Stocks & Bonds)
In general, most hedge funds attempt to reduce volatility (and risk) while attempting to deliver positive returns under all market conditions. For example:
CAGR)
Decrease Okay, now suppose we devote a fraction Our 2-asset portfolio would then have an Expected Return of Stat Stuff)
Here, r is the correlation between the two assets.
Suppose we've written (and sold) a Stock Price The Black-Scholes option price for 100 shares (that's one Option contract Price = $240 ... meaning the call option is worth $2.40, but we're talking >Black-Scholes? Where'd
>Mamma mia! I really don't think ...
>Wait! The B-S formula doesn't involve the expected stock return!
Sam will certainly NOT exercise the option immediately (asking us to sell him the stock at $50 when it's available on the market for $49). Aah, but suppose the stock price goes to $52 and Sam exercises the option.
Our gain would be:
>Sam's crazy, right? Why would he exercise the option ... and lose money?
S and the strike price is K and the option is worth C, our gain (or loss) is just:
See Figure 5A? For each $1 increase in stock price >And where does the hedging come in?
>How many additional shares would you buy?
>Or decrease when the stock tanks, right?
So here's the scenario: - We sell Sam a contract for $240 ... a 100-share contract at $2.40 option premium, as noted above.
- We calculate
**delta**= 0.52 from the current**parameters**... as given above. - We
__borrow__$2548 to buy 100***delta**= 52 shares of the stock ... 52 shares at $49 per share = 2548. - Suppiose that, in a week, the stock goes up to $52 and Sam asks for his 100 shares at $50.
- We run out and buy call options from Sally, at $4.16 ... the
**red**dot in Figure 5. - We exercise the contract we just purchased, get our 100 shares from Sally (worth $52) and pay Sam his 100 shares.
- We sell our 52 shares of the stock at the current price of $52 ... and that's a gain of $3 per share, hence a profit of 3*52 = $156.
- We repay the $2548 loan .
>How about the cost of borrowing?
Note that: - When the stock price increases and Sam exercises his option and asks for his shares, we buy an option at the going rate from Sally.
- We pay more to Sally than we got from Sam since the option increases by
**delta**for each $1.00 increase in stock price. - However, we also buy
**delta**shares of the stock ourself, so we make a profit when the stock goes up. - When the stock goes up by $
**X**we lose**delta*****X**on the option, but make**delta*****X**on the stock. - Then ...
>But won't > - We sell an option for $
**C**_{1}when the stock price is $**S**_{1}. - We borrow $
**D****S**_{1}to buy**D**shares at the current price of $**S**_{1}per share. Suppose the weekly interest rate is**i**(where**i**= 0.00089 means 0.089%). The weekly interest is then $**i D S**_{1}. **N**weeks pass and the stock price has increased to $**S**_{2}and somebuddy exercises the option we sold.- We then buy an option at price $
**C**_{2}... on the same stock. - We collect the shares from the option we bought to cover the option we sold.
We've just lost $**C**_{2}- $**C**_{1}on the buying and selling of the options. - We also sell the
**D**shares we bought at step 2, making**D**($**S**_{2}- $**S**_{1}) on the sale. So far our gain is:**D**($**S**_{2}- $**S**_{1}) - ($**C**_{2}- $**C**_{1}). - We pay the
**N**-week interest on the loan in step 2, namely $**N i D S**_{1} Our gain is now:**D**($**S**_{2}- $**S**_{1}) - ($**C**_{2}- $**C**_{1}) - $**N i D S**_{1}. - In order to make this strategy risk-free, we make this gain = $0; that is:
D (S_{2}- S_{1}) - (C_{2}- C_{1}) - N i D S_{1}= 0 - Hence our purchase of shares in step 2 should be for
**D**shares where:
**D = (C**+_{2}- C_{1}) / (S_{2}- S_{1})*stuff*
>Stuff?
Uh ... maybe. |