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 Call Put Symmetry
Issued on September 09 2010 par Strategies-options.com
Next steep after call-put parity for european options first.
We had seen Call-put Parity: A Typically European Relation that it was possible to express a call with a put and a put with a call.
There are other relationships between options which are useful. I - How to find a risk reversal that is worth 0 ? Simply using Bates (1991) options relation : If volatility smile is symmetric C ( S, X , T , r , q , σ) = ( X / ( S . exp ( ( r - q ) . T ) ) ) . P ( S , ( S . exp ( ( r - q ) . T ) ) ² / X , T , r , q , σ ) Where, S is the spot X is the strike T is the maturity r continuously compounded interest rate q continuously compounded dividend rate σ annualized volatility. That means : a call struck at X is worth the same value as a quantity of ( X / ( S . exp ( ( r - q ) . T ) ) ) Puts struck at ( S . exp ( ( r - q ) . T ) ) ² / X II - Example How many puts with which strike are to be sold in order to set a risk reversal that is worth 0 using a 1 year call priced at $14.23 using 30% for volatility, 5% continuously compounded interest rate, no dividends. It's just : 100 / (100 . exp ( 0.05 - 0 ) . 1 ) = 100 / 105.17 = 0.95 puts struck at : ( 100 . exp ( 0.05 - 0 ) . 1 ) ² / 100 = 11051.71 / 100 = 110.51 Selling 0.95 puts struck at 110.51 are enough. Check it out : For a spot at $100, a 1 year put struck at 110.51 using an annualized volatility at 30%, continuously compounded interest rate at 5% is worth $ 14.957 If one sells 0.95 of that put, it leads to a credit of 14.23 which is equal to the corresponding call struck at 100. Nota Bene Given the call put parity, if buying a call struck at 100 is worth the same value as for 0.95 puts struck at 110.51, then buying 0.95 calls struck at 110.51 is worth the same as for buying a put struch at 100. Check it out : A 110.51 call is worth 9.8363, then 0.95 calls struck at 110.51 are worth 0.95*9.8363=9.354 A 100 put is worth 9.354 Next : Call Put Symmetry #2 Previous : Call-Put Parity : American Style Issue Relationships Between Option Sensitivities - INDEXRelationships Between Option Sensitivities - CHAPTER I Relationships Between Option Sensitivities - CHAPTER II Relationships Between Option Sensitivities - CHAPTER III Strategies-options.com |
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