Range Accrual

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Range Accrual Valuation

FinPricing covers the following range accrual valuation models:

  • Range Accrual Note
  • Callable Range Accrual Note
  • Callable Range Accrual with Single Low or High Barriers
  • Callable Range Accrual with Multiple barriers)
  • Callable Range Accrual with Accrual inside or outside range
  • Callable Range Accrual with Variable coupons)
  • Callable Range Accrual with Partial mature and notional)
  • Autocallable with Range Accrual Payoff
  • Check FinPricing valuation models

1. Range Accrual Introduction

A range accrual note provides the holder with regular coupons that are linked to the performance of an underlying asset or index. It pays a coupon that accrues daily when the index is within the range specified in the product term sheet. The notes are principal-protected and also contain an embedded option that allows the issuer to exercise a call on the notes on a set schedule of dates prior to maturity. This feature can be used to limit the return that is paid to the holder.

Similar to a single range accrual, a dual-range callable range accrual note pays a fixed coupon that accrues daily when both the equity index is above a certain level and interest rate index is within a certain range.

The underlying instrument of a range accrual has a very digital profile, which is highly sensitive to the slope of the caplet smile, so you are interested in calibrating to out-of-money caplets. On the other hand the callable range accrual may be able to degenerated into Bermudan swaptions hence you have volatility exposure to co-terminal swaptions.

2. Range Accrual Valuation

Since the processes for the log-normal equity price and FX rate are very similar, the equity stock price can be seen as a degenerated case of the FX rate with the dividend curve, which substitutes to the foreign interest rate curve, except that the dividend rate is not stochastic

Volatility model is analytically calibrated using implied volatility surfaces for caps and swaptions as well as the equity stock implied volatilities.

A range accrual note written on a single stock provides the holder with a periodic coupon that accrues only when the spot price is within a specified range on a given reset date. On each day t, the value of the coupon is

Daily accrual

The coupon paid at a period is given by

range accrual coupon

The payoff of a range accrual note is

range accrual payoff

Range accrual payments can be replicated by digital floorlet.

digital floorlet

The value of the callable range accrual note is the value of the non-callable range accrual note minus the value of the call.

range accrual value

Callable range accrual is evaluated by discretizing the state variable, only on the exercise dates for the trade. Between such dates a roll-back algorithm, based on numerical integration, is used. This style of implementation scheme is also used by the so-called Markov-functional models.

At each point in the lattice two values, R(t, x), the value of holding the option at time t at the state x, and E(t, x) the value of exercising the option, are computed. To initialize the roll-back algorithm, the hold value R(Tlast) is set to 0, on the last exercise date. The value of E(Tlast) computed using the lattice discount factors b Z(T,X, ti).

To compute this integral, the cross-over point is identified; that is, the value of X such that E(X) = R(X). Away from this point the payoff function V (X, T) is smooth and one can evaluate the integral using a typical numerical quadrature method such as Gauss-Legendre. If one does not take care to avoid this ‘kink’ and simply applies a standard numerical quadrature directly on the non-smooth function V (T,X), unpleasant oscillations in the PV can occur. When evaluating the quadrature, estimates of E(X) and R(X) at points not at the nodes are required. For this implementation, a standard interpolation method is used within the state space. Beyond the boundary the functions are assumed to be constant.

3. Related Topics