# Density Implied by Shimko Method

### Description

`dshimko`

is the probability density function implied by the Shimko method.

### Usage

1 | ```
dshimko(r, te, s0, k, y, a0, a1, a2)
``` |

### Arguments

`r` |
risk free rate |

`te` |
time to expiration |

`s0` |
current asset value |

`k` |
strike at which volatility to be computed |

`y` |
dividend yield |

`a0` |
constant term in the quadratic polynomial |

`a1` |
coefficient term of k in the quadratic polynomial |

`a2` |
coefficient term of k squared in the quadratic polynomial |

### Details

The implied volatility is modeled as: *σ(k) = a_0 + a_1 k + a_2 k^2*

### Value

density value at x

### Author(s)

Kam Hamidieh

### References

D. Shimko (1993)
Bounds of probability.
*Risk*, 6, 33-47

E. Jondeau and S. Poon and M. Rockinger (2007):
*Financial Modeling Under Non-Gaussian Distributions*
Springer-Verlag, London

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 | ```
#
# a0, a1, a2 values come from Shimko's paper.
#
r = 0.05
y = 0.02
a0 = 0.892
a1 = -0.00387
a2 = 0.00000445
te = 60/365
s0 = 400
k = seq(from = 250, to = 500, by = 1)
sigma = 0.15
#
# Does it look like a proper density and intergate to one?
#
dx = dshimko(r = r, te = te, s0 = s0, k = k, y = y, a0 = a0, a1 = a1, a2 = a2)
plot(dx ~ k, type="l")
#
# sum(dx) should be about 1 since dx is a density.
#
sum(dx)
``` |

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