In this post we will review Structured Products.

\[\begin{aligned} \Pi_{t} &= -V_{t} + \Delta^{*}S_{t}\\ d\Pi_{t} &= -dV_{t} + \Delta^{*}dS_{t}\\ &= -f(S_{t}, t)dt - g(S_{t}, t)dW_{t} + \Delta^{*}dS_{t}\\ &= -f(S_{t}, t)dt - \Delta^{*}(S_{t}, t)\sigma S_{t}dW_{t} + \Delta^{*}(S_{t}, t)( \mu S_{t}dt + \sigma S_{t}dW_{t})\\ &= -f(S_{t}, t)dt - \Delta^{*}(S_{t}, t)\sigma S_{t}dW_{t} + \Delta^{*}(S_{t}, t)\mu S_{t}dt + \Delta^{*}(S_{t}, t)\sigma S_{t}dW_{t})\\ &= -f(S_{t}, t)dt + \Delta^{*}(S_{t}, t)\mu S_{t}dt \end{aligned}\]

Note that the random $dW_{t}$ part is gone, and so the portfolio should grow at the risk-free rate $r$.

The price of an option can also be obtained via the martingale equation:

\[\begin{aligned} V_{t} &= B_{t}E^{Q}\Big[\frac{V_{T}}{B_{T}}\Big]\\ B_{t} &= e^{rt} \end{aligned}\]

Evaluating the above equation leads to the Black-Scholes equation:

\[\begin{aligned} \frac{\partial V}{\partial t} + rS\frac{\partial V}{\partial S} + \frac{1}{2}\sigma^{2}\frac{\partial^{2} S}{\partial S^{2}} &= rV \end{aligned}\]

Self/Auto-Quantoes

For example, you are long a forward in EURUSD with forward price 1.2. This is naturally settled in USD. If, however, you want to be paid in EUR, this is a self-quanto forward.

The payoff in EUR at expiry is:

\[\begin{aligned} X_{T} - K \end{aligned}\]

To convert to USD:

\[\begin{aligned} X_{T}(X_{T} - K) \end{aligned}\]

But why do we need to convert the payoff to USD?

Introduction

Structured products are bespoke instruments that are designed to enable investors to tailor strategies to the view of the market. They can involve equities, interest rates, FX, commodities, credit or real estate. They exist because investors seek higher risk with better control of risk.

There are some caveats to take note for sturctured products. The risk of structured notes are not just based on market risk, but also the credit worthiness of the issuer. Structured notes are also usually illiquid. To get the issuing institution to cancel or restructure the contract would result in a higher margin as the institution has no incentive to price competitively.

The main themes of constructing structured products:

Principal Protection

Many structured notes are designed to be principal protected such that the investor gets back the principal at maturity regardless of the market. We can replicate that by simply buying a zero-coupon bond. Note that the principal repayment is subject to issuer’s credit unless the bonds are invested in government bonds.

A zero coupon bond will be cheaper that face value given positive interest rate. The difference between the face value and the price of the zero coupon bond can be invested in riskier investment. Investors might not want to protect all the principal but only a part of it. Hence more could be invested in a riskier investment.

Upside Participation

Options are acquired with an investment view with the remaining value from purchasing a zero coupon bond.

Protected Selling of Options for Yield

Capping the Coupon

By selling an option with a higher strike price with a long option with a lower strike price to enhance yield.

Barriers and Triggers

Barriers and triggers are provisions that cause a deal to terminate due to certain events.

Supposed you have a structured note that pays a series of coupons tied to the performance of EURUSD. A coupon payoff that is:

\[\begin{aligned} max(\text{EURUSD}_{t_{i}} - 1.4, 0) \end{aligned}\]

However, the note automatically terminates if EURUSD rises above 1.55. This is a trigger condition and will make the options cheaper to purchase. In the event of early termination, the principal is repaid immediately.

Callables

The note could contain a condition that allows the issuer the discretion to call it (in the issuer favor) and repay the principal immediately. The investor will be compensated with a higher coupon.

Tarns

The note could terminate when the total coupons received reach a certain value, say 15% of the notional. This limits the upside.

Conditional Payoffs

For example the payoff could be:

\[\begin{aligned} 0.3 \times max(\text{EURUSD}_{T} - 1.4, 0) \end{aligned}\]

But subject to Euribor being less than 2.5% at expiry date. With this added condition, the structure would be cheaper by having a higher coupon rate.

Range Accruals

For example, we could hae daily fixings and $n$ is the number of business days for which Euribor is under 2.5% out of $N$ business days. Then the payoff could be:

\[\begin{aligned} 0.25 \times max(\text{EURUSD}_{T} - 1.4, 0) \times \frac{n}{N} \end{aligned}\]

Carry Trade

A particular strategy is betting against the forward curve. For example, the USDJPY forward rate is downwards sloping. For example, given the following info:

\[\begin{aligned} \text{USDJPY}_{\text{Spot}} &= 120\\ \text{JPY 5Y} &= 1\%\\ \text{USD 5Y} &= 4\%\\ \end{aligned}\]

The 5Y forward rate would be:

\[\begin{aligned} \text{USDJPY}_{\text{5Y Fwd}} &= \frac{120 \times 1.01^{5}}{1 \times 1.04^{5}}\\ &= \frac{126.12}{1.2167}\\ &= 103.66 \end{aligned}\]

But most JPY investors do not think that the spot rate would drop to 103.66 in 5Y time and instead be close to 120 due to Japanese government intervention. This is known as the carry trade. Borrow JPY at lower interest rate, invest it in high yielding currency and betting the spot rate would not appreciate.

Diversification

Cross Asset Baskets

Bespoke baskets across asset classes. For example a basket containing equities commodities, and FX.

Asian Options

The payoff of an Asian option is based on the average of observations of the underlying rather than the value at expiration. Asian option is cheaper as it is cheaper to hedge as the average value is largely determined as the date draws closer to expiration.

No-Arbitrage Pricing

The basic idea in pricing desrivatives is that arbitrage should not be possible. There are two types of hedge. Static and dynamic. Dynamic hedge requires rebalancing the composition of the portfolio over time. Static replication involves no model assumptions, whereas dynamic replication requires a model to determine how to rebalance the hedging portfolio. We shall assume continuous trading, unlimited ability to buy or sell without affecting market price, no bid/offer spreads and infinite liquidity.

Let us consider a simple example. We would assume interest rates are 0 so no discounting cashflows back. There is a stock with spot at 100. Over a 1-month period, it can go up to 110 with probability $p$ or down to 90 with probability $1-p$. The value 90, 100 are calculated from implied volatility. Now, consider an option to buy the stock at strike 100 with expiry in 1-month.

We would like to construct a hedge comprising $\Delta$ amount of stock and $B$ amount of cash. This result in the following system of equations and solutions:

\[\begin{aligned} 110 \Delta + B &= 100\\ 90 \Delta + B &= 0\\ \Delta &= 0.5\\ B &= -45 \end{aligned}\]

Since the hedging portfolio is equal in value to the option at the expiry regardless what price, we can calculate the price of the options today as:

\[\begin{aligned} 100\Delta + B &= 100 \times 0.5 - 45\\ &= 5 \end{aligned}\]

The risk neutral probability $p$ can be calculated by computing:

\[\begin{aligned} p &= \frac{(1 + r) - R_{d}}{R_{u} - R_{d}}\\ &= \frac{1 - 0.9}{1.10 - 0.9}\\ &= 0.5 \end{aligned}\]

Due to the $r = 0$, the probability is exactly 0.5.

Martingale Equation

A numeraire is a unit to measure value, and any choice of numeraire must give the same price. Any domestic asset with price strictly greater than 0 can be used as an numeraire asset. For example, usually the numericaire asset is the money market account. In this example, we ware measuring value in units of the domestic currency. Given that $Q_{A}$ is the measure corresponding to the numeraire asset $A_{t}$, and both $V_{t}, A_{t}$ are domestic assets, the martingale equaion in a complete market given no-arbitrage is by:

\[\begin{aligned} \frac{V_{t}}{A_{t}} &= E^{Q_{A}}\Big(\frac{V_{T}}{A_{T}}\Big) \end{aligned}\]

A martingale is a process where the expected value at a future time is its value today.

The idea is that in a complete market, we should be able to replicate relative payoffs, such as the ratio of two assets.

We can link different measures via the common unique price of $V_{t}$:

\[\begin{aligned} V_{t} &= A_{t}E^{Q_{A}}\Big(\frac{V_{T}}{A_{T}})\\ &= B_{t}E^{Q_{B}}\Big(\frac{V_{T}}{B_{T}}) \end{aligned}\]

If $V_{t}$ is a foreign asset and $X_{t}$ is the spot rate, then the following modified martingale equation holds:

\[\begin{aligned} \frac{V_{t}X_{t}}{A_{t}} &= E^{Q_{A}}\Big[\frac{V_{T}X_{T}}{A_{T}}\Big] \end{aligned}\]

Barriers

Barriers differ from ordinary options as the payoff could change depending on whether the underlying crosses some specified level. We will discuss a few different kind of barriers.

Digitals

A digital call option pays a fixed amount if the underlying is above the digital barrier at expiry.

For example, if the barrier strike is set at $150 for a stock, the payoff will be $1. We can replicate an approximation hedge via 2 call options (bullish call spread).

Consider 2 call options with strikes:

\[\begin{aligned} K_{-} &= \$(150 - \delta)\\ K_{+} &= \$(150 + \delta) \end{aligned}\]

Suppose we long $\frac{1}{2\delta}$ with strike $K_{-}$ and short $\frac{1}{2\delta}$ with strike $K_{+}$. The payoff would be:

\[\begin{aligned} \frac{C(K - \delta) - C(K + \delta)}{2\delta} \end{aligned}\]

If the stock close below $149, both options expire worthless. If the stock closes above $151, the payoff will be:

\[\begin{aligned} \frac{S_{T} - 149}{2} - \frac{S_{T} - 151}{2} &= \frac{-149 + 151}{2}\\ &= \frac{2}{2}\\ &= 1 \end{aligned}\]

If the stock is 150, the long option will be worth $1 and short option -$1 and the total payoff is 0. As $\delta \rightarrow 0$, the payoff will approach the payoff of a digital.

In practice, the insitution tends to overhedge and sell the $150 strike rather than $151. In this case, if the stock close below $149, both options expire worthless like before. If the stock closes above $150:

\[\begin{aligned} \frac{S_{T} - 149}{2} - \frac{S_{T} - 150}{2} &= -149 + 150\\ &= \frac{1}{2} \end{aligned}\]

So the payoff would still be ITM when the stock is above $150.

A put digital is ITM if the underlying is below the strike price.

Asset-or-Nothing Options

An asset-or-nothing call is an option such that if the underlying trades above a certain level, it pays the value of the underlying rather than the difference between the value and the strike price. It has the following relationship:

\[\begin{aligned} \text{Call} &= \text{Asset-or-Nothing} - \text{Digital Call}\\ \text{Asset-or-Nothing} &= \text{Call} + \text{Digital Call} \end{aligned}\]

For example in a USDJPY FX, a digital call pays $1 if USDJPY is over 115 yen on expiry in 1y. The option is naturally valued in yen, but if the option is an asset-or-nothing, it would be paid in dollars.

Knockouts and Reverse Knockouts

A knockout option become worthless when it crosses the barrier. For a call option, it extinguishes if the underlying trades above the barrier and for a put option, it becomes worthless when the underlying trades below the barrier.

The knockout option tends to have negative vega near the barrier as a higher volatility will knock the options out. It does not always go down to negative but will be close to 0, because we still need volatility for the underlying to go as far from the barrier.

It is possible to construct a replicating portfolio using vanilla European options. The replicating portfolio has to have a value of 0 at barrier H. The replicating portfolio is said to be dynamic and would require adjustments. In addition, one need to predict forward skew in order to construct the replicating portfolio. For example, the general idea to replicate a knockout call option, one can long a call at the same strike as the knockout option and sell a put with lower strike.

A knockin option only comes alive if a barrier is breached prior to expiration. A combination of knockout and knockin will offset each other giving you back the vanilla option. For example:

\[\begin{aligned} \text{Call} &= \text{Knockin Call} + \text{Knockout Call} \text{Put} &= \text{Knockin Put} + \text{Knockout Put} \end{aligned}\]

A knock-out option in which the barrier is in-the-money with respect to the strike is called a reverse knock-out option. In other words, a reverse knockout is a knockout that extinguishes an ITM option. For example a reverse knockout put has barrier lower than the strike. The sharp discontinuity at the barrier level makes hedging expensive, as it require going from short the asset to long the asset. Usually reverse knockouts are packaged together with one-touches which provide some rebate if the barrier is breached.

One-Touches and No-Touches

For a one-touch call option, the option pays out at expiry if the underlying crosses above the barrier any time during the life of the option. A one-touch can be packaged with a reverse knockout so that if the barrier is reached, the one-touch will offer some rebate.

A one-touch option costs roughly twice that of a normal digital option. When the underlying right at the barrier, the digital option will have a value of 0.5 while one-touch is 1.

Double Barriers

A double barrier knockout has two barriers. If the underlying breaches either barrier, the option extinguishes. Adding 2 barriers make the options even cheaper than 1 barrier.

Quantoes

Quantoes are products that are paid in a non-natural currency without FX conversion. The underlying asset is denominated in one currency but settlement is made in another currency at a predtermined exchange rate.

This product depends on the correlation between the underlying and the FX rate. A quanto call option will cost less than a normal call optio if the correlation is positive.

Multiple-Currency Products

For example a structured note that pays, in 2 years’ time, 40% of the best return of the S&P 500 index, the STOXX 50 index, and the Nikkei 225 index, and pays out in USD. Another example would be FX quatoes, where the best performance of EURCHF, AUDJPY, USDCHF, and EURGBP is paid in USD. Interest rate quatoes, which pays in EUR the spread between 3M polish Libor and 3M Euribor for example.

NDF

For example, normally a forward contract to sell USDCNY forward at 6.5 means paying 6.5 CNY and receive 1 USD at maturity. The value of the forward would be (payable in CNY):