Delta Hedging In Foreign Exchange: Analysis And Comparison

Inputs and Delta Calculation

1. Calculate the delta hedge updates, adjustments and forward contract valuations in Appendix 2.

2. Do you think Scout Finch would view delta hedging as more or less risky for Dayton than an ordinary forward contract or purchased option hedge? Justify your answer with reference to literature.

3. How would you respond to the accusation that delta hedging is very subjective in its approach to viewing both the direction of an exchange rate movement and the proportion of hedge cover? Discuss with reference to literature.

4. If you were Scout Finch, what recommendation on the use of delta hedging would you make to your CFO? Justify your recommendation.?

Following are the inputs for the initial delta, as given in the case.

	Inputs
Spot rate	$1.3309/Euro
Strike price (same units as Spot)	$1.335/Euro
volatility (annualized)	10.00%
domestic interest rate (annualized)	3.30%
foreign interest rate (annualized)	2.00%
time to maturity in days	92
Put option value	$0.0265/Euro

Here delta is calculated using the formula

 

S0: Spot rate.

X: Strike rate.

r: domestic rate.

q: foreign exchange rate.

Sigma: annualized volatility.

t: time to maturity.

Delta calculation is done as illustrated by Jorion, (2009) in the book Financial Risk Manager Handbook.

The following gives the formula for put delta:

The initial vale of delta using this formula is (-0.4859). The negative sign indicates a put delta.

Using the above value of delta as initial value, proportionate hedging is done by buying forward contracts, while some part of the portfolio is kept uncovered. When 86 days to maturity are left, the delta is recalculated for the new spot price and time to maturity. The portfolio is re aligned with new values of delta. The process is repeated every week till maturity. Table 2 below lists down the deltas and forward contracts bought and sold at updated values of delta.

Update number	Number of Days to maturity	Spot Rate	Delta	Optimal hedge	Hedge adjustments	Remaining uncovered	Sold or bought Forward	Forward rate	Forward proceeds.
1	92	1.3309	-0.4859	-485900	0	514100	-485900	1.3353	648822.27
2	86	1.2956	-0.6986	-698600	-212700	301400	-212700	1.2996	276424.92
3	78	1.2893	-0.7455	-745500	-46900	254500	-46900	1.2929	60637.01
4	71	1.2908	-0.7501	-750100	-4600	249900	-4600	1.2941	5952.86
5	64	1.2926	-0.754	-754000	-3900	246000	-3900	1.2956	5052.84
6	57	1.3068	-0.6784	-678400	75600	321600	75600	1.3095	-98998.2
7	50	1.2919	-0.7917	-791700	-113300	208300	-113300	1.2942	146632.86
8	43	1.2834	-0.8594	-859400	-67700	140600	-67700	1.2854	87021.58
9	36	1.2643	-0.9513	-951300	-91900	48700	-91900	1.2659	116336.21
10	29	1.2555	-0.9817	-981700	-30400	18300	-30400	1.2568	38206.72
11	22	1.2568	-0.9909	-990900	-9200	9100	-9200	1.2578	11571.76
12	15	1.2227	-0.9992	-999200	-8300	800	-8300	1.2234	10154.22
13	8	1.2128	-0.9996	-999600	-400	400	-400	1.2132	485.28
14	1	1.2239	-0.9999	-999900	-300	100	-300	1.2239	367.17
15	0	1.2200

Table 3 calculates the final value of the portfolio at the end of the period. The uncovered proceeds are added to the forward proceeds, bought at different updates:

Delta hedge results
Net Proceeds from forwards	1308668
Proceeds from uncovered	122
Total dollar Proceeds	1308790

The four hedging strategies are compared in table 4. Here forward contract is used as benchmark and value of the portfolio at the end of the period using each of the strategies are compared.

Hedging Alternatives Comparison with Simple full forward covered
Remained uncovered	1220000	-115300
Forward covered	1335300	0
Put Option Cover	1308500	-26800
Delta Hedge	1308789.5	-26510.5

Table 5: Dollar movement Scenarios and strategy performance comparison:

	Dollar Stable	Dollar Strong	Dollar Strong	Dollar Strong
Strike Rate	1.75	no	1.75	yes	1.9	no	1.335	no
Volatility	5.06%		5.06%		10%		10%
Domestic interest rates	6%		6%		3.30%		3.30%
Foreign Interest rates	8%		8%		4%		2%
Forward Rate Range	1.754	1.7612	1.754	1.7371	1.905	1.8286	1.3353	1.2239
Spot Rate Range	1.764	1.7618	1.764	1.735	1.9111	1.8311	1.3309	1.22
Remained uncovered	1761800	2792.00	1735000	-10522	1831100	-41577	1220000	-88789.5
Forward covered	1754000	-5008.00	1754000	8478.00	1904960	32283.00	1335300	26510.50
Put Option Cover	1734625	-24383.00	1734635	-10887	1863818	-8859	1308500	-289.50
Delta Hedge	1759008	0.00	1745522	0.00	1872677	0.00	1308790	0.00

Note: here Delta hedge is kept as reference and other strategies are compared with that.

The above table compares the performance of Delta hedge with respect to 4 different strategies. The green indicates money is made and red indicates the money is lost with respect to the Delta hedge strategy. The comparison is done in the 4 scenarios given in the case.

 In case of Stable dollar, the forward contract loses money. For the Put option, as discussed by John H, (2013), the cost of purchasing a put option is very high and hence the strategy looses the highest in this case. Here delta hedging is the closest to the opening spot rate conversion of the accounts receivable for Dayton, which is by definition the motive behind hedging as discussed by Longo J (2009). This value of portfolio minimizes the uncertainty in portfolio performance with change in underlying value change. For stable dollar, uncovered strategy pays the most, as the exchange rate moves around a pegged value.

Dayton Manufacturing’s Delta Hedging Analysis

In scenario 2 as given in the case study the dollar is going to be stronger and thus taking a forward position on the entire portfolio is the best strategy. Put Option Cover and uncovered both losses a big sum of money, with respect to Delta hedge and forward rate contract.  The same thing repeats in the scenario 3 and scenario 4 as well.

In the current context, if we compare the 4 scenarios as given in the case study and rank the strategies:

Delta hedge ranks 2^nd best in all 4 cases. This makes the delta hedging strategy as the best of 4 strategies.

	Scenario 1	Scenario 2	Scenario 3	Scenario 4
	Dollar Stable	Dollar Strong	Dollar Strong	Dollar Strong
Remained uncovered	1	4	4	4
Forward covered	3	1	1	1
Put Option Cover	4	3	3	3
Delta Hedge	2	2	2	2

In the case study given, in 3 out of 4 scenarios, dollar was going to be stronger. While in only one case the dollar was predicted to be stable.

Delta hedging uses linear hedging (forward covered options) for hedging the position as discussed at optiontradingtips. Delta value tells the amount of the portfolio that should be covered to the total portfolio to make the risk of the portfolio to zero as discussed by Giovanni B (June 1987). Delta here is borrowed from the Black and Scholes model basics. So in a way, delta hedging is forward covered hedging, but a part of portfolio is forward covered, and the part is decided by delta.

Delta is kept to be updated, as the value of the underlying, (here the spot rates) changes the delta value changes, and the portfolio is no longer a risk free portfolio for new value of the delta. Hence the delta value needs to be updated and the value of the portfolio under the forward cover needs to be changed. When the number of the updates is increased the performance of the portfolio improves further, and in fully dynamically updated delta hedging, the portfolio delta is changed every day to achieve best results. The same thing is discussed by Rajiv S. (2014) in the book Derivative and Risk management.

When the movement of the underlying of the portfolio is known, then it is better to not to go for delta hedging, but simply go for a forward contract. The forward contract will fetch the best results, if the dollar is predicted to go stronger and uncovered contract if the dollar to go weak. But such kinds of cases are hardly available.

Secondly, every hedging scheme comes with a cost, while delta hedging suggests a value of optimum value of hedging required to make the portfolio risk free. This makes delta hedging a better strategy than standard forward contract hedging for 100% portfolio or buying put option cover. The cost of hedging is minimized by the

Delta hedging increases the flexibility for the hedgers. The losses made by delta hedging can be considered as reasonable price for the level of flexibility offered by delta hedging.

Conclusion:

For any arbitrary context, when the movement of the underlying is not predictable, Delta hedging, with the most number of updates is the most suitable strategy. Delta hedging takes a position making the portfolio risk free for that particular value of spot price, which is tuned by updating the value of delta every time, making the overall performance of the delta hedge better. Also the hedging cost is minimized by using delta hedging strategy.

Comparison of Hedging Alternatives

Delta hedging is a selective forward rate hedging technique, with a part of the portfolio only covered with forward contract, while remaining part uncovered.

To compare the subjectivity for the delta on either side changes in the underlying (Here spot price of the stock is the underlying.), Let us consider the following graph, which shows the put delta values for the underlying price.

The curve is a symmetric curve on both the sides with the value of delta changing the most in the middle and saturating at the ends. This clearly shows that the delta is not specific to the movements on either side of the exchange rates. The curve is taken from Dynamic Hedging. (2015) from riskencyclopedia.com

Now as delta hedging used these values of deltas, delta hedging is also symmetric for either side of exchange rate changes.

Sometimes, by observing the movement of the delta with respect to the change in the spot rate, it may seem that delta is changing more on one direction with the change in value of spot rate. The reason for this is the delta is a function of time and volatility, in addition to the spot rate. So as the time to maturity reduces, delta tends to move towards 0 in stable dollar case and -1 in stronger dollar case. This movements in delta with respect to time curtails the opposite side movement of delta due to exchange rate moving in opposite direction then the normal trend.

Thus it is clear that Delta is not subjective on the movement directions of the underlying spot rates. With symmetric delta, the hedge cover also becomes symmetric with respect to the movement in interest rates.

A scenario based hedging strategy, would be recommended to the CFO, where in for some predicted spot rate changes, forwards rate hedging would be suggested, while in case of times when exchange rate is moving randomly, a delta hedging strategy will be suggested.

As we saw in scenario 2, 3 and 4 in table 5, whenever the exchange rate is moving against the company, forward rate hedging becomes the best strategy. Forward rate hedging clips the losses happening due to domestic currency going stronger. So, whenever the domestic and foreign interest rates and other macroeconomic parameters are indicating strengthening of the dollar, than a plain forward rate hedging would be suggested as given by Banks (2006) in his book for a exporting form like Dayton.

But, when the exchange rate movements are totally unpredictable and may go up and down from its center value, as in case of the scenario 1, where exchange rates were stable, Delta hedging would be recommended. Moreover, more number of updates would be recommended for the delta hedging. The frequency of the updates will be decided by the volatility of the underlying spot rate.

As being delta hedged, the spot rates movements are not affecting the portfolio because the delta hedging is making it risk free every time whenever delta update is made. But as discussed by Antonio C. (2009) higher the value of volatility of spot rate, faster and larger is the movement of the spot rate. With every movement of spot rate the delta equilibrium is disturbed and an update is required.  Greeks like Vega can be used to monitor the movements of the volatility.

When it is expected that dollar is going to be weaker. As discussed by Alok D (2012) this condition is favorable for the company with receivables in the foreign currency; the dollar returns are going to be higher. In that case an uncovered portfolio will be the best option.Thus, a balanced mix of 3 strategies should be used for different kind of movements if exchange rate.

References:

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Giovanni B, Robert E.W. (June 1987).Efficient analytic approximation of American option values. Journal of Finance 42 (2)

Banks, Erik, Siegel, Paul (2006). The options applications handbook: hedging and speculating techniques for professional investors. McGraw-Hill Professional

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