Options Pricing: addon

1
Options Pricing addon

• Riccardo Colacito

2
Setup of the problem

• Stock price today is 100
• Two future dates
• At each date
• Stock price can go up by 10
• Stock price can go down by 5
• Call strike price is 110
• Risk free rate for each period is 5

3
Binomial Tree whats the price of the stock at
each date and state?
121
110
100
104.50
95
90.25
4
The recipe

• After reading this set of slides you should be
  convinced that pricing calls in this exercise is
  just identical to what we did in the simple
  binomial tree with only two dates.
• The only secret is that you have to decompose the
  problem into a bunch of simple two dates
  problems and apply the standard recipe that is
  reported on the next slide.

5
Option pricing using hedge ratio

1. Find hedge ratio
2. Find tomorrows value of portfolio made of H
   share and one written call
3. Find present value of portfolio (discount at risk
   free rate)
4. Set this value equal to todays cost of portfolio
   and solve for call price

6
Start from the end!
121
(call value 11)
110
100
104.50
(call value 0)
95
90.25
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Call price when stock is worth 110

1. Hedge ratio
2. Date 3 portfolio value
3. Discounted value
4. Rule out arbitrage

8
Start from the end!
121
110
100
104.50
(call value 0)
95
90.25
(call value 0)
9
Call price when stock is worth 95

• Here you can proceed brute force as we did
  before, or use a shortcut.
• Heres the shortcut if todays price is 95, the
  option tomorrow is going to expire out of the
  money no matter what happens. This option is
  worthless, hence its price should be zero!
• That is C950.

10
Now work on the initial date
We found these values before!
121
110
(call value 6.984)
100
104.50
95
(call value 0)
90.25
11
Call price when stock is worth 100

1. Hedge ratio
2. Date 2 portfolio value
3. Discounted value
4. Rule out arbitrage

12
Value of the call option at each date and state
11
6.984
4.434
0
0
0
13
Black and Scholes

• In a nutshell, the Black and Scholes option
  pricing formula is based on the binomial asset
  pricing model, by
• letting the distance between two consecutive
  dates shrink (think about two consecutive nodes
  being one millisecond away from each other!)
• Extending the tree to a large number of nodes