Again, the idea is that the skew of this portfolio is used in order to generate the skew of the barrier option. The procedure is in App. A.

8 Skew for other barrier options: Other single barrier options can be obtained using sum rules. For example, the down and out (DO) call can be obtained from the DO put and the DO forward. Up and in (UI) options can be obtained from the standard options and the DO options. Double-barrier option skew can be obtained approximately by using a perturbative approach modifying the standard double-barrier closed-form solutions by skew corrections involving a limited number of images (e.g. one image for each barrier), along with sum rules.

Some Numerical Results for Barrier Options with Skew

While naturally not identical, the results of this model are qualitatively similar to other approaches that have skew. For example, an up-out S+P call option9 was analyzed with the following results:

UO Call Price with No Skew (zeroth order approximation):

✓^Strike vol, No Skew _ cc o Spot vol. No Skew _ cq o academic ' academic

UO Call Price with Different Skew Models10:

_ zr-i r\ ^ Local Volatility _AI+OS^"' _ f\C\ 8

'-'Perturbative Skew — ^MC,Simulation — — U-J > "-Static Replication ~ OU,°

It is seen that the zeroth order approximation is better with the spot vol than the strike vol. The results including skew using three different model approaches are similar. We have included results from a MC local volatility simulation (see below), and from Derman's static replication approach, to which we now turn.

Static replication11' is a clever way of approximating a complicated path-independent option by a replicating portfolio consisting of simple options. This set replicates the payoff and boundary conditions of the original option. Because payoff and boundary conditions uniquely determine any option once the stochastic equation for the underlying variable is given, the original option can be replaced by the replicating portfolio. Derman noticed that this can be achieved as a function of time, so once the replicating portfolio is chosen, the same portfolio remains a replicating portfolio provided that the parameters in the stochastic equations do not change.

9 Barrier option example - parameters: Strike 800, barrier 1,000, spot 940, time to expiration 0.3 yrs, rebate 64 paid at touch, strike vol 35%, spot vol 27.5%, barrier vol 24.5%, interest rate 6 % (ctn365), dividend yield 1.7% (ctn365). Parameters rounded off.

10 Calculation Details for the Perturbative Skew Approach: The academic model (including rebate) with spot vol was used for the 0th order approximation. Multiplicative skew corrections with averaging over knockout times were used. The rebate was included in the digital option for the bare replicating portfolio used to construct the skew. A call-spread approximation was used for the digital. I thank Tom Gladd for assistance in the calculations.

The inclusion of skew using this approach is carried out by putting skew into each of the simple options in the replicating portfolio. Because the options are simple, the job of finding their skew is in principle relatively straightforward.

In general, the replicating portfolio has an infinite number of options; in practice, this is approximated by a finite portfolio. Therefore, the achieved replication is only approximate.

We next give a summary of Derman's static replication method using path integrals and Green functions for a continuous-barrier up-out European call option". Although the allowed region for paths contributing to this barrier option is below the barrier, the idea is to replace the existence of the barrier by a replicating set of OTM call options that have no payoff for any path below the barrier. The OTM call option payoffs above the barrier are propagated back using the standard Green function without the barrier. That is, an equivalent problem with no barrier is used to solve the original problem with the barrier. This is similar to using images; here the images can be thought of as the set of OTM call options replicating the zero boundary condition along the barrier.

The UO call C^0 ~Ca"^t*, E^j is equivalent to an ordinary call C[t*,E^ , both expiring at t* with strike E, plus the replicating portfolio VrepUcaljng for the barrier at K . An OTM call Cf = C(tJ,Ejl) in VrepUcatm,, has strike Ef > K ? expires at tf < t* with payoff Cj, and has weight w(,. The weights {w^} are chosen to enforce the zero boundary condition. At time tj, VrepUcaling consists of those {Cf] with tf > tj that have not expired. Exhibiting the variable Xj — In Sj, which is required to be below the barrier Xj < In K,

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