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# Derivatives, Week 8 Notes BU.232.710.W4.SP16

JHU

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This 3 page Class Notes was uploaded by Kwan on Saturday March 5, 2016. The Class Notes belongs to BU.232.710.W4.SP16 at Johns Hopkins University taught by Federico Bandi in Spring 2016. Since its upload, it has received 23 views. For similar materials see Derivatives in Finance at Johns Hopkins University.

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Date Created: 03/05/16

•" ! The!Greeks! •" (1)!(D)!Delta! •" Delta!(D)!is!the!rate!of!change!of!the!option!price!with!respect!to!the!price!of! the!underlying!(!!!!!!!!!)!! !! •" D!Hedging!(Short!call)! •" Example! !!!!Suppose!the!stock!price!is!$100!and!the!option!price!is!$10.!! !!!!!!An!investor!has!sold!(written)!20!call!option!contracts!(every!contract!is!for!100! shares!–!hence,!the!contracts!are!for!2,000!shares).!The!investor!hedges!the!option! position!by!buying!Δ×2,000!shares.!If!the!Δ!is!0.6,!the!investor!hedges!the!position!by! buying!1,200!shares!of!the!underlying.! !!!!!!If! the! stock! price! goes! up! by! $1! (producing! a! gain! of! $1,200! on! the! shares! purchased),!the!option!price!will!tend!to!go!up!by!0.6×$1=$0.6!(producing!a!loss!of! $0.6!×2,000!=!$1,200!on!the!options!written).! !!!!!!The!opposite,!if!the!stock!price!goes!down.! •" More!on!D!hedging!(Long!call)!! •" More!on!D!hedging!(Short!and!Long!Put)!! •" More!on!D!hedging! •" Once!an!option!position!is!ΔRhedged,!it!is!typically!called!ΔRneutral.! •" The!Δ!of!an!option!position!changes!every!time!the!price!changes.!Hence,!to! preserve!ΔRneutrality,!the!hedge!has!to!continuously!change!too.! •" The!Δ!of!a!portfolio!of!options!–!Consider!a!portfolio!consisting!of!a!quantity!w! i of!option!i.!The!Δ!of!the!portfolio!is!a!linear!function!of!the!individual!Δs!given! by:! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! Example!–!Suppose!a!financial!institutions!has!three!positions:! 1." A!long!position!in!100,000!call!options!with!strike!$55!and!expiration!3!months.! The!Δ!of!each!option!is!0.533.! 2." A!short!position!in!200,000!call!options!with!strike!price!$56!and!expiration!5! months.!The!Δ!of!each!option!is!0.468.! 3." A!short!position!in!50,000!put!options!with!strike!price!$56!and!expiration!2! months.!The!Δ!of!each!option!is!R0.508.! The!delta!of!the!portfolio!is! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!100,000×0.533!R!200,000×0.468!–!50,000×(R0.508)!=!R14,900!! The!portfolio!can!be!made!ΔRneutral!by!buying!14,900!shares.! •" D!and!BlackRScholes! (NonRdividend!paying!stocks)! •" The!delta!of!a!European!call!is!! ******************************! •" The!delta!of!a!European!put!is!! ! ! ! ! !!!!!!!Example:! Consider! a! European! call! option! with! strike! price! $50! and! time! to! maturity!20!weeks!(=0.3846!years).!Assume!the!stock!price!is!$49,!the!riskRfree!rate!is! 0.05,!and!volatility!is!20%.!In!this!case:! !!!!!!!!!!!!!!!!!!!!!!! •" (2)!(Θ)!Theta!! •" The!theta!(Q)!of!an!option!(or!portfolio!of!options)!is!the!rate!of!change!of!the! option!value!with!respect!to!the!passage!of!time! !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !!!!!!!!!! •" The!theta!of!a!European!call!or!a!European!put!!is!usually!negative.!This!means! that,!if!time!passes!with!the!price!of!the!underlying!asset!and!its!volatility! remaining!the!same,!the!value!of!the!options!declines.! •" !Θ!and!BlackRScholes! (NonRdividend!paying!stocks)! •" The!theta!of!a!European!call!is!! ******************************! •" The!theta!of!a!European!put!is! •" Time!is!measured!in!years.!To!measure!time!in!calendar!(trading)!days!divide! by!365!(252).!! ! ! ! ! !!!!!!!!Example:! Consider! a! European! call! option! with! strike! price! $50! and! time! to! maturity!20!weeks!(=0.3846!years).!Assume!the!stock!price!is!$49,!the!riskRfree!rate!is! 0.05,!and!volatility!is!20%.!In!this!case,!the!option’s!theta!is!R4.31.!The!theta!is!R 4.31/365!=!R!0.0118!per!calendar!day!or!R4.31/252!=!R!0.0171!per!trading!day.! •" (3)!(Γ)!Gamma! •" Gamma!(G)!is!the!rate!of!change!of!delta!(D)!with!respect!to!the!price!of!the! underlying!asset:! •" Equivalently,!Gamma!is!the!second!derivative!of!the!option!price!with!respect! to!the!price!of!the!underlying:!! •" Gamma!Addresses!Delta!Hedging!Errors!Caused!By!Curvature!! •" Γ!hedging! •" Example! (ΓRhedging!at!work)! •" !Γ!and!BlackRScholes! (NonRdividend!paying!stocks)! •" The!gamma!of!a!European!call!or!put!is!! ******************************! ! ! ! ! !!!!!!!Example:! Consider! a! European! call! option! with! strike! price! $50! and! time! to! maturity!20!weeks!(=0.3846!years).!Assume!the!stock!price!is!$49,!the!riskRfree!rate!is! 0.05,!and!volatility!is!20%.!In!this!case,!the!option’s!!gamma!is:! •" (4)!(ν)!Vega! •" Vega!(n)!is!the!rate!of!change!of!the!value!of!the!option!(or!of!the!option’s! portfolio)!with!respect!to!volatility:! •" Since!options!become!more!valuable!when!volatility!increases,!other!things! equal,!the!vega!is!positive!for!all!options.! !!!!!!!!!!!!!!!!!!!!!!!!!!!!! •" ν!hedging! •" Example! (νRhedging!at!work)! •" !ν!and!BlackRScholes! (NonRdividend!paying!stocks)! •" The!vega!of!a!European!call!or!put!is!! ******************************! ! ! ! ! !!!!!!Example:!Consider!a!European!call!option!with!strike!price!$50!and!time!to!maturity! 20!weeks!(=0.3846!years).!Assume!the!stock!price!is!$49,!the!riskRfree!rate!is!0.05,!and! volatility!is!20%.!In!this!case,!the!option’s!vega!is:! !!!!! •" (5)!(ρ)!Rho! •" Rho!is!the!rate!of!change!of!the!value!of!the!option!(or!of!a!portfolio!of!options)! with!respect!to!the!interest!rate:! ! ! ! ! ! ! ! ! ! •" For!currency!options!there!are!2!rhos.! •" !ρ!and!BlackRScholes! (NonRdividend!paying!stocks)! •" The!rho!of!European!calls!or!puts!is!! ******************************! ! ! ! ! !!!!!!!Example:! Consider! a! European! call! option! with! strike! price! $50! and! time! to! maturity!20!weeks!(=0.3846!years).!Assume!the!stock!price!is!$49,!the!riskRfree!rate!is! 0.05,!and!volatility!is!20%.!In!this!case,!the!option’s!rho!is:! •" Hedging!in!Practice! •" Traders!usually!ensure!that!their!options!portfolios!are!deltaRneutral!at!least! once!a!day!(as!discussed,!this!can!be!easily!done!by!investing!in!the!underlying).! •" Whenever!the!opportunity!arises,!they!!improve!gamma!and!vega!(by!investing! in!other!“nonlinear”!derivatives).! !

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