The following topics were discussed on February
6, 2007 we still have to finish Multiperiod
binomial option pricing
Arbitrage and Option Pricing
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The theory of put/call parity
Long Stock and hedging with options
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2
Calls
Puts
Risk Neutrality and implied branch probabilities
The binomial option pricing model
Multiperiod binomial option pricing
put/call parity
1.
Writing a call exposes you to risk, therefore you cover your
position with a long stock. What is the outcome of this strategy?
……………Synthetic short put…………
In chapter 5 you can read in detail the theory of put/call parity and how the equation was
derived. The PowerPoint presentation overhead 16-30 also deals with this relationship.
Option Risk less Hedge for Long Stock Position
The binomial option pricing model is the next part of chapter 5. The
PowerPoint presentation overhead 31-47
A) With Calls
Long stock and write N calls (for the lower branches intrinsic value is 0)
N=2
N=3
Conclude: Risk increases we need more options to hedge
-------------------------------------------------------------------Call price and risk relationship:
N=2
N=2
Rf = 10%
C= 9.09
C= 13.63
There was a very good question whether we can use risk neutral probability:
Note that if you go ahead and work with risk neutral probability you get the same results.
100% up; intrinsic value=10/1.1=9.09
75% up 25% down = .75*20/1.1=13.63
The consistency of the results shows their equivalence. Thank you for your attention and insights!
B) With Puts
Long stock and buy N puts (for the higher stock prices intrinsic value is 0)
N=2
N=3
Rf = 5%
104.76
P= 2.38
114.28
P=4.76
Note that if you go ahead and work with risk neutral probability you get the same results!
25% down
50% down
10*.25/1.05=2.38
10*.5/1.05=4.76
Risk Neutrality
1. Constructing risk less hedge entitles the investor to risk free return.
Therefore, we want to assign probabilities of up and down movements that
result in an expectation (mean; average) equal to the risk free return. As we
work in two state economy we have to set that Pup+Pdown=1. If we express
stock movements in respect to today’s price (U=Sup/S0 ; D= Sdown/S0 ) we
generalize the formula:
Risk neutrality and implied branch probability is discussed in the Appendix.
The PowerPoint presentation overhead 50-56.
If we use discrete compounding:
We are going to use these probabilities for multiperiod binomial option
pricing.
Stock price today is 100; in one period it is going up 15% or down 10%.
Calculate stock prices in three periods!
Multiperiod binomial option pricing:
Stock price is 100 and each period will go up 15% 0r down 10%.
Construct a three period binomial tree for this stock
Calculate intrinsic value of a call option K=100 at the end of the
third period!
Calculate the risk neutral probability of occurrence of
state “up” and state “down” if the risk free rate is 8%.
0.733148
0.266852
40.05992=(52.09*.7331+19.03*.2668)/1.08
12.91495 =(19.03*.7331+0*.2668)/1.08
30.3854=(40.05992*.7331+12.91495 *.2668)/1.08
8.7671=12.9149*.7331/1.08
22.79==(30.3854*.7331+8.7671*.2668)/1.08
The same example if the strike price is 90. In this case as a question was asked in class
we have intrinsic value at 93.15 and we include that intrinsic value in the calculations.