Derivatives

1. Hedging, Volatility
Chapters 19 & 20

2. • Consider a trader who sold 100,000 European call options on a non-
dividend-paying stock at $300,000 (i.e. $3 per call option) with S0 =
$49, K = $50, r = 5%, Standard Deviation = 20%, T = 20 weeks
• The Black-Scholes-Merton value of this call option is $2.40053, that
is $240,053 in total
• How does the trader hedge its risk to lock in a profit of approximately
$60,000?
• The initial value of delta of the call option is 0.5216, that is the total
delta of the (short) position is -52,160
• This means that 52,160 shares must purchased @ $49 each to
create a delta-neutral position
• This amount of $2,555,840 must be borrowed at 5%, which means
that interest paid in one week will be $2,458.72
2 Section
Dynamic Delta Hedging

3. • But a week later, if the stock declines to $48.13 and the delta
changes to 0.4587, the position is no longer delta neutral
> 6,290 shares must be sold to maintain a delta-neutral position
3 Section
Dynamic Delta Hedging

4. • This procedure is done week after week
• By the end of week 20, the option closes in-the-money, so the hedger
will receive $5,000,000 for delivery of 100,000 shares
> The cumulative cost is $5,263,087
> The net cost of hedging is therefore $263,087
> If discounted to time 0, it becomes $258,076 which is (not
exactly equal but) close to $240,053, BlackScholes-Merton
price
• In theory, if the hedge rebalancing is done continuously, the total net
cost of the hedge should be equal to the option price
4 Section
Dynamic Delta Hedging

5. • In practice, dynamic hedging is rebalanced several times a week
(maybe once a day) because of transaction costs and bid/ask spread
> In the previous example, the total cost would be close to
$240,053, that leaves the trader with a profit close to $60,000
• Maintaining a delta-neutral position for a small position will be too
expensive because of transaction costs incurred in the buy/sell
strategy
> The dynamic hedging is therefore more feasible for a large
portfolio of derivatives dependent on a single asset because
only one trade in the underlying asset is necessary to zero
out delta for the whole portfolio
5 Section
Dynamic Delta Hedging

6. • The Gamma of a portfolio of derivatives on a single asset is the rate
of change of the portfolio’s delta with respect to the price of the
underlying asset
• Gamma addresses delta hedging errors caused by curvature
> If gamma is small, adjustments to keep a portfolio delta
neutral need to be made only relatively infrequently because
delta changes slowly
> However, if gamma is large in absolute terms, delta is highly
sensitive to the price of the asset. It is then quite risky to
leave a delta-neutral portfolio unchanged for any period of
time
• Gamma is positive for a long position in an option (call or put). It is
greatest for at-the-money options
6 Section
Gamma

7. • The gamma reduces the delta-hedging errors
7 Section
Gamma

8. • In order to make a delta-neutral portfolio gamma-neutral, positions in
non-linear products (e.g. options) are required
• Suppose that a delta-neutral portfolio has a gamma of -3,000, and
that a given call option (on the same asset) available on the market
has a delta of 0.62 and a gamma of 1.5
> The portfolio can be made gamma-neutral by taking 2,000
(long) positions in the call option
> However, the delta of the new portfolio will then change from
0 to 2,000 x 0.62 = 1,240. Therefore 1,240 units of the asset
must be sold to keep its delta neutral
8 Section
Gamma Neutrality

9. • Another source of risk for a portfolio of derivatives is the volatility of the
underlying asset
• The Vega (v) of a portfolio is the rate of change of the value of the portfolio
with respect to volatility
• Like gamma, vega tends to be greatest for at-the-money options
• In practice, a trader must keep a delta-neutral portfolio with gamma and vega
within limits set by risk management
> They usually ensure that their portfolios are delta-neutral at least
once a day
> Whenever the opportunity arises, they improve gamma and vega
9 Section
Vega

10. • Consider a delta-neutral portfolio and two other options presented in the
table below
• What positions in option 1, option 2, and the asset are required to make
the portfolio delta, gamma, and vega neutral?
> First, we must take long positions in 400 option 1 and 6,000 in
option 2 to make it gamma and vega neutral
> Second, a short position in 3,240 (= 400 x 0.6 + 6,000 x 0.5)
units of the asset will make delta neutral
10 Section
Managing Delta, Gamma and Vega