Once again we assume that the beginning stock price is 100, that it can have a return of 1.07 or .9346 in one period. We also assumed here that R = 1.01, which is between the two returns of the stock.
We then asked two questions:
1. How much a call option with strike price 102 worth at t=1?
2. How will the price vary as p varies?
We were then introduced to a replication portfolio. A replicating portfolio is a portfolio with allocations that will reproduce the same payoff in either scenario (in this case the scenarios are that the stock price moves up to $107 or down to $93.46.
To create the replicating portfolio we consider that we buy x shares of stock and invest $y in R at time t=0.
At time t=1, the portfolio is worth;
107x + 1.01y when S =107 and
93.46x + 1.01y when S=93.46
We then looked at what the payout will be in each case:
107x + 1.01y = max{107-102,0} when S =107 and
93.46x + 1.01y= max{93.46-102,0} when S=93.46
OR
107x + 1.01y = 5
93.46x + 1.01y= 0
Here we have two equations with two unknowns and we solve to find;
x=0.3693 and y =-34.1708
This means that we should invest in .3693 shares of stock, and borrow out $34.17.
The cost of our portfolio at t=0 is: 0.3693*100 - 34.1708*1 = 2.76
Notice that our price does not depend on the buy or seller, but purely on the returns.
We then extended these results to the general case in which we do not know u,d, or R.
Our replicating portfolio is:
And we can solve for x and y to get the total cost:
This is the same as:
This is risk-arbitrage pricing and q and 1-q are the risk-neutral probabilities.
We then asked how the option price depends on p.
We looked at two stocks with different vales for p and 1-p:
And how should options with strike prices of $100 be priced?
Both of these will have payoffs of max{S(1) - 100,0} and by using the formulas above we find that both of these will have the same costs for the options. We will look into why this is so in future lessons.
93.46x + 1.01y= 0
Here we have two equations with two unknowns and we solve to find;
x=0.3693 and y =-34.1708
This means that we should invest in .3693 shares of stock, and borrow out $34.17.
The cost of our portfolio at t=0 is: 0.3693*100 - 34.1708*1 = 2.76
Notice that our price does not depend on the buy or seller, but purely on the returns.
We then extended these results to the general case in which we do not know u,d, or R.
Our replicating portfolio is:
And we can solve for x and y to get the total cost:
This is the same as:
This is risk-arbitrage pricing and q and 1-q are the risk-neutral probabilities.
We then asked how the option price depends on p.
We looked at two stocks with different vales for p and 1-p:
And how should options with strike prices of $100 be priced?
Both of these will have payoffs of max{S(1) - 100,0} and by using the formulas above we find that both of these will have the same costs for the options. We will look into why this is so in future lessons.
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