public class FeynmanKacEx5 extends Object
Solves for the "Greeks" of mathematical finance.
In this example, the FeynmanKac class is used to calculate the
so-called "Greeks" of mathematical finance. The "Greeks" are defined in terms
of various derivatives of Feynman-Kac solutions and have special
interpretations in the pricing of options and related financial derivatives.
In order to illustrate and verify these calculations, the Greeks are
calculated by two methods. The first method involves the Feynman-Kac (FK)
solution to the diffusion model for call options as described in example 2
FeynmanKacEx2, for the Black-Scholes case (\( \alpha \rightarrow
2\)). The second method calculates the Greeks using the closed-form
Black-Scholes (BS) evaluations, which can be found here:
Wikipedia:
The Greeks.
Example 5 calculates FK and BS solutions \(V(S, t)\) to the BS problem and the following Greeks:
Intrinsic Greeks include derivatives involving only S and
t, the intrinsic FK arguments. In the above list,
Value, Delta, Gamma, Theta, Charm, Color, and Speed are all
intrinsic Greeks. As is discussed in Hanson, R. (2008)
Integrating Feynman-Kac equations Using Hermite Quintic Finite
Elements, the expansion of the FK solution function \(V(S, t)\) in terms
of quintic polynomial functions defined on S-grid subintervals and
subject to continuity constraints in derivatives 0, 1, and 2 across the
boundaries of these subintervals allows Value, Delta, Gamma, Theta,
Charm, and
Color to be calculated directly by the FK class methods
getSplineCoefficients, getSplineCoefficientsPrime,
and getSplineValue.
Non-intrinsic Greeks are derivatives of V involving FK parameters other than the intrinsic arguments S and t, such as r and \(\sigma\). Non-intrinsic Greeks in the above list include Rho, Vega, Volga, and Vanna. In order to calculate non-intrinsic Greek (parameter) derivatives or intrinsic Greek S- derivatives beyond the second (such as Speed) or t- derivatives beyond the first, the entire FK solution must be calculated 3 times (for a parabolic fit) or five times (for a quartic fit), at the point where the derivative is to be evaluated and at nearby points located symmetrically on either side.
Using a Taylor series expansion of \(f(\sigma + \epsilon)\) truncated to m+1 terms (to allow an m-degree polynomial fit of m+1 data points):
$$ f(\sigma + \epsilon)\;=\;\sum_{n = 0}^m {\frac{f^{(n)}(\sigma)}{n!} \epsilon^n} $$
we are able to derive the following parabolic (3 point) estimation of first and second derivatives \(f^{(1)}(\sigma)\) and \(f^{(2)}(\sigma)\) in terms of the three values: \(f(\sigma - \epsilon)\), \(f(\sigma)\), and \(f(\sigma + \epsilon)\), where \(\epsilon = \epsilon_{frac} \sigma\) and \(0 \lt \epsilon_{frac} \lt\lt 1\):
$$ f^{(1)}(\sigma)\; \equiv \;\frac{\partial f(\sigma)}{\partial \sigma}\;\approx\; f^{[1]}(\sigma, \epsilon)\; \equiv \;\frac{f(\sigma + \epsilon)-f(\sigma - \epsilon)}{2 \epsilon} $$
$$ f^{(2)}(\sigma)\; \equiv \;\frac{\partial^2 f(\sigma)}{\partial \sigma^2}\;\approx\; f^{[2]}(\sigma, \epsilon)\; \equiv \;\frac{f(\sigma + \epsilon)+f(\sigma - \epsilon)-2f(\sigma)}{\epsilon^2} $$
Similarly, the quartic (5 point) estimation of \(f^{(1)}(\sigma)\) and \(f^{(2)}(\sigma)\) in terms of \(f(\sigma - 2\epsilon)\), \(f(\sigma - \epsilon)\), \(f(\sigma)\), \(f(\sigma + \epsilon)\), and \(f(\sigma + 2\epsilon)\) is:
$$ f^{(1)}(\sigma)\;\approx\;\frac{4}{3} f^{[1]}(\sigma, \epsilon) \;-\; \frac{1}{3}f^{[1]}(\sigma, 2\epsilon) $$
$$ f^{(2)}(\sigma)\;\approx\;\frac{4}{3} f^{[2]}(\sigma, \epsilon) \;-\; \frac{1}{3}f^{[2]}(\sigma, 2\epsilon) $$
For our example, the quartic estimate does not appear to be significantly
better than the parabolic estimate, so we have produced only parabolic
estimates by setting variable iquart to 0. The user may try the
example with the quartic estimate simply by setting iquart to
1.
As is pointed out in Hanson, R. (2008), the quintic polynomial expansion
function used by Feynman-Kac only allows for continuous derivatives through
the second derivative. While up to fifth derivatives can be calculated from
the quintic expansion (indeed class FeynmanKac method
getSplineValue will allow the third derivative to be calculated
by setting parameter ideriv to 3 as is done in this
Example), the accuracy is compromised by the inherent lack of continuity
across grid points (i.e. subinterval boundaries).
The accurate second derivatives in S returned by the FK method
getSplineValue can be leveraged into a third derivative estimate
by calculating three FK second derivative solutions, the first solution for
grid and evaluation point sets \(\{S,\; f^{(2)}(S)\}\) and the second and
third solutions for solution grid and evaluation point sets \(\{S+\epsilon,\;
f^{(2)}(S+\epsilon)\}\) and \(\{S-\epsilon,\; f^{(2)}(S-\epsilon)\}\), where
the solution grid and evaluation point sets are shifted up and down by
\(\epsilon\). In this example, \(\epsilon\) is set to \(\epsilon_{frac}
\bar{S}\), where \(\bar{S}\) is the average value of S over the range
of grid values and \(0 \lt \epsilon_{frac} \lt\lt 1\). The third derivative
solution can then be obtained using the parabolic estimate:
$$ f^{(3)}(S)\;=\;\frac{\partial f^{(2)}(S)}{\partial S}\;\approx\;\frac{f^{(2)}(S + \epsilon)-f^{(2)}(S - \epsilon)}{2 \epsilon} $$
This procedure is implemented in this example to calculate the Greek
Speed. (For comparison purposes, Speed
is also calculated directly in this example by setting the
getSplineValue input S derivative parameter iSDeriv
to 3. The output from this direct calculation is called
"Speed2".)
The average and maximum relative errors (defined as the absolute value of the
difference between the BS and FK values divided by the BS value) for each of
the Greeks is given at the end of the output. (These relative error
statistics are given for nine combinations of Strike Price and Volatility,
but only one of the nine combinations is actually printed in the output.)
Both intrinsic and non-intrinsic Greeks have good accuracy (average relative
error is in the range 0.01 -- 0.0001) except for
Volga, which has an average relative error of about 0.05. This is
probably a result of the fact that Volga
involves differences of differences, which will more quickly erode accuracy
than calculations using only one difference to approximate a derivative.
Possible ways to improve upon the 2 to 4 significant digits of accuracy
achieved in this example include increasing FK integration accuracy by
reducing the initial step size (using method
setInitialStepsize), by choosing more closely spaced S
and t grid points (by adjusting method
computeCoefficients input parameter arrays xGrid
and tGrid), and by adjusting \(\epsilon_{frac}\) so that the
central differences used to calculate the derivatives are not too small to
compromise accuracy.
| Constructor and Description |
|---|
FeynmanKacEx5() |
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