The cliché of a picture being worth a thousand words may be correct, but that doesn’t mean the same image doesn’t require explanation. Rather than take a screen shot of the myriad instances one can capture on Youtube and elsewhere, which might upset people, I’ve opted to go with an iconic, period image, and perhaps one reason for the madness:
There are many examples of this hand-above-the-head extension:
These are period images, mostly of foil play—to which I’ll return—and thus are “historical” in the sense that we normally mean it in historical fencing/martial arts. This said, there is an important distinction to be made between what we see in images like these and those from earlier works, equally historical but different in purpose. 
Every instructor has to make choices about what they want to include; they should be able to explain why as well. For me, lunging with one’s hand held so high is less covered than lunging with the hilt terminating at chin height as in earlier works for smallsword. If the goal is not to be hit, to reduce the chances of doubling, then a more conservative approach on the extension of the arm, the very first action made in the series of movements that comprise the lunge, makes sense.
The treatise from which I work most is P.J. F. Girard’s Traité des armes (1740), though much of my curriculum brings in ideas from Domenico Angelo’s L’École des armes/The School of Fencing (1763/1787) and de Liancour Le maître d’armes (1686), among others. I make a constant effort to read and compare the works I rely upon with others, incorporating material from some, merely noting others, and in some cases rejecting some ideas for inclusion in lessons. There are also times when I shelf a treatise for a time—with Le Perche du Coudray (L’Exercise des armes, 1750), there are sufficient reasons I’ve found to work on things and come back to it.
For example, the first lunge he covers, from quarte, depicts the hand higher than the rest of his work suggests. Is this so or an accident of the artist’s hand? The accompanying text has details, but isn’t specific as to hand height on the extension, he merely writes
Aprés sétre mes en garde et en mesure il faut dabord que ce soit la main qui parte la premiere en soutenant bien la poignet et baissant la pointe de l’Epée jusqu’a l’Estomach de l’Ennnemy, que les ongles soient tournés en En haut le bras bien etendue et bien soutenu…
After being in guard and in distance, the hand must move first, supporting the wrist well and lowering the point of the sword to the enemy’s stomach, so that the nails are turned upwards, the arm well-extended and well-supported… 
Here is the accompanying image:
In the illustration the lunging fencer’s hand is at his hairline, the lunge long, and nothing in the description offers an explanation as to why. Did Le Perche wish the fencer to lunge this way? Maybe. Parallels with other sources don’t offer much help. His lunge in tierce and seconde recall de la Touche’s (1670) super extended poses, the attacker’s head is much farther forward, and in some cases the attacker is not looking at target. 
The fencers in the plates all hold foils, but is that significant? In Plate 11, for example, the fencer attacking in quarte—his arm a bit lower than the hairline this time—uses the off-hand parry/check to prevent a double. This recalls similar images in Girard and suggests at least some attention to self-preservation. 
In sum, I’m not sure what to make of Le Perche yet and so, I’m noting what I see and will return to it later, hopefully armed with more information to help me read it more accurately. It’s possible that Le Perche is one of those texts that captures the subtle shift from practical swordplay to academic, salle play.
The hand in the air like one just doesn’t care… approach to the extension is popular in HEMA’s smallsword circles. While one can definitely point to sources for it, thus fulfilling the “H” of the acronym, the same cannot be said for the “MA” portion. To use a word I have come to detest, this raised hilt in the lunge is less “martial” than the custom preceding it.
Art vs. Practicality
It will come as no surprise that I do not teach this hand and hilt above the head approach to the extension. Graceful it may be, artful it may be, but from the standpoint of “don’t get hit” it’s less likely to protect one. Placed in this manner the weapon is more likely to fail to close the line as well as fall prey to an easy disarm.
Girard’s instruction illustrates well the more practical approach. In describing the guard, his very first point is revealing:
Il faut présenter la pointe de l’Epée droire, vis à vis la mamelle droite de l’ennemi, & que le demi tranchant regarde la terre.
The point of the sword must be presented straight, directed toward the right breast of the enemy, & so that the ridge of the blade is half turned to the ground 
This point forward, semi-extended position places the sharp tip as close to the opponent as possible while keeping the body as far back as possible. This compromise will be very familiar to students of Italian fencing—we see similar guard positions from Rosaroll & Grisetti to that of 20th century masters like Agesilao Greco. 
Taking Girard’s thrust in tierce as an example, the same conservatism is in evidence. To thrust in third, one
Coup de Tierce Haute tire Droit au dehors des Armes
Etant bien en Garde & en mesure, l’Epée engagée de tierce dehors les Armes, les ongles regardant la terre, je fais partir la main la premiere, les bras étendus en forme de croix, la main gauche également tournée de tierce, le genou gauche bien étendu, le pied à plat, & ferme & sur la terre, le genou droit plié; de forte qu’il soit vis à vis le milieu du pied droit & dans las ligne de l’ennemi, le corps soutenu, le côté droit panché au-dessus du genou droit, les deux épaules effacées, & la tête le long du bras à l’oposite de l’Epée, pour se garantir le visage. Le coup achevé dans cette Attitude se retirer en Garde, l’Epée devant soy, sans laisser baisser le poignet.
The Thrust of High Third pushed Directly Outside the Sword
Being in a good guard and in measure, the Sword engaged in Tierce outside the Sword, the nails facing the ground, I extend the hand first, arms extended in the shape of a cross, the left hand also turned in Tierce, the left knee well-extended, the foot flat and firm on the ground, the right knee bent; so that it is over the middle of the right foot & in line with the enemy, the body propped up, the right side over the right knee, shoulders profiled, & the head along the arm opposite to the Sword, to protect the face. The attack complete, in this this posture return to Guard, Sword out in front, without dropping the wrist. 
The image accompanying this section of text accords with it well, but the text itself is clear—hand first, so weapon first, opposition as one thrusts, and body positioned so that it is not only behind the guard, but so situated that the head is leaning away from that opposition, both for safety—as the master remarks—and so, I suspect, one can see the thrust and the opponent’s reaction better.
As a check on this, it’s worth taking a look at Girard’s first attack, the thrust in quarte. He writes
Le Coup de Quarte Haute tire’ Droit Au-dedans des Armes
Etant bien dans la Garde qu’il dit & en mesure, l’Epée engagée de quarte dans les Armes, je fais partir la main la premiere en levant le poignet, les oncles tournés dessus, regardant le Ciel, ainsi que le dedans de la main gauche; les bras étendus en croix, le corps panche du cote droit, & soutenu au-dessus du genou droit; les épaules effacées, la tête panché du côté de l’Epaule droite pour regarder le coup à l’oposite de l’Epée, de forte que le pommeau regarde l’oeil gauche, le bout du pied droit vis à vis l’ennemi, que le genou tombe perpendiculairement au dessus du milieu du pied avec le pied gauche à plat & ferme sur la terre, la jambe & la cuisse gauche élevées. Le coup achevé & tiré dans l’Attitude qu’il est dit, se retirer bien en Garde l’Epée devant soy, sans laisser baisser le poignet.
Being well in guard & in measure, the Sword engaged in quarte within the Swords, I start the hand first by raising the wrist, the nails turned up, looking at the Sky, the left hand following suit; the arms outstretched like a cross, the body leaning to the right side, & supported above the right knee; the shoulders profiled, the head titled on the side of the Right Shoulder, to see the blow against the Opposite Sword, so that the pommel is in line with the left eye, the tip of the right foot towards the enemy, the knee falls perpendicular over the middle of the foot with the left foot flat and firm upon the ground, the leg and left thigh straight. Having made the thrust in this position, as stated, return well in Guard with the Sword in front of you, without letting the wrist fall. 
It’s possible to regard the attacker in the image “pushing” fourth as having his hand at the hairline, but given the description that doesn’t follow. More likely, the artist, who knew their craft well, meant to show the head leaning right over the shoulder. The fact that the right hand is nearly as large as the head suggests that it’s meant to present closer to the onlooker, again helping create perspective. With the head so tilted, lining up the left eye with the pommel occurs at an angle and thus lower than the top of the head were the head upright. As a side note, while the opponent may have been hit in the chest, it’s possible he was hit in the arm—few smallsword or rapier texts for that matter spend much time on the forward target (a topic for another time), but it would be unwise to assume that any instructor would have looked down on such a strike.
Text, Awareness, & Choice
As with most things in HEMAland one can adopt whichever hand-height one wishes. However, I will maintain that knowing why we do what we do and where it comes from makes sense. It’s historical fencing, and part of any historical examination is understanding the context for things; it’s all the more important when the waters are muddy. The overlap between smallsword and foil play is significant, and thus, it’s a lot easier to treat them as one in the same. Did many smallsword fencers, who sought to learn to use a sword in earnest, learn via foils? Absolutely. But… they didn’t show up to Stephen’s Green or The Dueling Oak armed with one, and if they attempted to use moves better suited to impressing their fellow salle mates with their grace and poise than keeping them safe, they were likely to leave the Green or the Oak on their backs.
The hand held high versus at chin height reflects a change in culture. It coincides with the decline of dueling in some areas on the one hand, but with the increasing importance of fencing as elite accomplishment on the other. Both are historical, but reflect different contexts, different attitudes. The later our “smallsword” texts are in time, generally the more we see this reach for the sky business. De St. Martin, for example, whose L’art de faire des armes came out in 1804, has it, and while perhaps not in each plate, it’s there. In discussing the parry and riposte from third, de St. Martin specifies that “la main bien élevée au dessus de la tête,” that is, that one should have the “hand well raised above the head.”  We see the concern for grace, poise, and elegant execution of technique not only increase in the treatises, but in other areas as well. Many 18th century advertisements, even in the American colonies, often paired fencing and dancing as two pursuits for the genteel portions of society or those who wished to join their ranks. 
People pursue historical fencing for their own reasons and enjoy different aspects of it. That’s really pretty awesome. For those keen for smallsword or the study of early foil as a distinct track, some attention to these distinctions will only help one’s study. An awareness of the differences between foil as art form and foil as safer training weapon for smallsword is especially important for anyone purporting to study the MA aspects of HEMA. Few people will notice or care which one chooses, but it’s logical to know what we’re doing, where it comes from, and why it was done that way.
 Images are from:
Antonine Texier La Boëssière, Traité de l’art des armes, Paris: de l’Imprimerie de Didot, 1818, plate 1.
Mr. Olivier, Fencing Familiarized, or, A New Treatise on the Art of Sword Play, London: John Bell, 1771
John McArthur, The Army and Navy Gentleman’s Companion, or, A New and Complete Treatise on the Theory and Practice of Fencing, London: James Laver, 1780, plate 6.
 Le Perche du Coudray, L’Exercise des armes, Paris: 1750, from the pdf made available from the Biblioteca de la Univeridad Polltécnica Madrid, España, plate 6 and text, 29 and 31 in the pdf.
 Le Perche, L’Exercise des armes, plate 7, p. 33 of the pdf. Cf. plates 8 (37) and 9 (41).
 Le Perche, L’Exercise des armes, plate 11, p. 49.
 Girard, Traité des armes, La Haye: Chez Pierre de Hondt, 1740, 5-6. In note IV of the same section he specifies that the hand and hilt are turned to half-quarte (Avoir le bras droit, & le poignet flexible & tourné demi quarte…de forte que le demi tranchant de la main droite regarde le Ciel…) so I have translated note I’s last clause, que le demi trenchant regarde la terre, as “half turned” to reflect this, demi tranchant meaning “half edge.” [Pagination is that of the BnF pdf]
 See Giuseppe Rosaroll & Pietro Grisetti, The Science of Fencing, Milan, 1803, trans. by Christopher A. Holzman, 99-101; table 2, figure 2.
Of note, in the Italian edition, § 66. Descrizione della stessa [guardia], explains this about the weapon and arm:
Il braccio destro disteso verso il nemico rappresenta la linea di offesa colla spada, la punta della quale dee essere diretta all’occhio del nemico, parlando di spade nuda, ed al centro del petto nelle accademie; il gomito del detto braccio dee essere volto alla terra, e propriamente sulla direttrice. [p.45 in the pdf]
The arm, much like Girard advocates, is more or less straight out, only pointing at the eye for serious encounters (diretta all’occhio del nemico) versus the chest.
Agesilao Greco, on the right as one looks, some timebefore 1900–not the arm:
 Girard, Traité des armes, p. 16.
 Girard, Traité des armes, p. 14-15.
 M. J. de St. Martin, L’art de faire des armes, Vienne: de l’Imprimerie de Janne Schrämble, 1804, p. 30. NB: this is not an isolated mention. When de St. Martin first mentions the direct thrust (le coup droit), he mentions that the wrist should be “well elevated” (le poignet bien élevé), p. 29. It’s always dangerous to take illustrations at face value, but overall most of his depictions of the lunge show the hand higher than Girard and other, earlier masters advise.
 See for example, https://adverts250project.org/tag/fencing-master/page/2/