Chapter Sixteen
Financial Engineering and Risk Management
Answers to Problems and Questions
1. Finance is based on the premise that the easy access to information will
result in few free lunches. People prefer more wealth to less, everything
else being equal, and will always seek to improve their own welfare.
Studies of pricing relationships, fair value, and optimum strategies are all
based on the assumption that the best strategy is the one that achieves a
desired goal with the least risk.
2. Financial engineering refers to the construction of asset portfolios that have
precise technical characteristics, particularly when those characteristics are
not conveniently available in an existing exchange product.
3. Writing calls provides downside protection that is limited to the option
premium received. Puts provide much more complete protection, as they
become more valuable as prices continue to fall.
4. Subsequent increases in the stock market, which should cause the value of
the stock portfolio to increase, will be largely offset by losses from the
marking to market of the futures contracts.
5. We measure theta in dollars. Dollars are fungible across all option
premiums. Economically, one option that will lose 10 cents per day has the
same effect on portfolio value as two options that will each lose a nickel per
day. Option deltas, however, are particular to the underlying asset. While
you can add dollars, it is not as meaningful to add a GM delta to a GE delta.
6. Writing options may reduce the cost of engineering some option portfolio,
but short options increase gamma risk and leave you open to substantial
losses from sharp changes in the prices of the underlying assets.
7. Negative gamma can hurt you in a rapidly changing market. A person may
not want an engineered product with a large negative gamma.
8. As time passes or the other Black-Scholes variables change, delta, gamma,
and theta also change. The engineered product will “loosen up” and
become inaccurate if it is not periodically fine-tuned.
60
Chapter Sixteen. Financial Engineering and Risk Management
9.
Delta
Gamma
Theta
JUN 45 call
0.972
0.010
-0.007
JUN 55 call
0.551
0.063
-0.020
JUN 60 call
0.260
0.052
-0.015
a. position delta: (500 x 0.972) + (500 x 0.551) + (-500 x 0.260) = 632
b. position theta: (500 x - 0.007) + (500 x -0.020) + (-500 x -0.015) = -6
c. position gamma: (500 x 0.010) + (500 x 0.063) + (-500 x 0.052) = 11
(We do not normally express position derivatives with more precision than
the nearest whole point. These values are rounded accordingly.)
10. The new option derivatives would be
Delta
Gamma
Theta
JUN 45 call
0.969
0.012
-0.008
JUN 55 call
0.478
0.070
-0.020
JUN 60 call
0.188
0.047
-0.013
position delta: (500 x 0.969) + (500 x 0.478) + (-500 x 0.188) = 630
position theta: (500 x - 0.008) + (500 x -0.020) + (-500 x -0.013) = -8
position gamma: (500 x 0.012) + (500 x 0.070) + (-500 x 0.047) = 18
11. a) Because these are European options, the put delta equals the call delta
minus 1.0; if you know the call deltas, you can immediately determine the
put deltas without further use of the Black-Scholes model.
b) You would need to recalculate the put theta.
relationship between these two values.
There is no consistent
c) No. Given a set of initial conditions, the put gamma and the call gamma
are the same.
12. The Black-Scholes inputs are T = (365 x 3) = 1,085, R = 4%, σ = 25%, S = 55,
K = 55. Using the CBOE options calculator, the theoretical value is $6.11.
13. We want to keep the input variables the same except for time. There is some
time until expiration that will predict a put value of 90% of the original value,
or 0.90 x $6.11 = $5.50. The simplest way to find the associated time period is
via trial and error with the CBOE options calculator. With 733 days until
expiration the theoretical put value is $5.50, so you could keep the option
1,085 – 733 = 352 days.
61
Chapter Sixteen. Financial Engineering and Risk Management
14. You can show this with the options calculator. A six month option costs
more than 50% of the premium of a one-year option. Using the input
variables at the start of the Problem section in this chapter, a one-year call
would cost $6.51 while a six-month call would cost $4.40. While you
could prove this relationship with mathematics, you can get good intuition
into why this must be from binomial option pricing. Consider the example
in Figure 5-6. To solve for the value of a one year option, we have (75 –
2C)(1.10) = 50. To solve for a six-month option, we would have (75 –
2C)(1.10)0.5 = 50. Chopping the option premium in half would be too
great a reduction for the equation to hold.
15. Individual response. Implied volatility can logically be different for
different time periods.
16. This option is out-of-the-money. The plot will be generally similar to the
lower line in Figure 7-1.
17. a. If the index falls to 200 the call delta, gamma, and theta will all go to
zero; the put delta will go to –1, the gamma to zero, and theta to zero.
Position delta: being long 3,082 put contracts, the position delta would
be 3,082 x 100 x (-1) = -308,200.
Position gamma: zero
Position theta: zero
b. If the index rises to 500 the put delta, gamma, and theta will all go to
zero; the call delta will go to 1, the gamma to zero, and theta to zero.
Position delta: being net long 431 call contracts, the position delta
would be 431 x 100 x 1 = 43,100.
Position gamma: zero
Position theta: zero
18. As time passes the option delta will approach zero.
position delta will approach the delta of the stock alone.
Therefore, your
19. As prices fall the put delta will become increasingly negative. Added to
the constant delta of the stock, the position delta will become smaller and
62
Chapter Sixteen. Financial Engineering and Risk Management
will approach zero if the number of puts matches the quantity of stock
held.
20. This position is a hedge wrapper. With both options out of the money, as
expiration approaches, the position delta will approach that of the stock
alone and the position gamma and delta will both approach zero.
63