Esteban Volkov remembers sensing the worst as soon as he saw unusual activity outside the Mexico City home he shared with his exiled grandfather, Leon Trotsky. Bitter experience and a scientific inclination left little room to hope for anything else in the teenager's mind.

"I felt the anxiety rise. The fear," says Volkov, recalling that day 72 years ago on 20 August when he came home from school to find that his grandfather had been struck in the head with an ice pick – the culmination of Joseph Stalin's long campaign to assassinate him. "It would have been going against the laws of probability for us to be lucky again. It would have defied mathematics."

That afternoon – 20 August 1940 – Volkov entered the garden, passing an agitated guard with his pistol drawn, to find the assassin wailing in a corner, beaten up by police. Entering the house, he caught sight of his fatally wounded grandfather on the floor, before being shooed away.

As his life slipped away, the founder of the Red Army who had become a thorn in the side of Stalin's regime, had managed to choke out an order to prevent his grandson getting too close to the bloody scene. "His was a very complete life," the 86-year-old Volkov says, summing up what it all means to him now. "He dedicated it to his ideas and, above all, to putting them into practice."

Delivered in the same pleasant garden where so much happened that day, Volkov's account of the last chapter of his grandfather's life provides a glimpse of the person behind the legend, as well as a hint of everyday normality that somehow survived the ideological fervour, historical drama and family tragedy of those extraordinary years.

Trotsky arrived in Mexico in 1937 after nearly a decade of wandering exile on the other side of the Atlantic, granted political asylum by the post-revolutionary government of President Lázaro Cárdenas at the behest of muralist Diego Rivera. By then Trotsky had not only lost the battle to direct the course of socialism in the Soviet Union to Stalin, but also all four of his children, and many other relatives besides. But Volkov, who got to Mexico two years later at the age of 13, says his grandfather never appeared downcast. "He was always full of life and energy," he recalls. "He was filled with a quite extraordinary and contagious confidence that the coming of [authentic] socialism was inevitable, inescapable and absolute."

Volkov also remembers Trotsky – the hard-nosed Bolshevik blamed for the brutal suppression of the Kronstadt rebellion – as a loving and attentive grandfather, when he wasn't shut away with his work. Most of his working hours were spent on a biography of Stalin, because he had been given a substantial advance in dollars, or planning the one he really wanted to write about Lenin.

The Old Man, as he was known in the household, would rise early to feed the chickens and rabbits he kept at the bottom of the garden, and then submerge himself in his work, stopping only for short naps and collective meals. Sometimes Volkov says he would go into the study to be with his grandfather for a while. He smiles at the memory of a game of chess in which he was roundly beaten. But Trotsky deliberately kept his grandson away from the political debates that reverberated around the house, as if to protect him from the terrible costs they had already exacted on his young life.

By the time he arrived in Mexico, Volkov had lived in Yalta, Moscow, Berlin, Vienna, Paris and on a Turkish island. Along the way his father was shot and his mother killed herself, his aunt had died of TB, and his grandmother was deported to Siberia. Volkov's uncle, with whom he lived for a while, was also killed. As well as Trotsky and Natalia Sedova, his second wife, Volkov shared his new Mexican home with about seven secretaries and guards he describes as "a large family". Most of them were American and all of them foreign, to avoid any suggestion that Trotsky was breaking his promise to stay out of local politics.

By then the household no longer had the bohemian air once provided by Rivera and Frida Kahlo. Volkov says the split was rooted in "political differences", rather than the much-rumoured romance between the revolutionary and the painter. He deems such talk irrelevant, given Kahlo's penchant "for sleeping with everyone".

But there were still games in the garden on Sundays, trips to the countryside to collect cacti, and outings to the cinema – at least until 24 May 1940, when about 20 gunmen burst into the house, led by Mexican muralist and Stalinist David Alfaro Siqueiros.

Volkov cowered behind his bed while bullets flew from three sides. In the next room Sedova pushed her husband – doped up with sleeping pills – into a corner, shielding him with her body.

The house was subsequently fortified with metal doors and watchtowers and the outings were banned, but some of the good humour remained. "Natasha, they have given us one more day of life," Volkov remembers Trotsky joking most mornings. "My grandfather knew that the next attack would be soon. The question was where from."

It came from inside. A Stalinist agent had wormed his way into the house through an affair started years before with a trusted member of the inner circle. "It was very well done," Volkov says with a hint of admiration. "We were like children compared to them."

Volkov is suddenly mournful as he remembers the "lonely and empty" period that followed, but describes himself as a lucky man who saw history unfold before him and then the chance of a tranquil and rewarding career as a research chemist and a happy marriage with children. Three of his four daughters – a poet, a statistician and a doctor – still live in Mexico, while the fourth is now the head of the national institute of drug abuse in the US.

Volkov says he is at peace with having let the family revolutionary tradition drop, but he believes that radical change along the lines laid out by his grandfather remains necessary: "Capitalism is a total disaster, completely unable to solve humanity's problems. It is an obsolete system." This year, as on every anniversary of Trotsky's death, a small commemoration is planned beside the revolutionary's small tomb decorated with a hammer and sickle and set in the garden of the old house. Now a museum, it does not boast the queues that form outside Kahlo's house around the corner. Not that the modesty of it all seems to bother Volkov, who predicts that future generations will give Trotsky his rightful place. "We are in no hurry," he says with the kind of confidence in inexorable historical processes that would do his grandfather proud.