Refresh History
  • MillerTime: [link]
    Yesterday at 09:15:30 PM
  • MillerTime: 9, you may want to read it.
    Yesterday at 09:16:03 PM
  • h4cheng: Verduccis article is interesting but I don't think it's a fair comparison
    Yesterday at 09:18:30 PM
  • h4cheng: Not many Latin pitchers are starters
    Yesterday at 09:18:44 PM
  • 9inches: More people need to sign Pitchers like RA Dickey... no UCL.
    Yesterday at 09:19:23 PM
  • MillerTime: Howe agreed.  I think every high school SP should read, 88-90mph is acceptable.
    Yesterday at 09:20:59 PM
  • h4cheng: Most are rp or positional players
    Yesterday at 09:21:15 PM
  • 9inches: MT, saw that Cahill was missing up in teh zone a bit.
    Yesterday at 09:21:17 PM
  • MillerTime: Typically cause secondary pitches are under development.  I thought all Latin players played SS.
    Yesterday at 09:22:09 PM
  • MillerTime: At his velocity, recipe for disaster
    Yesterday at 09:22:53 PM
  • MillerTime: Especially in ARZ
    Yesterday at 09:23:04 PM
  • h4cheng: Also it doesn't explain why guys like beachy needs rj
    Yesterday at 09:23:11 PM
  • h4cheng: Tj
    Yesterday at 09:23:14 PM
  • h4cheng: Beachy if I remember correctly did not even come through as a pither
    Yesterday at 09:23:35 PM
  • 9inches: yup.
    Yesterday at 09:23:48 PM
  • 9inches: I would tend to agree with the idea that the search for Velo must have something to do with it. I also question whether today's increased focus on weightlifting makes a significant difference.
    Yesterday at 09:25:53 PM
  • h4cheng: Velocity is definitely a factor but it's hard to say if it's the overriding one
    Yesterday at 09:26:47 PM
  • 9inches: got cheap prices on these fellas. LMK if you wanna stab at one or more before they hit the open market. Some will surely get dropped, others not. If you want 'em before you have to bid on them, lmk.
    Yesterday at 09:27:00 PM
  • 9inches: Name    L    POS    AB    R    H    2B    3B    HR    RBI    BB    K    SB    CS    LOB    E    Avg    OBP    SLG    OPS [X]     Ronald Torreyes    AAA    2B    52    9    16    4    1    0    7    3    3    0    0    19    4    0.308    0.345    0.423    0.768 [X]     Justin Bour    AAA    1B    56    9    21    5    0    2    11    7    3    1    0    22    1    0.375    0.444    0.571    1.015 [X]     Eric Campbell    AAA    1B    58    18    19    5    0    2    11    8    6    1    0    17    0    0.328    0.409    0.517    0.926 [X]     Junior Arias    A+    CF    33    13    13    1    0    2    5    3    5    8    2    12    1    0.394    0.459    0.606    1.065 [X]     Dan Butler    AAA    DH    38    4    10    5    0    1    5    3    12    0    0    21    0    0.263    0.317    0.474    0.791 [X]     Charles Cutler    AA    C    10    1    2    2    0    0    0    3    4    0    0    8    0    0.200    0.385    0.400    0.785 [X]     Chris Dickerson    AAA    CF    36    7    10    4    0    2    6    3    13    0    2    17    0    0.278    0.333    0.556    0.889 [X]     Kevin Frandsen    MLB    PH    21    2    7    1    0    0    1    3    4    0    0    6    0    0.333    0.417    0.381    0.798 [X]     Willy Garcia    AA    RF    45    4    7    1    1    3    7    1    20    0    0    21    2    0.156    0.174    0.422    0.596 [X]     Tzu-Wei Lin    A    SS    40    3    12    2    0    0    6    9    9    2    0    12    1    0.300    0.412    0.350    0.762 [X]     Corban Joseph    AAA    1B    34    3    9    2    0    1    5    1    9    0    0    16    0    0.265    0.286    0.412    0.698 [X]     Brandon Martin - DNP                                                           [X]     Francisco Martinez    AA    3B    27    1    3    0    0    0    0    2    6    0    0    11    1    0.111    0.172    0.111    0.283 [X]     Harold Martinez    A+    3B    52    3    11    3    0    0    3    1    14    0    0    23    1    0.212    0.222    0.269    0.491 [X]     Wilton Martinez - DNP                                                           [X]     Kent Matthes    AAA    RF    50    8    11    1    1    0    7    3    21    1    0    30    0    0.220    0.278    0.280    0.558 [X]     Wilson Betemit    AAA    DH    46    7    10    1    0    3    9    10    18    0    0    25    0    0.217    0.351    0.435    0.786 [X]     Ernesto Mejia    AAA    1B    53    9    19    2    0    6    15    5    15    0    0    21    1    0.358    0.433    0.736    1.169 [X]     Todd Cunningham    AAA    CF    54    6    13    3    0    0    5    4    8    2    2    24    0    0.241    0.305    0.296    0.601 [X]     Jake Elmore - DNP                                                           [X]     Yeison Asencio    AA    RF    40    1    5    1    0    0    2    1    10    0    0    13    1    0.125    0.146    0.150    0.296 [X]     Jett Bandy    AA    C    28    5    6    2    0    1    5    4    8    0    0    12    0    0.214    0.333    0.393    0.726 [X]     Jason Bourgeois    AAA    RF    47    7    12    3    0    1    4    2    3    1    0    17    1    0.255    0.286    0.383    0.669 [X]     Keon Broxton    AA    LF    23    2    3    0    0    0    1    4    9    1    1    12    0    0.130    0.259    0.130    0.389 [X]     Alex Buchholz    AAA    3B    30    7    11    3    0    0    10    2    1    0    0    16    0    0.367    0.412    0.467    0.879 [X]     Ramon Cabrera    AA    C    36    1    13    0    0    0    2    2    0    0    0    13    0    0.361    0.395    0.361    0.756 [X]     Mark Canha    AAA    PH    46    9    12    4    1    1    5    4    12    0    0    19    2    0.261    0.333    0.457    0.790 [X]     Ezequiel Carrera    AAA    RF    55    10    18    4    1    0    8    5    12    7    0    19    1    0.327    0.377    0.436    0.813 [X]     Alexi Casilla    AAA    SS    39    2    8    2    0    0    4    5    9    3    1    15    2    0.205    0.295    0.256    0.551 [X]     Adron Chambers    AAA    CF    3    1    2    0    0    1    3    0    0    0    1    0    0    0.667    0.667    1.667    2.334 [X]     Chun-Hsiu Chen - DNP                                                           [X]     Pin-Chieh Chen    A+    RF    44    5    12    6    0    0    7    5    11    0    2    17    0    0.273    0.347    0.409    0.756 [X]     Chih-Hsien Chiang    AA    RF    44    3    9    4    0    0    8    2    5    0    0    27    0    0.205    0.234    0.295    0.529 [X]     Thomas Coyle    A+    2B    42    6    12    2    0    0    3    7    10    1    0    17    2    0.286    0.388    0.333    0.721 [X]     Conner Crumbliss    AA    2B    40    4    8    2    0    1    3    7    10    1    0    23    2    0.200    0.319    0.325    0.644 [X]     Vicmal De La Cruz - DNP                                                           [X]     Austin Dean    A    LF    46    7    12    2    0    1    3    7    14    1    0    26    1    0.261    0.370    0.370    0.740 [X]     Brandon Dulin - DNP                                                           [X]     Shawon Dunston    A    DH    27    3    4    0    1    0    3    2    2    1    1    12    0    0.148    0.207    0.222    0.429 [X]     Adam Ehrlich    A+    C    4    0    0    0    0    0    0    0    2    0    0    2    0    0.000    0.000    0.000    0.000 [X]     Thairo Estrada - DNP                                                           [X]     Eric Farris    AAA    RF    39    6    13    1    0    0    5    1    3    2    1    25    0    0.333    0.333    0.359    0.692 [X]     Cameron Flynn    A+    RF    50    6    12    1    1    1    3    0    10    1    0    19    1    0.240    0.240    0.360    0.600 [X]     Adonis Garcia    AAA    CF    31    3    6    1    0    1    4    2    2    0    0    20    0    0.194    0.235    0.323    0.558 [X]     Chris Gimenez    AAA    1B    19    3    4    0    0    0    2    4    5    0    0    8    2    0.211    0.348    0.211    0.559 [X]     Alex Glenn    A+    LF    40    7    11    2    1    2    9    4    5    2    0    9    0    0.275    0.333    0.525    0.858 [X]     Jake Goebbert    AAA    LF    38    7    14    5    2    0    5    4    7    0    0    19    0    0.368    0.442    0.605    1.047 [X]     Jack Hannahan - DNP                                                           [X]     Juan Herrera    A    SS    37    8    9    2    0    0    4    3    2    5    3    20    2    0.243    0.300    0.297    0.597 [X]     Mike Hessman    AAA    3B    45    7    14    2    0    2    5    5    14    1    1    20    1    0.311    0.385    0.489    0.874 [X]     John Hicks    AA    C    29    4    6    1    0    0    6    4    12    0    0    14    1    0.207    0.303    0.241    0.544 [X]     Justin Higley - DNP                                                           [X]     Tyler Holt    AA    LF    30    1    7    1    0    0    4    7    7    3    1    13    0    0.233    0.368    0.267    0.635 [X]     Cedric Hunter    AA    PH    26    5    5    0    1    1    3    5    4    0    2    9    0    0.192    0.353    0.385    0.738 [X]     Ender Inciarte    AAA    CF    60    16    17    2    2    0    3    7    11    5    1    18    1    0.283    0.358    0.383    0.741 [X]     Paul Janish    AAA    2B    35    4    8    0    0    0    1    2    3    0    0    14    0    0.229    0.289    0.229    0.518 [X]     James Jones    AAA    RF    42    7    13    2    1    0    4    3    9    2    1    23    0    0.310    0.370    0.405    0.775 [X]     Kyeong Kang    AA    DH    27    5    5    0    0    0    1    2    8    0    0    13    0    0.185    0.241    0.185    0.426 [X]     Matt LaPorta    AAA    PH-1B    37    9    10    1    0    3    9    7    6    0    0    22    0    0.270    0.400    0.541    0.941 [X]     Che-Hsuan Lin - DNP                                                           [X]     Gabriel Lino    A    C    34    7    8    4    0    1    4    7    15    0    0    15    1    0.235    0.357    0.441    0.798 [X]     Matt Lipka    AA    CF    47    5    11    3    0    0    4    3    7    6    1    13    0    0.234    0.294    0.298    0.592 [X]     Jack Lopez    A+    2B    45    4    9    3    0    0    6    3    3    4    0    22    2    0.200    0.250    0.267    0.517 [X]     Rafael Lopez    AA    C    31    6    10    5    0    1    5    7    7    0    0    10    3    0.323    0.447    0.581    1.028 [X]     Zach Lutz    AAA    3B    53    11    17    3    1    1    12    11    11    0    0    20    1    0.321    0.478    0.472    0.950 [X]     Luis Marte    A    SS    27    4    8    1    0    0    3    0    5    2    0    11    2    0.296    0.296    0.333    0.629 [X]     Hersin Martinez - DNP                                                           [X]     David Masters    A    SS    38    5    5    0    1    0    2    4    7    1    1    21    1    0.132    0.233    0.184    0.417 [X]     Shilo McCall - DNP                                                           [X]     Michael McKenry    AAA    C    30    4    8    3    0    1    7    1    4    0    0    8    0    0.267    0.281    0.467    0.748 [X]     Angel Morales    A+    CF    40    8    11    3    0    1    6    2    13    3    1    18    0    0.275    0.310    0.425    0.735 [X]     Austin Nola    AA    SS    47    4    10    1    0    0    6    8    7    1    0    21    2    0.213    0.351    0.234    0.585 [X]     Justin OConner    A+    C    38    3    6    2    0    1    4    1    12    0    0    26    4    0.158    0.200    0.289    0.489 [X]     Francisco Pena    AAA    C    31    4    8    0    0    2    3    3    9    0    1    15    0    0.258    0.324    0.452    0.776 [X]     Fernando Perez    A    DH    47    7    15    5    0    1    11    0    7    0    0    21    2    0.319    0.313    0.489    0.802 [X]     Daniel Pertusati - DNP                                                           [X]     Cord Phelps    AAA    2B    31    1    7    0    0    0    0    4    6    1    0    12    0    0.226    0.314    0.226    0.540 [X]     Shawn Pleffner - DNP                                                           [X]     Jason Pridie    AAA    CF    51    11    16    3    1    2    7    4    7    3    0    16    0    0.314    0.364    0.529    0.893 [X]     Juremi Profar    A    3B    22    2    6    0    0    1    4    2    3    1    0    8    3    0.273    0.333    0.409    0.742 [X]     Andrew Pullin    A    2B    47    3    9    2    0    0    1    3    15    1    0    25    1    0.191    0.250    0.234    0.484 [X]     Jeremy Rathjen    A+    RF    55    8    14    4    0    2    5    3    17    1    1    21    2    0.255    0.328    0.436    0.764 [X]     Trayvon Robinson    AAA    LF    42    2    7    1    0    0    1    2    13    0    1    13    0    0.167    0.200    0.190    0.390 [X]     Gabriel Rosa - DNP                                                           [X]     Adam Rosales    AAA    DH    41    7    7    0    0    2    6    7    15    0    0    18    1    0.171    0.306    0.317    0.623 [X]     Tyler Saladino    AAA    SS    48    9    18    4    1    1    7    7    9    2    1    15    4    0.375    0.446    0.563    1.009 [X]     Alex Sanchez - DNP                                                           [X]     Dave Sappelt - DNP                                                           [X]     Silento Sayles - DNP                                                           [X]     Rock Shoulders    A+    LF    47    4    10    2    0    1    6    5    17    0    0    19    0    0.213    0.288    0.319    0.607 [X]     Kevan Smith    AA    C    38    5    12    2    0    0    4    1    6    0    0    10    0    0.316    0.341    0.368    0.709 [X]     Tyler Smith    A+    SS    54    9    14    2    1    0    3    6    12    0    0    22    3    0.259    0.344    0.333    0.677 [X]     Eric Stamets    AA    SS    42    5    11    0    0    0    3    1    10    2    0    20    1    0.262    0.295    0.262    0.557 [X]     Cal Towey    A+    DH    54    8    13    3    2    1    7    7    16    4    0    20    1    0.241    0.359    0.426    0.785 [X]     J.R. 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Author Topic: Delta-Gamma-Theta Approximation  (Read 2793 times)

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Offline CRS245

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Delta-Gamma-Theta Approximation
« on: March 23, 2010, 08:50:52 PM »

Delta-Gamma-Theta Approximation

The definition of the Taylor Series is:

f(x)=f(x_0)+f'(x_0)(x-x_0)+\frac{1}{2}f''(x_0)(x-x_0)^2+\ldots

Let's do some one-to-one substitutions to make the Taylor Series fit our subject, option pricing.  Let  x=S_t and E=S_t-S_0  then if the "output" is the Option Price (f(x)=C(s_t)), the 1st derivative with respect to stock price will be Delta and the 2nd derivative with respect to stock price will be Gamma.  The infinite amount of terms following the 3rd term would, in most cases, be relatively small.  Therefore, in this case, those terms can be replaced by one simple error term.

Rewriting the Taylor Series equation gives us:

C(S_t)=S_0+\Delta E + \frac{1}{2}\Gamma E^2 + \text{error term}

Example
Using the same parameters from the 2nd example, estimate the change in call's value if the stock price increases to 210.

Recall that S_0=200, r=0.05,\sigma=0.2  and \Delta=0.8554 , so the approximate value of the call is:

C(S_t)=C(S_0)+\Delta E + \frac{1}{2}\Gamma E^2
C(S_t)=27.95+0.8554(10)+0.5\Gamma(100)=36.504+50\Gamma

The value of the call is highly dependent upon Gamma, the 2nd derivative of option price with respect to stock price.  Concavity and convexity will only forecast if the stock will level off or change even more in value, so those aspects of Calculus are important for forecasting stock prices.  What can we expect Gamma to be in this example?

There really isn't enough information given to calculate or predict the Gamma.  All that we know is that Gamma will be the same whether it is a call or a put.  Assuming Gamma to be zero makes the option follow a more linear pattern which is not a good estimation of the option itself, so an arbitrarily small value for Gamma will suffice.

Let's assume that:  \Gamma=0.02

 36.504+50\Gamma=37.504

The call value is expected to increase by 9.554, which is less than 10, the change in stock price.  This follows the laws of arbitrage and the increase in the call can be expected with the increase in the stock price.

The Error Term in Hedging
The approximation used in the last example is actually Delta-Gamma approximation.  To apply Delta-Gamma-Theta approximations to option values, the parameter of time must be introduced to the equation.  Specifically, the Greek of Theta must be used.  The additional error term, albeit small, will most often reduce the approximate option value because Theta is usually negative.  Why is Theta usually negative?  Theta measures the increase in option price with respect to the decrease in time to maturity, and options increase in value as maturity time increases due to extra room for volatility.

Less maturity time => Lower variance => Decreased expected value

The error term is measured in days, and time in the Delta-Gamma-Theta approximation is measured in years, so in order to keep all time variables equal, it must be converted like so:

C(S_t)=S_0+\Delta E+\frac{1}{2}\Gamma E^2 + \frac{t\theta}{365}

The number of days in a year is a matter of convention.  The banker may use 360 days whereas the actuary would use 365.25 days.  The investor may use actual number of days, 365 or 366, depending whether the current year is a leap year or not.

References

W. McDonald, R.L., Derivatives Markets (Second Edition), Addison Wesley, 2006

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« Last Edit: November 05, 2010, 06:35:16 PM by Colby »
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