role of the expected return on the stock in the option valuation using black scholes

Answer 6

None. Doesnt need it. Only need the risk free rate.

We can basically assume that we are in the world where it is easiest to value the option, which would be the risk neutral world.

Question 7

what is the challenge with jumps in regards to options?

Answer 7

The relience on delta hedging is not suitable. Delta hedge only works for small movements.

Question 8

what assumption is made by Merton to handle the jumps?

Answer 8

Assume that the jumps are idiosyncratic and therefore diversifiable.