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I am trying to find the industry accepted method on how to price a long term American call option (maturities 5 to 10 years) on an underlying which is an accumulation fund (so no dividend payouts) which is not directly available for the retail market through any exchange. As, to my knowledge, also these option quotes are not readily available in the market, I am a bit stuck on how to determine the long-term volatility (5-10 years) to be used in the Black Scholes pricing model or alternatively which parties in the market could deliver these quotes or even the software and approaches used by market makers to price these kind of options (we would need to price the options 5 to to 6 times per year with different maturities).

I would greatly appreciate any help on this.

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The equity options "embedded" in convertible bonds are frequently of similarly long tenor to yours, so you may want to consider reviewing some convertibles literature.

In my professional experience with convertibles, by far the most common approach for dealing with volatility was a jump-adjusted Black-Scholes with a premium, which is a slight variation on what @AKdemy proposes.

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I doubt anyone will provide or have a vol surface for this. Well, if it is sold, your market maker will price that but that is a different story in my opinion. It's like buying insurance for your car vs a Bugatti Chiron. The latter will not follow standard logic and likely have a hefty premium to a "theoretical" price of insurance. Long term vol for 10 years is even tough for the most liquid options on say S&P500.

It's usually all extrapolated. If no listed options are available, you can only search for proxy vol surfaces (closest available asset in terms of co-movement).

In terms of American, I would simply ignore it. Early exercise will without dividends essentially never make sense apart for maybe some obscure reasons.

Generally for such long dated options, hybrid models with stochastic interest rates are often considered but for this case, when not even vol is well defined, you can safely ignore that too.

Long story short, find a proxy vol and use vanilla European pricer.

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Options with very long maturities (more than 3 years) can be traded on OTC markets. Globally, the longer the maturities, the harder are such options to find. Indeed, market makers are reluctant to take such positions in book because the hedging can be hard to establish, and also they have no volatility benchmark available... You can find some counterparties willing to enter a contract but that's rare. That's even more true with 5 to 10 years options... Intuitively, imagine the risk you have to bear over 5 to 10 years...

Furthermore, do not forget Market makers are supposed to make money while hedging, but without any clue about how to hedge, making money becomes less likely... Thus, it does not interest them that much...

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