The Black Scholes Formula

1
The Black Scholes Formula
2
Option Value Determinants

• Call Put
• Stock price
• Exercise price
• Interest rate
• Volatility in the stock price
• Expiration date
• The value of a call option C0 must fall within
• max (S0 EX, 0) lt C0 lt S0. Intrinsic
  value Current Stock Price
• The precise position will depend on these factors.

3
Expanding the binomial model to allow more
possible price changes
Binomial to Black Scholes
1 step 2 steps 4 steps
(2 outcomes) (3 outcomes) (5
outcomes) etc. etc.
4
Binomial Option Pricing Model
The most important lesson (so far) from the
binomial option pricing model is

• the replicating portfolio intuition.

Many derivative securities can be valued by
valuing portfolios of primitive securities when
those portfolios have the same payoffs as the
derivative securities.
5

• The Black-Scholes formula is

Where C0 the value of a European option at
time t 0
r the risk-free interest rate.
N(d) Probability that a standardized, normally
distributed, random variable will be less than or
equal to d.
The Black-Scholes Model allows us to value
options in the real world just as we have done in
the one or two period world.
6
Important Points

• The rate of return is not explicitly in the
  formula
• This is analogous to the point that in the
  two-period model I didnt tell you the
  probability of jumping up or down.
• However, the variable S does implicitly capture
  the expected rate of return of the stock.

7

• www.cboe.com
• Click on trading tools then on volatility
  optimizer and finally on options calculator

8
Implied Volatility

• The BS formula gives us the predicted stock
  price, given historical parameters.
• But option prices are set by supply and demand!
• Taking the opposite approach, we could use the
  actual market price to infer what the market
  thinks future volatility will be.
• This is called implied volatility