can somebody explain intuitively why the below is true and what difference the hedge ratio being -ve or +ve makes to buying or selling the underlying thanks (CFA EOC questions 3 reading50 (2013) on options)
In order to create a hedge portfolio:
call option is underpriced => buy calls and sell underlying
call option is overpriced => sell calls and buy underlying
hedge ratio -ve
put option is underpriced => buy puts and buy underlying
put option overpriced=> sell puts and sell underlying
I’m not sure what you mean by the –ve or +ve, but I think I can address the first part of your question.
Basically option pricing uses the idea that the correct price is the one that does not allow for any arbitrage (best defined, in my opinion, as riskless profit without capital outlay).
The hedge ratio tells you the ratio of options (let’s assume calls) and underlying (let’s assume shares of stock) you need to buy and sell in order to lock in your end of period return. You will always buy and sell at the hedge ratio, it does not matter if an option is correctly priced or not. By doing this you create a hedged portfolio. If a stock increases in price an option also increases in price, and therefore we need to buy one and sell the other in order to hedge.
Here is a simplified example:
Current Stock Price: $100
Future Stock Price: either $150 or $50
Risk Free Rate: 0% (to make the concept easier, I’ll talk about how it’s different with a risk free rate)
Therefore the Hedge Ratio is 0.5 and the correct price of the call option is $25 (I skipped the calculation because that’s just formulas to memorize, and the focus is the logic of it, but on the exam you will need to do these yourself so make sure you’re able to)
In order to illustrate what would happen if the option was correctly priced:
If you buy one stock (costing $100) and sell two calls (giving you $50 total) you will need to borrow $50 in order to not spend any money yourself. This means you need to make sure at the end of this period, when the stock price is either $150 or $50 you have at least $50. If the stock increases it will be worth $150, giving you $100 more than the $50 you owe, which is just enough to cover the $100 you owe on the two calls ($50 each). If the stock decreases you will owe nothing on the calls and have the $50 to cover the $50 you owe.
Now, assume you can sell calls for $30 (calls are overpriced). You will buy a stock for $100, sell calls for $60 and only need to borrow $40 (instead of $50). You will still receive $50 at the end of the period (even though the calls are more expense the payout at the end remains the same). After paying back the $40 you will have $10 left over and BOOM, you have arbitrage (riskless profit without any capital outlay).
If we assume calls are $20 you would do something similar. Short the stock getting $100, while buying two calls, spending $40 (meaning you have $60 in the bank). At the end of the period either you will need to pay $150 for the stock, while receiving $100 from the calls, for a total of a $50 net payment, or just pay $50. Either way you had $60, paid $50, and have a riskless $10 profit. As you can see the hedge ratio of half a share to each call (or in this case one share to two calls) works either way.
To review, when a call is either overpriced or underpriced you lock in the profit by borrowing less up front than you normally would, and only paying back the normal amount. In both examples we earn $10 and spend nothing. The hedge ratio is the same if calls are correctly priced, under-priced, or overpriced, this is becasue the hedge ratio does not use the current option price, it uses the final payouts.
Puts are similar logic, except because they decrease in value when a stock increases you will either buy both the stock and the put or sell both in order to get a hedged portfolio.
If there is a risk free rate you would need to earn whatever you borrowed plus the risk free rate (for example if the risk free rate was 10% you would pay back $55 on the original $50 you borrowed). The binomial pricing formulas take risk free rates into account, so if you do the calculations correctly it should work out.
All of these concepts are based on the idea of no-arbitrage, so every strategy should earn you the risk free rate, if you can earn more there is an arbitrage opportunity, if you can’t even earn the risk free rate, you did something wrong.
Hopefully this helps, while the logic behind option pricing isn’t exactly rocket science, it’s not easy. I would suggest building it out on excel to help visualize it better (that’s what I did when I first learned it and I found it helpful).
thanks much clearer