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OptionsFinancial Options
• A financial option gives its owner the right (but not the obligation)
to purchase or sell an asset at a fixed price at some future date.
• Puts
• Calls
• Strike price/Exercise price
• American/European
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OptionsFinancial Options
Option Pricing• Binomial - Two state single period - Law of one price - Replicating portfolio Call option, SP 50, No dividend, Stock will either rise by 10 or fall by 10 Risk free rate is 6%
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OptionsFinancial Options/Pricing
Stock Bond Call
60 1.06 max (60-50,0) = 10
0 1
Stock 50 Bond 1 40 1.06 max (40 -50,0) = 0
60 = up and 40 = down S = Share price and t = number of shares and B = investment in the bond. SP = 50
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OptionsFinancial Options/Pricing
• Value of portfolio containing the stock and the bond must = the value of the portfolio in each state.
• 60St + 1.06B = 10• 40St + 1.06B = 0• So St = .5• And B = - 18.8679• 60 x .5 - 1.06 x 18.8679 = 10• 40 x .5 – 1.06 x 18.8679 = 0
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OptionsFinancial Options/Pricing
• Generalising we get
• St = Cu – Cd and B = Cd – Sdt
Su – Sd 1+rf
This gives us the replicable portfolio
The Call option price then follows
C = St + B or 50x.5 – 18.8679(1) = 6.13
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OptionsFinancial Options/Pricing
• But what about multi period models?
• Strike price of 50, Rf = 6%
0 1 2 Periods
50
30
40
60
40
20
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OptionsFinancial Options/Pricing
• We start at the end and work back
1 2
50
60
40
Max(60 -50,0) = 10
Max 40 -50,0) = 0
This is the same as before therefore St = .5 and B = -18. 87 and the call value at time 1 is 6.13
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OptionsFinancial Options/Pricing
• What if share dropped to 30 in the next period (period 1)?
30
40 Max (40 – 50, 0) = 0
20 Max (30 – 50, 0) = 0
The option is worthless in both states so no portfolio value
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OptionsFinancial Options/Pricing
• Now move back a period
0 1
40
Stock Call
50 6.13
30 0
Now work out replicating portfolio at time 0
St = Cu – Cd = 6.13 – 0 = 0.3065
Su – Sd 50 – 30
B = Cd – Sdt = 0 - 30(0.3065) = - 8.67
1 + rf 1.06
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OptionsFinancial Options/Pricing
• So the Call value at Time 0 is
• C = St + B = 40(0.3065) +(-) 8.67 = 3.59
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OptionsFinancial Options/Pricing
• For European options if we let each period shrink to ‘zero’ and have an infinite number of periods then we may use the Black-Scholes formula to calculate the binomial pricing…………..
• but we won’t
• But remember the important factors in the pricing
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OptionsFinancial Options/Pricing
• The strike price
• The stock price
• The exercise date
• The risk free interest rate
• The volatility
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OptionsReal Options
• And there are Real Options
The right to take a particular business decision e.g. a capital investment decision.
Main distinction is that the asset is normally not traded
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OptionsReal Options
• Until now we have considered a stream of cash flows during the project, starting from today, to determine the NPV
• But what about alternatives such as delaying the start or abandoning the project after a while?
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OptionsReal Options
• To analyse the alternatives we need Decision Trees
A graphical representation of future decisions and uncertainty resolution (B & DeM)
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OptionsReal Options
• Meet (re meet) Megan
• Goes to markets
• Sells, average profit 1,100
• Costs of Booth 500, in advance
Go to meet
Stay at home
profit
1,100 – 500 = 600
0
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OptionsReal Options
• Now add some uncertainty
• If it rains (25% chance) she will make a loss = -100
• If it is sunny her profit is higher = 1,500
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OptionsReal Options
Go to meet
sunshine 75%
1,500
Stay at home
Rain 25% - 100
Decision node
Information node
0
- 500
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OptionsReal Options
• Go to meet or not?
Don’t pay for booth
Pay for booth - 500
Sunshine 75%
Rain 25% Go to meet
Stay at home
Stay at home
Go to meet1,500
0
-100
0
0
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OptionsReal Options
• So what is the ‘value’ of this real option to Megan?
Expected profit without choice i.e. go to meet regardless
= 1,500 x .75 +-100 x .25 = 1,100
Expected profit with choice
= 1,500 x .75 = 1,125
So choice/option worth 25
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OptionsReal Options
• Should Megan pay for the booth?
Expected profit will be 1125 -500 = 625
So Yes
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OptionsReal Options
• When else may they be used?- Option to delay Invest now only where NPV is
substantially greater than zero But What are costs of delay? What is volatility? What are costs of investment?