124x Filetype PDF File size 0.11 MB Source: www.iam.uni-bonn.de
Option Volatility and Pricing Mouna HADDADI (PhD student) Faculty of Science and Technology of Marrakesh, Cadi Ayyad university, Morocco. YOUNGWOMENINPROBABILITY2014 Abstract Part 1. Historical volatility Part 2 : Implied volatility This poster discusses three types of volatility : the historical, implied Thehistorical volatility reflects the past price movements of the under- The implied volatility is often interpreted as estimation of the future and stochastic volatility, and the concept of local volatility. Finally the lying asset, it is calculated as a standard deviation of a stock’s returns volatility. It means that this volatility is a volatility anticipated by the pricing formula of Vanilla options with stochastic volatility model such over a fixed number of days. market maker. In other words it is the value of σ that equalizes the as the model of Heston is presented. The estimate of historical volatility starting from data : price calculated by Black & Scholes model with the observed prices on n+1:thenumberof observations; the market S : the price at the time t; observed Black−Scholes impl t C (S ;T;K) = C (S ;T;K;σ ) u : the return at time t; t t t t Introduction t τ : duration of the time intervals in year. Then Estimation of implied volatility S Wecan find the value of an European call, by using the Risk-Neutral The study of the volatility of certain financial assets became very im- u =ln t t S portant subject in finance. This importance comes from the possibility t−1 valuation method i.e by considering that price of a call corresponds to to measuretheuncertaintyofevolutionoftheyieldonanasset(shareor for t = 1;2;:::;n its expected value of future return discounted : Theestimate s of the standard deviation of u is given by this formula : index) and the fact that the fluctuations of prices can not be neglected. t −rT b −rT b + C0 = e E[max(S −K;0)]=e E (S −K) Nowadays, any investor is conscious of these fluctuations which intro- v T T u n duce an element of risk into its portfolio. That is why the investors wish u 1 X 2 s = t (u −u) b to choose the degree ”of exposure” at the risk compatible with their n−1 t where E is the expectation operator under the probability risk–neutral. level of tolerance to this risk. Thus the study of the volatility plays an t=1 essential role in evaluation and hedging of risk of an investment. where u is the average of u The value of call and put option a time 0 is given by : t The Black and Scholes model assumes the following dynamic of the C = S Φ(d )−KerTΦ(d ) 0 0 1 2 stock price : P = KerTΦ(−d )−S Φ(−d ) dS =µSdt+σSdz 0 2 0 1 The definition of the volatility t t t t where z is a Wiener process where t √ S σ2 dS ln( 0) + (r + )T √ Since t behaves like a normal distribution N(µdt;σ dt), the stan- d = K √ 2 ,d =d −σ T St √ 1 2 1 Thevolatilityisameasureforvariationofpriceofafinancialinstrument dard deviation of the return is equal to σ dt σ T over time. Therefore s is an estimator of σ√τ. Thus we estimate σ by σb where : and Φ is a the normal probability distribution function. The types of volatility : s It is not possible to reverse the preceding equation and to express σ σb = √ according to S , K, r, T and C . However, it is possible to determine –Historical volatility τ 0 0 –Implied volatility the value of this implied volatility by using methods of interpolation –Stochastic volatility like the method of Newton & Raphson. Part 3 : Smile volatility Part 4 : Local volatility Part 5 : Heston model The implied volatility of an option evolves according to the strike and Dupire formula : Heston model (1993) the maturity of the option, now when we draw the implied volatility The Heston stochastic volatility model is based on the following stock according to strike for a given maturity, generally we do not obtain a Fokker-Planck equation : price and variance dynamics horizontal line, which corresponds to the assumption of consistency of (∂f + r ∂ (xf) − 1 ∂2 (x2σ2(x;T)f) = 0 dS(t) = µ(t)S(t)dt+pv(t)S(t)dZ implied volatility. ∂T ∂x 2∂x2 1 f(x;t) = δ(S −x) sur [0,+∞]x[t,+∞] t p where δ is the Dirac function dv(t) = κ(θ −v(t))dt +σ v(t)dZ 2 where hdZ ;dZ i = ρdt , θ : the long-run average of v(t), Theorem : for every (t,s) fixed the function : 1 2 κ : controls the speed by which v(t) returns to its long-run mean and σ : the volatility of volatility. Thefundamental partial differential equation (PDE) verified by option −r(T−t) Q + C(T;K)=e E [(S −K) =S =s] price is : T t is a solution of Dupire equation : ∂C 1 ∂2C ∂2C 1 ∂2C ∂C ∂C ∂C 1 2 2∂2C + CS2 +ρσvS + σ2v +rS −rC ∂T −r(C −K∂K)−2σ (T;K)K ∂K2 =0 ∂t 2 ∂S2 ∂S∂v 2 ∂v2 ∂S In particular : s ∂C ∂C+r(C−K∂C) −∂v[κ(θ−v)−λv]=0 σ(T;K) = 2∂T ∂K K2∂2C We seek to solve the preceding PDE in the case of a European call ∂K2 option of strike K and maturity T, by analogy with the Black & Scholes formula, the solution of this option is of the form : C(S;v;t) = SP −Ke−r(T−t)P 1 2 Part 4 : The link between local volatility and implied volatility By injecting it in the PDF, and From the Fourier inversion theorem, we have : Z " # P =1+1 +∞Re e−iφln(K)fj dφ Implied volatility smile j 2 π iφ Let us consider a model with non-deterministic volatility. That is the 0 risk-neutral dynamics of S is written as : For j = 1;2 , P are probabilities , and f are characteristics functions j j The Smile of volatility is a phenomenon observed on the markets of such as dS options vanillas which contradicts the assumption of Black and Scholes t = µdt +σ dW S t t f (x;v;T;φ) = exp(C(τ;φ)+D(τ;φ)v +iφx) according to which the volatility of an option is constant and is not t j influenced by the value of other parameters. From a statistical point Instantaneous volatility σ is a process such as : h dri and C(τ;φ) = rφiτ + a (b −ρσφi+d)τ −2ln 1−ge of view such a form of curve of volatility according to the strike price Z σ2 j 1−g corresponds to a value of Kurtosis higher than 3, therefore risk-neutral T σ2ds < ∞;∀ T > 0 " # dynamics of Black & Scholes and Merton are not compatible with the s bj −ρσφi+d 1−edr 0 D(τ;φ) = phenomenon of smile which exists on all markets options. σ2 1−gedr Thus we can define the local variance as conditional expectation of the future instantaneous variance bj −ρσφi+d q h i g = and d = ρσφi−b 2−σ2 2u φi−φ2 2 Q 2 b −ρσφi−d j j σ (T;K) = E σ S =K j L T T u = 1 , u = −1 , a = κθ , b = κ+λ−ρσ , b = κ+λ This definition and that based on Dupire’s formula are equivalent. 1 2 2 2 1 2
no reviews yet
Please Login to review.