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Volatility Smiles

VIEWS: 29 PAGES: 20

									Volatility Smiles

   Chapter 18


                    1
     Put-Call Parity Arguments
   Put-call parity p +S0e-qT = c +X e–r T
    holds regardless of the assumptions
    made about the stock price distribution
   It follows that
    pmkt-pbs=cmkt-cbs




                                              2
           Implied Volatilities

   The implied volatility calculated from a
    European call option should be the same
    as that calculated from a European put
    option when both have the same strike
    price and maturity
   The same is approximately true of
    American options

                                               3
             Volatility Smile

   A volatility smile shows the variation of
    the implied volatility with the strike
    price
   The volatility smile should be the same
    whether calculated from call options or
    put options


                                                4
The Volatility Smile for Foreign
      Currency Options

        Implied
        Volatility




                     Strike
                     Price


                                   5
Implied Distribution for Foreign
       Currency Options
   The implied distribution is heavier in both
    tails than for the lognormal distribution.
   It is also “more peaked” than the
    lognormal distribution




                                                  6
The Volatility Smile for Equity
            Options
        Implied
        Volatility




                     Strike
                     Price


                                  7
Implied Distribution for Equity
           Options

The right tail is less heavy and the left tail is heavier
than the lognormal distribution




                                                            8
        Other Volatility Smiles?

    What is the volatility smile if
   True distribution has a less heavy left tail
    and heavier right tail
   True distribution has both a less heavy left
    tail and a less heavy right tail




                                               9
Possible Causes of Volatility
           Smile

 Asset price exhibiting jumps rather than
  continuous change
 Volatility for asset price being stochastic

 (One reason for a stochastic volatility in
  the case of equities is the relationship
  between volatility and leverage)

                                            10
        Volatility Term Structure

   In addition to calculating a volatility smile,
    traders also calculate a volatility term
    structure
   This shows the variation of implied
    volatility with the time to maturity of the
    option



                                                 11
   Volatility Term Structure

The volatility term structure tends to be
downward sloping when volatility is high
and upward sloping when it is low (mean
reversion)




                                            12
Example of a Volatility Surface

                     Strike Price
            0.90   0.95    1.00     1.05   1.10

   1 mnth   14.2   13.0    12.0     13.1   14.5
   3 mnth   14.0   13.0    12.0     13.1   14.2
   6 mnth   14.1   13.3    12.5     13.4   14.3
   1 year   14.7   14.0    13.5     14.0   14.8
   2 year   15.0   14.4    14.0     14.5   15.1
   5 year   14.8   14.6    14.4     14.7   15.0



                                                  13
Brief Review




               14
   Consider a 3-month put (European) on a stock
    with K = $20. Over the next 3 months the
    stock is expected to either rise by 10% or drop
    by 10%. The risk-free rate is 5%.
   Q1: What is the risk-neutral probability of the
    up-move?
                                   0.051 / 4
                  e d e
                      rT
                                0.9
               p                    0.5629
                   ud   1.1  0.9
   Q2: what if the stock pays a dividend yield of
    3% per year?
             e( r q )T  d e( 0.050.03)1 / 4  0.9
         p                                           0.5251
                 ud               1.1  0.9

                                                             15
   Q3: What position in the stock is necessary to
    hedge a long position in 1 put option (assume
    no dividends)?
                    pu  pd   02
                                  0.5
                    Su  Sd 22  18
   Buy 0.5 shares
   Q4: How would the answer to Q3 change if the
    stock paid a dividend yield of 3%?
   Q5: What is the value of the put?




                                                     16
   Assume now that the put has 6 months to
    maturity and there are 2 periods on the tree
    (each period is 3-moths long). Everything else
    is the same.
   Q6: Compute the value of the put option.

   Q7: Answer question 6 for an American put.

   Q8: The cost of hedging which option is higher
    – American or European put? Compute it for
    both. Compute Gamma of the European put
    option. Can you hedge your Gamma-exposure
    by trading futures?
                                                     17
   Consider a European call option on FX. The
    exchange rate is 1.0000 $/FX, the strike price is
    0.9100, the T is 1 year, the domestic risk-free
    rate is 5% the foreign risk-free rate is 3%. If
    the call is selling for 0.05 $/FX, is there
    arbitrage and, if so, how would you exploit it?
                         rf T         rT
     c  0.05  Se                Ke         0.1048
                                   rf T
   Hence, borrow and sell e            units of FX,
    invest the PV of K and buy the call today.
   At maturity, you have $K which is more than
    enough to buy the currency.
                                                       18
   A trader in the US has a portfolio of derivatives
    on the AUD with the delta of 450. The USD
    and AUD risk-free rates are 5% and 7%.
   Q1: What position in the AUD creates a delta-
    neutral position?
   Short 450 AUD
   Q2: What position in 1-year futures contract
    on the AUD creates a delta-neutral position?
         ( r  rf ) T          ( 0.05.07)1
    e                    e                      450  459
   Redo Q2 for the case of forwards.

                                                           19
   A portfilio of derivatives on a stock has a
    delta of 2000 and a gamma of -100.
   Q1: What position in the stock would
    create a delta-neutral position?
   Short 2,000 shares
   Q2: If an option on the stock with a delta
    of 0.6 and a gamma of 0.04 can be traded,
    what position in the option and the stock
    creates a portfolio that is both gamma and
    delta neutral?
   Short 3,500 shares and buy 2,500 options
                                             20

								
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