![The Moneyness properties
The moneyness function m(t,T,r,K,St) properties
are:
are:
1. m ∈ C 2 [ 0 ,T ] × ( t ,T ] × » + × » + + × » + +
2. lim m < ∞
t →T
∂ 2m
3. lim <∞
t →T ∂t 2](https://image.slidesharecdn.com/theconvenienceyieldimpliedbyquadraticvolatilitysmiles-presentationcompatibilitymode-100723045714-phpapp02/85/The-convenience-yield-implied-by-quadratic-volatility-smiles-presentation-compatibility-mode-16-320.jpg)
The convenience yield implied by quadratic volatility smiles presentation [compatibility mode]
- 1. The convenience yield implied by quadratic volatility smiles (2001, 2002) 2001, 2002) By Prof. Haim Reisman Presented by Yigal Ben Tal & David Feldman
- 2. Introduction The Implied Convenience Yield (ICY) The Moneyness, what is it? Moneyness, The Implied Volatility and its Smile ☺ Black & Scholes Model representation
- 3. Basic definitions - ICY The Implied Convenience Yield of an illiquid option is the rate of locally risk-less profit obtained from risk- hedging this option using other liquid at-the-money at-the- (ATM) options as hedging instruments. instruments. (Prof. Haim Reisman, 06/2002)
- 4. Basic definitions - Moneyness Let the function m(t,T,r,K,St) be a function of time, maturity date, interest rate, strike price and underlying price. Then the moneyness X=Xt at time t (0 ≤ t<T) is price. X=X t<T) generally defined as X= m(t,T,r,K,St). m( The function m(·) is referred to as the moneyness function. function. It is required that the moneyness to be increasing in K. (Reinhold Hafner, 2004)
- 5. Basic definitions - IV The Implied Volatility is the value of the expected volatility imputed from an option pricing model (such as B&S), given the option price, the asset’s price, the exercise price, the time to maturity, and the risk- risk- free interest rate. rate. (OECD Economic Outlook Glossary)
- 6. Basic definitions - IV (the math definition) If C = f(σ,·) is a theoretical value of an option, and option, f(·) is a pricing model that depends on volatility σ f(·) plus other inputs, and f(·) f(·) is monotonically increasing in σ, than if exists some inverse function f -1(·), ·), such that σC* = f -1(C*,·), (C*,·), where C* is the market price of an option, than the value σC* is the implied volatility by the market price C*. C*. (Reinhold Hafner, 2004)
- 7. Basic definitions - Smile For any fixed maturity date t (t ≤ T), the function σ(K,·) of implied volatility against strike price K K,·) (K>0) is called the volatility smile or just smile (for K>0 maturity T) at date t (0 ≤ t<T). t<T) IV X = Ke − r ∆t S t ATM Call ITM Call OTM Put OTM Put ITM 1 X
- 8. The economic assumptions • The ICY of the liquid options is zero and one of the non-liquid options isn’t zero. non- zero. • The stock index and European options are traded continuously. continuously. • The interest rate is constant over the time. time. • The ATM options are traded at no transaction costs, but those that are away from the money are traded with it. it. • The paper analyzes a set of fix expiration options. options.
- 9. The mathematic assumptions The Volatility Smile is quadratic with coefficients as Ito's processes. processes. This is the IV of ATM option. option. This is the slope of the volatility smile. smile. This is curvature measure of the smile. smile. This is moneyness. moneyness.
- 10. Black & Scholes Model changes The standard formula for European Call option is: C ( t ,T , S t , K ) = S t N ( d 1 ) − Ke − r ∆t N ( d 2 ) ln ( S t K ) + ( r + σ 2 2 ) ∆ t d1 = , d 2 = d1 − σ ∆t σ ∆t ∆t The changed B&S formula, that used in the paper is: C ( t ,T , St , K ) = St N ( d1 ) − XN ( d2 ) , X = Ke−r∆t St − ln X V ( t ,T , X ) ∆t d1 = + , d2 = d1 −V ( t ,T , X ) ∆t V ( t ,T , X ) ∆t 2
- 11. The target of the paper The ICY for non-liquid options may be explained as non- a stream of the cash (received/paid) for the discomfort of the option holding. holding. The non-liquid options may have non-zero ICY. non- non- ICY. The target of the article is a creation of an exact formula for the ICY and for its hedging coefficients. coefficients.
- 12. The getting formula dC − rC dt = ∆ * ⋅ ( dS t − rS t dt ) + A B 2 + ∑ V ega * ⋅ ( d z k − α k d t ) + k k =0 C + ε ( t ,T , K ) dt D Where µ is very complicated expression of the B&S model’s partial derivatives. ∆* = ∆ − X ⋅ (Vega S ) z1 + 2 ( X − 1) z2 , Vega* = ( X − 1) ⋅ Vega k k ∂ 1 ∂2 α0 ( X = 1) = µ Vega , α1 ( X = 1) = ( µ Vega ) , α2 ( X = 1) = 2 ( µ Vega ) ∂X 2 ∂X ε = µ − ∑αk ⋅ Vega* , ε ( t,T ,K ) = o ( ( X − 1) ) 2 3 k k =0
- 13. The advantage remarks • The received formulas are simple computation and depend just on currently observable parameters. parameters. • There is no need for any historical data or some arbitrary assumption on the behavior of processes in the future. future.
- 14. There are some question points • The model has many different initial parameters (zk, cov(dw,dwk), ets.). cov( ets. • There are many undefined expressions used by the author (cov(dw,dwk), coefficients of zk). cov( • Various economic and mathematical assumptions, that are not clear (ICY, the formula of the hedging portfolio options).
- 15. The end. Thank you for your attention!!!
- 16. The Moneyness properties The moneyness function m(t,T,r,K,St) properties are: are: 1. m ∈ C 2 [ 0 ,T ] × ( t ,T ] × » + × » + + × » + + 2. lim m < ∞ t →T ∂ 2m 3. lim <∞ t →T ∂t 2