Abstract
In the current paper, we develop a methodology to price lookback options for cryptocurrencies. We propose a discreetly monitored window average lookback option, whose monitoring frequencies are randomly selected within the time to maturity, and whose monitoring price is the average asset price in a specified window surrounding the instant. We price these options whose underlying asset is the CCI30 index of various Cryptocurrencies, as opposed to a single cryptocurrency, with the intention of reducing volatility, and thus, the option price. We employ the Normal Inverse Gaussian (NIG) and Rough Fractional Stochastic Volatility (RFSV) models to the cryptocurrency market, and using the Black-Scholes as the benchmark model. In doing so, we intend to capture the extreme characteristics such as jumps and volatility roughness for cryptocurrency price fluctuations. Since there is no availability of a closed-form solution for lookback option prices under these models, we utilize the Monte Carlo simulation for pricing, and augment it using the antithetic method for variance reduction. Finally, we present the simulation results for the lookback options, and compare the prices resulting from using the NIG model, RFSV model with those from the Black-Scholes model. We find that the option price is indeed lower for our proposed window average lookback option, than for a traditional lookback option. We found the Hurst parameter to be H=0.09 which confirms that the cryptocurrency market is indeed rough.
Highlights
We find that BS lies in-between Normal Inverse Gaussian (NIG) and Rough Fractional Stochastic Volatility (RFSV)
When comparing option prices between the Monte Carlo and Antithetic Monte Carlo simulations for the Black-Scholes Model, we find that employing the latter significantly reduces standard error
We notice a clear decrease in price as time spacing between the monitor instances increases for both Black-Scholes and NIG implementations, a result of a lower number of asset return values being considered for pricing the option
Summary
Lookback options belong to a class of path dependent options Using these options, a trader is able to exploit or hedge extreme movements in the underlying asset price. We apply the Monte Carlo simulation to price lookback options under both a Black-Scholes, a pure jump process Normal Inverse Gaussian (NIG) and in the case for rough fractional stochastic volatility model. Following the method proposed in [6], we employ the Normal Inverse Gaussian (NIG) model, to capture extreme variations and jumps in the CCI30 returns.
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