How Traders ManageTheir RisksChapter 7RiskManagementandFinanciallnstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
How Traders Manage Their Risks Chapter 7 Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 1
A Trader's Gold Portfolio. How ShouldRisks Be Hedged? (Table 7.1, page 138)PositionValue ($)Spot Gold180.000-60,000ForwardContracts2,000FuturesContracts80,000SwapsOptions-110,000Exotics25,000Total117,0002RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
A Trader’s Gold Portfolio. How Should Risks Be Hedged? (Table 7.1, page 138) Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 2 Position Value ($) Spot Gold 180,000 Forward Contracts – 60,000 Futures Contracts 2,000 Swaps 80,000 Options –110,000 Exotics 25,000 Total 117,000
DeltaDelta of a portfolio is the partial derivative of a portfoliowith respect to the price of the underlying asset (gold inthis case)Suppose that a so.1 increase in the price of gold leads tothe gold portfolio decreasing in value by $100The delta of the portfolio is -1000The portfolio could be hedged against short-termchanges in the price of gold by buying 1000 ounces ofgold. This is known as making the portfolio delta neutral3RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
Delta ⚫ Delta of a portfolio is the partial derivative of a portfolio with respect to the price of the underlying asset (gold in this case) ⚫ Suppose that a $0.1 increase in the price of gold leads to the gold portfolio decreasing in value by $100 ⚫ The delta of the portfolio is −1000 ⚫ The portfolio could be hedged against short-term changes in the price of gold by buying 1000 ounces of gold. This is known as making the portfolio delta neutral Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 3
Linear ys Nonlinear Products When the price of a product is linearlydependent on the price of an underlyingasset a ""hedge and forget"' strategy canbe usedNon-linear products require the hedge tobe rebalanced to preserve delta neutralityRiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull20124
Linear vs Nonlinear Products ⚫ When the price of a product is linearly dependent on the price of an underlying asset a ``hedge and forget’’ strategy can be used ⚫ Non-linear products require the hedge to be rebalanced to preserve delta neutrality Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 4
Example of Hedging a NonlinearProduct (page140)A bank has sold for $300.000 aEuropean call option on 100,000 sharesof a nondividend paying stock S。 = 49, K = 50, r = 5%, = 20%,T = 20 weeks, μ= 13%The Black-Scholes-Merton value of theoption is $240,000 How does the bank hedge its risk to lockin a $60,000 profit?RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull20125
Example of Hedging a Nonlinear Product (page 140) ⚫ A bank has sold for $300,000 a European call option on 100,000 shares of a nondividend paying stock ⚫ S0 = 49, K = 50, r = 5%, s = 20%, T = 20 weeks, m = 13% ⚫ The Black-Scholes-Merton value of the option is $240,000 ⚫ How does the bank hedge its risk to lock in a $60,000 profit? Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 5
Deltaof the OptionOptionpriceSlope = △BAStockpriceRiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull20126
Delta of the Option Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 6 Option price A B Slope = D Stock price
Delta HedgingInitially the delta of the option is 0.522The delta of the position is -52,200This means that 52.200 shares mustpurchased to create a delta neutral positionBut. if a week later delta falls to 0.458, 6.400shares must be sold to maintain deltaneutrality Tables 7.2 and 7.3 (pages 142 and 143)provide examples of how delta hedgingmight work for the option.7RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
Delta Hedging ⚫ Initially the delta of the option is 0.522 ⚫ The delta of the position is -52,200 ⚫ This means that 52,200 shares must purchased to create a delta neutral position ⚫ But, if a week later delta falls to 0.458, 6,400 shares must be sold to maintain delta neutrality ⚫ Tables 7.2 and 7.3 (pages 142 and 143) provide examples of how delta hedging might work for the option. Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 7
Table 7.2: Option closes in themoneyWeekDeltaSharesStockPricePurchased049.000.52252,200148.120.458(6,400)20.40047.37(5,800)350.250.59619,600191.00055.871,0000201.00057.258RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
Table 7.2: Option closes in the money Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 8 Week Stock Price Delta Shares Purchased 0 49.00 0.522 52,200 1 48.12 0.458 (6,400) 2 47.37 0.400 (5,800) 3 50.25 0.596 19,600 . . . . 19 55.87 1.000 1,000 20 57.25 1.000 0
Table 7.3: Option closes out ofthemoneyWeekDeltaSharesStock PricePurchased049.000.52252,200149.750.5684,600252.000.70513,700350.000.579(12,600)1946.630.007(17,600)200.00048.12(700)9RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
Table 7.3: Option closes out of the money Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 9 Week Stock Price Delta Shares Purchased 0 49.00 0.522 52,200 1 49.75 0.568 4,600 2 52.00 0.705 13,700 3 50.00 0.579 (12,600) . . . . 19 46.63 0.007 (17,600) 20 48.12 0.000 (700)
Where the Costs Come From. Delta hedging a short option position tendsto involve selling after a price decline andbuying after a price increaseThis is a “sell low, buy high" strategy.The total costs incurred are close to thetheoretical price of the option10RiskManagementandFinancialInstitutions,3e,Chapter7,CopyrightJohnC.Hull2012
Where the Costs Come From ⚫ Delta hedging a short option position tends to involve selling after a price decline and buying after a price increase ⚫ This is a “sell low, buy high” strategy. ⚫ The total costs incurred are close to the theoretical price of the option Risk Management and Financial Institutions, 3e, Chapter 7, Copyright © John C. Hull 2012 10