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The Greek Letters Chapter 15

The Greek Letters Chapter 15. Example. A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock S 0 = 49, X = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% The Black-Scholes value of the option is $240,000

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The Greek Letters Chapter 15

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  1. The Greek LettersChapter 15

  2. Example • A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock • S0 = 49, X = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% • The Black-Scholes value of the option is $240,000 • How does the bank hedge its risk?

  3. Naked & Covered Positions Naked position Take no action Covered position Buy 100,000 shares today Both strategies leave the bank exposed to significant risk

  4. Stop-Loss Strategy This involves: • Buying 100,000 shares as soon as price reaches $50 • Selling 100,000 shares as soon as price falls below $50 This deceptively simple hedging strategy does not work well

  5. Option price Slope = D B Stock price A Delta (See Figure 15.2) • Delta (D) is the rate of change of the option price with respect to the underlying

  6. Call Delta • Call Delta = N(d1) • Always positive • Always less than one • Increases with the stock price

  7. Delta Hedging • This involves maintaining a delta neutral portfolio • The delta of a European call on a stock isN (d1) • The delta of a European put is N (d1) – 1 Put delta is always less than one and increases with the stock price

  8. Example • Suppose c = 10, S = 100, and delta = .75 • If stock price goes up by $1, by approximately how much will the call increase? • Answer: $.75 • If you write one call, how many shares of stock must you own so that a small change in the stock price is offset by the change in the short call? • Answer: .75 shares

  9. Check • Suppose stock goes to $101. Then the change in the position equals .75x$1 - .75x$1 = 0 • Suppose the stock goes to $99. Then the change in the position equals (.75)(-1) – (.75)(-1) = -.75 + .75 = 0 • This strategy is called Delta neutral hedging

  10. Delta Hedgingcontinued • The hedge position must be frequently rebalanced • Delta hedging a written option involves a “buy high, sell low” trading rule • See Tables 15.3 (page 307) and 15.4 (page 308) for examples of delta hedging

  11. Gamma • Gamma (G) is the rate of change of delta (D) with respect to the price of the underlying asset • See Figure 15.9 for the variation of G with respect to the stock price for a call or put option

  12. Gamma Addresses Delta Hedging Errors Caused By Curvature (Figure 15.7, page 313) Call price C’’ C’ C Stock price S S’

  13. Interpretation of Gamma • For a delta neutral portfolio, DP»QDt + ½GDS2 DP DP DS DS Positive Gamma Negative Gamma

  14. Vega • Vega (n) is the rate of change of the value of a derivatives portfolio with respect to volatility • See Figure 15.11 for the variation of n with respect to the stock price for a call or put option

  15. Managing Delta, Gamma, & Vega • Delta, D, can be changed by taking a position in the underlying asset • To adjust gamma, G, and vega, n, it is necessary to take a position in an option or other derivative

  16. Rho • Rho is the rate of change of the value of a derivative with respect to the interest rate • For currency options there are 2 rhos

  17. 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

  18. Scenario Analysis A scenario analysis involves testing the effect on the value of a portfolio of different assumptions concerning asset prices and their volatilities

  19. 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

  20. 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

  21. 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

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

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