A type of structured product that has become fashionable these days is the structured share repurchase program. When a corporate client wants to buy back its own shares, it normally uses its broker or investment bank to buy shares on the open market. The repurchase program will be announced before the start of the program and the announcement may have an impact on the share price.

In a normal share repurchase program, the broker behaves as an agent and buys a fixed amount per day up to a certain notional amount or number of shares. The amount purchased per day depends on the deadline and the target notional or target number of shares. The deadline for the completion of the repurchase program is often quite important for the corporate client and is normally set before the end of a quarterly accounting period or announcement. Depending on whether the corporate client targets a notional amount or a number of shares, the share repurchase program is called fixed notional or fixed shares.

A new structure proposed by investment banks has recently become popular as it can provide cheaper prices for the corporate client. Under these new structures, the maturity and target number of shares or notional amount are fixed, but the total number of shares that are repurchased can vary. The investment bank has the optionality on how many shares to purchase per day as long as it commits itself to finishing the repurchase program by the deadline and not buying back more than a certain number of shares each day. With such programs the investment bank is able to give a discount to the client. At the same time the investment bank will be paid the volume weighted average price (VWAP) over the period minus the discount. A good trader taking advantage of the daily optionality can beat the VWAP average price. Hence in many circumstances it is desirable for the investment bank to not delta hedge its position and keep the option naked in the book so that it can make money out of daily price movements.

The Dupire equation for local volatility is surely one of the biggest technological discoveries in the pricing of equity derivatives. It has significantly changed how the market analyses and manages the risk of structured products. Despite having many deficiencies, the model is being used in an industrial way. By industrial, I really mean like a factory. Although it doesn’t account for many of the associated risks, the local volatility model is the benchmark for pricing any structured product. The local volatility model is also the most widely used approach for risk management. But why is the local volatility model so widely used? The main reasons are that the model is very easy to implement and the simulations using the model are so precise. Nevertheless the local volatility model has many deficiencies and does not accurately price products with non-European options. Furthermore the standard Dupire formula is not the best choice for actually implementing the model. Instead Jim Gatheral’s implementation of local volatility is surely more practical for the simple reason that we tend to see prices as a function of implied volatilities rather than of actual vanilla option prices. Gatheral suggests the following formulation:

where is the local variance, is the strike, and is the maturity. is the log moneyness strike where is the forward and is the implied total variance. As seen in the above equation, the local volatility is a function of the implied volatility rather than call/put prices. This approach is much more useful in practice.

It’s been a while since I looked at variable annuity products so I am writing this post to refresh my own memory as well as provide readers with an overview!

A variable annuity is an insurance product, most often used to provide life insurance. People invest their money and are supposed to receive periodic payments that depend on the performance of the investment portfolio selected. Let’s assume that Mr. A has bought a variable annuity from a life insurance company. The insurance company purchases a basket of equity, bond, and money market instruments on behalf of Mr. A. The basket can be fixed over time or can be rebalanced periodically in order to maintain a target set of weights among the different assets. The insurance company will pay a coupon based on the performance of the basket each year until death. In the event of death the insurance company pays an amount that depends on the value of the basket to the beneficiaries of Mr. A. However the insurance company also guaranties that the coupons will be above some minimum level and also that the death benefit will be at least some specified amount. So the insurance protects Mr. A against the risk of having his income fall below a certain level and provides Mr. A’s beneficiaries with protection in case of his death. With these products, the client also has the right to exit the contract and receive a lump sum based on the value of the basket less a haircut. Depending on the contract, this lump sum can be also floored so that the client will receive at least his initial investment back.

So Mr. A, having invested in a variable annuity, can receive three types of income:

1. A retirement income equal to the minimum coupon plus a variable amount depending on the performance of the investment portfolio.
2. At least the guaranteed lump sum in case of death.
3. A lump sum with a hair cut applied if he exits the contract.

The lump sum is normally calculated based on a maximum drawdown. It is a weighted lookback or a level that depends on the maximum and average value of the variable annuity portfolio since inception.

These life insurance products are generally in demand for many reasons, including providing an extra pension for the purchaser and covering the cost of inheritance tax for the beneficiaries. This overview of variable annuities is a very simplistic one. There are three main risks associated with variable annuities: the client exit risk, the death risk, and the market risk. The first two are actuarial risks. The market risk arising from selling a variable annuity is a complicated structured product.

The insurance company normally uses a bank to cover its market risk, which is the risk that the portfolio income cannot cover the floored level for the lump sum. Both the death and client exit risks can be quantified using deterministic models based on actuarial assumptions or stochastic models.

One of the risks of the variable annuity product is its sensitivity to the correlation between stocks and bonds. When rates go high, they normally have a slightly negative correlation to equities and a strong negative correlation to bond values. So in case of a positive move in rates, both the equity and the bond components of the portfolio will tend to go down in value. This dynamic will magnify the portfolio losses and will need to be taken into account when pricing the put option embedded in the variable annuity.