Quanto Himalayan Option Valuation

Himalayan options are a form of European-style, path-dependent, exotic option on a basket of equity underliers, in which intermediate returns on selected equities enter the payoff, while the equities

Quanto Himalayan Option Analytics

Himalayan options are a form of European-style, path-dependent, exotic option on a basket of equity underliers, in which intermediate returns on selected equities enter the payoff, while the

equities are subsequently removed from the basket.

Like other exotic option, Quanto Himalayan may also have cap floor constraints and barrier conditions. The barrier option may knock out the trade before maturity, referred to as https://finpricing.com/lib/EqBarrier.html

For a set of d equities (stocks or equity indexes), let their respective prices at time t

be Y1(t), . . . , Yd(t). Given a collection of fixing dates, t0, t1, . . . , td, let the equity returns

Ri,j , for i = 1, . . . , d, j = 1, . . . , d be defined as

Moreover, let the integer-valued function k : {1, . . . , d} 7! {1, . . . , d} be defined so as to

satisfy the following condition, for i = 1, . . . , d

In other words, k(i) is the index of the equity with the highest return, at the fixing

date ti, among those equities, whose returns at the previous fixing dates were not the

highest2. Given a strike level, K, the Himalayan option payoff is

The delta hedge ratio for i-th equity is defined as

where V is the option price and Yi(t) is the price of the i-th equity on valuation date.

The cross-gamma hedge ratio for i-th and j-th equities is defined as

The vega hedge ratio for i-th equity is defined as

where vB,i(tj) is the tj-term Black’s volatility for equity i.

This definition is consistent with the one in the case of domestic equities, that is, in the absence of a quanto adjustment. In the quanto case, the effect of the change in the quanto adjustment resulting from a change in the volatility is ignored.

The rho hedge ratio for i-th equity is defined as

where Ri(tj) is the tj-term spot interest rate for the currency of equity i.

The theta hedge ratio is defined as

where V (t + 1) is the option premium computed with the valuation date, t, shifted

forward by one day.

With the exception of the theta, the hedge ratios are computed using Malliavin weights.

The presence of the volatility calibration step requires an adjustment in the calculated

Malliavin hedge ratios, according to the chain rule.

Let X(_ ) be the d-dimensional vector of logarithms of the d gains processes. The vector process X(_ ) will be assumed to follow the SDE

Let the known dividend schedule for equity i consist of payments _i,1, . . . , _i,mi made at

times _i,1, . . . , _i,mi , all of which lie between the valuation date t and the last relevant

fixing tn. Furthermore, let the discount curve in the native currency of equity i be di,

that is, for any time s greater than the valuation time t, let the discount factor be

di(s − t).For times t _ s1, t _ s2, let us define di(s1, s2) = di(s2)/di(s1). We shall define

We employ the definitions of the respective hedge ratios, as stated in the section on

the WM pricing method. With the exception of the theta, calculated through the

finite difference technique, all hedge ratios are computed using the Malliavin weight

approach. Additional considerations arise from the impact of the calibration procedure

on the sensitivity ratios, as describe in the next section.

Moreover, as can be seen from the test results below, the relative error in the gamma

hedge ratios can be quite large. These quantities are generally very small, making the

relative error a poor indicator of agreement. Moreover, it appears that the Monte Carlo

sampling error is quite large for the gamma hedge ratio.

For a large number of paths, this ratio should be approximately normally distributed with mean zero and unit variance, if the means in the two models are equal. We believe that this is a more

meaningful way of comparing results whose Monte Carlo sampling error is large relative

to the size of their expectation.