FRM 2008 Practice PII question 7 - pfol insurance using index futures

Discussion in 'P1.T3. Financial Markets & Products (30%)' started by fullofquestions, Oct 19, 2009.

  1. fullofquestions

    fullofquestions New Member

    You want to implement a portfolio insurance strategy using index futures designed to protect the value
    of a portfolio of stocks not paying any dividends. Assuming the value of your stock portfolio decreases,
    which strategy would you implement to protect your portfolio?
    a. Buy an amount of index futures equivalent to the change in the call delta x original portfolio value.
    b. Sell an amount of index futures equivalent to the change in the call delta x original portfolio value.
    c. Buy an amount of index futures equivalent to the change in the put delta x original portfolio value.
    d. Sell an amount of index futures equivalent to the change in the put delta x original portfolio value.
    Answer: d
    a. Incorrect. For portfolio insurance strategy to work, index futures should be sold in an amount
    corresponding to the change in the put delta x original portfolio value.
    b. Incorrect. For portfolio insurance strategy to work, index futures should be sold in an amount
    corresponding to the change in the put delta x original portfolio value.
    c. Incorrect. For portfolio insurance strategy to work, index futures should be sold in an amount
    corresponding to the change in the put delta x original portfolio value.
    d. Correct. Portfolio insurance strategy is accomplished by selling index futures contracts in an amount
    equivalent to the proportion of the portfolio dictated by the delta of the required put option. When a
    decrease in the value of the underlying portfolio occurs, the amount of additional index futures sold
    corresponds to the change in the put delta x original portfolio value.

    Where in Hull sixth edition is this? I looked in the 'Use of Index Futures' section (p366) and did not find anything. Can someone point me to the page or give me an explanation? Clearly you have to sell to hedge the long exposure (the fact that you are long is barely obvious here), but I really don't know how the call vs. put delta provide a different answer. Any thoughts?
  2. chiranjiv

    chiranjiv New Member

    check page 380...if you already haven't found it... :)
  3. hsuwang

    hsuwang Member

    I have the same question here,

    my logic is, we are long the stock, so to hedge it, if we were to use options, is to buy put options (protective put), so we can guarantee our payoff on the stocks, and that's why we use put delta here.... maybe?

    by the way I checked p380 and found volatility smile..

  4. chiranjiv

    chiranjiv New Member


    You have the right intuition. The correct answer choice here is nothing but a description of portfolio insurance by creating a synthetic put option using index futures.

    Sorry about the too glib answer! of course the pagination differs across editions!!

    Portfolio insurance is described in chapter 17 of the 6th edition of Hull's book ("The Greek Letters". Section 17.13 ).It's also a part of the required readings.

  5. fullofquestions

    fullofquestions New Member

    Sorry but I am not convinced yet. The answer, " Sell an amount of index futures equivalent to the change in the put delta x original portfolio value." Actually gives you a long position, i.e. sell future in the amount of the change in put delta is a positive position. Put delta/change in put delta is negative...

    The answer is clearly differentiating between the call and put delta, i.e. these are different values. We want to hedge by selling futures in the amount proportional to abs(some delta) x portfolio value. I am not clear as to why it is the put delta and not the call delta.

    By the way, I have searched in Hull 6th edition and no luck yet. The "Greek Letters" section suggested above is not in chapter 17 but in 16.5 (according to my book). This section talks about the sticky delta and strike rules. Could we give it another shot and have someone else chime in on where this information is located? Please provide chapter number and section name (the numbers don't matter as much as the section names).

    Thanks again.
  6. rahul.goyl

    rahul.goyl Member

    Hi FOQ,

    The Sell Index futures doesn't mean (Call or Put), they mean to say sell S&P 500 Futures of Equivalent Amount to the change in the Put Delta.

    So if you are long 100 share of MS put delta is say negative (0.60)
    Then simply sell 100*-0.60/250 * Index value (i.e 5)
    - 0.048
    That means sell 0.048 S&P Future 500 index.

    I hope all the confusion is clear now.

  7. ajsa

    ajsa New Member

    Hi Rahul,

    So I still wonder why a is incorrect? Or what is the difference between hedging using call and put?

  8. Hi asja,

    Building on Rahul's helpful example, the idea with this "synthetic put" via an index is:
    to have a cumulative short position in the index future that is proportional to the "as if" put option delta
    (and the put delta, 1 - (Nd1), must be between 0 and -1).
    (we can rule out buying the index: that amplifies the long position)

    it may be helpful to think in extremes:
    e.g., as the portfolio is dropping in value. Then the put delta is tending toward -1; in the extreme, this is converging toward a short index notional position that equals the long portfolio value... at this point, when the portfolio is tending toward zero (i.e., long portfolio is deep OTM and short index/short put is deep ITM) the "synthetic insurance" has accumulated an offsetting short position in the index and it now resembles the pure hedge of short futures index
    ...and the call delta does not work: as portfolio drops in value, call delta is tending toward zero which implies zeor index futures

    as the portfolio increases, in the extreme, put delta tends toward 0. Cumulative, the short index position is evaporating (this is fine and assymetrical, per the insurance concept, and unlike the pure-hedge-with-forward): very high long portfolio value --> delta put almost 0 --> unravel the short index future position
    (and again, call delta fails b/c here call delta approaching 1.0)

  9. ...sorry to belabour, I just thought the "connection" between the the portfolio hedge (Hull Ch 3) and synthetic insurance is helpful.

    Given long portfolio (P)

    1. To hedge is simply to short P/F futures contracts (or, beta*P/F)

    2. But to synthesize portfolio insurance, if I just ignore momentarily the discounting (which is modest, anyway), the position is roughly the same but multiplied by put delta:
    i.e., be short (put delta)*A1/A2 ~ be short (put delta)*P/F index futures

    so you can see the similarity and also the assymetrical difference:
    as portfolio drops in value, and put delta tends to -1, the dynamic insurance position is tending toward the hedge: P/F
    but the insurance is more dynamic. On increase, put delta*P/F tends to 0 and the short is not needed

  10. ajsa

    ajsa New Member

    Hi David,

    Thanks! so if i have a short position on the underlying I should long a future using the delta of the call option instead, is it right?

  11. Hi asja,

    good thought! We rounded of the (modest) discounting functions, but otherwise, yes this does appear to follow: e.g., as short position experiences losses, call option delta increases (ITM tending to 1.0) and synthetic insurance is to accumulate long index position; as short position gains, call option delta tends toward zero (OTM) and we "unravel" the index futures...David
  12. ajsa

    ajsa New Member

    Hi David,

    I have one more confusion..

    so we long call to hedge short underlying.. the other way to view this is we short underlying to hedge long call.. so how can we hedge a short call? I am asking because in the delta neutral hedging (using S to hedge C), it seems we do not care about if the call is long or short.. Lastly this portfolio insurance approach seems to be consistent with delta neutral, is it right?

  13. Hi asja,

    we'd hedge a short call with a long equity position ... i agree, the delta netural hedge can apply to long/short equity and long/short call ....

    Re: Lastly this portfolio insurance approach seems to be consistent with delta neutral:
    it's similar but not the same, the straight futures hedge (i.e., hedge the equity portfolio by going short P/F futures) is roughly delta neutral, but this synthetic insurance is not delta value neutral: as the portfolio drops in value, the index futures position, being only a fraction of (i.e., put delta times), is offseting gain but not entirely hedging. So, the delta concept is applied, but this here is not delta neutral.

  14. ajsa

    ajsa New Member

    Hi David,

    I see.. so this is called insurance, than hedging.. right?
  15. ...yes, this whole question asks about synthetic insurance!

    as opposed to "hedge" by shorting a greater number of futures. Frankly, an actual exam question is just as like to ask "how many futures to hedge?" (a greater numer, yes?)
    and as opposed to "actual" insurance which, here, is to go long the put.

  16. ajsa

    ajsa New Member

    Hi David,

    If we need to delta-neutral hedge a long stock, we can either short call or long put with their corresponding delta, right? so is there any preference which option to use in this case?

  17. Hi ajsa,

    I think long stock + long put, since long put has negative delta, could be used in the delta-hedge framework; I think the reason long stock + short call is preferred is simply that it partially self-funds: the intial cost of long stock + long put = S + p, but the initial cost of long stock + short call is only = S - c. first glance, they seem to me equivalent from a dynamic delta hedge perspective, but they aren't exactly the same: position Gamma is negative for the short call and positive for the long i suppose that, in a way, with the extra premium cost of the long put, what you are getting for your money is positive Gamma

  18. hsuwang

    hsuwang Member

    Hello David,

    In the delta-hedging sense, I see how long a stock and short a call can delta-hedge the position (+delta -delta), but in the profit/loss sense, I really don't see how long stock +short call (covered call) can be a hedge for the underlying stock... when stock prices fall, we lose on the stock and the premium on the short call can only hedge the loss for so much before we start losing money... also, when stock prices fall, how do we keep delta-hedging the long position, do we sell more call options?

    by the way, I think normally the hedge would be long call options and short stock instead of long stock and short call right?

  19. hsuwang

    hsuwang Member

    sorry to append,

    Here's what I would like to make sure I got the logic.. long call, short stock (delta-neutral hedge), when stock price drop, the long call loses value (delta decreases), but if we neutralize delta immediately (ie. buy shares), we gain from this because we're buying "low". Am I correct?

    A trader takes an overnight position that is long a call on an underlying asset with exact Delta equivalent hedge on the Asset itself. Now the asset drops down 10% while the trader does not have time to adjust the Hedge. The P&L should show a
    a) Gain
    b) Loss
    c) Zero affect
    d) Cannot be determined

    in this case the trader would suffer a loss because the call drops in value while the trader did not have a chance to take profit from the long stock position by adjusting the hedge?

  20. Hi Jack,

    Yes, you are correct but there is a nuance (i think). Please see:
    ...and i copied the same image here:


    say stock = $10 and delta = 0.6
    then the delta hedge = long one call and short 0.6 shares
    if stock drops by $1, by the approximation (!), short gains +0.6 (i.e., +1$ * 0.6 shares)
    and the call loses by -0.6
    ...however, the curvature (the gamma) is working in favor of our position: the wider the move the less exact is the -0.6 loss; the wider the move, due to the gamma/curvature, our actual (exact) loss on the long call is < -0.6
    ...on the upside, the curvature also works in our favor: +1 gain implies short loss of -0.6, but *precise* long call gain of > +0.6.

    so the nuance (i think) refers to:
    * in a static sense, we have referred here to fact that this delta hedge (i.e., short delta units of stock plus long one call) is profit-neutral only for an instantenous change; this is the same theme of the linear approximation (aka, only "locally accurate"). The gamma/curvature implies that, beyond the approximation, the actual position profits under large moves in either direction (i.e., long volatility)

    * in a dynamic sense, the delta-neutral hedging is the constant re-balancing (the automatic "buy low, sell high" that is implied by the positive gamma)

    in regard to your question, i am frankly a bit confused by it: I would have answered (a) because I would assume the P&L is mark to market and includes the unrealized gain on the short position. Or, put another way, why would the loss on the call be counted but not the short share; it seems to me the net unrealized impact is gain (per the above)


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