Derivatives and Risk Management Delta Risk and Reward

Question: 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.? Answer: 1. Delta hedge updates, adjustments and forward contract valuations As per the case the data related for the scenario 4 is as follows. Exhibit 1: Inputs 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 Calculation for the value of Delta as a put option 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. 2: Daytons Delta Hedging updates US Dollar /Euro Spring 2005 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. Exhibit 3: Hedge results 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: Exhibit 4: Comparison if different strategies 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 Performance results of different strategies for Dayton manufacturing: 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. Case1: Dollar to pound stable exchange rates 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. Case2: Dollar to pound and euro exchange rates going weak (Dollar strong) 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. 3. Dilemma of Increase in Risk in Delta Hedging for Dayton Manufacturing: 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. Exhibit 6: Rank wise comparison of the 4 strategies: 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 Justification in favor of delta: 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. Delta hedging a special case of forward contract: 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. More updates required for increasing the performance: 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 cost is reduced: 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. Flexibility in management of portfolio: 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. 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 hedge is subjective in its movement for changes in exchange rates in either direction and same is true for hedged covered. 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. Figure 1: Put Delta values Vs 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. 4. Strategic Recommendations to CFO of Dayton manufacturing: 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. Dollar/Euro and Dollar/Pound rates falling (Dollar getting stronger): 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. Bounded Random movement of exchange rate around a pegged value: 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. Dollar /Euro and dollar/Pound rate increasing (Dollar getting weaker): 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. References: Suma John. (2015) Options Greeks: Delta Risk and Reward. [Online] Available from: https://www.investopedia.com/university/option-greeks/greeks2.asp [Accessed: 7th June 2015]. Rajiv S. (2014). Derivatives and risk management. New Delhi: OUP India. Alok D, Surendra Y, P.K. Jain. (2012). 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(2015) Currency options pricing explained. [Online] Available from: https://www.derivativepricing.com/blogpage.asp?id=22 [Accessed: 7th June 2015]. Jorion, Philippe (2009).Financial Risk Manager Handbook(5 ed.). John Wiley and Sons. Giovanni B, Robert E.W. (June 1987).Efficient analytic approximation of American option values.Journal of Finance42(2) Alok D, Surendra Y, P.K. Jain. (2012). Derivative Markets in India: Trading, Pricing, and Risk. India: McGraw Hill Education. Banks, Erik, Siegel, Paul (2006). The options applications handbook: hedging and speculating techniques for professional investors. McGraw-Hill Professional