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The Greek Letters Chapter 17. Goals. OTC risk management by option market makers may be problematic due to unique features of the options that are not available on exchanges. Many dimensions of risk (greeks) must be managed so that all risks are acceptable.

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  • OTC risk management by option market makers may be problematic due to unique features of the options that are not available on exchanges.
  • Many dimensions of risk (greeks) must be managed so that all risks are acceptable.
  • Synthetic options and portfolio insurance
  • Delta (D) is the rate of change of the option price with respect to the underlying:


  • Remember, this creates a hedge (replication of option payoff):

*S - f = 0

  • Two ways to think about it:
    • If S =$1 => f = $
    • If short 1 option, then need to buy  shares to hedge it
  • Let c=0.7. What does this mean?
  • -c + 0.7S = 0
  • Short call can be hedged by buying 0.7 shares or
  • For $1 change in S, C changes by 70 cents.
example cost of hedging
Example: Cost of hedging
  • Assume the following conditions:
    • r = 10%,
    • T = 6 months;
    • S0 = 20; u=1.1;
    • d = 0.9;
    • K = 20.
  • Compute the cost of hedging a short call position on a two-period binomial tree (each period = 3 months).
example cost of hedging6
Example: Cost of hedging
  • Suu=24.2;
  • Sud=Sdu=19.8;
  • Sdd=16.2
  • Cuu=4.2; Cud=Cdu=Cdd=0
  • Risk-neutral probability:
    • p = (e0.1/4 – 0.9)/(1.1-0.9)=0.62658
  • Cu=e-0.1/4*p*4.2 = 2.56667; C0 = 1.5685
example cont d
Example (cont’d)
  • The delta is changing over time:
    • u = (4.2-0)/(24.2-19.8) = 0.95455
    • 0 = (2.56667-0)/(22-18) = 0.64167
    • d = 0
  • Assume that the stock ends up going up twice in a row: uu move
example cont d8
Example (cont’d)
  • Hedging cost can be computed as follows:
    • t=0 : the risk-free portfolio is - C + 0.64167*S
    • Cost(t=0) = 0.64267*20 = 12.8334 (borrow it)
    • t=1: u = 0.95455 => buy (u - 0) = 0.31288 more shares
    • Cost(t=1) = 0.31288*22 = 6.88336
    • t=2 : option is exercised by the long position holder
    • You get:-C + u*S = 0.95455*24.2 – 4.2 =18.90011 (inflow)
example cont d9
Example (cont’d)
  • Present Value of the costs:
    • PVt=0 = 12.8334 + 6.88336e-0.1/4

– 18.90011e-0.1/2 = 1.5685 = call premium

  • In practice, transaction costs and discreteness of hedging lead to imperfect hedging
example cont d10
Example (cont’d)
  • What if the path is du?
    • d = 0
  • No shares needed to hedge the short call as there is no uncertainty about its value any more (it is OTM).
  • At home confirm that in this case, i.e., du path, the cost of hedging is also equal to call premium and that it does not depend on the path of the underlying asset.
example cont d11
Example (cont’d)
  • What if the stock pays a dividend yield q = 4%? Tree is the same. What are the ’s now?
  • Need to take into account both the cap. gains and the dividends now:
  • - Cu + u Su

-Cuu + u *Suueqt

Cud + u *Sudeqt

example cont d12
Example (cont’d)
  • Thus,
  • Delta is smaller. Is option price bigger or lower in this case? What about costs of hedging?
deltas for currency and futures options
Deltas for Currency and Futures Options
  • Let f be the price of an option.
  • Currency options:
  • Futures options (F is the futures price):
two special cases
Two Special Cases


Su – Ke-r(T-t)=fu

  • Forwards:


  • For S – f to be a hedge,  = 1(perfect short hedge)
  • It is all cap gains here, no income.


Sd – Ke-r(T-t)=fd

two special cases cont d
Two Special Cases cont’d


Fu – F0

  • Futures:


  • For S – f to be a hedge,  = er(T-t) (perfect short hedge); can ignore t, if small.
  • It is all income here, no cap gains.


Fd – F0

using futures for delta hedging
Using Futures for Delta Hedging
  • Futures can be used to make an option position -neutral.
  • The delta of an option contract on a dividend-paying asset is e-(r-q)T times the delta of a spot contract:
  • The position required in futures for delta hedging is therefore e-(r-q)T times the position required in the corresponding spot contract
using futures for delta hedging17
Using Futures for Delta Hedging
  • Example: hedging a currency option on £ requires a short position in £458,000. If r=10% and rf = 13%, how many 9-month futures contracts on £ would achieve the same objective?
  • One contract is for £62,500 => need to sell about 7 contracts
  • Theta (Q) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time
  • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset
  • In our previous example:
  • Assets linear in S (futures, stock) do not affect Gamma.
interpretation of gamma
Interpretation of Gamma
  • For a delta neutral portfolio,

dP»Qdt + ½GdS2





Positive Gamma

Negative Gamma

  • Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility
  • See Figure 17.11 for the variation of n with respect to the stock price for a call or put option
managing delta gamma vega
Managing Delta, Gamma, & Vega
  • D can be changed by taking a position in the underlying or the futures on it
  • To adjust G & n it is necessary to take a position in an option or other derivative
  • The Greeks of a portfolio are computed as follows:
delta and gamma of a protective put
Delta and Gamma of a Protective Put
  • Protective put = long in the put and the stock. The put has a delta of -0.4. Let a call on the stock have a delta of 0.6 and a gamma of 2.
  • Delta of the protective put:
    •  = p + S = -0.4+1 = 0.6 (why is it equal to call delta?)
    • Gamma:  = 2+0=2
    • You cannot change gamma buy trading shares
  • Rho is the rate of change of the value of a derivative with respect to the interest rate
  • For currency options there are 2 rho’s
hedging in practice
Hedging in Practice
  • Traders usually ensure that their portfolios are delta-neutral at least once a day
  • Whenever the opportunity arises, they improve gamma and vega
  • As portfolio becomes larger hedging becomes less expensive
hedging vs creation of an option synthetically
Hedging vs Creation of an Option Synthetically
  • When we are hedging we take positions that offset D, G, n, etc.
  • When we create an option synthetically we take positions that match D, G, &n
portfolio insurance
Portfolio Insurance
  • In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically
  • This involves initially selling enough of the portfolio (or of index futures) to match the D of the put option
  • In our earlier example: at t=0 we need to sell -p = 1 – 0.64167 = 0.35833 = 35.83% of the portfolio and invest the proceeds into riskless assets.
portfolio insurance29
Portfolio Insurance
  • Can achieve the same with index futures. One benefit is lower transactions costs.
  • Recall
  • Remember, T is the maturity of the futures.
portfolio insurance continued
Portfolio Insurancecontinued
  • As the value of the portfolio increases, the D of the put becomes less negative and some of the original portfolio is repurchased
  • As the value of the portfolio decreases, the D of the put becomes more negative and more of the portfolio must be sold
portfolio insurance continued31
Portfolio Insurancecontinued

The strategy did not work well on October 19, 1987...