This paper traces the survival of FX structured products in the private banking business when all other classes of structured products have suffered setbacks during the credit crisis. Different categories of FX products were showcase to illustrate the applications and risks of these derivative structures. This paper explains the reasons for the resilience of this strain of structured products in the face of adversity. The FX accumulator was chosen to demonstrate the principles and considerations behind designing, pricing and hedging structures from an issuer's perspective. This paper concluded with lessons learned from structuring, pricing and risk managing for structured products.
Since the collapse and bankruptcy of Lehman Brothers in September 2008, most investors have steered clear of structured products. Gone were the glorious days when bankers could easily package and peddle them by the truckloads and investors are attracted to them like ants to honey.
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Equity and commodity-linked structured products were particularly popular during the boom years of 2005 to 2007 when interest rates were low while the global stock markets rallied and commodity prices were breaching historical highs consistently.
When the markets tanked with the credit crisis, these structured products, with payoffs and principals pegged to bullish market performance, brought excruciating financial pain to millions of investors who bought them without fully appreciating the risks. Since then, derivatives and structured products were frowned upon as "weapons of mass destruction".
However, in the less publicised world of private banking, a class of evergreen products continue to thrive in this post-crisis era. These were the foreign exchange (FX) structured products. FX is a unique asset class, which is less influenced by movements in the global stock markets. The FX market is a highly liquid Over-The-Counter (OTC) market that operates 24 hours a day, almost 7 days a week, across major cities of the world. According to the Bank for International Settlements (BIS), the FX market is the world's largest financial market with an average daily turnover exceeding USD 2 trillion.
The reasons why private banking clients continue to favour FX structured products are as follows. First, FX structures continue to serve a real hedging need in the daily conduct of their affairs and businesses, which span the globe. Second, interest rate differentials between the developed and emerging markets continue to offer investment opportunities when the stock and commodity markets have come to a standstill. Third, FX derivatives and structured products allow these cash rich customers to speculate on directional view of currency movements influenced by macroeconomic and political events.
2. Private Banking and Structured Products
According to regulatory definitions, High Net Worth (HNW) individuals are clients with investible assets in excess of USD 1 million. By industrial norms, a client is usually worth servicing only if he or she deposits assets above USD 5 million with the bank. As such, you can imagine the cash and holding power this group of investors have over the average retail investors.
Private banking customers have a whole arsenal of investment tools and advisors at their disposal. The more mundane ones would include brokerage facilities, trust administration, custodian services, tax planning, offshore banking, portfolio management, investment advisory, succession planning, private equity and hedge funds investments, while the more exotic ones would include art and wine collection, jet financing and real estate procurement. Some clients emphasize wealth preservation while others look for ways to enjoy the fruits of their labour. Whatever their needs, the shrewd private bankers would always have a personalized solution for them.
As such, structured products that were initially targeted at large corporations soon found their way into the private banking world. These products, which started off as highly customised solutions, soon became commoditized due to the high margin they generated for the banks and their rising popularity among private banking clients. Some strains of these structured products eventually flowed to the retail market. That is where the trouble began.
3. Distribution and Marketing of Structured Products
A private bank typically provides structured products or solutions to its clients through the following channels.
For global banks that combine investment banking, private banking, retail banking and asset management under one roof, the products will usually be conceived and customized by the private bank, structured and hedged by the investment bank, distributed internally to private banking customers first followed by retail customers for the vanilla ones, the asset management arm may also take on the custodian of the structured products.
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Marked to Standard
For the boutique private banks that do not have the capability to structure and hedge these products in the capital markets, they will usually source these products from the investment banks that market them or act as co-lead managers to create the product and underwrite the issue.
Most of the time, private banks do not have the capability to hedge complex structures. As such, they usually act as distributors for the issuers. In cases where they need to customize derivative solutions for their clients, they will execute back-to-back cover deals with their investment banks against their client deals. They will usually only earn the spreads and not take any proprietary positions.
4. FX Structured Products for Private Banking Clients
Structured products in the private banking world can be broadly classified into 3 main categories, namely Participation products, Yield products and Hedging products.
Participation products are for investors who have a directional view (bullish, bearish or ranging) on the market or underlying and they would like to speculate on this trend.
Yield products are for investors seeking guaranteed coupon or yield enhancement. These products usually offer a coupon that can be fixed, conditional or both.
Hedging products are for investors wishing to hedge their portfolio.
Participation Product 1: Twin-Win
How it works?
Twin Win is a structure that allows the investor to participate in the upside and the downside of the underlying from a strike level. This product is suitable for an investor who believes the underlying is set to rise (bullish view) or fall (bearish view) until a certain level but is not sure of the direction the movement will take. The product adopts a strategy similar to a Straddle with barrier on both sides of the strike.
Redemption at maturity (with continuous barrier)
If the underlying has not breached the lower barrier and has never breached the upper barrier during the product life, the investor recovers his capital plus a cash gain equal to the absolute performance of the underlying. If the underlying has traded at a level equal to or less than the lower barrier or has traded at a level equal to or higher than the upper barrier during the product life, the investor receives only its capital back.
1) Capital is protected at maturity.
2) Investor does not need to have a directional view on the underlying.
3) Investor benefits from both rise and fall of the underlying up to a certain level.
1) Opportunity cost if the underlying breaches barrier or does not move.
1) Risk Indicator on Capital: Capital-Protected.
2) Market Scenario Indicator: Bullish or Bearish.
3) Risk Profile Indicator: Low.
1) Best case scenario
The barriers have not been breached during the life of the product. Investor receives 100% of his capital back plus the absolute performance of the Underlying.
2) Worst case scenario
The Upper barrier has been breached during the life of the product. Even if the final underlying performance is positive, investor receives only 100% of its capital back.
Participation Product 2: Coupon and Upside (CUP)
How it works?
The CUP is a participation product for investors willing to take a directional view on the underlying. If the barrier has not been breached, then a minimum return will be guaranteed at maturity.
Redemption at maturity (with continuous barrier and bullish trend)
If the underlying has never traded at a level less than its barrier level during the product life, the investor receives the best return between a bonus level and the positive performance of the underlying. (See Best-case scenario)
If the underlying has traded at a level equal to or less than their barrier level during the product life, a capital loss may occur, the investor receives 100% + the performance of the underlying whether such performance is negative or positive
(See Intermediate & Worst-case scenarios)
1) If the barrier has not been breached, a minimum return is guaranteed, the bonus coupon.
1) Capital is at risk.
1) Risk Indicator on Capital: Not Capital-Protected
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2) Market Scenario Indicator: Bullish or Bearish
3) Risk Profile Indicator: High
Intermediate case scenario:
Yield products are for investors seeking guaranteed coupon or yield enhancement. These products usually offer a coupon that can be fixed, conditional or both. Some common yield enhancement products include range accrual notes, binary notes, notes based on a basket of currencies and reverse convertible notes.
Yield Product 1: Range Accrual Notes
How it works?
A Range Accrual note pays the investor an attractive coupon for each day that the reference index fixes within a pre-defined range, comprising a lower barrier and/or an upper barrier, over a given maturity. The investor is taking the view that the reference index will not change much, or will remain within specified levels. The capital is protected at maturity. Investor is short volatility while issuer is long volatility. Range accrual notes typically have life spans of 6 to 24 months.
On observation date (t), the coupon is accrued and calculated as follows:
Coupon Rate x (n / N) where n is the number of days the Underlying is in the range, and N the total number of days over the period
Early redemption possibilities:
The call option allows the issuer to early redeem the structure before maturity, at par, under some specific market conditions, on each observation date (t), subject to a non-call period.
Redemption at maturity:
If the structure was not previously redeemed, the note is redeemed at 100% at maturity, and the last coupon is paid.
1) Capital is protected at maturity.
2) Opportunity to earn higher than market yield.
3) Flexibility in the choice of calendar observation periods.
1) Opportunity cost if the underlying is above the barrier most of the time and no or low coupon yield.
1) Risk Indicator on Capital: Capital-Protected.
2) Market Scenario Indicator: Stable.
3) Risk Profile Indicator: Low.
Calculation of Coupon: (T x n/N) p.a.
Effective coupon at end-period: 4.5266% p.a.
Redemption at maturity = EUR 1,022,633.
Yield Product 2: Binary Notes
How it works?
A Binary note is a structure with a protection on capital at maturity that pays an attractive coupon, above the risk free rate, if a condition on the underlying is fulfilled.
It gives the investor the flexibility to choose the condition(s) based on his view of the market.
There are 2 approaches to structure the conditions:
1) European-style option structures: those with observation at expiry date
2) American-style option structures: those with continuous observation
Redemption at maturity
At maturity, if the underlying has respected the condition, the investor receives 100% of its capital back and the pre-fixed coupon. Otherwise, the investor receives only 100% of its capital back without the coupon.
1) Capital is protected at maturity.
2) Coupons and levels chosen by the investor according to his expectations.
3) Benefit from an enhanced potential coupon.
4) Simple pay-off.
1) "All or nothing" payout.
2) Opportunity cost of risk-free rate foregone if condition not met.
1) Risk Indicator on Capital: Capital-Protected.
2) Market Scenario Indicator: Applicable to any market condition.
3) Risk Profile Indicator: Low.
Possible Permutations of the product:
The product depends on the levels chosen by the investor to establish his strategy:
1) Purely directional strategy
2) Opportunity directional strategy
3) Volatility strategy
4) Stability strategy
European-style options structures are ideally suited to purely directional strategies and operate as follows:
a) If on the product's expiry date the spot rate fulfils the condition underlying the strategy, the investor earns the potential coupon upon maturity.
b) If on the product's expiry date the spot rate does not fulfil the required condition, the investor earns no coupon and is repaid his guaranteed nominal upon maturity.
1) Purely directional strategies
The Digital Note
The investor plays a purely upward or downward strategy on the underlying and fixes a strike, which reflects his expectations. He earns a coupon only if the spot rate on the maturity date is at the same level as or is above or below the pre-defined strike.
American-style options structures suit all other strategies and operate in the following manner:
a) If the spot rate fulfils the underlying condition during the whole life of the product, the investor earns the potential coupon.
b) If the spot rate never fulfils the required condition during the whole life of the product, the investor earns no coupon and is repaid his nominal.
2) Opportunity directional strategies
The investor has no precise idea about the future trend of the underlying but thinks that it may reach a certain limit (upward or downward) during the course of its trend.
2a) The One-Touch Note
The investor fixes a strike, at the level of this limit, which reflects his expectations. He earns a coupon only if the spot rate hits the pre-defined strike at least once during the whole life of the product.
2b) The No-Touch Note
The investor fixes a strike, at the level of this limit, which reflects his expectations. He earns a coupon only if the spot rate never hits the pre-defined strike during the whole life of the product.
3) Volatility strategies
The Double-Touch Note
The investor has no precise idea about the future trend of the underlying (upward or downward) but reckons on the underlying moving within a wide range. He fixes a range (a lower limit and an upper limit), which reflects his expectations. He earns a coupon only if the spot rate hits either of the limits at least once during the whole life of the product.
4) Stability strategies
The Double No-Touch Note
The investor has no clear idea of the future trend of the underlying (upward or downward) but reckons on the underlying remaining stable. He fixes a range (a lower limit and an upper limit), which reflects his expectations. He earns a coupon only if the spot rate never hits either of the limits during the whole life of the product.
Hedging products are for investors wishing to hedge their portfolio. Usually the investor would have an existing deposit or portfolio denominated in a specific currency with the intention to convert it into another currency, likely his home currency or safe G7 currency, in the foreseeable future.
Hedging Product 1: Capped / Floored Certificate
How it works?
Certificates replicate the performance of an underlying asset or theme. A Capped/Floored certificate replicates the Bull-Spread/Bear-Spread strategies of the underlying currency pair. The spread enables the investor to participate in the strengthening/weakening of one currency relative to the other.
For both strategies, the investor benefits from the flexibility to choose any combination of strike prices according to his directional view of the market and risk appetite.
1) Access to speculative strategies at a cost that is lower than that of purchasing a vanilla call or put option.
2) Flexibility to choose the strike prices.
3) The downside volatility risk is limited should his directional view be wrong.
1) The upside gain for the investor is limited even if his directional view is correct.
2) Should the underlying trend become unfavourable, the certificate may lose some or all of its value at maturity.
1) Risk Indicator on Capital: Not Capital-Protected
2) Market Scenario Indicator: Applicable to Bullish or Bearish market
3) Risk Profile Indicator: Medium
Pay-off at maturity of the Capped Certificate
1) Purchase a call option at strike 100%.
2) Sell a call option at strike 120%.
Pay-off at maturity of the Floored Certificate
1) Purchase a put option at strike 120%.
2) Sale of a put option at strike 100%.
Hedging Product 2: FX Accumulator Forward
How it works?
An FX Accumulator Forward is a structure that allows the investor to hedge his currency exposure through an accrual mechanism. The investor is able to secure a more favourable conversion rate than the outright forward rate for the same period. There are many variations of this product in the market, but in general they abide by the same basic framework. The time frame of such products can range from 3 to 24 months.
The product depends on 3 parameters:
1) A conversion rate / strike price at which the investor hedges his exposure
2) A barrier / knock-out level that defines whether a condition is fulfilled or not
3) A calendar schedule that determines the fixing and settlement dates
The parameters are decided at inception.
During the life of the product:
1) For each observation period that the spot rate fulfils the condition underlying the structure relative to the knock-out level, the investor accumulates a portion of his capital to be converted at maturity.
2) For any observation period that the spot rate does not fulfil the required the condition, the accumulation ceases and the investor knows what portion of his capital he will convert in due course at the pre-determined conversion rate.
1) The product can be priced as a zero-cost structure.
2) Allows the investor to beat the forward rate for the period.
1) Capital is at risk.
2) In the situation where there is a rise in volatility and/or strong movement in the underlying currency pair, the hedge may loose its value.
1) Risk Indicator on Capital: Not Capital-Protected.
2) Market Scenario Indicator: Applicable to any market conditions.
3) Risk Profile Indicator: High.
Accumulator Forward EUR Put / USD Call
In this example, the investor has a EUR deposit of 1 million and wishes to convert the deposit to USD at a more favourable rate than the existing outright forward rate.
Calculation of amount converted to USD: [Amount in EUR x strike] x n/N.
Redemption in USD upon maturity: USD 915,342.
Calculation of amount in EUR: Amount in EUR x [(N-n)/N + 0.05% p.a].
Redemption in EUR upon maturity: EUR 287,815.
Other possible variations of this structure can take the following forms:
1) Accumulator Forward Double Knock-Out
In order to remove the risk of the capital being converted at a worse rate compared to the spot rate upon maturity and to enable the investor to benefit from any favourable movement in the underlying currency pair, the investor may choose to add a second knock-out level in the direction that he wishes to speculate.
During the product's life, as long as the spot rate remains within the range created by the two knock-out levels, the investor accumulates on a permanent basis, at each observation period, a portion of the nominal to be converted.
If, during the products life, the spot rate hits either of the knock-out barriers at least once, the accumulation ceases and the investor knows from then on what amount he will have to convert and what amount will remain in his deposit currency upon maturity.
2) Fade Forward
In order to increase the probability of converting the nominal, the investor may opt for a temporary knock-out instead of a permanent knock-out. In this case, he sets a Fade Forward level instead of a knock-out.
If the spot fulfils the condition relating to the structure, relative to the Fade Forward level, at each observation period the investor accumulates a portion of the nominal to be converted.
If for one observation period, the spot rate does not fulfil the required condition, the accumulation ceases temporarily. The accumulation resumes as soon as the spot rate fulfils the condition relating to the structure again at any subsequent observation periods.
5. Evaluation of FX Structured Products by Category
In the following sections, the different types of products under the 3 categories described above are illustrated. Each product will be evaluated based on 3 indicators namely Risk Indicator on the Capital (Capital Protected and Not Capital Protected), Market Scenario Indicator (Bullish, Stable, Bearish) and Risk Profile Indicator (Low, Medium, High).
6. Designing and Pricing an FX Accumulator from an Issuer's perspective
In the previous section, a brief description of an FX Accumulator Forward and its variations has been explained from an investor's perspective. The following section will attempt to design, price and hedge the product from an issuer's perspective.
Unglamorous History of Accumulators
An accumulator forward is a highly path dependant product that can be structured as a zero-cost forward enhancement without a guaranteed worst case. As such it tends to be speculative in nature. At the beginning, it was a very popular product among many corporates in Europe, particularly France, Italy and the UK. Subsequently, the product was adapted to suit the private banking and retail customers.
Equity Accumulators were very popular among Asian retail investors, particularly in Hong Kong and Singapore, during the pre-crisis days when the stock markets were bullish. They were viewed as safe vehicles to tap the market rallies. These products enabled investors to purchase stocks at a discount when the market was bullish but exposed them to unlimited downside risk when the market turned south. Many high profile court cases highlighting the negative aspects of Equity Accumulators were reported in the media throughout Asia when the crisis started to unfold.
In spite of the bad press, for Accumulators in general, FX accumulators continue to enjoy success with private banking clients due to the following reasons. First, an FX Accumulator can be used as a hedge that offers a better rate than an outright forward. Second, an FX Accumulator can be structured as a zero-cost product, which makes it an attractive alternative to purchasing a costlier option. Most private clients have a genuine need to swap a deposit or portfolio from one currency into another at the maturity of the contracts. Finally, unlike stocks, which are popular during bull markets and quiet during bear markets, due to regulatory constrains on short-selling, currencies offer directional plays all year round. This enables ambitious clients to take speculative views using accumulators.
6a Factors to Consider in Designing the FX Accumulator
In designing an FX accumulator, the following parameters have to be careful considered as they influenced the pricing as well as risk profile of the product.
Number of observations: 240 business days (n)
Frequency of settlement: Monthly
The type of payoff will determine the barrier level, strike, leverage and type of building blocks to use.
In practice, different combinations of ranges can be set and for each range a notional amount has to be specified. In this example, there are 3 ranges to consider. Range 1: Above knock-out barrier (1.5000), Range 2: Between knock-out barrier (1.5000) and strike (1.3492) and Range 3: Below the strike (1.3492).
At observation date, if spot falls in Range 1, the structure can be permanently or temporarily knocked-out, zero amount is accumulated for that day. Accumulation will stop if the structure is built with a non-resurrecting range.
If spot falls within Range 2, 1 time the notional divided by total number of observation days is accumulated. The investor gets to accumulate USD10,000 to sell at the strike rate on settlement date.
If spot falls within Range 3., X times the notional divided by the total number of observation days is accumulated. X represents the leverage or risk the investor wishes to take. In this example the leverage is 2 times. The investor accumulates twice the daily notional of USD10,000 (i.e. USD20,000) to sell at the strike rate on settlement date. If the investor is willing to take on higher leverage, the issuer will be able to offer a more favourable knock-out barrier and/or strike price.
3) Knock-Out Barrier
Knock-out barriers are added to improve the strike and reduce the cost of the structure. Without a barrier, the structure will be more expensive, offer a less attractive strike or suffers a higher leverage. It is important to specify whether the barrier will be monitored continuously or discretely.
The amount the investor wishes to hedge is USD2.4Million. In this case, the daily notional is USD10,000 and the leverage is 2 times. In this example, the investor will have to accumulate twice the daily notional, i.e. USD20,000, should the spot move moves below the strike. The client improves his exchange rate by taking the risk to multiply his loss by 2 times should the rates move against him. Any positive number is possible, though a common factor would be 1 or 2. If the investor is aggressive, he can improve his strike substantially by taking on more risk.
5) Resurrecting or Non-resurrecting Knock-Out conditions
Each barrier level has to be declared either resurrecting or non-resurrecting. For resurrecting barriers, any knock-out and cease of accumulation is only temporary. In this case, the structure will resume as soon as the spot falls back below the knock-out level at any of the subsequent observation date again. The condition for the resurrecting structure is usually less favourable than the non-resurrecting ones.
6) Amount kept in case of knock-out
The design also has to take into consideration whether at knock-out, all of the accumulated amounts will be kept or none at all, i.e. "keep all" or "keep nothing".
It is important to be exact on the fixing schedule and the fixing source. The fixing schedule will determine the date and time, with reference to which time zone and market. The fixing source will determine which reference source to extract the spot price for comparison. It could be a public source such as Bloomberg and Reuters or an alternative agreed source. It is also important to pre-define the steps to resolve any conflicts or doubts when both parties do not agree on the fixing rate.
The settlement frequency has to be decided upfront. It could be daily, weekly, monthly or at maturity.
9) Extra Features
There are many ways to improve the attractiveness of the product to appeal to investors. Additional features such as rebates, stripped settlement, bounds on amounts and improved rates on early knock-outs can also help to mitigate the clients' exposure.
6b Steps in Pricing the FX Accumulator
According to Wystup, an accumulative forward consists of fade-in calls and fade-in puts, possibly with extra knock-out ranges. Faders are second generation exotic options, whose nominal is directly proportional to the number of fixings the spot stays inside or outside a pre-defined range. A fade-in option progressively activates the nominal while a fade-out option does the opposite. Wystup proposed to price Fader contracts using closed form solutions in the Black-Scholes model.
An approach to pricing equity accumulators was proposed by Lam, Yu and Ling. In the paper, they decomposed the accumulator into a summation of pairs of long up-and-out barrier call options and short up-and-out barriers put options with different expiration time. By adapting the results derived by Harrison (1985) and by Rubinstein and Reiner (1991), they were able to formulate a closed form solution for the accumulator under immediate settlement. They went on to modify the solution to handle delayed settlement by taking into discount factors. However, both of the formulas were only adequate for continuously monitored barriers. To cater to accumulators with discrete barriers, the team made use of the proposition put forward by Broadie, Glasserman and Kou (1997). Essentially, a correction term was used to shift the barrier to approximate discretely monitored barrier option values. Finally, the team compared the results derived from the analytical solutions against a parallel Monte Carlo simulation.
In essence, a fader is made up of a basket of barrier options. Hence both the approaches by Wystup and Lam, Yu and Ling should theoretically arrive at the same result if applied on the same underlying instrument and using the same assumptions.
In this paper, I have taken the Lam, Yu and Ling's approach by modifying the closed form solution to cater to the change in the underlying instrument, from stock to currency. The risk-free rate of the foreign currency (Rf) has to be included in addition to the risk-free rate of the domestic currency (Rd).
The following procedure was carried out to establish an initial price for the FX accumulator before further calibration was done to adjust the barrier and strike price to achieve a zero-cost structure. A zero-cost structure, is not a pre-requisite for an accumulator product, but makes it more appealing to investors.
1. Equity Accumulator Model
An analytical model was built using the exact premise and parameters described in the Lam, Yu and Ling's paper. The results obtained from my model were compared to those published by the authors. As the codes for the paper were not published, I had to take this necessary step to ensure that my initial model has correctly captures the principles highlighted in the paper.
2. FX Accumulator Analytical Model
Once the results in the Equity Accumulator model were satisfactory, I went on to build a model for tan FX accumulator by modifying the first model to include the impact of the risk-free rate of the second currency. The parameters in this second model were replaced to reflect the change in the underlying instrument from a stock to a currency pair, EUR/USD. A closed form solution provides speed while a simulation compromises on speed but produces a more accurate solution.
3. Monte Carlo Simulation for FX Accumulator
Next a Monte Carlo model was built with the same knock-out barrier level, strike price, spot price, volatility, interest rates and time frames as the second model. A series of simulations ranging from 1,000 to 1,000,000 were executed to provide a more accurate basis for comparison. The results generated by the Monte Carlo model were compared to those obtained from the Analytical model. Within a certain margin of error, the prices seem to converge.
4. Zero-Cost Structure Calibration
With some level of comfort, I went on to calibrate the barrier levels and strike prices of the FX accumulator to achieve zero-cost structures for the various combinations of strikes, barrier level and volatility.
Step1: Building of Equity Accumulator
An analytical model for pricing the Equity Accumulator, described in the by Lam, Yu and Ling, was built based on the following assumptions and closed form formulas.
1) Volatility, s is constant
2) Risk-free interest rate, r is constant
3) Payout rate, q is constant
Closed-form Formulas for Barrier Options (Black-Schole's framework)
Formulas derived by Harrison (1985) and Rubinstein and Reiner (1991).
The results generated by the replication model closely resembles the results published in Table III in the paper.
Step2: Building of FX Accumulator
The parameters used in this model are as follows:
1) Volatility, s is constant at 20%.
2) Risk-free interest rate of domestic currency (USD), Rd is constant at 0.25%.
3) Risk-free interest rate of foreign currency (EUR), Rf is constant at 1%.
From the results generated by the analytical model, we can see a clear segregation of positive (green) and negative (red) prices. The positive values suggests that the strike prices favour the investors while the negative values benefits the issuers.
Step3: Monte Carlo Simulation of FX Accumulator
The Monte Carlo simulation was built on the following geometric Brownian motion equation:
St is the spot exchange rate,
Rd is the continuous domestic interest rate,
Rf is the continuous foreign interest rate,
s is the constant volatility,
Wt is a standard Brownian motion.
Applying Ito's rule to InSt results in the following equation for the process St:
which demonstrates that St follows a lognormal distribution.
1) There is no arbitrage.
2) Trading is frictionless, no transaction costs.
3) Any position can be taken at any time, short, long, arbitrary fraction, no liquidity constraints.
This is a standard model widely adopted in practice to communicate prices in currency options. The objective here is to provide a simplified sanity check on the analytical model, hence this straight forward model should suffice.
However, I do recognize that more sophisticated stochastic volatility models such as the Heston model and SABR may be more appropriate for pricing exotic options and constructing volatility smile.
Simulation completed with the same parameters used in the analytical model.
H=1.5000, S=1.4000, K=1.3492, s=20%, Rd=0.25% and Rf=1%.
The Monte Carlo simulation demonstrated convergence towards the analytical model with some margin of error. As such, the analytical model is relative accurate for pricing the FX Accumulator.
Step4: Calibrate parameters to achieve
From the results we can clearly identify the strike range to calibrate for each volatility level to obtain a zero-cost structure. To achieve a zero cost structure for an accumulator with volatility of 10%, the selected strike price has be approximately 1.3384, between 1.3290 and 1.3391.
6c Considerations in Hedging the FX Accumulator
As highlighted above, the FX Accumulator is made up of a basket of barrier options with different expiry dates. As such, hedging the product requires hedging a basket of barrier options. The perfect hedge is achieved through perfect replication, i.e. the payoff of the barrier option and the hedge matches exactly for all outcomes. However, replication of the barrier option payoff is often impossible due to market imperfections.
To hedge a barrier option position requires market liquidity, optimal position sizing in the hedge portfolio, minimizing number of positions in hedge portfolio within acceptable hedging error level and minimizing realization risks for static hedging.
The two classic hedging approaches for barrier options are dynamic or delta hedging, which requires frequent rebalancing of the underlying, and static hedging, which requires setting up an initial portfolio of vanilla options that will not require any further adjustment.
Dynamic hedging is executed by continuously ensuring that the delta, the first order sensitivity of the option to the underlying price, is neutral. For example, for each short up-and-out call option (issuer is short one up-and-out option when client is long one up-and-out option), the issuer receives the option premium and set up a portfolio by buying delta EUR and places the remaining premium in a bank account. Over time, the issuer will adjust the hedge portfolio continuously in order to maintain delta-neutrality. The delta of barrier options is highly sensitive to the price of the underlying, especially around the barrier level, hence the hedge needs to be frequently rebalanced. This results in huge transaction costs and operational difficulties when managing large barrier option positions.
Static hedging is carried out by constructing a portfolio of vanilla options with varying strikes, maturities and fixed weights at inception. No adjustments are required throughout the life of the product. The two well-known static hedging approaches were the Calendar-spread method, invented by Derman et al.  , and the Strike-spread hedging method, created by Carr and Chou . The Calendar-spread method hedges the payoff of the barrier options using vanilla options with varying maturities. The Strike-spread approach hinges on the idea of converting the problem of replicating a barrier option to a problem of replicating a European security with a non-linear payoff function, known as the adjusted payoff function. The adjusted payoff is hedge by constructing a portfolio of finite number of vanilla options with different strikes. In general the hedge quality of static strategies worsen over time.
Many attempts to circumvent the limitation of the above classical approaches were proposed in academic literature by Taleb N.(1996), Nalholm and Poulsen (2006), Maruhn and Sachs (2005) and numerous other researchers. However, none of them can achieve the perfect hedge for barrier options. Hence the choice of hedging approach for an FX Accumulator is subject to the issuer's skill and competence.
This paper attempted to explain the continued popularity of FX structured products among private banking clients when all other classes of structured products have fallen out of favour as a result of the credit crisis. This clearly shows that derivatives and structured products are not destined for extinction as long as they can innovate to serve the real and changing needs of investors.
The FX accumulator was singled out to demonstrate this point. Despite the downfall of its close cousin, the Equity Accumulator, it continued to thrive. This shows that the same type of structure works well for one asset class but not the others. Financial Engineers should be careful not to blindly encapsulate the same structure across every asset class without first understanding the peculiarities of each of the underlings.
In the final sections of this paper, an attempt was made to design, price and hedge an FX accumulator from an issuer's perspective. The process is built on multiple assumptions and limitations. This highlights the importance of having good validation and control processes in place to reduce market and model risks for the issuers. Without which, the issuers would be subjected to disproportional amount of risks for the price they are charging for a product.
Appendix 1 - VBA Codes for Pricing Barrier Options
Appendix 2 - VBA Codes for Equity Accumulator
Appendix 3 - VBA Codes for FX Accumulator Analytical Model
Appendix 4 - VBA Codes for Monte Carlo Simulation for FX Accumulator
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