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Diff btwn Binomial, Black Scholes Merton, Geometric Brownian Motn



Can anyone give a detailed concept of Binomial, Black Scholes Merton, Geometric Brownian Motion, Monte CArlo for Valuation? Which is applicable in which circumstances?

I'm taking Level 1 this year so I'm not sure if all will be tested and I'm pretty confused by them.

My understanding is:

Simply probabilities of up and down

e-2rt (p2U + 2p(1-p)UD + (1-p)2D) = option price

Black Scholes Merton
- Include lognormal properties

d1 = [ln (S/K) + (r+σ2/2)T] ÷σT

d2 = [ln (S/K) + (r-σ2/2)T] ÷σT = d1 - σT

CALL PRICE = S0N(d1) – Ke -rT N(d2)
PUT PRICE = Ke -rT N(-d2) - S0N(-d1)

Monte Carlo
- Factors into stochastic part of price mvmt
which formula do I need to memorize?

Geometric Brownian Motion
- no idea?



Well-Known Member
Hi skoh,
Binomial from its name implies two states up or down.Binomial model determines future payoffs of a bond or stock using trees where each node points to two possible states.Based on future cash flows and interest rates based on certain probability we derive the price of bond/option or other instrument using backward induction methodology where we discount all the payoffs in the future to present. We need to determine magnitude of up or down movement and than probability of up and down move using several params of volatility and time. We subsequently calculate the payoffs or interest rates in future using these size of movement or probability. And discount these payoffs to present to get the value.
Black scholes merton model is used to value european options(call and put). BSM is derived from delta hedged portfolio earning risk free yield. It is based on arbitrage principle. There are few assumptions like stock price can vary from zero to infinity, constant volatility, constant risk free rate and no dividends etc.
The Monte carle simulation is used to find a value of a option or callable bonds etc. by considering thousands of scenarios in which future can emerge. We can assume distributions of inputs for a certain output with a mean and volatility. The output is itselfa distribution with mean value providing an estimate of the concerened quantity which is to be find.
Finally the Geometric Brownian motion is an stochastic process which is defined by a constant drift and a volatility determined random motion.



I'm still a bit confused by the answer for 13.08(a). How can we derive N(0.008) as N(d2)? Also 13.08C?

And the difference between absolute and relative VaR (13.08d and 13.08e)?