In [[Principles of Corporate Finance]] a replicating portfolio is defined as the combination of:
* Borrowing money
* Buying a ratio of stock
Such that you replicate exactly the payoff from one call option. The number of shares need to replicate one call is called the ''option delta'' or ''hedge ratio''.

Take the example of a stock with has a current stock price of \$80,- with a maximum downside potential of 25% drop and an upside of 33% increase. This means that the stock can move between 
* \$60,-
* \$106,67,-

A call option for \$80,- in this case would be moving between:
* \$0,- (in the case there the stock price was below \$80,-)
* Between \$0,- and \$26,67 if the stock price was moving between and  \$106,67,-
* \$26,67 (in the case the stock price was exactly \$106,67,-)

How should we value such a call option? There are two methods of calculation:

!Calculate option delta
The option delta can be calculate as follows:

$\Large \text{Option delta}$ = $\Large \frac{\text{Spread of option prices}}{\text{Spread of share prices}}$=$\LARGE \frac{26.67 - 0}{106.67 - 60}$=$\LARGE \frac{26.67}{46.67}$=$\LARGE \frac{4}{7}$

$\LARGE \frac{80 - ((4/7) \cdot 80)}{1.025}$= $\Large 33.45$

A more intuitive way of calculating the amount of money that needs to be borrowed is to simply subtract 

The value of the call is then:

$\Large \text{Value of call}$ = $\Large \text{Value of (4/7) shares}$ -  $\Large \text{33.45 bank loan}$ =
 = $\Large 80 \cdot (4/7) - 33.45$  =  $\Large 12.26$

!Risk Neutral Valuation
<<tiddler [[Risk-Neutral Valuation]]>>
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finance_public
created
Sat, 14 Jan 2012 21:03:36 GMT
creator
dirkjan
modified
Sat, 14 Jan 2012 21:03:36 GMT
modifier
dirkjan
tags
M12
Principles of Corporate Finance
Term
creator
dirkjan