MARIANNA MOROZOVA IEIE SB RAS, Novosibirsk, Russia

Transcription

1 MARIANNA MOROZOVA IEIE SB RAS, Novosibirsk, Russia

2 1 clue of ineffectiveness: BS prices are fair only in case of complete markets FORTS is clearly not complete (as log. returns are not Normal) Market prices are usually close to BS FORTS is not effective?

3 2 clue of ineffectiveness: Limited number of traded instruments and market participants Weak rate of trades Low liquidity FORTS is not effective?

4 Low liquidity of Russian option market as well as the fact of market option prices closeness to Black-Sholes prices indicates that market option prices cannot be considered as fair prices

5 VERIFICATION OF OPTION PRICING EFFECTIVENESS ON FORTS

6 1. Development of the fair option prices estimation methodic at illiquidity markets, which Russian option market refers to 2. Suggest reasonable models for underlying assets prices behavior and calibrate them statistically 3. For a vast variety of present options calculate corresponding fair prices and compare to market ones

7 Analyzed Instruments: future options on index RTS, AO Gazprom and AO SberBank stocks. Underlying assets: daily closing future prices for the period between December 2006 and February Derivatives: daily closing American call option prices for the period between January 2007 and April 2010.

8 Before crisis Crisis recession Market Activity Log returns on RTSI RTS Index (RTSI)

9 Underlying asset price, St Objective probability of future scenarios S t = S 0 exp( X t ) fact ω3 ω1 ω ω0 ω2 ω0 Risk-neutral probability measure The subjective view of an investor t, time Marianna Morozova

10 Objective pro obability of future scenarios ω 0 Option payoffs H( S) = h( S ( ω), t [0, T]) t Risk-neutral pricing rule Π = I ( ) ( ( )) Q H exp r t T E [ H ] t The subjective view of an investor available information at the moment t0 Risk-neutral pricing rule represents the price of option in an arbitrage-free market as its discounted expected payoff under appropriate probability measure (risk-neutral measure Q)

11 Calibrating Problem Assumption: Derivative market is effective, thus option prices are dynamically arbitrage-free Pricing Problem Goal: To evaluate the degree of market option prices deviation from arbitrage-free prices Measure estimation: F*(S) F(S Q) min Q distribution Calibration law of underlying problem of the model option prices to market prices asset price Market estimation: П*(Н) П(Н Q) min Q Calibration problem of underlying the value of market dynamics to risk-neutral an option

12 Calibration Problem Assumption:Derivative market is effective, thus option prices are dynamically arbitrage-free Pricing Problem Goal: To evaluate the degree of market option prices deviation from arbitrage-free prices Measure estimation: F*(S) F(S Q) min Q Measure estimation: П*(Н) П(Н Q) min Q Calibration problem of the model option prices to market prices Calibration problem of underlying market dynamics to risk-neutral

13 Jump- Diffusion Processes Self- Similar Processes NORMAL Mean-Reverse Models

14 Reality Asymmetric distributions with heavy tails and high kurtosis Unexpected price movements due to the arrival of new information - jumps

15 N t X t = γ t Drift + σ W t Diffusion + + t= 1 J t Diffusion Part Jump Part

16 Diffusion process Jump diffusion process Finite activity Pure jump process Infinite activity Infinite variation Market situations Normal shocks Compound Poisson jumps Large and rare events Both small and large jumps Many small jumps Examples of Levy processes Brownian Motion Merton, Kou Gamma Variance (GV), Meixner CGMY, Modified Tempered Stable (MTS), Kim and Rachev(KR) Alfa-stable, Normal Inverse Gaussian (NIG), Hyperbolic Market movements of all magnitudes: from small movements to market crashes

17 H1 H0 H2 P t << Pˆ P P t Pˆ t > Pˆ >> Pˆ < Pˆ P t P t Risk arbitrage Short-term arbitrage opportunities Long-term arbitrage opportunities Market illiquidity Market Effectiveness Market Ineffectiveness

18 Rub. price

19 Rub. price log return (r) S t lns t

20 Эмпирическая плотность Плотность нормального распределения (a=0.0011, σ=0.0208) Плотность MTS распределения (a=0; C=39.4; l+= 25.6; l-= 28.6; g= -3.65) f(x) lnx Empiric density Normal density MLE density lnx x

21 # Distributions RTS Index Future (total-100) GazpromStock Future (total-73) SberbankStock Future(total-73) 1. Normal Merton JD Kou JD Gamma Variance Normal Inverse Gaussian Hyperbolic Meixner Stable CGMY Modified Tempered Stable Kim-Rachev t-student - - -

22 # Distributions RTS Index Future (total-100) GazpromStock Future (total-73) SberbankStock Future(total-73) 1. Normal Merton JD Kou JD 5-3 Exist several possible alternative 4. Gamma Variance distributions, which can give 5. Normal Inverse Gaussian Hyperbolic very similar results in the regions 7. Meixner of interest 8. Stable CGMY Modified Tempered Stable Kim-Rachev t-student - - -

23 Strike= Option on RTS Index Future, Strike= Fair Price Underlying Price Theor. Price BS Market Price Days before expiration

24 Hy ypothesis Options Index RTS GazpromStock SberbankStock Futures,% (100 pcs.) Futures, % (73 pcs.) Futures, % (73 pcs. ) Total number of options, % (246 pcs.) H H H

25 The actual market prices differ more from the arbitrage-free ones with the distributions of logreturns of underlying assets differ more from the normal

26 The jumps in the logreturns of underlying assets dynamics are the reason for the market to underestimate the out-of-themoney options and overestimate the in-the-money ones;

27 nevertheless, the resulting arbitrage opportunities somehow do not force prices to change in the direction of the fair ones, thus the market is systematically ineffective

28 according to the rules of options pricing the prices all along tend to converge with an option s maturity expiring, which can be mistaken for effective market work;

29 still this convergence doesn t seem to be resulted from arbitrage-driven demand and supply changing forces;

30 this convergence is due to the role of diffusion part o logreturns process, as the closer expiration is the lower probability of extreme jumps that can change market trajectory significantly

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