_{1}

^{*}

To capture the impact of skewness and increase kurtosis on Black’s [1] European put values, we first substitute a Gram-Charlier (GC) distribution and next a Johnson distribution for Black’s Gaussian one. We introduce next each distribution in the option payoff and develop until the closedform expression of each put is arrived at. Finally, we estimate by simulations GC, Johnson and Black put options, choosing the latter one as benchmark. Simulation estimates encompassing both skewness and kurtosis show that, for at-the-money (ATM) or slightly in-the-money put values, 1) Black’s overvaluation with respect to Johnson puts is very significant and 2) its undervaluation with respect to GC ones remains moderate. Yet, by using the same skewness values for both GC and Johnson puts, we highlight the differences induced by increasing kurtosis between the two models. In this case, the GC overvaluation for ATM values is explained by value differences in the put time component. Yet, while both Black and GC values exhibit significant time decay close to expiry, Johnson’s ones remain stable up to maturity.

This paper offers a solution to the following problem: how to account for skewness and extremely large Pearso- nian kurtosis (up to 39) when estimating a European put option embedded in a non-traded asset (a bank’s credit- line commitment) subject to Basel III capital sufficiency? There are two steps to the solution: first to determine the four-parameter analytical solution of the put option and then to use it to value the credit risk of banks’ loan commitments.

In option finance, numerous empirical studies have shown that the log-return distribution exhibits positive or negative skewness coupled with various degrees of positive excess kurtosis. It thus makes sense to consider moving away from the Gaussian distribution underlying Black’s [^{1}. Secondly, the truncated expansion may not converge to the true value and may only approximate the unknown distribution. Thirdly, adding more terms in an orthogonal expansion may not necessarily mean greater accuracy: it is known that higher-order approximations may become more and more oscillatory.

So, as first improvement, let us consider replacing the GC expansion by a “true” GC distribution, namely a generalized GC distribution of which the density function is of the same form as the expansion―a normal den- sity times a polynomial. The latter true probability density function (pdf) has the advantage to be nonnegative, integrate to one and be of an even order―if it were odd the polynomial would take negative values for some x. Among the generalized GC distributions, we then focus on the GC four-moment family: The latter is then intro- duced in the option payoff which is developed until the closed-form expression of the European put option is ar- rived at. This expression will be referred to as the GC put option. The procedure is attractive as long as the lep- tokurtic distribution presents moderate kurtosis, namely no larger than 7. Yet, many of the distributions of well-known indices underlying options do exhibit positive or negative skewness combined with more severe kurtosis, in the range of 10 to 25. To wit, kurtosis values up to 10 are reported by Jha and Kalimipalli [

In view of this empirical evidence, our quest then narrows down to finding another distribution that accom- modates more severe excess kurtosis. This distribution is based on Johnson’s translation method [^{2} by which the transformed variate becomes at least approximately normal. Matching frequency curves is used as follows. Select the appropriate translation system so that the first four moments of the true distribution of variable x match those of an approximated distribution, say z. Compute next the four parameters of the latter distribution, introduce it in the option payoff and develop so as to arrive at a closed-form expression of the European put op- tion. This analytical expression will be referred to as Johnson’s put option. The latter analytics show that excess kurtosis mainly affects the time component of the put value.

To assess the benefits of GC and Johnson put options, the real-world case of the European put option embed- ded in banks’ credit line commitments is examined. It is first explained how the commitment value (to be re- ferred to as the indebtedness futures value) gives rise to the implicit European futures put option. The latter put is next valued by simulation under the Gaussian distribution, the true GC distribution and Johnson’s mo- ment-matching distribution, respectively. These put values are computed as a function of both indebtedness val- ue and option term, with Black’s put chosen as benchmark. Regarding the combined effect of skewness and kurtosis, the simulations reveal that, for at-the-money (ATM) or slightly in-the-money (ITM) indebtedness values, Black’s put values are greater than Johnson’s ones but smaller than GC’s ones. In the sequel, we speak of over- or undervaluation for the sake of simplicity and clarity. For deeper ITM indebtedness values, however, Black’s over- valuations with respect to both Johnson and GC estimates is minimal, though slightly more pronounced in GC’s case. The sole effect of increasing kurtosis is captured by the differences between GC and Johnson put values when the values of skewness and the other parameters are kept the same for both options. Here again, the GC over- valuation is greatest for ATM or slightly ITM values and an insightful explanation is provided. It is also worth- while examining how the put value is decaying over the option last two months for ATM or slightly ITM values. Does the rapid time decay of Black’s put value also extend to the GC and Johnson’s values?

The layout of the paper is as follows. Section 2 is devoted to the derivation of the European put options based on GC or Johnson four-parameter distributions. Section 3 examines the real-world situation of the European fu- tures put option embedded in a credit-line commitment. In the first subsection, the credit-line indebtedness value is derived and the statistical evidence regarding its log-returns is presented. The second one explains the choice of simulation parameters as well as the meaning of the put estimates. The third one assesses value differences between GC, Johnson and Black put options arising from skewness and excess kurtosis. Short concluding re- marks close the paper in Section 4.

We choose Black’s [_{B} (a choice conditioned by the case-study illustration) as starting point as well as benchmark for future comparisons. Namely:

with

where F_{0} denotes the date-0 underlying futures value; K, the exercise value; d_{1}, the standard moneyness with ^{3}; s, the standard deviation of the futures value; and T, the put expiration date. The log-return distribution underlying Black’s put option being Gaussian, skewness is nil and Pearsonian kurtosis is equal to three.

Our quest thus is: How can we improve on Black’s put option by varying skewness and kurtosis away from their Gaussian values? While a rich literature exits regarding option models with stochastic volatility, stochastic interest rate, with or without jumps (see Bakshi, Cao and Chen [

where

and

with He_{k} referring to the Hermite polynomial of order k. We then introduce the GC distribution in the option payoff and derive the analytical form of the four-parameter GC European futures put value that satisfies the martingale constraint (see for instance Corrado [

where

with

and

where μ_{3} and μ_{4} denote the centered moments of order i, for i = [3, 4], and ω accounts for these moments in the put standard moneyness. In addition

and

More concretely, the GC put value P_{GC} in Equation (2) is a Black European put option, P_{B}, minus terms for non-normal skewness and kurtosis. Here the GC distribution does influence directly the skewness and kurtosis coefficients.

In practice, while skewness values oftentimes are falling within the constrained interval, kurtosis values range way beyond seven, the upper-bound value of the constrained GC distribution. A first improvement can be found in a subset of the order-2m GC family introduced by Leon, Mencia and Santana [

Our quest thus then narrows down to: Does a distribution other than the generalized GC distribution exist that accounts for kurtosis values larger than 7 let alone 20? The solution is to be found in a particular family of fre- quency curves generated by Johnson’s translation system [_{J}:

where γ, δ, ξ, and λ are the four parameters of the unbounded translation system S_{U} defined in Appendix B,

^{‒1} the inverse of the hyperbolic sine function, the other terms having

been defined previously. In contrast with the GC distribution, Johnson’s S_{U} distribution does not influence di- rectly the coefficients of skewness and kurtosis. Yet, Johnson’s put is appealing since it accounts for all kurtosis values versus a maximum of 7 for the constrained Gram-Charlier distribution. There is no pretense on our part that moment fitting be regarded as providing the “best” solution in any sense. The more modest claim is that fitting by moments improves on the distribution-based GC approach and produces some significant put-value differences as evidenced from the credit-commitment case study of the next section.

Since Thakor, Hong and Greenbaum [^{4}.

A bank credit-line commitment allows a borrower to draw, say, over a one-year period [0, T] up to K = $100 at a floating prime rate defined as ^{5} value F_{j} as:

where ^{th}), F_{6} denotes a six- month-old indebtedness value which still has a remaining six-month term to maturity. The F_{j}s are the values of the banking instrument underlying the put in Equations (1), (2), and (6), of which the date-T payoff is

Suppose next that:

1) At year end, namely the date at which the bank’s audit under Basel III regulation takes place (see Basel Committee on Banking Supervision [^{6}.

2) At valuation date j, suppose that the fluctuations in the spot markup of the floating prime rate on demand loans result in_{j} is less than the option strike price K. When

Given Equation (7), let us now consider the distribution of the monthly log returns of the indebtedness value. Namely

where _{t}F(j) is the date-t value of the j-month-old indebtedness value. Expression (8) thus generates a time series of monthly change relatives from an indebtedness value that remains continuously j-month-old. The log-F rela- tives from the third to the ninth month are listed in Exhibit 1. From the statistical evidence presented in the third column, the volatility of the empirical distributions fluctuates in the narrow range [1.51% p.a. to 1.63% p.a.] for log-F relatives computed for the selected months. The confidence intervals for the normal sample skewness and kurtosis coefficients with 300 observations are computed in the note at the bottom of the exhibit. According to the statistics shown in the fourth and fifth columns, several positive and negative skewness coefficients as well as all kurtosis values fall outside their respective confidence intervals. This indicates that the empirical distribu- tions present mostly weak asymmetry coupled with an extremely strong leptokurtic pattern (very severe Pearso- nian kurtosis). Since the indebtedness value is a non-traded banking instrument, the historical values of the vola- tility, asymmetry and kurtosis coefficients from Exhibit 1 will be used in the next subsection, in which the put option reflecting commitment credit or spread risk is priced.

The embedded put values are estimated by simulations based on the statistical evidence presented in Exhibit 1. From the information reported in its columns 6 and 7, nearly all indebtedness values vary between $96.2 and $104.1, with $100 being par value; we thus set F_{j} at $100, $99.5, $99, $98.5, $98, and $97.5^{7} for a commitment put that moves progressively in the money. For these indebtedness values, the simulations are performed for commitments with strike price K = $100, short-term risk-free rate r = 3% p.a., remaining terms to maturity (T ? j) from 3 to 9 months, and volatility and skewness values from Exhibit 1. Regarding Gram-Charlier puts, we choose a kurtosis value of 6.5 which, while just remaining under the Jondeau-Rockinger upper bound, still accommodates the skewness estimates of Exhibit 1. As for Johnson’s put values, we use Tuenter’s [

Before commenting on the simulation patterns, we clarify the meaning of our slightly in-the-money reference scenario, namely the European put value of which the entries are F = $99 and T ? j = 6 months in the first matrix of _{J} = $0.985, the Johnson put has an equilibrium value of slightly less than 1% per $100 of commitment if the floating prime-rate commitment with say a 2.0% p.a. fixed markup is priced when the same-date spot-loan stochastic markup is 3.0% p.a. On the other hand, Black and Gram-Charlier corresponding put values, P_{B} = $1.101 and P_{GC} = $1.092 in the table second and third matrices respectively, are larger and thus most likely overvalued in terms of P_{J}. ^{8}. For illustrative purpose, the first (Johnson) matrix is shown in Chart 1. Its visual

Exhibit 1. Statistical analysis of the 300 monthly observations of the time series of indebtedness-value change relatives computed with Equation (8) for the period from 1988.01 to 2012.12.

^{A}Monthly unbiased (divided by n-1) value × √12 = sigma in percent per annum. Note: For a sample size n = 300 observations, the 95% confidence intervals for normal sample coefficients of skewness and kurtosis are: ±1.96 (6/300)^{1/2} = ±0.277 and 3 ± 1.96 (24/300)^{1/2} = 3 ± 0.554, respectively. Source: Statistics Canada monthly time series V122495 and V122504 of the prime credit rate and one-month banker’s acceptance rate of chartered (commercial) banks, respectively.

Risk ¯ | Term: ® | 3 ms | 4 ms | 5 ms | 6 ms | 7 ms | 8 ms | 9 ms | |
---|---|---|---|---|---|---|---|---|---|

P_{j} | F = 100.0 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 | 0.005 | |

99.5 | 0.492 | 0.495 | 0.494 | 0.493 | 0.491 | 0.49 | 0.49 | ||

99.0 | 0.993 | 0.99 | 0.988 | 0.985 | 0.983 | 0.98 | 0.98 | ||

98.5 | 1.488 | 1.485 | 1.481 | 1.478 | 1.474 | 1.47 | 1.47 | ||

98.0 | 1.985 | 1.98 | 1.975 | 1.97 | 1.965 | 1.96 | 1.96 | ||

97.5 | 2.481 | 2.475 | 2.469 | 2.463 | 2.457 | 2.45 | 2.45 | ||

P_{B } | F = 100.0 | 0.309 | 0.371 | 0.394 | 0.447 | 0.485 | 0.511 | 0.51 | |

99.5 | 0.618 | 0.669 | 0.668 | 0.735 | 0.769 | 0.792 | 0.79 | ||

99.0 | 1.028 | 1.058 | 1.069 | 1.101 | 1.125 | 1.142 | 1.139 | ||

98.5 | 1.496 | 1.506 | 1.509 | 1.525 | 1.539 | 1.548 | 1.544 | ||

98.0 | 1.986 | 1.985 | 1.983 | 1.987 | 1.991 | 1.994 | 1.989 | ||

97.5 | 2.481 | 2.476 | 2.470 | 2.468 | 2.466 | 2.463 | 2.457 | ||

P_{GC } | F = 100.0 | 0.393 | 0.473 | 0.501 | 0.568 | 0.616 | 0.649 | 0.647 | |

99.5 | 0.687 | 0.746 | 0.763 | 0.805 | 0.87 | 0.893 | 0.875 | ||

99.0 | 1.036 | 1.067 | 1.073 | 1.092 | 1.158 | 1.171 | 1.143 | ||

98.5 | 1.478 | 1.475 | 1.47 | 1.463 | 1.509 | 1.511 | 1.481 | ||

98.0 | 1.975 | 1.956 | 1.943 | 1.917 | 1.935 | 1.926 | 1.90 | ||

97.5 | 2.478 | 2.462 | 2.448 | 2.419 | 2.415 | 2.399 | 2.38 |

Note: Matrix P_{J}: Johnson’s European futures put values from Equation (6). Matrix P_{B}: Black’s European futures put values from Equation (1). Matrix P_{GC}: Gram-Charlier’s futures put values from Equation (2). Parameter definition: F = indebtedness futures value in $ computed from Equation (7); K = $100 the credit-line exercise value; r = short-term risk-free rate of interest, in % per annum; T = commitment maturity date; and T ‒ j = time remaining to commitment put expiry, in months. Common parameters: L = 100; r = 0.03; T ? j = 3,…, 9 months with T = 12 months. The volatility, skewness and kurtosis parameter values are from columns 3 to 5 of Exhibit 1.

inspection reveals that the straight put-value surface is down-sloping implying that Johnson’s put values are far more driven by risk changes than by maturity changes (namely not significantly tilted in the term-to-maturity dimension). Similar but slightly curved down-sloping surfaces (not shown here) depict the GC and Black put- value matrices.

Value differences due to the joint impact of skewness and kurtosis on put values are examined first. Selecting Black’s put value as benchmark, the differences [P_{B} ? P_{J}]/P_{B} and [P_{B} ? P_{GC}]/P_{B}, namely Black’s percentage dif- ferences with respect to Johnson’s and GC put values, highlight the departures from the Gaussian distribution. These are shown in the first and second matrices of

Chart 1. Surface of Johnson’s at- and in-the-money European futures put values as a function of risk and term.

_{ } | Term ® Risk ¯ | 3 ms | 4 ms | 5 ms | 6 ms | 7 ms | 8 ms | 9 ms |
---|---|---|---|---|---|---|---|---|

(P_{B} ‒ P_{J})/P_{B}_{ } | F = 100.0 | 98.38 | 98.61 | 98.76 | 98.90 | 98.95 | 99.02 | 99.07 |

99.5 | 19.66 | 26.08 | 28.27 | 32.99 | 36.08 | 38.09 | 38.13 | |

99.0 | 3.49 | 6.40 | 7.59 | 10.51 | 12.65 | 14.15 | 14.18 | |

98.5 | 0.51 | 1.41 | 1.86 | 3.13 | 4.21 | 5.03 | 5.04 | |

98.0 | 0.06 | 0.26 | 0.39 | 0.85 | 1.29 | 1.67 | 1.68 | |

97.5 | 0.00 | 0.04 | 0.07 | 0.20 | 0.36 | 0.51 | 0.52 | |

(P_{B} ‒ P_{GC})/P_{B}_{ } | F = 100.0 | −27.29 | −27.19 | −27.12 | −26.92 | −27.09 | −27.00 | −26.84 |

99.5 | −11.21 | −11.34 | −10.79 | −9.53 | −13.21 | −12.82 | −10.69 | |

99.0 | −0.70 | −0.89 | −0.44 | 0.81 | −2.93 | −2.58 | −0.29 | |

98.5 | 1.24 | 2.04 | 2.60 | 4.10 | 1.91 | 2.35 | 4.09 | |

98.0 | 0.58 | 1.50 | 2.02 | 3.51 | 2.83 | 3.39 | 4.45 | |

97.5 | 0.13 | 0.59 | 0.90 | 1.95 | 2.05 | 2.59 | 3.14 | |

(P_{GC} ‒ P_{J})/P_{GC}_{ } | F = 100.0 | 98.73 | 98.91 | 99.02 | 99.13 | 99.18 | 99.23 | 99.21 |

99.5 | 27.76 | 33.61 | 35.26 | 38.83 | 43.54 | 45.13 | 44.11 | |

99.0 | 4.17 | 7.23 | 8.00 | 9.77 | 15.14 | 16.31 | 14.43 | |

98.5 | −0.74 | −0.65 | −0.76 | −1.01 | 2.34 | 2.74 | 0.99 | |

98.0 | −0.53 | −1.26 | −1.66 | −2.75 | −1.58 | −1.77 | −2.90 | |

97.5 | −0.12 | −0.55 | −0.84 | −1.78 | −1.72 | −2.13 | −2.71 |

Note: Matrix (P_{B} ‒ P_{J})/P_{B}: Johnson’s differentials with respect to Black’s put values. Matrix (P_{B} ‒ P_{GC})/P_{B}: Gram-Charlier’s differentials with respect to Black’s put values. Matrix (P_{GC} ‒ P_{J})/P_{GC}: Johnson’s differentials with respect to Gram-Charlier’s futures put values. Parameter definition: P_{B}: Black’s European futures put values from Equation (1); P_{GC}: Gram-Charlier’s futures put values from Equation (2); P_{J}: Johnson’s European futures put values from Equation (6); F = indebtedness futures value in $ computed from Equation (7); and Term = time remaining to commitment put expiry, in months.

or slightly ITM indebtedness values, Black’s put values overestimate Johnson’s ones but underestimate GC’s ones. Percentage wise, Black’s overvaluation with respect to Johnson values is much greater (ranging up to 99.07%) than the undervaluation with respect to GC values (ranging up to 27.29%). For deeper ITM indebted- ness values, Black’s overvaluations with respect to both Johnson and GC are minimal, with those with respect to GC’s values slightly larger than those with respect to Johnson’s values. By way of contrast, the table third matrix highlights the kurtosis-induced differences between GC and Johnson put values (skewness and the other parameter values being then kept identical for both puts), namely [P_{GC} ? P_{J}]/P_{GC}. This matrix is also mapped into Chart 2 below. Recall that for GC put values, kurtosis was fixed at 6.5 while it ranges from 24.16 to 39.34 for Johnson’s ones in Exhibit 1. In Chart 2, the GC largest overvaluation (99.23%, close to the front upper left corner of the chart) takes place when the indebtedness value is ATM with 8-month re- maining to maturity. Yet, overvaluation turns into slight undervaluation for deeper ITM indebtedness values, witness the 2.71% undervaluation for the longest remaining term (9 months)―in the front lower corner of the chart.

Now why are the differences between Johnson and Black and Gram-Charlier put values largest for ATM in- debtedness values? According to the developments in Appendix B, it is the time value of the Johnson put that makes all the difference. As for any option, this time value usually happens to be largest for the ATM value. But as the ATM time value is smallest for the Johnson put relative to the Black and Gram-Charlier ones, this is where their value differences are greatest. But for deeper ITM put values, the time-induced value differences between the three procedures dwindle significantly. Moreover, changes in the ATM put values over the last two months before expiry provide some additional insight. For Black’s put, it is well-known that the ATM values are rapidly decaying over the option last two months. Exhibit 2 presents for the reference scenario (the put with six months left to maturity in

While both Black’s and GC’s ATM values decay rapidly over the two last months, Johnson values remain remarkably stable throughout at the much lower level of about half a cent. For the slightly in-the-money values, Johnson’s ATM put values also remain stable about 50 and 99 cents respectively. Yet, while Black’s and GC’s

Chart 2. Over- and undervaluation of Gram-Charlier’s European futures put values with respect to Johnson’s corresponding ones: Values differences as a function of risk and term.

Exhibit 2. Black, GC and Johnson at- or slightly in-the-money put values, expressed in dollars.

values remain larger than Johnson’s ones, they are slowly converging to Johnson’s ones, with two weeks re- maining to maturity. Practically, other things remaining constant, long-position holders of Black’s and GC puts should sell their positions preferably two months before maturity so as to avoid losing time value. On the other hand, holders of long Johnson’s puts can maintain theirs without any loss of time value.

So, we suggest the following heuristic explanation for such ATM time differences. With a very leptokurtic distribution, Johnson put values are bunched close to the peak of a density function that does not exhibit much volatility, and may or may not have thick tails. Thus using a distribution that allows for increasing kurtosis val- ues reduces the time value of ATM or slightly ITM European futures put options; this is the case of Johnson’s approach as it is able to accommodate higher levels of kurtosis. In other words, Johnson distributions may be well-suited to time series in which value changes over time take place in a narrow band as for interest rates or are range-bound as in the case of many default-free bonds. Needless to say, additional empirical studies cover- ing traded options with different underlying financial instruments should be considered so as to corroborate the patterns evidenced for an embedded put option on a non-traded underlying bank instrument.

To capture the impact of skewness and increasing kurtosis on values of Black’s [

Next Black, GC and Johnson estimates of the European put option embedded in credit line commitments are obtained by simulations, with Black’s put option selected as benchmark. Regarding the combined effect of skewness and kurtosis, the simulations reveal that, for at-the-money (ATM) or slightly in-the-money (ITM) in- debtedness values, Black’s European put values overestimate Johnson’s ones but underestimate GC’s ones. For deeper in-the-money indebtedness values, however, Black’s overvaluations with respect to both Johnson and GC are minimal, although slightly more pronounced in GC’s case. The sole effect of increasing kurtosis is captured by the differences between GC and Johnson put values when the values of skewness and the other parameters are kept the same for both options. Here again, the GC overvaluation is most significant for at- or slightly in-the-money indebtedness values because the time component of Johnson’s put is smaller than that of Black’s or GC’s. Thus using a distribution that allows for increasing kurtosis values reduces the time value of ATM or slightly ITM European futures put estimates; this is more evident for Johnson’s approach since it is able to ac- commodate higher levels of kurtosis. This pattern seems characteristic of very peaked distribution with low vo- latility. Such distributions can be found for default-free bonds or interest-rate time series in normal circums- tances. Additional empirical studies covering traded options with different underlying financial instruments should be considered so as to corroborate the patterns evidenced in this case study. This constitutes one of the avenues for further study.

For helpful comments and discussions, I thank Daniel Dufresne, Anatole Joffe, Steven Lo, Lewis Tam, and par- ticipants to a research seminar at the University of Macau, China, the 2013 Conference in Mathematical Finance in Montreal, Canada, and the 2013 Conference on Quantitative Methods in Finance in Sydney, Australia.