Hedging ... and its relatives
motivated by e-mail from Ron M.
Suppose we invest in some stock or mutual fund or option or real estate or ...

>Yeah, so?
Unless the investment is risk-free, it may be good to invest in some other asset to counter the possibility that our first investment may go South.

>Huh?
Suppose we invest in a stock currently worth $36.
Suppose, further, that we estimate future stock prices and get something like Figure 1.

>How do we do that?
It doesn't matter. Just listen.
Our estimate says there's a 30% probabillity that we'll lose money in N months.

>And 70% that we'll make money. Not bad, I'd say.

What we want to do is invest in some other asset that is likely to go UP when our stock price goes down.

We want an asset so that the distribution of our two-asset portfolio looks like the purple graph in Figure 2.

>What's the green curve?
That's the distribution of portfolio values if we did not buy that second asset.
That second asset (which tends to counteract losses in the first asset) should then reduce the standard deviation of our portfolio.

>Reduce the standard deviation without destroying your portfolio return, right?
Well ... yes, tho' you may be willing to accept a somewhat smaller expected return in order to increase your probability of making money.


Figure 1


Figure 2

Note that if we add an asset whose correlation with our stock is negative, the standard deviation will decrease. (See 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:
Suppose the stock had an Expected (or Mean) Annual Return of R and Standard Deviation of S, then a reasonable approximation for your compound annual growth rate is: CAGR = R - (1/2)S2 (See CAGR)

Decrease S and you get a larger CAGR ... as we've noted above.

Okay, now suppose we devote a fraction x of our portfolio to the stock and y = (1-x) to a second asset with Expected Return and Standard Deviation of A and B respectively.

Our 2-asset portfolio would then have an Expected Return of x R + y A and (Standard Deviation)2 of x2S2 + y2B2+ 2 r x y S B (See Stat Stuff)
Here, r is the correlation between the two assets.
Okay, for our 2-asset portfolio our CAGR estimate would then be:
      CAGR = x R + y A - (1/2) { x2S2 + y2B2+ 2 r x y S B }

Figure 3 shows one situation where the stock parameters are:
R = 10%     S = 25%
and our second asset has:
A = 4%     B = 8%
and the correlation varies from -100% to +100% (or -1.0 to +1.0).

Note that, when the correlation is quite negative, we can reduce the volatility and increase the CAGR

> I like r = -1
Yes. Good luck in finding such a second asset ...


Figure 3

However, if one could find such an asset and we kept 80% in our stock and put 20% into that second asset
(see the dot in Fig. 3?) we'd reduce the volatility of our portfolio from 25% to about 18% (see the dot in Fig. 4?).

>And you'd have a smaller expected return?
Yes, it'd be reduced from R = 10%   to   80%*R + 20%*A = 80%*10% + 20%*4% = 8.8%

>I prefer the larger expected return ... that R = 10%.
The CAGR is a better representation of what you'd get. Did you notice that it just went up?

>Uh ... yes.


Figure 4

Delta Hedging ... for Call Options

Suppose we've written (and sold) a call option on a stock where the current stock parameters are:

      Stock Price S = $49
      Strike Price K = $50
      Stock Volatility V = 20%
      Expected Stock Return = 13%
      Weeks to Expiry T = 20 ... so Years to Expiry T = 20/52
      Risk-free Rate R = 5%

The Black-Scholes option price for 100 shares (that's one contract) would be:

      Option contract Price = $240 ... meaning the call option is worth $2.40, but we're talking 100 shares, eh?

>Black-Scholes? Where'd that come from?
It doesn't matter ... however the Excel formula is:

B-S Call Option price = S*NORMSDIST((LN(S/K)+(R+V^2/2)*T)/(V*SQRT(T))) - K*EXP(-R*T)*NORMSDIST((LN(S/K)+(R+V^2/2)*T)/(V*SQRT(T))-V*SQRT(T))

>Mamma mia! I really don't think ...
Good idea. Don't think, just listen.

>Wait! The B-S formula doesn't involve the expected stock return!
Interesting, eh?
Anyway, suppose Sam buys our contract ... paying us $240.

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.
We'd sell Sam 100 shares at $50 and he could then sell those shares at $52. Sam's gain would be:
[a]   $5200 (from the sale of 100 shares @ $52) - $5000 (the cost of buying your 100 shares @ $50) - $240 (the cost of buying the contract) = - $40.

Our gain would be:
[b]   $5000 (from the sale of 100 shares @ $50 to Sam) - $4900 (the cost of buying our 100 shares @ $49) + $240 (from the sale of the contract to Sam) = $340.

>Sam's crazy, right? Why would he exercise the option ... and lose money?
He probably wouldn't.
However, Sam can sell the contract at any time, so he keeps his eye on the stock price and notes how the value of the contract changes with the stock price. As the option price increases, the probability of Sam exercising his option increases.

Figure 5 shows the option price for various stock prices, one week later (with 19 weeks to expiry)

>One week later?
Yes. We use the B-S formula above, but with 19 weeks instead of 20.

However, suppose we never really owned any shares ... so we wrote a "naked" option.
If Sam exercises the option we must sell him 100 shares at $50.
If the shares are worth $52 when Sam exercises the option, we've got to buy at $52 and sell to Sam at $50. Now our gain would be:


Figure 5

[c]   $5000 (from the sale of 100 shares @ $50) - $5200 (the cost of buying your 100 shares @ $52) + $240 (from the sale of the contract) = $40

>You make what Sam loses?
Yes, for a naked option.
Note that the terms in equation [a] are the negative of those in equation [c].

Now comes the big problem: What if the price goes to, say $55 or $56 or ...

>You lose a bundle, eh?
Yes, as in Figure 5A which shows our gains (well ... losses), if we have to buy the stock at $55 or $56 ... and sell it at $50.


Figure 5A

Note that, if the stock price is S and the strike price is K and the option is worth C, our gain (or loss) is just:

      Gain = K + C - S

See Figure 5A? For each $1 increase in stock price S, our 100 shares give us a gain which decreases by $100.

>And where does the hedging come in?
Good question. We want to hedge against the possibility that the stock price will go up.
If we actually bought some additional stock, say N shares, then we could sell those at the higher price.
Then we'd make some gains on the stock sale and that'd offset some of our option losses (as indicated in Figure 5A).

>How many additional shares would you buy?
Aah, that's where Delta Hedging comes in.
First, recall that there's a number called Delta that measures how rapidly the option price will increase when the stock price goes up.

>Or decrease when the stock tanks, right?
Uh ... yes, though we're mostly interested in having to buy stock at a higher price to cover our option commitment.
(There's some stuff on delta here.)

So here's the scenario:

  1. We sell Sam a contract for $240 ... a 100-share contract at $2.40 option premium, as noted above.
  2. We calculate delta = 0.52 from the current parameters ... as given above.
  3. We borrow $2548 to buy 100*delta = 52 shares of the stock ... 52 shares at $49 per share = 2548.
  4. Suppiose that, in a week, the stock goes up to $52 and Sam asks for his 100 shares at $50.
  5. We run out and buy call options from Sally, at $4.16 ... the red dot in Figure 5.
  6. We exercise the contract we just purchased, get our 100 shares from Sally (worth $52) and pay Sam his 100 shares.
  7. 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.
  8. We repay the $2548 loan .
So what's our gain (or loss)?
+$240 from the sale of the 100-share contract to Sam
-$416 for the purchase of the 100-share contract from Sally
+$156 from the sale of the 52 shares we bought at $49 and sold for $52
= -$20 our net gain (or loss)

>How about the cost of borrowing?
Yes. We'll get to that in a moment.
>But you just lost $20!
Yes. We'll get to that too ... in a moment.

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 delta change? After all, it depends upon ...
Yes, it depends upon volatility, time to expiry, etc. That means we change our stock position as the parameters change.
That's called ...

>Dynamic hedging, right?
You got it ... and if you just sit there with your original stock purchase, doing nothing, it's static hedging.
If we play our cards right we should be able to generate an almost risk-free strategy, like so:

  1. We sell an option for $C1 when the stock price is $S1.
  2. We borrow $D S1 to buy D shares at the current price of $S1 per share.
          Suppose the weekly interest rate is i   (where i = 0.00089 means 0.089%).
          The weekly interest is then   $ i D S1.
  3. N weeks pass and the stock price has increased to $S2 and somebuddy exercises the option we sold.
  4. We then buy an option at price $C2 ... on the same stock.
  5. We collect the shares from the option we bought to cover the option we sold.
          We've just lost   $C2 - $C1 on the buying and selling of the options.
  6. We also sell the D shares we bought at step 2, making D ($S2 - $S1) on the sale.
          So far our gain is:   D ($S2 - $S1) - ($C2 - $C1).
  7. We pay the N-week interest on the loan in step 2, namely $N i D S1
          Our gain is now:   D ($S2 - $S1) - ($C2 - $C1) - $N i D S1.
  8. In order to make this strategy risk-free, we make this gain = $0; that is:
          D (S2 - S1) - (C2 - C1) - N i D S1 = 0
  9. Hence our purchase of shares in step 2 should be for D shares where:
          D = (C2 - C1) / (S2 - S1) + stuff

>Stuff?
Note that (C2 - C1) / (S2 - S1) is the (average) rate of change in option price as the stock prices changes.
For small changes, that's just delta ... and that's the number of shares we should buy in step 2.
Hence the name delta hedging.
>Stuff?
Uh ... yes, that stuff is:   N i D S1 / (S2 - S1).
For our example above, it'd be about 0.75 shares.

>That'd cover the borrowing cost?

Roughly, but remember that (C2 - C1) / (S2 - S1) is the average rate of change in option price when the stock changes from S1 to S2 ... shown in Figure 5B, for our example.

We don't know that average in advance ... so we buy delta shares.

>And maybe a bit more.
Why not?  
>So can you calculate delta immediately, without waiting to see ...?
Yes:

Delta = NORMSDIST((LN(S/K)+(R+V^2/2)*T)/(V*SQRT(T)))


>And that delta hedging is risk-free?
Well ... close.
>So, you think you understand this deta hedging stuff?
Uh ... barely.


Figure 5B

>And is there a spreasheet?
Uh ... maybe.

for Part II