Hedging Exotic Products

Selling and hedging exotic options, and setting position limits and valuations for them, run into the cases where the delta hedging of vanilla options breaks down.

What vanilla call hedging assumes

Delta hedging holds the underlying asset in line with the amount by which an option's value moves when the price of the underlying changes (the delta). See Delta Hedging for the procedure and profit and loss. The method breaks down for some exotic options, such as those with discontinuous payoffs or rights that are extinguished at a barrier.

For a vanilla call option, delta, gamma, and vega are always positive and move smoothly with the price of the underlying¹. For exotic options, this property breaks down for the following reasons.

• The payoff is discontinuous, and the delta jumps to a very large value near maturity.
• For a barrier option, the rights are extinguished when the barrier is touched, and the signs of delta, gamma, and vega change near the barrier.
• The price depends on how the volatility smile moves and on the choice of model, and the required hedge changes with them.
• Risks that vanilla options cannot easily hedge remain, such as the correlation among several underlying assets.

Consider a hypothetical example in which the Black-Scholes assumptions hold (constant volatility and interest rate, no dividends, and continuous trading without costs), with an interest rate of 1% and volatility of 20%².

Digital options near maturity and pin risk

Suppose Q Corp has sold 100 million units of a digital option (call) with a strike price of 100 yen that pays 10 yen per unit if the price of the underlying at maturity exceeds 100 yen. The most Q Corp can pay is 1 billion yen.

The delta of this digital call is given by the following formula³.

$$\Delta = \frac{Q\, e^{-rT}\, \phi(d_2)}{S\, \sigma \sqrt{T}}, \qquad d_2 = \frac{\ln(S/K) + (r - \sigma^2/2)\,T}{\sigma \sqrt{T}}$$

$Q$ is the payment per unit (10 yen), $K$ is the strike price, $S$ is the price of the underlying, $T$ is the time to maturity in years, $r$ is the interest rate, $\sigma$ is the volatility, and $\phi$ is the density function of the standard normal distribution. Because $\sqrt{T}$ is in the denominator, the delta grows without bound as maturity approaches while the price stays near the strike. One day before maturity (1/250 of a year), with the price moving from 98 yen to 102 yen, Q Corp's delta hedge position is as follows².

Figure description

The chart plots the deltas in the table above as a line, with the price of the underlying one day before maturity (98 to 102 yen) on the horizontal axis and the delta per unit on the vertical axis. The delta rises from 0.894 at 98 yen to 3.154 at the strike of 100 yen and falls back to 0.912 at 102 yen, forming a peak. The delta of a vanilla call stays between 0 and 1, whereas the delta of this digital exceeds 3 near the strike. The chart shows that the hedge position swells sharply near the strike.

Because Q Corp has sold 100 million units, the quantity of the underlying held (in million shares) is the delta multiplied by 100. The amount is the quantity times the price at that moment. When the price is 100 yen, Q Corp holds about 31.5 billion yen of the underlying, more than 30 times the maximum payment of 1 billion yen. While the price rises from 98 yen to 100 yen, Q Corp buys an additional 226.0 million shares. Assuming each trade costs 0.05 yen per share, this additional buying alone costs 11.3 million yen, and with the selling when the price falls back to 98 yen the total is 22.6 million yen, or 2.3% of the maximum payment. If the price is above 100 yen at maturity, Q Corp pays 1 billion yen, and if not, it pays nothing. Until it is decided which, Q Corp keeps holding a large position.

When the price of the underlying moves back and forth around the strike near maturity, the theoretical hedge therefore calls for repeated large and rapid trades. Real markets do not allow instantaneous trading, so losses from small price moves and trading costs can exceed the maximum liability of the digital that was sold. This is called pin risk⁴.

Sensitivities reverse near the barrier

Consider a hypothetical example in which Q Corp sells an up-and-out call with a strike price of 100 yen, a barrier of 120 yen, and one month (1/12 of a year) to maturity. The barrier is monitored continuously, and there is no rebate when the rights are extinguished. The table below shows the price and sensitivities per unit of this option, next to the delta of a vanilla call². Vega is the change in price when volatility rises by 1 percentage point (from 20% to 21%).

Figure description

The chart plots the two deltas in the table above as lines, with the price of the underlying (100 to 119 yen) on the horizontal axis and the delta per unit on the vertical axis. The barrier is at 120 yen, so the right edge of the chart is just short of it. The points are joined by straight lines, so the lines differ from the actual movement. The delta of the vanilla call rises from 0.517 toward 1, reaching 0.999. The delta of the up-and-out call falls from 0.499 at 100 yen to 0.158 at 110 yen, passes −0.845 at 115 yen, and falls further to −1.350 at 119 yen. The chart shows that the two lines start at close values near 100 yen and spread apart in opposite directions as the price approaches the barrier.

The delta of a vanilla call never exceeds 1, but the delta of the up-and-out call falls below −1 near the barrier. The closer the price is to the barrier, the more likely a rise in price is to extinguish the rights. Gamma also changes from a positive value at 100 yen to a negative value near the barrier. Vega becomes negative near the barrier because higher volatility makes it more likely that the price crosses the barrier, and the value falls by the amount of the rights that are extinguished¹.

For Q Corp, the seller, all the signs are reversed. The delta hedge position is a purchase of 0.499 units at 100 yen, but a sale of 0.845 units at 115 yen. As the price rises from 100 yen to 115 yen, Q Corp sells 1.344 units (the sum of 0.499 and 0.845). In terms of the sensitivity of the option sold, delta is a short 0.499 units at 100 yen and a long 0.845 units at 115 yen, so the hedge trades are the opposite of that. Vega is described the same way: at 100 yen Q Corp is short 0.099 per percentage point, but at 115 yen it is long 0.429. If you hedge vega with vanilla options, whether to buy or sell switches depending on the price level.

This table assumes continuous trading without costs. Real markets have trading costs, discrete trading, and a lack of liquidity, and the large changes in delta and gamma near the barrier turn directly into hedging error and cost¹.

Smile, model, and correlation change the hedge amount

The implied volatility obtained from vanilla options differs by strike (the volatility smile). The price and the hedge amount of an exotic option therefore depend on how the smile moves as the price moves. The paper from the Institute for Monetary and Economic Studies of the Bank of Japan divides the movement of the smile into three types. In sticky strike, the smile does not move when the price changes. In sticky delta, it moves by the same amount in the same direction as the price. In sticky tree (the term used in the Japanese paper), it moves in the opposite direction. The delta needed to hedge a short call position is larger than the Black-Scholes delta under sticky delta and smaller under sticky tree⁵.

The price of a digital option also includes the slope of the smile, because the payoff of a digital corresponds to the derivative of the call price with respect to the strike. Consider a hypothetical example in which a digital call with a strike of 100 yen, a payment of 10 yen, and one year to maturity is approximated by a combination of 5 units each of the calls at 99 yen and 101 yen. With no smile (constant volatility of 20%), the price is 4.75 yen. If the slope is such that implied volatility falls 0.5 percentage points for each 10-yen rise in the strike, the price is 4.95 yen, and if it rises 0.5 percentage points, the price is 4.56 yen²⁶. For Q Corp, which sold 100 million units, compared with no smile, the slope makes a price difference of about 20 million yen (+0.20 yen for the falling slope and −0.19 yen for the rising slope, times 100 million units).

For an exotic option whose rights depend on the path, as with a barrier option, liquidity is low and the market price is less reliable. The fair value is therefore estimated with a model fitted to the prices of vanilla options. The fair value is determined as the cost of a hedging strategy built with that model, so it depends on the model used⁷.

Another risk that vanilla options cannot easily hedge is correlation. The risk of an individual stock or a basket of stocks is often hedged with stock index futures or options, and this hedge relies on the correlation between the index and the stocks being as expected. If it is not, the hedge does not work, and market risk can remain⁸.

Replacement and static hedges

A dynamic hedge, which trades the underlying repeatedly, incurs trading costs at every rebalancing⁹. There are two alternatives.

The first is to replace the digital option with a combination of two vanilla calls with strikes on either side of the digital's strike (a call spread). The payoff of a call spread is smooth, so it avoids a delta that grows without bound¹⁰. In the previous example, if you combine one unit each of the calls with strikes of 95 yen and 105 yen, the delta one day before maturity stays between 0.989 and 1.000 anywhere from 98 yen to 102 yen². In exchange, the payoff at maturity does not match the digital. If Q Corp sells the digital and buys this call spread, the result at maturity is as follows (premiums excluded).

These figures are per unit. For 100 million units, the differences are 300 million yen, 500 million yen, and −300 million yen. The digital in this example does not pay at exactly 100 yen, so the receipt of 5 yen is left over at 100 yen. If you set the strikes at 95 yen and 100 yen and combine 2 units each, the receipt is at least the digital payment at every price. However, at a price of 100 yen one day before maturity, the spread costs 8.99 yen, about 4 yen more than the digital at 4.99 yen². In other words, you pay a high cost up front in exchange for removing the cases where the receipt at maturity falls short.

The second is the static hedge. When a barrier option trade starts, a combination of vanilla options is set up, and it is not rebalanced except when the barrier is touched and the rights are extinguished¹¹. Because continuous trading is not needed, trading costs can be kept down¹². However, existing methods that aim at perfect replication rely on strong assumptions, namely that options with particular strikes and maturities can be traded and that the underlying asset follows a particular distribution¹³. A combination built on the assumption of constant volatility has a large hedging error when volatility changes. A combination that also allows for changes in volatility needs many illiquid options, which makes it hard to implement at all⁹.

Every method trades off a cost or a remaining error. For a dynamic hedge it is the trading cost at each rebalancing, for a call spread replacement it is a high up-front cost or the difference in the receipt at maturity, and for a static hedge it is the limits of its assumptions and of liquidity.

Limits, reserves, and the valuation price

The risk that remains after hedging is managed with sensitivity limits, reserves, and the valuation method. This section is based on the descriptions in the versions dated 1998 to 2006 of the guidance that the US Federal Reserve System gave to examiners. It does not describe current regulation or the standard practice of all financial institutions.

For option positions, limits are set on each of delta, gamma, vega, theta, and rho to contain the risk of trading¹⁴. When markets are turbulent or liquidity is low, trading itself may also be restricted¹⁴.

A position that relies on dynamic hedging has a further risk, a decline in liquidity. If the hedge can no longer be rebalanced, the actual risk of such a position becomes higher than expected. A financial institution that has sold options is short volatility and gamma, so it is affected most when market liquidity falls¹⁵.

In valuation, model risk, which comes from errors in a model or from using it in ways not intended, has to be managed. It arises easily when complex, illiquid securities are valued and hedged with a model¹⁶. A reserve that defers part of the day-one profit of a trade is set up to cover the uncertainty of the price and the risk of managing the position. The level of the reserve should not be left to the trader or the trader's department to decide¹⁷.

The prices used for valuation include the difference between bid and offer (the bid-ask spread). There are two methods: valuing a long position at the lower of bid and offer and a short position at the higher, or valuing at the mid between them. As of 2003, many financial institutions used the mid¹⁸. Some institutions that value at the mid instead value at a price discounted or marked up to the more conservative side of the mid, and set the difference aside as a reserve¹⁸. Because the bid and the offer are the prices at which a position can be closed out, the value on the books should not fall outside that range, even after the reserve is deducted¹⁸.

Notes

1. Dominique Y. Dupont, "Hedging Barrier Options: Current Methods and Alternatives" (University of Twente, February 2002), the introduction (pages 2–3) and Figure 1 (page 30). It describes the delta and gamma of an up-and-out call taking large negative values near the barrier when expiry is close, vega turning negative near the barrier (with the reason), the always-positive sensitivities of a vanilla option, and frictions such as trading costs. ↩ ↩² ↩³

2. Calculated under the Black-Scholes assumptions, with analytical solutions for the digital call and for the continuously monitored up-and-out call. Delta and gamma are finite differences from moving the price up and down by 0.01 yen, and vega from moving volatility up and down by 0.5 percentage points, and the call spreads use differences of the same calculation. ↩ ↩² ↩³ ↩⁴ ↩⁵ ↩⁶

3. OpenGamma Quantitative Research, "Digital Forex Options", Section 2 "Pricing in Garman-Kohlhagen model" (page 2). It shows that the delta is the density function of the standard normal distribution divided by $S\sigma\sqrt{T}$. The foreign exchange notation is replaced with that for a foreign interest rate of 0, and the formula is multiplied by the payment $Q$. ↩

4. Alison Etheridge, "A Course in Financial Calculus" (Cambridge University Press, 2002), Chapter 6, Section 6.1, Example 6.1.1 (page 141). It states that when the price moves around the strike near maturity, risk from small price moves and trading costs can exceed the maximum liability of the digital that was sold, and calls this "pin risk". ↩

5. Yuki Shinbara, "確率局所ボラティリティ・モデルのもとでのヘッジ戦略：最尤経路を利用したバリア・オプションの静的ヘッジ" (Hedging strategies under stochastic local volatility models: static hedging of barrier options using the most likely path), Kinyu Kenkyu, Institute for Monetary and Economic Studies, Bank of Japan (October 2011, in Japanese), Section 2 (1) イ "オプションのデルタ・ヘッジ" (Delta hedging of options) (pages 128–131). ↩

6. OpenGamma Quantitative Research, "Digital Forex Options", Section 3 "Call spread approximation" (pages 2–3) and the end of Section 2 (page 2). It treats the payoff of a digital as the derivative of the vanilla price with respect to the strike, and says that pricing a digital with only the implied volatility at the same strike is inconsistent with the vanilla prices. The slope example is a fictional value. ↩

7. Yuki Shinbara, cited above, Section 2 (1) ロ "エキゾチック・オプションの時価評価と静的ヘッジ" (Fair value of exotic options and static hedging) (pages 131–132). ↩

8. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (April 2003), Section 4345.1 "Equity Derivatives", "Hedging" (page 3). ↩

9. Yuki Shinbara, cited above, Section 1 "はじめに" (Introduction) (page 126). It states that dynamic hedging incurs large transaction costs from frequent rebalancing, that a static hedge built on constant volatility has a large hedging error, and that a static hedge that allows for changes in volatility needs many illiquid options. ↩ ↩²

10. OpenGamma Quantitative Research, "Digital Forex Options", Section 3 "Call spread approximation" (pages 2–3). It lists as advantages that the call spread does not depend on a model and has a smooth payoff that avoids a delta growing without bound, and as a disadvantage that the payoff does not match near the strike. It adds that the position can be placed to approximate from above (super-replication). ↩

11. Yuki Shinbara, cited above, Section 2 (1) ロ (page 132). The definition of a static hedge. ↩

12. Dupont, cited above, the introduction (page 4). It states that with market frictions dynamic hedging becomes too costly, and in practice rebalancing is done less often and static hedging is adopted. ↩

13. Dupont, cited above, page 1 (abstract). It states that current static hedging methods aim at perfect replication and rely on strong assumptions on the availability of traded options with certain strikes or maturities, or on the distribution of the underlying asset. ↩

14. Board of Governors of the Federal Reserve System, "Trading and Capital-Markets Activities Manual" (February 1998), Section 2010.1 "Market Risk", "Limits on options positions" (page 7). ↩ ↩²

15. The same source, Section 2030.1 "Market Liquidity Risk of Trading Activities" (September 2006), "Dynamic Hedging Risks" (page 2). ↩

16. The same source, Section 2100.1 "Financial Performance" (April 2003), "Pricing Models" (page 4). ↩

17. The same source, Section 2100.1, "Holdback Reserves" (page 7). ↩

18. The same source, Section 2100.1, "Revaluation" (page 3). The same passage also states that some institutions value on the more conservative side of the mid and treat the discount or markup to the mid as a "holdback reserve". It also states that the bid and offer prices are regarded as limits on the value before reserves. Because they are the prices at which a position can be closed out, the value on the books should not go outside them. ↩ ↩² ↩³