By Jan H. Maruhn
This e-book provides hedging concepts for a category of monetary concepts. The emphasis is on theoretical and numerical features, i.e., the glory of acceptable lifestyles, duality and convergence effects. The mathematical options diversity from monetary arithmetic, stochastic and semi-infinite optimization, convex research, partial differential equations to semi-definite optimization.
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Extra resources for Robust Static Super-Replication of Barrier Options (Radon Series on Computational and Applied Mathematics)
For all ! k 2 . t. 6) A feasible point of this problem will guarantee the super-replication property for the M considered scenarios. 1) which in turn increases 36 Chapter 3 An Optimization Approach to Static Super-Replication the computational effort to solve the problem. 2 the set of static knock-out trading strategies SKO is isomorphic to RnC1 . ˛0 ; : : : ; ˛n /T 2 RnC1 denotes the unique vector corresponding to the trading strategy 2 SKO . / W RnC1 ! R is linear 8 t 2 Œ0; T ; 8 ! 2 .
This call guarantees a perfect replication of the target option payoff if the barrier D is not hit during the lifetime of the option. / < T this call has a significant positive value, whereas the up-and-out call expires worthless. The idea is now to offset this positive value of the call at a discrete set of time points 0 D t1 < t2 < : : : < tn t < T along the barrier by adding calls with varying maturity to the portfolio. e. tj ; D/ denotes the Black–Scholes value of the i th hedge instrument at time tj if the underlying S tj equals the barrier D.
1), we consider the following example of hedging an up-and-out call in Heston’s stochastic volatility model which we will refer to in later sections. 2. 3) where r is the constant riskfree rate and ı is the constant dividend yield. The variance Y of S is modeled by a square-root process, where denotes the mean reversion rate, is the long-run mean of the variance and the volatility of volatility. W 1 and W 2 are standard Brownian motions with correlation coefficient . e. W 1 ; W 2 /. ST K/C 1¹max t2I S t