A modified static hedging method for continuous barrier options (original) (raw)
2010, Journal of Futures Markets
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A Numerical Scheme Based on Semi-Static Hedging Strategy
2012
In the present paper, we introduce a numerical scheme for the price of a barrier option when the price of the underlying follows a diffusion process. The numerical scheme is based on an extension of a static hedging formula of barrier options. For getting the static hedging formula, the underlying process needs to have a symmetry. We introduce a way to "symmetrize" a given diffusion process. Then the pricing of a barrier option is reduced to that of plain options under the symmetrized process. To show how our symmetrization scheme works, we will present some numerical results applying (path-independent) Euler-Maruyama approximation to our scheme, comparing them with the path-dependent Euler-Maruyama scheme when the model is of the Black-Scholes, CEV, Heston, and $ (\lambda) $-SABR, respectively. The results show the effectiveness of our scheme.
Semi-Static Hedging of Barrier Options Under Poisson Jumps
International Journal of Theoretical and Applied Finance, 2011
We show that the payoff to barrier options can be replicated when the underlying price process is driven by the difference of two independent Poisson processes. The replicating strategy employs simple semi-static positions in co-terminal standard options. We note that classical dynamic replication using just the underlying asset and a riskless asset is not possible in this context. When the underlying of the barrier option has no carrying cost, we show that the same semi-static trading strategy continues to replicate even when the two jump arrival rates are generalized into positive even functions of distance to the barrier and when the clock speed is randomized into a positive continuous independent process. Since the even function and the positive process need no further specification, our replicating strategies are also semi-robust. Finally, we show that previous results obtained for continuous processes arise as limits of our analysis.
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