清华大学金融学LessonStochasticVolatility.docVIP

1. Implied volatility volatility smile The volatility implied by option price observed in the mkt. -In B-S , is assumed to be constant : -In reality , -Some empirical facts: (1) Options on eqity indices : one-sided “smile” or “skew” (2) Options on FX: more or less symmetric “smile” lumpier volatility (3) Options on interest rates: more monotonous one-sided skew (Caps/ Floors /Swaptions) 2. Arbitrage Pricing and Hedging with Smile -The goal: To build a spot process that: a) is compatible with the observed Smiles at all maturity. b) keeps the model complete Given European calls of all strikes K and maturity T: C(K,T) to construct a risk-neutral process for the spot price : where (S, t) is a deterministic function of spot prices and time t. -The problem: Assume r=0 , fT(S) is the risk-neutral density function of the spot price at time T. Fokker- Planck equation : Integrate twice in S for const t: Assume: 3. Arbitrage pricing with stochastic volatility Assume: a continuum of all CK,T are traded. 3.1 Log contract For simplicity assume r=0 Def: A Log contract is a contingent claim on S, with payoff at time T equals log(ST). (Log contract can be synthesized ) Let its price be denoted LT(t). 3.2 Forward variance by Ito’s lemma: 3.3 Arbitrage free price of Fwd variance: 2(LT1(t)-LT2(t)) Def: Fwd time Time T, instantaneous variance (observed at t): 3.4 Stochastic assumptions on fwd variance and risk-neutral process 3.5 risk-neutral processes for instantaneous variance and volatility Define instantaneous variance at time t 3.6 Contingent claim pricing Risk-neutral process for joint Let A be a contingent daim that delivers in T a payoff dependent on the paths followed by spot S and variance, i.e., a square-integrable -measurable random variable on . Payoff Out of money puts out of money calls k3 k2 k1 St=k0 k4 k5 underlying s