Replicating Portfolio

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:

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

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

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:

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

= $80\cdot (4/7)-33.45$  =  $12.26$

Risk Neutral Valuation

In Principles of Corporate Finance the risk neutral valuation is calculated by assuming that investors are indifferent to risk and expect return on stock to be equal to the risk-free rate of interest. Furthermore it assumed that stock have a certain downside potential $d$ and and and upside potential $u$. There is a certain chance $p$ that the stock will go up with percentage $u$ and a chance of $1-p$ that the stock will go down with percentage $d$. This gives:

${\text{Expected Return}}$ = $p\cdot u+(1-p)\cdot d=r$

This gives the general probability of rise of the stock:

$p$ = ${\frac {(1+r)-d}{u-d}}$

Valuing a call this way gives ($k$ is the exercise price):

${\text{Expected value of a call}}$ = ${\frac {p\cdot max(Su-k,0)+(1-p)\cdot max(Sd-k,0)}{1+r}}$

Using the same example as in Replicating Portfolio we get:

$p$ = ${\frac {(0.025)-(-0.25)}{0.33-(-0.25)}}$ = $.471$

${\text{Expected value of a call}}$ = ${\frac {0.471\cdot max(106.67-80,0)+(1-0.471)\cdot max(60-80,0)}{1+r}}$ = $12.26$