The document discusses principles of option pricing, specifically related to puts. It covers: 1) The minimum value a put can have is 0, as it cannot be negative. The maximum value of a European put at expiration is the exercise price times 1 plus the interest rate to the power of time to maturity, while the maximum value of an American put is simply the exercise price. 2) Higher exercise prices and longer times to maturity result in higher put prices, as they provide more value. Interest rates and volatility also impact put prices, with puts having an inverse relationship with interest rates. 3) At expiration, a put's value equals its intrinsic value of max(0, exercise price - stock price

1. Principles of Option Pricing

2. Outline
Minimum values of puts
Maximum values of puts
Values of puts at expiration
Effect of Exercise Price, Time to Maturity,
Interest Rates, Volatility

3. Minimum Value of a Put
A put cannot have a negative value
– P(S,T,E)  0
An American put can be exercise early.
Therefore, Pa  Max [0, (E-S)]
Minimum value also known as intrinsic
value

4. Maximum Value of a Put
At expiration, the payoff from a Put is Max
[0, (E-S)]. What is the best outcome that a
put holder can expect at put expiration?
Maximum value of American put E
Maximum value of European put E(1+r)-T

5. Lower bound of a European Put
Lower bound of a European Put
– Pe  Max [0, E(1+r)-T – S]

6. Time to Expiration and Put
Value
Longer lived American put must always be
worth at least as much as a shorter-lived
American put with the same terms.
Time and Put Price—more complex

7. Exercise Price and Put
 The price of a European put must be at least as high as the
price of otherwise identical European put with a lower
exercise price
– What if Pe(S,Ehigh,T) < Pe(S,Elow,T)
 The price of an American put must be at least as high as
the price of an otherwise identical American put with a
lower exercise price.
 The difference in the prices of two European puts that
differ only by the exercise prices cannot exceed the
present value of the difference in their exercise prices
 The difference in the prices of two American puts that
differ only by exercise price cannot exceed the difference
in their exercise prices

8. Interest Rates and Puts
Put options vary inversely with put interest
rates—Why?
A put is like deferring the sale of stock.
When you sell the stock, you will receive E
dollars. If interest rates rise, the present
value of E dollars will be lower.

9. Early Exercise of American Puts
Does it pay to exercise an American put
before maturity?
If the put is deep in-the-money, it should
be exercised early even if the stock does
not pay dividend

10. Value of Put at Maturity
No time value left
P = Max (0, E-ST)

11. THANK YOU