Quiz: Hedging

This is a draft, shown for review before publication.

6 questions to check your understanding of the chapter “Hedging”. Choose an answer and select “Check answer” to see the answer and explanation. You can also open “Answer and explanation” to see the answer first.

1. Question 1

   As a fictional example, you sell 10 units of a call option (call) with an underlying asset price of 100, a strike price of 100, 0.5 years to expiry, and an interest rate of 1% (continuously compounded). The delta of one unit, found with the Black-Scholes model at an implied volatility of 20%, is 0.5422. To make the position delta-neutral, how do you trade the underlying?

   1. A.Sell 5.422 units of the underlying
   2. B.Buy 5.422 units of the underlying
   3. C.Buy 10 units of the underlying
   4. D.Buy 0.5422 units of the underlying
   Answer and explanation

   Answer: B

   You buy 5.422 units of the underlying. When you hold many units of an option, you multiply the delta of one unit by the quantity to find the amount of the underlying to hedge with: 0.5422 × 10 = 5.422. A short call loses when the underlying price rises, so you hold the underlying in the opposite direction by buying it. This way of using delta to limit the risk of option price moves is called delta hedging. A position whose delta has been brought to zero is delta-neutral.

   Selling 5.422 units adds to a position in the same direction as the short call, so the delta does not become zero. Buying 10 units means holding the underlying for the whole quantity sold, which takes the delta above zero. 0.5422 units is the delta of one unit, without multiplying by the quantity.

   Under the Black-Scholes model, if the underlying price rises from 100 to 101, the call price rises by 0.5562 per unit, from 5.8760 to 6.4322. For the 10 units sold, that is a loss of 5.562. The 5.422 units of the underlying bought rise in value by 5.422 in total, which is a gain. Together, all that remains uncanceled is the difference, a loss of 0.140. When the underlying price falls, the short call gains and the underlying loses, so the directions are reversed. However, delta changes as the underlying price moves and time passes, so the amount of the underlying is adjusted at set times (rehedging).

   Learn more: Delta Hedging

2. Question 2

   As a fictional example, you sell 1 unit of a call with an underlying price of 100, a strike price of 100, 0.5 years to expiry, and an interest rate of 1% (continuously compounded). You sell it at 5.88, its Black-Scholes price at an implied volatility of 20%. You delta hedge it many times. The underlying grows on average at the same rate as the interest rate. Realized volatility is 25%, and the price at a volatility of 25% is 7.28. What value does the mean profit or loss approach?

   1. A.A mean profit or loss of about zero
   2. B.A profit of about 5.88
   3. C.A loss of about 13.16
   4. D.A loss of about 1.40
   Answer and explanation

   Answer: D

   The mean profit or loss approaches the difference between the option price at realized volatility (the size of the price moves that actually occur) and the price at which the option was sold. Subtracting the price at the realized volatility of 25%, 7.28, from the sale price of 5.88 gives 5.88 − 7.28 = −1.40. Option prices are higher when volatility is higher, so if realized volatility is greater than the implied volatility at which the option was sold, a seller who delta hedges loses on average.

   The mean is about zero when realized volatility equals the implied volatility of 20%. Even with delta hedging, the mean profit or loss is not zero when the two volatilities differ. A profit of 5.88 assumes that delta hedging removes all profit or loss from price moves, so the premium received is kept in full. A loss of 13.16 adds 5.88 and 7.28 instead of subtracting (5.88 + 7.28 = 13.16).

   For a party that buys the call and hedges it, the sign of profit or loss is reversed, and the mean is a gain of about 1.40. If realized volatility is 15%, the price at 15% is 4.4730. Written to four decimal places, the sale price is 5.8760, so the seller's mean profit or loss is 5.8760 − 4.4730 = 1.4030, a gain of about 1.40. In every case these are means: the profit or loss on each path varies around the mean depending on the path the underlying price takes.

   Learn more: Delta Hedging

3. Question 3

   As a fictional example, you buy 1 unit of a call with an underlying price of 100, a strike price of 100, 1 year to expiry, an interest rate of 1% (continuously compounded), and an implied volatility of 20%, and you delta hedge it. The Black-Scholes gamma is 0.0197. The underlying price falls from 100 to 98.5 in one day (with 365 days in a year). Using the approximation that splits profit or loss into gamma and theta, what is the profit or loss?

   1. A.A loss of about 0.0114
   2. B.A profit of about 0.0222
   3. C.A profit of about 0.0114
   4. D.A loss of about 0.0108
   Answer and explanation

   Answer: C

   The position makes a profit of about 0.0114. For a delta-hedged long option position, the profit or loss over a short period $\Delta t$ can be approximated by the following formula. $\Gamma$ is gamma, $S$ is the underlying price, $\Delta S$ is its move, and $\sigma$ is the implied volatility.

   $$\text{profit or loss} \approx \frac{1}{2}\,\Gamma S^2 \left[\left(\frac{\Delta S}{S}\right)^2 - \sigma^2 \Delta t\right]$$

   $\frac{1}{2}\Gamma S^2$ is 0.5 × 0.0197 × 100 × 100 = 98.5. The squared return is 0.015 × 0.015 = 0.000225. The part expected by the implied volatility is 0.2 × 0.2 ÷ 365 = 0.00010959. Multiplying the difference, 0.00011541, by 98.5 gives 0.011368, a profit of about 0.0114.

   Because the return enters squared, the formula gives the same profit or loss for a rise and a fall of the same size. The call loses value when the underlying price falls, but the gain on the underlying sold for the delta hedge offsets it, so the result is not a loss of 0.0114. A profit of 0.0222 leaves out the part expected by the implied volatility (the part of theta's loss of value that remains after removing the part matched by interest on the delta hedge's cash): 98.5 × 0.000225 = 0.0222. A loss of 0.0108 is the profit or loss on a day with no move, which assumes that delta hedging removes all profit or loss from price moves.

   With an underlying price of 100, the move that gives a profit or loss of zero is about 1.05. On a day with a smaller move, such as 0.5, the result is 98.5 × (0.000025 − 0.00010959) = −0.0083, a loss of about 0.0083. For a position that sells the call and delta hedges it, the sign of the same day's profit or loss is reversed.

   Learn more: Managing Gamma and Vega

4. Question 4

   As a fictional example, calls have been bought and sold so that the vega of the whole position is close to zero. The vega by expiry (per 1% of implied volatility), found with the Black-Scholes model, is 5.629 at 3 months, −39.448 at 1 year, and 33.813 at 2 years. If only the 2-year implied volatility falls by 1 percentage point, what is the profit or loss estimated with vega?

   1. A.A profit or loss of about zero
   2. B.A loss of about 33.81
   3. C.A profit of about 39.45
   4. D.A profit of about 33.81
   5. E.A loss of about 39.45
   Answer and explanation

   Answer: B

   The position loses about 33.81. Multiplying the 2-year vega of 33.813 by the change in volatility of −1 gives 33.813 × (−1) = −33.813. The volatilities of the other expiries do not move, so the 3-month and 1-year vegas produce no profit or loss.

   The profit or loss is about zero when the implied volatilities of all three expiries move by 1 percentage point together. Bringing total vega to zero still leaves vega by expiry, so a profit or loss arises when the volatilities of different expiries move differently. A profit of 33.81 is the result when the 2-year volatility rises by 1 percentage point. A profit of 39.45 and a loss of 39.45 come from taking the 1-year expiry as the one that moves: if only the 1-year volatility falls by 1 percentage point, the result is a profit, and if it rises by 1 percentage point, a loss.

   Implied volatility is priced in the market for each expiry, so the vega of a position is measured separately for each expiry.

   Learn more: Managing Gamma and Vega

5. Question 5

   What kind of risk is the pin risk borne by a financial institution that sells a digital option near expiry and delta hedges it?

   1. A.The risk that, when the price moves near the strike price, hedging losses or costs can exceed the maximum payout
   2. B.The risk that the slope of the volatility smile can change the price of the digital
   3. C.The risk that the correlation between an index and a stock is not as expected, so the hedge fails and market risk remains
   4. D.The risk that the choice of model changes the quantity needed to hedge, so it differs from what was expected
   Answer and explanation

   Answer: A

   Pin risk is the risk that, when the underlying price is near the strike price close to expiry, hedging losses or trading costs can exceed the maximum payout of the digital that was sold. The delta of a digital grows without limit when the price stays near the strike price as expiry approaches, and a hedge that follows the theory requires repeatedly trading large quantities in a hurry.

   The difference in price from the slope of the smile and the difference in hedge quantity from the choice of model both come from differences in the model assumed, not from the price moving back and forth near the strike price. Hedging error from the correlation between an index and a stock not being as expected is a problem of hedges that depend on correlation.

   Learn more: Hedging Exotic Products

6. Question 6

   As a fictional example, you sell 1 unit of an up-and-out call (a barrier option; the barrier is monitored continuously, and there is no rebate) with a strike price of 100 yen, a barrier of 120 yen, and 1 month to expiry, and delta hedge it. The Black-Scholes delta is 0.158 at 110 yen and −0.845 at 115 yen. As the underlying price rises from 110 yen to 115 yen, how do you change the hedge position on a net basis?

   1. A.Sell about 0.845 units of the underlying
   2. B.Buy about 0.687 units of the underlying
   3. C.Sell about 1.003 units of the underlying
   4. D.Buy about 0.038 units of the underlying
   Answer and explanation

   Answer: C

   You sell about 1.003 units of the underlying. The hedge position of the party that sold the call is the same quantity as the delta of the call it sold, held in the opposite direction. At 110 yen it holds 0.158 units long; at 115 yen the delta is negative, so it holds 0.845 units short. Moving from a long position of 0.158 units to a short position of 0.845 units, the net quantity sold is 0.158 + 0.845 = 1.003 units. The closer the price gets to the barrier, the more likely a rise in the price is to extinguish the right, and the sign of the delta changes.

   Selling 0.845 units is the quantity you get by looking only at the position at 115 yen and forgetting the long position held at 110 yen. Buying 0.687 units is the difference of the two deltas' sizes with the signs ignored (0.845 − 0.158 = 0.687). Buying 0.038 units is the difference between the deltas of a vanilla call (0.955 at 110 yen and 0.993 at 115 yen), the vanilla intuition that you buy more as the price rises.

   Combined with the instrument being hedged, small moves cancel out. For example, when the price rises 0.01 yen from 110 yen, the sold call loses 0.158 × 0.01 = 0.00158 and the long underlying gains 0.00158, so the total is 0. When the price rises 0.01 yen from 115 yen, the sold call gains 0.845 × 0.01 = 0.00845 and the short underlying loses 0.00845, so the total is again 0. However, this assumes continuous trading without costs; in practice, large changes in delta near the barrier become hedging error and cost as they are.

   Learn more: Hedging Exotic Products