Exotic Options - Continued

1
Exotic Options - Continued

• Chris Lamoureux

2
The Exercise

• Several observations
• None of the 3 option alternatives are true hedges
  in the sense of reducing volatility of cash
  flows. (And note that there is no quantity risk
  in the statement of the case.)
• The Options strategies allow upside participation
  in appreciation of the DM.
• Note too that the case makes an assumption about
  pass-through. Why?

3
The Exercise Contd.

• The 1-shot European and Asian options provide no
  (direct) protection against scenarios where the
  DM depreciates for the first half of the period,
  and then appreciates over the second half back
  to where it started.
• But again, it is sub-optimal to just wait to the
  end with the European option selling some at a
  gain each month may be optimal.

4
Beyond Implied Volatilities

• As Shimko notes, there is much more information
  in option prices than just the implied
  volatility. Remember that if options are priced
  as if they are redundant, then the density of the
  stock price that is implicit in option prices is
  the risk-neutralized density.

5
Beyond Implied Volatilities

• The first thing Shimko notes is that we can imply
  the underlying asset value (net dividends) and
  the risk-free rate using put-call parity. This
  may seem redundant, but it is not in the case of
  a large index (that pays a continuous dividend),
  due to the synchroneity issue.

6
Regression of c-p on X
7
Interpretation

• The intercept is the implied index value net
  dividends, and the slope is the appropriate
  discount factor (e-rT).

8
The Volatility Smile

• Next, Shimko notes that even if the Black-Scholes
  model is not correct, implied volatilities from
  B-S are a useful way to summarize option prices.
• To this end, we imply volatilities from each of
  the call options.

9
The Smile
10
Information in the Smile

• If this polynomial regression were a perfect fit,
  then we could state the option pricing formula
  as
• c BS(S,X, s,T,r),
• Where s is the polynomial expression in X.
• Note that while this uses the B-S structure, this
  option pricing formula is more general than Black
  Scholes.

11
A Digression

• In order to get more information from the option
  prices about the markets risk-neutral density of
  the underlying asset, we need to establish a
  fact.
• Specifically, any option pricing formula may be
  thought of as the sum of 2 parts
• The expected value of the stock conditional on
  it being greater than the strike price and
• The present value of the strike times the
  probability that the option expires in the money.

12
The digression continues

• So the derivative of the option price with
  respect to the strike price includes the
  risk-neutral probability that the option will
  expire in the money.
• In B-S, we have

13
The digression continues

• So
• Of course, under B-S, this would simply provide
  the information from the lognormal distribution.
• This result is useful in the extended option
  pricing model.

14
The Information

• In the general model, Shimko shows
• n(d2) is N(d2) the normal density evaluated
  at d2.
• I have a macro for this function.