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1 The Greek Letters Chapter 17
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2 Goals OTC risk management by option market makers may be problematic due to unique features of the options that are not available on exchanges. 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. Many dimensions of risk (greeks) must be managed so that all risks are acceptable. Synthetic options and portfolio insurance Synthetic options and portfolio insurance
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3 Delta Delta ( ) is the rate of change of the option price with respect to the underlying: Delta ( ) is the rate of change of the option price with respect to the underlying: = f/ S Remember, this creates a hedge (replication of option payoff): Remember, this creates a hedge (replication of option payoff): * S - f = 0 * S - f = 0 Two ways to think about it: Two ways to think about it: If S =$1 => f = $ If S =$1 => f = $ If short 1 option, then need to buy shares to hedge it If short 1 option, then need to buy shares to hedge it
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4 Example Let c =0.7. What does this mean? Let c =0.7. What does this mean? - c + 0.7 S = 0 - c + 0.7 S = 0 Short call can be hedged by buying 0.7 shares or Short call can be hedged by buying 0.7 shares or For $1 change in S, C changes by 70 cents. For $1 change in S, C changes by 70 cents.
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5 Example: Cost of hedging Assume the following conditions: Assume the following conditions: r = 10%, r = 10%, T = 6 months; T = 6 months; S 0 = 20; u=1.1; S 0 = 20; u=1.1; d = 0.9; d = 0.9; K = 20. K = 20. Compute the cost of hedging a short call position on a two-period binomial tree (each period = 3 months). Compute the cost of hedging a short call position on a two-period binomial tree (each period = 3 months).
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6 Example: Cost of hedging S uu =24.2; S uu =24.2; S ud =S du =19.8; S ud =S du =19.8; S dd =16.2 S dd =16.2 C uu =4.2; C ud =C du =C dd =0 C uu =4.2; C ud =C du =C dd =0 Risk-neutral probability: Risk-neutral probability: p = (e 0.1/4 – 0.9)/(1.1-0.9)=0.62658 p = (e 0.1/4 – 0.9)/(1.1-0.9)=0.62658 C u =e -0.1/4 *p*4.2 = 2.56667; C 0 = 1.5685 C u =e -0.1/4 *p*4.2 = 2.56667; C 0 = 1.5685
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7 Example (cont’d) The delta is changing over time: The delta is changing over time: u = (4.2-0)/(24.2-19.8) = 0.95455 u = (4.2-0)/(24.2-19.8) = 0.95455 0 = (2.56667-0)/(22-18) = 0.64167 0 = (2.56667-0)/(22-18) = 0.64167 d = 0 d = 0 Assume that the stock ends up going up twice in a row: uu move Assume that the stock ends up going up twice in a row: uu move
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8 Example (cont’d) Hedging cost can be computed as follows: Hedging cost can be computed as follows: t=0 : the risk-free portfolio is - C + 0.64167*S t=0 : the risk-free portfolio is - C + 0.64167*S Cost(t=0) = 0.64267*20 = 12.8334 (borrow it) Cost(t=0) = 0.64267*20 = 12.8334 (borrow it) t=1: u = 0.95455 => buy ( u - 0 ) = 0.31288 more shares t=1: u = 0.95455 => buy ( u - 0 ) = 0.31288 more shares Cost(t=1) = 0.31288*22 = 6.88336 Cost(t=1) = 0.31288*22 = 6.88336 t=2 : option is exercised by the long position holder t=2 : option is exercised by the long position holder You get:-C + u *S = 0.95455*24.2 – 4.2 =18.90011 (inflow) You get:-C + u *S = 0.95455*24.2 – 4.2 =18.90011 (inflow)
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9 Example (cont’d) Present Value of the costs: Present Value of the costs: PV t=0 = 12.8334 + 6.88336e -0.1/4 PV t=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 In practice, transaction costs and discreteness of hedging lead to imperfect hedging
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10 Example (cont’d) What if the path is du? What if the path is du? d = 0 d = 0 No shares needed to hedge the short call as there is no uncertainty about its value any more (it is OTM). 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. 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.
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11 Example (cont’d) What if the stock pays a dividend yield q = 4%? Tree is the same. What are the ’s now? 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: Need to take into account both the cap. gains and the dividends now: - C u + u S u - C u + u S u -C uu + u *S uu e q t C ud + u *S ud e q t
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12 Example (cont’d) Thus, Thus, Delta is smaller. Is option price bigger or lower in this case? What about costs of hedging? Delta is smaller. Is option price bigger or lower in this case? What about costs of hedging?
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13 Deltas for Currency and Futures Options Let f be the price of an option. Let f be the price of an option. Currency options: Currency options: Futures options (F is the futures price): Futures options (F is the futures price):
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14 Two Special Cases Forwards: Forwards: S 0 For S – f to be a hedge, = 1(perfect short hedge) For S – f to be a hedge, = 1(perfect short hedge) It is all cap gains here, no income. It is all cap gains here, no income. S u S u – Ke -r(T- t) =f u S d S d – Ke -r(T- t) =f d
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15 Two Special Cases cont’d Futures: Futures: S 0 For S – f to be a hedge, = e r(T- t) (perfect short hedge); can ignore t, if small. For S – f to be a hedge, = e r(T- t) (perfect short hedge); can ignore t, if small. It is all income here, no cap gains. It is all income here, no cap gains. S u F u – F 0 S d F d – F 0
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16 Using Futures for Delta Hedging Futures can be used to make an option position - neutral. 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 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 The position required in futures for delta hedging is therefore e -(r-q)T times the position required in the corresponding spot contract
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17 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? 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 One contract is for £62,500 => need to sell about 7 contracts
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18 Theta Theta ( ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time Theta ( ) of a derivative (or portfolio of derivatives) is the rate of change of the value with respect to the passage of time
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19 Gamma Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset Gamma ( ) is the rate of change of delta ( ) with respect to the price of the underlying asset In our previous example: In our previous example: Assets linear in S (futures, stock) do not affect Gamma. Assets linear in S (futures, stock) do not affect Gamma.
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20 Gamma Addresses Delta Hedging Errors Caused By Curvature S C Stock price S’S’ Call price C’ C’’
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21 Interpretation of Gamma For a delta neutral portfolio, For a delta neutral portfolio, t + ½ S 2 SS Negative Gamma SS Positive Gamma
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22 Vega Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility Vega ( ) is the rate of change of the value of a derivatives portfolio with respect to volatility See Figure 17.11 for the variation of with respect to the stock price for a call or put option See Figure 17.11 for the variation of with respect to the stock price for a call or put option
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23 Managing Delta, Gamma, & Vega can be changed by taking a position in the underlying or the futures on it To adjust & it is necessary to take a position in an option or other derivative To adjust & it is necessary to take a position in an option or other derivative The Greeks of a portfolio are computed as follows: The Greeks of a portfolio are computed as follows:
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24 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. 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: Delta of the protective put: = p + S = -0.4+1 = 0.6 (why is it equal to call delta?) = p + S = -0.4+1 = 0.6 (why is it equal to call delta?) Gamma: = 2+0=2 Gamma: = 2+0=2 You cannot change gamma buy trading shares You cannot change gamma buy trading shares
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25 Rho Rho is the rate of change of the value of a derivative with respect to the interest rate 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 For currency options there are 2 rho’s
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26 Hedging in Practice Traders usually ensure that their portfolios are delta-neutral at least once a day Traders usually ensure that their portfolios are delta-neutral at least once a day Whenever the opportunity arises, they improve gamma and vega Whenever the opportunity arises, they improve gamma and vega As portfolio becomes larger hedging becomes less expensive As portfolio becomes larger hedging becomes less expensive
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27 Hedging vs Creation of an Option Synthetically When we are hedging we take positions that offset , ,, etc. When we are hedging we take positions that offset , ,, etc. When we create an option synthetically we take positions that match & When we create an option synthetically we take positions that match &
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28 Portfolio Insurance In October of 1987 many portfolio managers attempted to create a put option on a portfolio synthetically 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 of the put option This involves initially selling enough of the portfolio (or of index futures) to match the 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. 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.
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29 Portfolio Insurance Can achieve the same with index futures. One benefit is lower transactions costs. Can achieve the same with index futures. One benefit is lower transactions costs. Recall Recall Remember, T is the maturity of the futures. Remember, T is the maturity of the futures.
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30 Portfolio Insurance continued As the value of the portfolio increases, the of the put becomes less negative and some of the original portfolio is repurchased As the value of the portfolio increases, the of the put becomes less negative and some of the original portfolio is repurchased As the value of the portfolio decreases, the of the put becomes more negative and more of the portfolio must be sold As the value of the portfolio decreases, the of the put becomes more negative and more of the portfolio must be sold
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31 Portfolio Insurance continued The strategy did not work well on October 19, 1987...
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