Delta hedge updates, adjustments, and forward contract valuations in Appendix 2.

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.?

As per the case the data related for the scenario 4 is as follows.

	Inputs
Spot Price in $/euro	1.3309
Strike price $/euro	1.3350
volatility (annualized) in percentages	10
domestic interest rate (annualized)	3.30
foreign interest rate (annualized)	2.00
time to maturity in days	92
Put option value $/euro	0.0265

Here delta is calculated using the formula

Here d1 is as follows:

S0: Spot price.

X: Strike price.

r: domestic price.

q: foreign exchange rate.

Sigma: annualized volatility.

t: time to maturity in years.

The formula for Delta calculation is as illustrated by Jorion, (2009) in Financial Risk Manager Handbook.

The seed/initial value of delta using the above formula is (-0.4859). The delta for a put is negative. Using the seed value of delta, proportionate hedging is done with forward contracts, while remaining part of portfolio is kept uncovered. Using the new spot price and time to maturity at 86 days, the delta value is recalculated and portfolio hedging recalibrated. The process of updating in continued every week till maturity. Exhibit 2 lists the deltas with forward contracts at different updates 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

Using the value of covered and uncovered proceeds from exhibit 2, we get the delta hedge results as in Exhibit 3.

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

Using the forward contract as reference the performance for the 4 strategies are listed in Exhibit 4:

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

Exhibit 5: Potential Dollar movement scenarios and performance:

	Dollar Stable	Dollar Strong	Dollar Strong	Dollar Strong
Strike Rate	1.75		1.75		1.9		1.335
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	1735000	-10522	1831100	-41577	1220000	-88789.5
Forward covered	1754000	-5008	1754000	8478	1904960	32283	1335300	26510.5
Put Option Cover	1734625	-24383	1734635	-10887	1863818	-8859	1308500	-289.5
Delta Hedge	1759008	0.00	1745522	0.00	1872677	0.00	1308790	0.00

Assumption: Delta hedge is reference for all the calculations.

The above exhibit compares the performance of the 4 strategies. The green cells indicates the strategy performed better than delta hedging , while red indicates, that strategy performed bad with respect to delta hedging. The outcome is based on the 4 scenarios given in the case.

In case of Stable dollar with respect to pound or euro, the forward contract loses money as entire portfolio was 100% forward hedged at forward rates at the starting of the period, and benefits of weaker dollar during the term of contract could not be taken. Second for Put option a hefty sum is to be given as option price as discussed by John H, (2013), hence the strategy looses the highest in this case. Here the value of portfolio at the end of term for delta hedging is the closest to the spot rate at the time of opening that is why hedging is done, to have minimum uncertainty with respect to current positions as discussed by Longo J (2009).

For uncovered strategy for stable dollar, the exchange rate moves around a predefined pegged value only. Thus uncovered strategy makes the profit, but there is no certainty to this case and strategy may lose a lot of money too.

Comparison of delta hedging with other hedging strategies

For scenario 2 in the case study the dollar to pound and dollar to euro for scenario 4 the exchange rates are going weaker, this means that dollar is going strong and Dayton loses money in this scenario. Taking a full forward position is the best strategy for such predicted strong dollar case. Delta hedge being a modified version of Forward hedge looses little money in this case. While the put option and uncovered strategy looses the maximum amount in this scenario.

The 4 strategies are compared on the basis of their rankings in 4 different scenarios given in the case. As can be seen, the Delta hedge strategy was the second best performer for all the cases. Other strategies have worked with varied level of performance for different scenarios.

	Scenario 1/ Dollar stable	Scenario 2/ Dollar Strong	Scenario 3/ Dollar strong	Scenario 4/ 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

The Scenarios that the case discussed, in that 3 out of 4 scenarios were showing a dollar strengthening trend, and for that case, forwards cover is always a very good strategy for an exporter in U.S. , to get money in foreign currency.  Delta strategy does not perform that well, to increase its performance, the number of updates needs to be increased.

As per Giovanni B (June 1987), forward contract is a linear hedging technique with underlying moving in almost a linear fashion. This technique is used in delta hedging. Delta hedging finds an optimal delta value making portfolio risk free, and the part of the portfolio equivalent to the delta proportion, has to be hedged using forward contract. Delta used here is the same delta used in black Scholes model to make the portfolio risk free. Thus it can be said that Delta hedging is a special case of forward contract hedging technique.

Delta needs to be updated, as the spot rates – the underlying, changes. For new value of spot rate, the portfolio is no longer a risk free and hence a new value of the delta needs to be found. Portfolio needs to be aligned with covered and uncovered parts using new delta values. With increase in the number of the updates the performance of the portfolio improves, and in fully dynamically updated delta hedging, the portfolio delta is changed many times in a day. As discussed by Rajiv S. (2014) a dynamically updated delta hedging produces the best result.

Hedging comes with a cost, as premium is to be paid for hedging. Now for full forward contract and Put option, an upfront price is to be paid for hedging 100% of the portfolio. While in case of Delta hedging every time only part of portfolio is hedged that makes the portfolio risk free. This makes the hedging cost to minimum.

During the course of entire period, the hedgers have full flexibility in updating and reorienting the portfolio and change what part of the portfolio needs to be hedged through forward contract. The loss in this strategy in some of the cases for delta hedging is worth for the level of flexibility the strategy provides.

Subjectivity of delta hedging and its approach

Conclusion:

For exchange rate taking a random walk with no fixed direction of movement, the delta hedging is the best strategy. When the number of updates is increased the performance of the delta hedging improves. Delta hedging makes the portfolio independent of the movement of spot price, thus the volatility in the spot price only matters for deciding the portfolio performance.

Delta hedging uses forward contract as the underlying hedging technique. The delta value controls the portfolio to be kept under forward cover. To compare the delta movements on either side changes in underlying spot exchange rate, let us consider the graph, which plots the put delta values against the underlying price.

For either side changes in exchange rate the curve is a symmetric, the value of delta changing the most in the middle and saturates at either ends. The curve is taken from Dynamic Hedging. (2015) from riskencyclopedia.com. The curve shows that, the delta moves equally for either side changes in exchange rate. Delta hedging uses these values of Delta for reorienting the delta and hence delta hedging is symmetric for either side changes in Exchange rates or portfolio under hedge cover is also symmetric for either side changes in spot exchange rate.

Thus it is clear that delta hedging is not subjective to either side movements in spot exchange rates. On observing the delta movements with change in the spot rates, sometimes it seems that delta moves more in one direction with change in the spot rate, while changes very little or do not change at all with opposite movement in spot exchange rate. The reason for this one side movement is, Delta is a function of spot exchange rate, time and underlying volatility also. So, Delta tends to move in one direction as time to maturity decreases, towards -1 in case of dollar getting stronger and 0 in stable dollar case. The movement by time curtails the movement of delta in opposite direction due to exchange rate moving in opposite direction.

Thus it can be said that delta hedging moves symmetrically on both the sides with either side movements in exchange rates.

A scenario based hedging strategy needs to be followed to achieve the best results for Dayton manufacturing. The 3 strategies for different kind of movements of underlying spot rate is as follows.

For an exporting firm like Dayton manufacturing with large part of receivables in foreign currency, should use a full covered forward contract, whenever it is predicted that dollar is going to get stronger. Banks (2006) suggests that whenever the macroeconomic parameters like interest rate differential between domestic and foreign interest rates etc indicates that domestic currency is going to be stronger; a fully covered forward contract works the best. The 100% hedged portfolio using forward contract clips the loss to the forward rate at the time of the signing the contract. As the spot rate is bound to fall further, this strategy pays the best in such cases.

When exchange rates are stable with small random movements around a pegged value, delta hedging technique provides the best result. The portfolio is hedged in such a way that makes it risk free every time. Frequent updates are done to re align the portfolio making it risk free. With increased number of updates the, the performance of the portfolio improves.

Moreover, Delta hedging makes the portfolio independent of the underlying spot rate, as with delta hedging the portfolio is made risk free. But the volatility in spot rates decides how fast the delta needs to be changed. So Delta hedging becomes a function of volatility of underlying spot rate. As discussed by Antonio C. (2009) higher the value of volatility of spot rate, faster and larger is the movement of the spot rate and more delta updates are required. An option Greek called Vega helps to monitor, how frequent delta changes need to be made.

Whenever the macroeconomic parameters show that dollar is going to be weak with respect to the currency of inflow, than it is most advisable to keep the portfolio fully uncovered. In this weak dollar case, all the exchange rates movements is going to be beneficial for Dayton, with larger dollar income every time.

Thus Dayton manufacturing should use all the strategies.

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