Target Redemption Forward (TARF)

A TARF is an FX derivative sold to investors, and the seller (the issuer) holds the obligations on the side opposite the buyer. It is built from a series of forwards, one per observation date (see FX Forwards), and what follows is written from the side that buys dollars (the side that gains when the yen weakens).

The payment on each date

A TARF holds one FX forward per observation date. If the spot rate on the observation date is on the favorable side of the strike, the buyer receives the gap. If it is on the unfavorable side, the buyer pays the gap multiplied by a constant α. α is often 1 or 2, and 20 to 60 observation dates is typical¹.

Here the boundary between the favorable and unfavorable sides is set at the strike price $X$. If the spot rate on observation date $k$ is $S_k$, the buyer's payment per unit of currency, $c_k$, is as follows.

$$c_k = \begin{cases} S_k - X & (S_k > X) \\ \alpha\,(S_k - X) & (S_k \le X) \end{cases}$$

When $S_k > X$, $c_k$ is the buyer's receipt. When $S_k \le X$, it is negative and is the buyer's payment. With α set to 2, the payment on the unfavorable side for the same gap doubles. This has the same result as doubling the amount delivered on the unfavorable side.

The definition in the source adds two conditions to this form. One is a band between the favorable and unfavorable sides in which the payment is zero. The other is that when the spot rate reaches a level $b$ above the strike, all remaining forwards are canceled, and the payment for that date is also forfeited¹. The latter is a condition equivalent to the knockout of a barrier option.

How the target is counted and three types of knockout

A TARF ends when the accumulated gain reaches or exceeds the target $U$. Here the total does not subtract payments on the unfavorable side. The sources also include a form that does subtract them (see the next paragraph). The accumulated total per unit of currency, $A_k$, is the total through the previous date plus the favorable gap on that date.

$$A_k = A_{k-1} + \max(S_k - X,\ 0)$$

The example of Chakrabarti et al. (a weekly TARF with 52 dates) also adds payments on the unfavorable side into the total, so a loss moves the total away from the target¹. The form that Luo and Shevchenko treat adds only the receipts on the favorable side and has no payments on the unfavorable side². What the total counts is set by the terms of the contract.

When a date arrives that takes the total past the target, the following three types decide how the receipt on that date is treated. They are called full gain, no gain, and part gain². Here only the treatment of the receipt is compared.

The paper deals not with the TARF itself but with the TARN (Target Accumulation Redemption Note). A TARN is a product whose capped sum of payments can end early because of a target placed on the accumulated total³. Because the payment structure is close to that of a TARF, it is used here to compare the three types.

With the three types, the buyer's receipt differs even for the same rate path. The example in the next section sets out the numbers.

Three types and losses in an L Corp example

Consider a hypothetical example in which L Corp, which pays for USD 50,000 of imports every month, enters into the following TARF.

The receipt is measured as the difference from buying at spot.

If the yen weakens, the transaction ends early. In the full gain type, suppose the spot rate is 154 yen on date 1 and 155 yen on date 2.

On date 2 the total exceeds the target of 8 yen, so the transaction ends and the forwards for dates 3 to 6 are canceled. L Corp's receipt is (4 + 5) × USD 50,000 = 450,000 yen. For dates 3 to 6, the dollars for the payments are bought at the spot rate on each day.

When the date that crosses the target is the last one, the difference between the types shows up in the numbers. Suppose the spot rates on dates 1 to 6 are 151, 152, 151, 151, 150, and 156 yen.

The total through date 5 is 5 yen, which is 3 yen short of the 8-yen target. On date 6 the favorable gap is 6 yen, so the full gain type recognizes 6 yen, the no gain type recognizes 0, and the part gain type recognizes 3 yen. L Corp's receipt is 11 yen × USD 50,000 = 550,000 yen in the full gain type, 5 yen × USD 50,000 = 250,000 yen in the no gain type, and 8 yen × USD 50,000 = 400,000 yen in the part gain type. The receipt on the favorable side is capped at the target in the part gain type, at 8 yen × USD 50,000 = 400,000 yen.

If the yen strengthens, the total does not grow and the transaction continues for all six dates. Suppose the spot rates on dates 1 to 6 are 147, 144, 142, 146, 148, and 145 yen. L Corp buys USD 100,000 at 150 yen each time. The difference from buying at spot is the difference in yen multiplied by USD 100,000.

Figure description

L Corp bears a difference of 300,000 yen on date 1, 600,000 yen on date 2, 800,000 yen on date 3, 400,000 yen on date 4, 200,000 yen on date 5, and 500,000 yen on date 6, for a total of 2.8 million yen. The difference is the gap between 150 yen and the spot rate multiplied by USD 100,000, and it is not an actual payment. The legend of the figure reads "↑ Favorable gap　↓ Adverse gap", and the downward arrows are not payments; they are the adverse difference compared with buying at spot. The length of each arrow is a rough guide to the size of the amount.

The total difference over the six dates is −2.8 million yen. The cost of buying USD 100,000 at 150 yen each time is 2.8 million yen more in total than buying the same USD 100,000 at spot. Of each purchase, USD 50,000 exceeds the monthly need, and L Corp holds those dollars. In the yen-weakening examples the gain stops at 450,000 or 550,000 yen, while in the yen-strengthening example the difference keeps widening as the yen strengthens.

The shape of the loss

The loss on a TARF continues on every observation date while the rate is on the unfavorable side, because the obligation to trade at the set rate remains. It widens in proportion to the gap on the unfavorable side and has no ceiling. In the example above, the gaps on unfavorable dates were 2 to 8 yen, and the differences, after multiplying by USD 100,000, were −200,000 to −800,000 yen. With 20 to 60 observation dates, the total becomes larger than this the longer the unfavorable side continues.

The Securities and Futures Commission (SFC) of Hong Kong pointed out that the terms of FX TARFs sold to investors resemble those of accumulators and decumulators, and that investors' losses have no limit⁴. The same source also cites a condition that changes the amount delivered (leverage) as one that can greatly enlarge losses⁴. The problems that arose in sales to small and midsize companies are covered in Overview of Exotic FX Products.

Valuation tracks the accumulated total as a state

A TARF's payment depends on both the rate on that day and the total through the previous dates. Valuation therefore holds the total as a state variable and updates it on each observation date. It is a kind of path-dependent option³.

The price is the expected value of the discounted payments under a risk-neutral stochastic process. The numerical experiments of Luo and Shevchenko use a model in which the exchange rate follows a lognormal distribution and volatility is constant. In this case the Monte Carlo method needs to simulate only the rates on observation dates⁵. In a local volatility model, where volatility also depends on the rate, the simulation must also cover the time between observation dates, which increases the computation⁶. Maran et al. value a TARF in a stochastic local volatility model⁷.

There are two ways to solve numerically: Monte Carlo and partial differential equations. For path-dependent products such as TARFs, Monte Carlo is normally used⁸. Luo and Shevchenko presented a method that solves the partial differential equation by the finite difference method. In 12 examples under one setting (20 observation dates, strike 1.0, volatility 20%), for the same computation time the standard error of Monte Carlo was about 0.1% and the relative error of the finite difference method was about 0.02%⁶. This comparison is a result for the setting of one paper and does not necessarily hold for every TARF. A closed-form solution can be found only when there is a single observation date or when the target is very large⁶.

Deep diveSolving the partial differential equation by tracking the total on a grid

In the form of Luo and Shevchenko (a TARN that adds only the receipts on favorable dates), the total is observed only on observation dates, so between observation dates the same partial differential equation as for an ordinary option holds. Here $V(S, t, A)$ is the value at time $t$ with rate $S$ and total $A$, $\sigma$ is volatility, $r_d$ is the yen interest rate, and $r_f$ is the dollar interest rate⁹.

$$\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 V}{\partial S^2} + (r_d - r_f)\, S\, \frac{\partial V}{\partial S} - r_d V = 0$$

On observation date $t_k$, the total jumps by the amount of the receipt. If the receipt is $C_k$, the value satisfies the following relation⁹.

$$V(S, t_k^-, A) = V(S, t_k, A + C_k) + C_k$$

The total is tracked on a grid $A_1, \dots, A_J$ from 0 up to the target $U$, with a one-dimensional solution at each grid point. On an observation date the total falls off the grid points, so the value after the jump is interpolated with a cubic spline. Applying the jump in the direction of time in a straightforward way cannot represent the case where the total exceeds the target. The source therefore swaps the direction of the jump and subtracts the value at the point jumped to from the grid point⁹.

In hedging, discontinuous sensitivities are the problem

The seller of a TARF holds the obligations on the opposite side from the buyer, so its position has to be rebalanced as the sensitivities change. Derivatives traders use sensitivities such as delta and gamma, part of the Greeks, every day to rebalance positions¹⁰.

The delta and gamma of a TARF become discontinuous near levels such as the knockout. Maran et al. show, in an example of a TARF on EUR/USD (payments set weekly, with knockout and knock-in levels), that singularities appear near the two levels⁷. When gamma is computed by Monte Carlo with finite differences, the smaller the shift in the rate (the bump), the less the bias of the approximation but the more the Monte Carlo estimate varies. The problem is large for second-order sensitivities, and many paths are needed to obtain a reliable value¹⁰.

Notes

1. Shouvanik Chakrabarti et al., "A Threshold for Quantum Advantage in Derivative Pricing" (Quantum, 2021), Appendix A.5 "Target Accrual Redemption Forwards" (page 25). Equation (54) gives the payment on each date, and Algorithm A.2 gives the counting of the total. ↩ ↩² ↩³

2. Xiaolin Luo and Pavel V. Shevchenko, "Pricing TARN Using a Finite Difference Method" (2013), Section 2 (pages 3–4). Equations (1) to (5) define the receipt and the total, and the three types (full gain, no gain, part gain). The paper treats one form of TARN, which has no payments on the unfavorable side. ↩ ↩²

3. Luo and Shevchenko, Section 1 (page 2). A TARN is a capped sum of payments that can end early because of a target placed on the accumulated total, and the payment on each date depends on both the rate and the total. ↩ ↩²

4. Securities and Futures Commission, "Findings of concurrent SFC-HKMA thematic review of the distribution of non-exchange traded investment products", Circular 24EC21, Annex 1, page 4, item (ii) and footnote 3. ↩ ↩²

5. Luo and Shevchenko, Equation (7) in Section 3 (pages 4–5), and the discussion in Section 5 (page 13). With constant or time-dependent volatility, the transition between observation dates is lognormal in closed form, so Monte Carlo needs only the observation dates. ↩

6. Luo and Shevchenko, Section 5 (pages 13–15), Table 1 and the text. The settings and errors of the 12 examples are in Table 1, and the conditions for a closed-form solution are on page 14. ↩ ↩² ↩³

7. Andrea Maran et al., "Chebyshev Greeks: Smoothing Gamma without Bias" (2021), Section 3 and Section 3.1 (pages 7–9) and Figure 1 (page 9). The TARF is valued in a stochastic local volatility model, and the singularities appear at spot levels of 1.135 and 1.19 (the knockout and knock-in levels). ↩ ↩²

8. Chakrabarti et al., Section 3.1 (page 6). Path-dependent derivatives such as autocallables and TARFs are normally priced by Monte Carlo or quasi-Monte Carlo methods. ↩

9. Luo and Shevchenko, Section 4 (pages 5–10), Equations (8) to (12). The grid of totals $A_1, \dots, A_J$, the cubic spline interpolation, and the swap of the jump direction are covered in Sections 4.1 and 4.2. ↩ ↩² ↩³

10. Maran et al., Section 1 (page 3). The paper states that derivatives traders use Greeks every day to prepare positions for market moves, and that combining finite differences with Monte Carlo creates a trade-off between bump and variance, especially for second-order Greeks, which can undermine the reliability of hedge ratios. ↩ ↩² ↩³