The Greek Letters Chapter 17. The Greeks are coming! The Greeks are coming! Parameters of SENSITIVITY Delta = Theta = Gamma = Vega = Rho = . c = SN(d 1 ) – Ke –r(T – t) N(d 2 ) p = Ke –r)T –t) N(-d 2 ) – SN(-d 1 ) Notationally: c = c(S; K; T-t; r; σ )

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The Greek Letters Chapter 17

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The Greeks are coming! The Greeks are coming! Parameters of SENSITIVITY Delta = Theta = Gamma = Vega = Rho =

c = SN(d1) – Ke–r(T – t)N(d2) p = Ke–r)T –t)N(-d2) – SN(-d1) Notationally: c = c(S; K; T-t; r; σ) p = p(S; K; T-t; r; σ) Once c and p are calculated, WHAT IF?

The GREEKS are measures of sensitivity. The question is how sensitive a position’s value is to changes in any of the variables that contribute to the position’s market value.These variables are: S, K, T-t, r and . Each one of the Greek measures indicates the change in the value of the position as a result of a “small” change in the corresponding variable. Formally, the Greeks are partial derivatives.

Delta = In mathematical terms DELTA is the first derivative of the option’s premium with respect to S. As such, Delta carries the units of the option’s price; I.e., $ per share. For a Call: (c) = c/S For a Put: (p) = p/S Results: (p) = (c) - 1 For the (S) = S/S = 1

THETA Theta measures are given by: (c) = c/(T-t) (p) = p/(T-t) s are positive but the they are reported as negative values. The negative sign only indicates that as time passes, t increases, time to expiration, T – t, diminishes and so does the option’s value, ceteris paribus. This loss of value is labeled the option’s “time decay.” Also, (S) = 0.

GAMMA Gamma measures the change in delta when the price of the underlying asset changes. Gamma is the second derivative of the option’s price with respect to the underlying price. (c) = (c)/S = 2c/ S2 (p) = (p)/S = 2p/ S2 Results: (c) = (p) (S) = 0.

VEGA Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return. (c) = c/ (p) = p/ Thus, Vega is in terms of $/1% change in . (S) = 0.

RHO Rho measures the sensitivity of the option’s price to “small” changes in the rate of interest. (c) = c/r (p) = p/r Rho is in terms of $/%change of r. (S) = 0.

Call price Slope = D C Stock price A Call Delta (See Figure 15.2, page 345) • Delta is the rate of change of the call price with respect to the underlying

THETA Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration: (c) = c/(T-t) (p) = p/(T-t) Theta is given in terms is $/1 year. (c) = - $12.2607/year if time to expiration increases (decreases) by one year, the call price will increase (decrease) by $12.2607. Or, 12.2607/365 = 3.35 cent per day.

GAMMA Gamma measures the change in delta when the price of the underlying asset changes. = 0.0181. (c) = .6151; (p) = -.3849. If the stock price increases to $101: (c) increases to .6332 (p) increases to -.3668. If the stock price decreases to $99: (c) decreases to .5970 (p) decreases to -.4030.

VEGA Vega measures the sensitivity of the option’s market price to “small” changes in the volatility of the underlying asset’s return. = .268416 (Check on Computer)

RHO Rho measures the sensitivity of the option’s price to “small changes in the rate of interest. Rho = CallPut .252515 -.221559 Rho is in terms of $/%change of r. (check on computer)

DELTA-NEUTRAL POSITIONS A market maker wrote n(c) calls and wishes to protect the revenue against possible adverse move of the underlying asset price. To do so, he/she uses shares of the underlying asset in a quantity that GUARANTEES that a small price change will not have any impact on the call-shares position. Definition: A portfolio is Delta-neutral if (portfolio) = 0

DELTA neutral position in the simple case of call-stock portfolios. Vportfolio = Sn(S) + cn(c;S) (portfolio) = (S)n(S) + (c)n(c;S) (portfolio) = 0 n(S) + (c)n(c;S) = 0. n(S) = - n(c;S)(c). The call delta is positive. Thus, the negative sign indicates that the calls and the shares of the underlying asset must be held in opposite direction.

EXAMPLE: call - stock portfolio We just sold 10 CBOE calls whose delta is $.54/shares. Each call covers 100 shares. n(S) = - n(c;S)(c). (c) = 0.54 and n(c) = -10. n(c;S) = - 1,000 shares. n(s) = - [ - 1,000(0.54)] = 540. The DELTA-neutral position consists of the 10 short calls and 540 long shares.

The Hedge Ratio c Definition: Hedge ratio. In the example: Hedge ratio = 540/1,000 = .54 Notice that this is nothing other than (c).

In the numerical example, Slide 9: The hedge ratio: (c) = 0.6151 With 100 CBOE short calls: n(S) = -(c)n(c;S). n(c;S) = -10,000. n(S) = -(.6151)[-10,000] = +6,151 shares The value of this portfolio is: V = -10,000($10.3044) + 6,151($100) V = $512,056

Suppose that the stock price rises by $1. SNEW = 100 + 1 = $101/share. V = - 10,000($10.3044 + $.6151) +6,151($101) V = - 10,000($10.3044) + 6,151($100) - 10,000($.6151) + $1(6,151) V = $512,056 - $6,151 + $6,151 V = $512,056.

Suppose that the stock price falls by $1. SNEW = 100 - 1 = $99/share. V = - 10,000($10.3044 - $.6151) +6,151($99) V = - 10,000($10.3044) + 6,151($100) - 10,000( - $.6151) - $1(6,151) V = $512,056 + $6,151 – $6,151 V = $512,056.

In summary: The portfolio consisting of 100 short calls and 6,151 long shares is delta- neutral. Price/share: +$1-$1 shares +$6,151 -$6,151 calls +(-$6,151)-(-$6,151) Portfolio $0 $0

DELTA neutral position in the simple case of put-stock portfolios. Vportfolio = Sn(S) + pn(p;S) (portfolio) = (S)n(S) + (p)n(p;S) (portfolio) = 0 n(S) + (p)n(p;S) = 0. n(S) = - n(p;S)(p) Since the put delta is negative, then the negative sign indicates that the puts and the underlying asset must be held in the same direction.

EXAMPLE: put – stock portfolio. We just bought 10 CBOE puts whose delta is -$.70/share. Each put covers 100 shares. n(S) = - n(p;S)(p). (p) = -.70 and n(p) = 10. n(p;S) = 1,000 shares. n(S) = - 1,000(-.70) = 700. The DELTA-neutral position consists of the 10 long puts and 700 long shares.

In the numerical example, Slide 9: The hedge ratio: (p) = -0.3849 The Delta neutral position with 100 CBOE long puts requires the holding of: n(S) = -(p)n(p;S) n(S) = -(-.3849)[10,000] = +3,849shares The value of this portfolio is: V = 10,000($6.4360) + 3,849($100) V = $449,260.

Suppose that the stock price rises by $1. SNEW = 100 + 1 = $101/share. V = 10,000($6.4360- .3849) +3,849($101) V = 10,000( $6.4360) – 3,849($100) - 10,000(.3849) + $1(3,849) V = $449,260- $3,849 + $3,849 V = $449,260.

Suppose that the stock price falls by $1. SNEW = 100 - 1 = $99/share. V = 10,000($6.4360+ $.3849) +3,849($99) V = 10,000( $6.4360) + 3,849($100) 10,000($.3849) - $1(3,849) V = $449,260+ $3,849 - $3,849 V = $449,260.

In summary: The portfolio consisting of 100 long puts and 6,151 long shares is delta- neutral. Price/share: +$1-$1 shares +$3,849 -$3,849 calls +(-$3,849)-(-$3,849) Portfolio $0 $0

An extension: calls, puts and the stock position Vportfolio = Sn(S) + cn(c;S) + pn(p;S) portfolio = (S)n(S) + (c)n(c;S) + (p)n(p;S). But S = 1. Thus, for delta-neutral portfolio: portfolio= 0 and n(S) = -(c)n(c;S) -(p)n(p;S).

EXAMPLE: We short 20 calls and 20 puts whose deltas are $.7/share and -$.3/share, respectively. Every call and every put covers 100 shares. How many shares of the underlying stock we must purchase in order to create a delta-neutral position? n(S) = -(c)n(c;S) + [-(p)]n(p;S). n(S) = -(.7)(-2,000) – [-.3](-2,000) n(S) = 800.

Example continued The portfolio consisting of 20 short calls, 20 short puts and 800 long shares is delta- neutral. Price/share: +$1-$1 shares +$800 -$800 calls -$1,400 +$1,400 Puts+$600-$600 Portfolio $0 $0

EXAMPLES The put-call parity: Long 100 shares of the underlying stock, long one put and short one call on this stock is always delta-neutral: (position) = 100 + (p)n(p;S) + (c)n(c;S) = 100 + [(c) – 1](100) + (c)(-100) = 0.

EXAMPLES A long STRADDLE: Long 15 puts and long 15 calls (same underlying asset, K and T-t), with: (c) = .64; (p) = - .36. (straddle) = 15(100)[.64 + (- .36)] =$420/share. Long 420 shares to delta neutralize this straddle.

Results: • The deltas of a call and a put on the same underlying asset, (with the same time to expiration and the same exercise price) must satisfy the following equality: • (p) = (c) - 1 • 2. Using the Black and Scholes formula: • (c) = N(d1) 0 < (c) < 1 • (p) = N(d1) – 1 -1 < (p) < 0

THETA Theta measures the sensitivity of the option’s price to a “small” change in the time remaining to expiration: (c) = c/(T-t) (p) = p/(T-t) Theta is given in terms is $/1 year. Thus, if (c) = - $20/year, it means that if time to expiration increases (decreases) by one year, the call price will increase (decreases) by $20. Or, $20/365 = 5.5 cent per day.

GAMMA Gamma measures the change in delta when the market price of the underlying asset changes. (c) = (c)/S = 2c/ S2 (p) = (p)/S = 2p/ S2 Results: (c) = (p) (S) = 0.

GAMMA In general, the Gamma of any portfolio is the change of the portfolio’s delta due to a “small” change in the underlying asset price. As the second derivative of the option’s price with respect to S, Gamma measures the sensitivity of the option’s price to “large” underlying asset’s price changes. May be positive or negative.

Interpretation of GammaThe delta neutral position with 100 short calls and 6,151 long shares has Γ= -$181 Position value $512,056 S 75 100 125 More negative Γ Negative Gamma means that the position loses value when the stock price moves more and more away from it initial value.