Local Volatility Pricing Models for Long-Dated Derivatives in Finance and Insurance

Transcription

1 Local Volatility Pricing Models for Long-Dated Derivatives in Finance and Insurance G. Deelstra Université Libre de Bruxelles Joint work with G. Rayee (ULB) Berlin, 29th of April 10th Scientific Day (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

2 Outline of the talk Outline of the talk 1 Introduction 2 The model for long-dated FX derivatives The forward PDE for the probability density The local volatility function Calibration 3 The model for GAO s Definition and introduction The model Pricing methods Calibration Numerical results Future research Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

3 Introduction Introduction See e.g. Recent years, the long-dated (maturity > 1 year) foreign exchange (FX) option s market has grown considerably Vanilla options (European Call and Put) Exotic options (barriers,...) Hybrid options (PRDC swaps) Foreign Exchange Risk, Uwe Wystup and Jürgen Hakala, 2002, Risk Publications, London. FX Options and Structured Products, Uwe Wystup, 2006, Wiley, England. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

4 Introduction Introduction A suitable pricing model for long-dated FX options has to take into account the risks linked to: domestic and foreign interest rates by using stochastic processes for both domestic and foreign interest rates dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ(t, S(t))]dt + σ f (t)dwf DRN (t) the volatility of the spot FX rate (Smile/Skew effect) by using a local volatility σ(t, S(t)) for the FX spot ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dWS DRN (t), by using a stochastic volatility ν(t) for the FX spot and/or jump ds(t) = (r d (t) r f (t))s(t)dt + ν(t)s(t)dws DRN (t), dν(t) = κ(θ ν(t))dt + ξ ν(t)dwν DRN (t) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

5 Introduction Introduction Stochastic volatility models with stochastic interest rates: A. Antonov, M. Arneguy, and N. Audet, Markovian projection to a displaced volatility Heston model, Available at (2008). L.A. Grzelak and C.W. Oosterlee, On cross-currency models with stochastic volatility and the interest rate smile, Available at (2010). A. van Haastrecht, R. Lord, A. Pelsser, and D. Schrager, Pricing long-maturity equity and fx derivatives with stochastic interest rates and stochastic volatility, IME, 45 (3), , A. van Haastrecht and A. Pelsser, Generic Pricing of FX, Inflation and Stock Options Under Stochastic Interest Rates and Stochastic Volatility, Forthcoming in Quant. Fin., Local volatility models with stochastic interest rates: V. Piterbarg, Smiling hybrids, Risk, 66-71, (May 2006). Duy Minh Dang et al., A PDE Pricing Framework for Cross-Currency Interest Rate Derivatives, working paper, (May 2010). Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

6 Introduction Introduction Local volatility models for equity: Derman & Kani (1994), Dupire (1994),... the local volatility σ(t, S(t)) is a deterministic function of both the spot and time. It avoids the problem of working in incomplete markets in comparison with stochastic volatility models and is therefore more appropriate for hedging strategies has the advantage to be calibrated on the complete implied volatility surface. local volatility models usually capture more precisely the surface of implied volatilities than stochastic volatility models Also some disadvantages, depending on the situation. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

7 Introduction Introduction The calibration of a model is usually done on the vanilla option market local and stochastic volatility models (well calibrated) return the same price for these options. But calibrating a model to the vanilla market is by no means a guarantee that all types of options will be priced correctly example: We have compared short-dated barrier option market prices with the corresponding prices derived from either a Dupire local volatility or a Heston stochastic volatility model both calibrated on the vanilla smile/skew. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

8 Introduction Introduction A FX market characterized by a mild skew (USDCHF) exhibits mainly a stochastic volatility behavior, A FX market characterized by a dominantly skewed implied volatility (USDJPY) exhibits a stronger local volatility component. See Bossens et al. (2010), IJTAF. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

9 Introduction Introduction The market dynamics could be better approximated by a hybrid volatility model that contains both stochastic volatility dynamics and local volatility ones. Lipton (2002), Lipton and McGhee (2002), Madan et al. (2007) example: ds(t) = (r d (t) r f (t))s(t)dt + σ LOC2 (t, S(t)) ν(t)s(t)dws DRN (t), dr d (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dwd DRN (t), dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ(t, S(t))]dt + σ f (t)dwf DRN (t), dν(t) = κ(θ ν(t))dt + ξ ν(t)dwν DRN (t). The local volatility function σ LOC2 (t, S(t)) can be calibrated from the local volatility that we have in a pure local volatility model! Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

10 The model for FX Derivatives The model for FX Derivatives (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

11 The model for FX Derivatives The three-factor model with local volatility The spot FX rate S is governed by the following dynamics ds(t) = (r d (t) r f (t))s(t)dt + σ(t, S(t))S(t)dW DRN S (t), (1) domestic and foreign interest rates r d and r f follow a Hull-White one factor Gaussian model defined by the Ornstein-Uhlenbeck processes { drd (t) = [θ d (t) α d (t)r d (t)]dt + σ d (t)dw DRN d (t), (2) dr f (t) = [θ f (t) α f (t)r f (t) ρ fs σ f (t)σ(t, S(t))]dt + σ f (t)dw DRN f (t), (3) θ d (t), α d (t), σ d (t), θ f (t), α f (t), σ f (t) are deterministic functions of time. Equations (1), (2) and (3) are expressed in the domestic risk-neutral measure (DRN). (WS DRN (t), Wd DRN (t), Wf DRN (t)) is a Brownian motion under the domestic risk-neutral measure Q d with the correlation matrix 1 ρ Sd ρ Sf ρ Sd 1 ρ df. ρ Sf ρ df 1 Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

12 Forward PDE Forward PDE for the probability density References: Dupire (1994), Derman and Kani (1994),... Overhaus et al. (2006) Equity Hybrid Derivatives (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

13 Forward PDE Forward PDE for the probability density Consider a derivative that pays off V (S(t), r d (t), r f (t), t) at time t. V (S(0), r d (0), r f (0), t = 0) = E Q d [e t 0 r d (s)ds V (S(t), r d (t), r f (t), t)] = P d (0, t)e Qt [V (S(t), r d (t), r f (t), t)] = P d (0, t) V (x, y, z, t)φ F (x, y, z, t)dxdydz (4) where Q t is the t-forward neutral probability and φ F (x, y, z, t) corresponds to the t-forward probability density. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

14 Forward PDE Forward PDE for the probability density Following the same approach as in Equity Hybrid derivatives (Overhaus et al., 2006), one derives the following Fokker-Plank equation for φ F (x, y, z, t): 0 = φ F t + (r d (t) f d (0, t)) φ F + [(r d (t) r f (t))s(t)φ F ] x + [(θ f (t) α f (t) r f (t) ρ fs σ f (t)σ(t, S(t)))φ F ] z 1 2 [σd 2(t)φ F ] 2 y [σ(t, S(t))S(t)σ f (t)ρ Sf φ F ] x z + [(θ d (t) α d (t) r d (t))φ F ] y 1 2 [σ 2 (t, S(t))S 2 (t)φ F ] 2 x 2 2 [σf 2(t)φ F ] z 2 2 [σ(t, S(t))S(t)σ d (t)ρ Sd φ F ] x y 2 [σ d (t)σ f (t)ρ df φ F ]. (5) y z This PDE is solved forward in time with the initial condition at time t = 0 given by φ F (x, y, z, t) = δ(x x 0, y y 0, z z 0 ), where δ is the Dirac delta function and x 0, y 0 and z 0 correspond to the values at time t = 0 of the spot FX rate, the domestic and foreign interest rates respectively. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

15 The local volatility function standard approach The local volatility derivation : standard approach References: Dupire (1994), Derman and Kani (1994),... Overhaus et al. (2006) Equity Hybrid Derivatives Other approach: Atlan (2006) by using Tanaka s formula Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

16 The local volatility function standard approach The local volatility derivation : standard approach Consider the forward call price C(K, t) of strike K and maturity t, defined (under the t-forward measure Q t) by C(K, t) = C(K, t) P d (0, t) = EQt [(S(t) K) + ] = Differentiating it with respect to the maturity t leads to + (S(t) K)φ F (S, r d, r f, t)dsdr d dr f. K C(K, t) t = + K (S(t) K) φ F (S, r d, r f, t) t dsdr d dr f we have shown that the t-forward probability density φ F satisfies the following forward PDE: φ F t = (r d (t) f d (0, t)) φ F [(r d (t) r f (t))s(t)φ F ] x [(θ f (t) α f (t) r f (t))φ F ] z + 2 [σ(t, S(t))S(t)σ d (t)ρ Sd φ F ] x y [σ 2 (t, S(t))S 2 (t)φ F ] x [σ(t, S(t))S(t)σ f (t)ρ Sf φ F ] x z [(θ d (t) α d (t) r d (t))φ F ] y 2 [σ 2 d (t)φ F ] y [σ 2 f (t)φ F ] z [σ d (t)σ f (t)ρ df φ F ]. (6) y z Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

17 The local volatility function standard approach The local volatility derivation : standard Integrating by parts several times we get C(K, t) t + = f d (0, t) C(K, t) + [r d (t)k r f (t)s(t)]φ F (S, r d, r f, t)dsdr d dr f K (σ(t, K)K)2 φ F (K, r d, r f, t)dr d dr f = f d (0, t) C(K, t) + E Qt [(r d (t)k r f (t)s(t))1 {S(t)>K} ] (σ(t, K)K)2 2 C(K, t) K 2. This leads to the following expression for the local volatility surface in terms of the forward call prices C(K, t) σ 2 (t, K) = C(K,t) f t d (0, t) C(K, t) E Qt [(r d (t)k r f (t) S(t))1 {S(t)>K} ]. 1 K 2 2 C(K,t) 2 K 2 Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

18 The local volatility function standard approach The local volatility derivation : standard approach The (partial) derivatives of the forward call price with respect to the maturity can be rewritten as C(K, t) t = [ C(K,t) P d (0,t) ] t = C(K, t) t 2 C(t, K) K 2 = 2 C(K,t) [ P d (0,t) ] 1 2 C(t, K) K 2 = P d (0, t) K 2. 1 P d (0, t) + f d (0, t) C(t, K), Substituting these expressions into the last equation, we obtain the expression of the local volatility σ 2 (t, K) in terms of call prices C(K, t) σ 2 (t, K) = C(K,t) P t d (0, t)e Qt [(r d (t)k r f (t) S(t))1 {S(t)>K} ]. 1 2 K 2 2 C(K,t) K 2 Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

19 Calibration Calibration (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

20 Calibration Calibration Before pricing any derivatives with a model, it is usual to calibrate it on the vanilla market, determine all parameters present in the different stochastic processes which define the model in such a way that all European option prices derived in the model are as consistent as possible with the corresponding market ones. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

21 Calibration Calibration The calibration procedure for the three-factor model with local volatility can be decomposed in three steps: 1 Parameters present in the Hull-White one-factor dynamics for the domestic and foreign interest rates, θ d (t), α d (t), σ d (t), θ f (t), α f (t), σ f (t), are chosen to match European swaption / cap-floors values in their respective currencies. 2 The three correlation coefficients of the model, ρ Sd, ρ Sf and ρ df are usually estimated from historical data. 3 After these two steps, the calibration problem consists in finding the local volatility function of the spot FX rate which is consistent with an implied volatility surface. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

22 Calibration Calibration σ 2 (t, K) = C(K,t) P t d (0, t)e Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ] 1 2 K. 2 2 C K 2 Difficult because of E Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ] : there exists no closed form solution it is not directly related to European call prices or other liquid products. Its calculation can obviously be done by using numerical methods but you have to solve (numerically) a three-dimensional PDE: 0 = φ F t + (r d (t) f d (0, t)) φ F + [(r d (t) r f (t))s(t)φ F ] x + [(θ f (t) α f (t) r f (t))φ F ] z [σ(t, S(t))S(t)σ d (t)ρ Sd φ F ] x y 2 [σ 2 (t, S(t))S 2 (t)φ F ] x [σ(t, S(t))S(t)σ f (t)ρ Sf φ F ] x z + [(θ d (t) α d (t) r d (t))φ F ] y 2 [σ 2 d (t)φ F ] y [σ 2 f (t)φ F ] z 2 2 [σ d (t)σ f (t)ρ df φ F ]. (7) y z Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

23 Calibration first method Calibration : Comparison between local volatility with and without stochastic interest rates Reference:Atlan (2006) In a deterministic interest rates framework, the local volatility function σ 1f (t, K) is given by the well-known Dupire formula: C(K,t) σ1f 2 (t, K) = t + K(f d (0, t) f f (0, t)) C(K,t) + f K f (0, t)c(k, t). 1 2 K 2 2 C(K,t) K 2 If we consider the three-factor model with stochastic interest rates, the local volatility function is given by C(K,t) σ3f 2 (t, K) = P t d (0, t)e Qt [(Kr d (t) r f (t)s(t))1 {S(t)>K} ]. 1 2 K 2 2 C(K,t) K 2 We can derive the following interesting relation between the simple Dupire formula and its extension σ 2 3f (t, K) σ2 1f (t, K) = KP d (0, t){cov Q t [r f (t) r d (t), 1 {S(t)>K} ] + K 1 CovQ t [r f (t), (S(t) K) + ]}. 1 2 K 2 2 C K 2 (8) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

24 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models Reference: Derman and Kani (1998), Dupire (2004), Atlan (2006) If there exists a local volatility such that the one-dimensional probability distribution of the spot FX rate with the diffusion ds(t) = (r d (t) r f (t)) S(t) dt + σ(t, S(t)) S(t) dws DRN (t), is the same as the one of the spot FX rate with dynamics ds(t) = (r d (t) r f (t)) S(t) dt + γ(t, ν(t)) S(t) dws DRN (t) for every time t with (ν t) t a stochastic process, then this local volatility function has to satisfy σ 2 (t, K) = EQ d [γ 2 (t, ν(t))e t0 r d (s)ds S(t) = K] E Q d [e t 0 r d (s)ds S(t) = K] = E Qt [γ 2 (t, ν(t)) S(t) = K]. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

25 The model for GAO s The model for GAO s References: Pelsser (2003), Ballotta & Haberman (2003), Boyle & Hardy (2003) and local volatility papers Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

26 Definition The model for GAO s Introduction GAO definition : Consider at time 0 an x 0 = x T year old policyholder who receives at the age x at time T the payout of his capital policy which is linked to a fund with value S(T ) at T (an index or of a portfolio). The policyholder has the right to choose at time T either a cash payment equal to the investment in the equity fund S(T ) or an annual payment of S(T )rx G, with rx G being a fixed rate called the Guaranteed Annuity rate The cash payment equal to the investment in the equity fund S(T ) can imply and therefore can be considered as an annual payment of S(T )r x (T ), with r x (T ) being the market annuity payout rate over an initial single premium of 1, namely r x (T ) = 1 ä x (T ) and ä x (T ) = ω x n=0 np x P(T, T + n) with n p x denoting the probability that the remaining lifetime of the policyholder is greater than n Therefore the guaranteed annuity contract with GAO can be seen as follows: Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

27 Introduction The model for GAO s Introduction At time T the value V (T ) of the guaranteed annuity contract with GAO is given by ω x V (T ) = S(T ) max(rx G, rx (T )) = S(T ) + S(T ) max((r G x n=0 ω x n=0 since r x (T ) = 1 ä x (T ) and äx (T ) = ω x n=0 np x P(T, T + n) np x P(T, T + n) np x P(T, T + n)) 1, 0) Very popular in the 1970 s and 1980 s in the United Kingdom (r G x = 11%). Since the interest rate levels decreased in the 1990 s and the (expected) mortality rates increased, the value of the GAO s increased rapidly. Similar options in US and Japan as part of variable annuity products: GMIB (Guaranteed Minimum Income Benefit). Given the long maturities of these insurance products, volatility modelling is important. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

28 Literature review The model for GAO s Introduction Pricing GAO in a constant volatility framework (with stochastic interest rates): Dunbar (1999), Yang (2001), Wilkie et al. (2003), Boyle and Hardy (2003), Pelsser (2003), Ballotta and Haberman (2003), Chu and Kwok (2006),... Pricing GAO in a stochastic volatility framework (with stochastic interest rates): A. van Haastrecht, R. Plat and A. Pelsser, Valuation of Guaranteed Annuity Options Using a Stochastic Volatility Model for Equity Prices, IME, 2010, 47(3), A. Kling, F. Ruez, and J. Russ, The Impact of Stochastic Volatility on Pricing, Hedging, and Hedge Efficiency of Variable Annuity Guarantees, working paper, (14 August 2009) Pricing GAO in a local volatility framework (with stochastic interest rates):? Pricing GAO in a hybrid volatility framework (with stochastic interest rates):? Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

29 The model for GAO s The model The model Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

30 The model for GAO s The model The two-factor model with local volatility The spot S (capital policy) is governed by the following dynamics ds(t) = (r(t) q)s(t)dt + σ(t, S(t))S(t)dW Q S (t), (9) σ(t, S(t)) is a local volatility function which allows to take into account the risk linked to the volatility of the spot. Interest rates denoted by r(t) follow a Hull-White one factor Gaussian model defined by the Ornstein-Uhlenbeck processes dr(t) = [θ(t) α(t)r(t)]dt + σ r (t)dw Q r (t), (10) θ(t), σ r (t) and α(t) are deterministic functions of time in general. In our calibration we will take σ r and α constant. Equations (9) and (10) are expressed in the (domestic) risk-neutral measure. W Q S Q (t) and Wr (t) are correlated Brownian motions dw Q S (t)dw r Q (t) = ρ Sr dt Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

31 The model The model for GAO s The model Remark The market dynamics could be better approximated by a hybrid volatility model that contains both stochastic volatility dynamics and local volatility ones. example: ds(t) = (r(t) q)s(t)dt + σ LOC2 (t, S(t)) ν(t)s(t)dw Q S (t), dr(t) = [θ(t) α(t)r(t)]dt + σ d (t)dwr Q (t), dν(t) = κ(θ ν(t))dt + ξ ν(t)dw Q ν (t). The local volatility function σ LOC2 (t, S(t)) can be calibrated from the local volatility that we have in a pure local volatility model. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

32 Pricing methods Pricing methods (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

33 Pricing methods Pricing methods At time T the value V (T ) of the guaranteed annuity contract with GAO is given by ω x V (T ) = S(T ) max(rx G, rx (T )) = S(T ) + S(T ) max((r G x n=0 ω x n=0 since r x (T ) = 1 ä x (T ) and äx (T ) = ω x n=0 np x P(T, T + n) np x P(T, T + n) np x P(T, T + n)) 1, 0) At time t = 0 the value of the guaranteed annuity contract with GAO (entered by an x 0 = x T year old policyholder) V (x, 0, T ) is given by V (x, 0, T ) = E Q [e T 0 r(s)ds V (T ) 1 (τx0 >T ) F 0 ] = E Q [e T 0 r(s)ds V (T )] E Q [1 (τx0 >T )] where τ x0 is a random variable which represents the remaining lifetime of the policyholder aged x 0, and where we assume that the mortality risk is unsystematic and independent of the financial risk Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

34 Pricing methods Pricing methods Using E Q [ 1 (τx0 >T )] = T p x0, V (x, 0, T ) = T p x0 E Q [e T 0 r(s)ds V (T )], = T p x0 {E Q [e T 0 r(s)ds S(T )] + C(x, 0, T )} (11) GAO = T p x0 C(x, 0, T ) = T p x0 E Q [e T 0 r(s)ds S(T )[ max((r G x ω x n=0 np x P(T, T + n)) 1, 0)]] where np x is the probability that the remaining lifetime of the policyholder is strictly greater than n Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

35 Pricing methods Pricing methods At time t = 0 the value of the GAO entered by an x 0 = x T year old policyholder is given by GAO = T p x0 C(x, 0, T ) = T p x0 E Q [e T 0 r(s)ds S(T )[ max((r G x ω x n=0 np x P(T, T + n)) 1, 0)]] By the density process ξ T = dq S dq F T = e T 0 r(s)ds S(T ) (where the numeraire is the S(0) equity price), we define a new probability measure Q S equivalent to the measure Q such that (Geman et al., 1995) T p x0 C(x, 0, T ) = T p x0 E Q [e ω x T r(s)ds 0 rx G S(T )( with K = 1. rx G ω x = T p x0 rx G S(0)E QS [( n=0 n=0 np x P(T, T + n) K) + ] np x P(T, T + n) K) + ] Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

36 Pricing methods Pricing methods The model under Q S, with σ r and α constant: ds(t) = [r(t) q + σ 2 (t, S(t))]S(t)dt + σ(t, S(t))S(t)dW Q S S (t) dx(t) = [ αx(t) + ρ rs σ r σ(t, S(t))]dt + σ r dw Q S r (t) where we have rewritten stochastic interest rates as a sum of a stochastic and a deterministic part (see e.g. Brigo and Mercurio): r(t) = x(t) + x(t), (12) where the deterministic part obeys the dynamics : dx(t) = (θ(t) αx(t))dt (13) and moreover (see e.g. Brigo and Mercurio) x(t) = f mkt (0, t) + σ2 r 2α 2 (1 e αt ) 2, (14) with f mkt (0, t) the market instantaneous forward rate at time 0 for the maturity t. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

37 Pricing methods Pricing methods In this model, the time T price of a zero-coupon bond P(T, T + n) maturing at time T + n is given by where B(T,T +n)x(t ) P(T, T + n) = A(T, T + n)e A(T, T + n) = Pmkt (0, T + n) P mkt e 1 [V (0,T +n) V (0,T ) V (T,T +n)] 2 (0, T ) B(T, T + n) = 1 α (1 e α(t +n T ) ) V (u, v) = σ2 r α 2 [v u + 2 α e α(v u) 1 2α e 2α(v u) 3 2α ] with P mkt (0, T ) denoting the market s discounting factor maturing at T. ω x GAO = T p x0 C(x, 0, T ) = T p x0 rx G S(0)E QS [( n=0 np x A(T, T + n)e B(T,T +n)x(t ) K) + ] Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

38 Calibration Calibration (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

39 Calibration Calibration Introduction Before pricing any derivatives with a model, it is usual to calibrate it on the vanilla market, determine all parameters present in the different stochastic processes which define the model in such a way that all European option prices derived in the model are as consistent as possible with the corresponding market ones. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

40 Calibration Calibration Introduction The calibration procedure for the two-factor model with local volatility can be decomposed in three steps: 1 Parameters present in the Hull-White one-factor dynamics for the interest rates, θ(t), α and σ r, are chosen to match European swaption / cap-floors values. 2 The correlation coefficient of the model, ρ rs, is usually estimated from historical data. 3 After these two steps, the calibration problem consists in finding the local volatility function of the spot which is consistent with the implied volatility surface. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

41 Calibration Calibration Introduction σ 2 (T, K) = C(K,T ) + qc(k, T ) qk C(K,T ) + KP(0, T )E T K Q T [r(t )1 {S(T )>K} ] 1 2 K. 2 2 C K 2 Difficult because of E Q T [r(t )1 {S(T )>K} ] : there exists no closed form solution it is not directly related to European call prices or other liquid products. Its calculation can obviously be done by numerical integration methods but you have to solve (numerically) a two-dimensional PDE: 0 = φ F T + (r(t ) f (0, T ))φ F + [(r(t ) q)s(t )φ F ] + [(θ(t ) α(t )r(t ))φ F ] x y [σ 2 (T, S(T ))S 2 (T )φ F ] x [σ 2 r (T )φ F ] y 2 2 [σ(t, S(T ))S(T )σr (T )ρ Sr φ F ] x y Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

42 Calibration Monte Carlo simulations Monte Carlo simulations σ 2 (T, K) = C(K,T ) + qc(k, T ) qk C(K,T ) + KP(0, T )E T K Q T [r(t )1 {S(T )>K} ] 1 2 K. 2 2 C K 2 Monte Carlo techniques are possible for determining E Q T [r(t )1 {S(T )>K} ] This expectation is expressed under the T -forward neutral measure Q T and therefore we have to use the dynamics of S(t), r(t) and thus x(t) under that measure. { ds(t) = (r(t) q σ(t, S(t))σr B(t, T )ρ Sr )S(t)dt + σ(t, S(t))S(t)dWS TF (t), (15) dx(t) = [ αx(t) σr 2 TF B(t, T )]dt + σr dwr (t), (16) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

43 Calibration Monte Carlo simulations The Euler discretisations of equations (15) and (16) are the following: S(t k + 1) = S(t k ) + (r(t k ) q σ(t k, S(t k ))σ r B(t k, T )ρ Sr )S(t k ) t (17) +σ(t k, S(t k ))S(t k ) tz S (18) where x(t k+1 ) = x(t k ) + [ αx(t k ) σr 2 B(t k, T )] t + σ r t[ρsr Z S + 1 ρ 2 Sr Zr ] r(t k+1 ) = x(t k+1 ) + x(t k+1 ) (19) x(t) = f mkt (0, t) + σ2 r 2α 2 (1 e αt ) 2, (20) and Z r and Z S are two independent standard normal variables. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

44 Calibration Monte Carlo simulations First step: The goal is to determine the local volatility function at the first time step T = T 1 for all strike K. In this step we assume that the local volatility is given by the local volatility associated to the deterministic interest rate case (Dupire). Knowing that function we can simulate S(T 1 ) and r(t 1 ) by using the Euler decompositions (18) and (19): S(T 1 ) = S(0) + (r(0) q σ(0, S(0))σ r B(0, T )ρ Sr )S(0)T 1 + σ(0, S(0))S(0) T 1 Z S (21) where x(t 1 ) = x(0) + [ αx(0) σr 2 B(0, T )](T 1 0) + σ r (T1 0)[ρ Sr Z S + 1 ρ 2 Sr Zr ] r(t 1 ) = x(t 1 ) + x(t 1 ) (22) x(t 1 ) = f mkt (0, T 1 ) + σ2 r 2α 2 (1 e αt 1 ) 2. (23) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

45 Calibration Monte Carlo simulations Then we can compute the expectation E Q T [r(t 1 )1 {S(T1 )>K}] for all K by using: E Q T [r(t 1 )1 {S(T1 )>K}] = 1 n n r i (T 1 )1 {Si (T 1 )>K} (24) i=1 This allows us to get the local volatility expression σ(t 1, K) at time T 1 for all strike K. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

46 Calibration Monte Carlo simulations Following the same procedure we can easily calibrate the local volatility at time T 2 by using the local volatility obtained at time T 1. More precisely, S(T 2 ) = S(T 1 ) + (r(t 1 ) q σ(t 1, S(T 1 ))σ r B(T 1, T )ρ Sr )S(T 1 )(T 2 T 1 ) (25) +σ(t 1, S(T 1 ))S(T 1 ) (T 2 T 1 )Z S x(t 2 ) = x(t 1 ) + [ αx(t 1 ) σr 2 B(T 1, T )](T 2 T 1 ) + σ r (T2 T 1 )[ρ Sr Z S + 1 ρ 2 Sr Zr ] r(t 2 ) = x(t 2 ) + x(t 2 ) (26) where x(t 2 ) = f mkt (0, T 2 ) + σ2 r 2α 2 (1 e αt 2 ) 2. (27) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

47 Calibration Monte Carlo simulations Then we can compute the expectation E Q T [r(t 2 )1 {S(T2 )>K}] for all K by using: E Q T [r(t 2 )1 {S(T2 )>K}] = 1 n n r i (T 2 )1 {Si (T 2 )>K} (28) i=1 This allows us to get the local volatility expression σ(t 2, K) at time T 2 for all strike K. Between time T 1 and T 2, we can use an interpolation method (bicubic splines,...). Following this procedure we can generate the local volatility expression up to time T = T k. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

48 Calibration Monte Carlo simulations Second method : by adjusting the Dupire formula Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

49 Calibration By adjusting the Dupire formula Calibration : Comparison between local volatility with and without stochastic interest rates In a deterministic interest rates framework, the local volatility function σ 1f (T, K) is given by the well-known Dupire formula: C(K,T ) σ1f 2 (T, K) = + K(f (0, T ) q) C(K,T ) + qc(k, T ) T K. 1 2 K 2 2 C(K,T ) K 2 If we consider the two-factor model with stochastic interest rates, the local volatility function is given by C(K,T ) σ2f 2 (T, K) = + qc(k, T ) qk C(K,T ) KP(0, T )E T K Q T [r(t )1 {S(T )>K} ] 1 2 K. 2 2 C K 2 We can derive the following interesting relation between the simple Dupire formula and its extension σ 2 2f (T, K) σ2 1f (T, K) = P(0, T )CovQ T [r(t ), 1 {S(T )>K} ] 1 2 K. 2 C K 2 (29) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

50 Calibration By adjusting the Dupire formula Third method : by mimicking stochastic volatility models Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

51 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models Consider the following risk-neutral dynamics for the spot ds(t) = (r(t) q)s(t)dt + γ(t, ν(t))s(t)dw Q S (t) ν(t) is a stochastic process which provides the stochastic perturbation for the spot volatility. Common choices: 1 γ(t, ν(t)) = ν(t) 2 γ(t, ν(t)) = exp( ν(t)) 3 γ(t, ν(t)) = ν(t) The stochastic process ν(t) is generally modelled by a Cox-Ingersoll-Ross (CIR) process as for example the Heston stochastic volatility model: dν(t) = κ(θ ν(t))dt + ξ ν(t)dw Q ν (t) a Ornstein-Uhlenbeck process (OU) as for example the Schöbel and Zhu stochastic volatility model: dν(t) = k[λ ν(t)]dt + ξdw Q ν (t) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

52 Calibration By mimicking stochastic volatility models Calibrating the local volatility by mimicking stochastic volatility models Applying Tanaka s formula to the non-differentiable function e t r(s)ds 0 (S(t) K) +, where ds(t) = (r(t) q)s(t)dt + γ(t, ν(t))s(t)dw Q S (t) we obtain E Q [γ 2 (t, ν(t))e t 0 r(s)ds S(t) = K] E Q [e t 0 r(s)ds S(t) = K] = C(K,T ) t + qc(k, t) qk C(K,t) KP(0, t)e Q t [r(t)1 K {S(t)>K} ] 1 2 K 2 2 C K 2 } {{ } σ 2 (t,k) Therefore, if there exists a local volatility such that the one-dimensional probability distribution of the spot with the diffusion ds(t) = (r(t) q)s(t)dt + σ(t, S(t))S(t)dW Q S (t), is the same as the one of the spot with dynamics ds(t) = (r(t) q)s(t)dt + γ(t, ν(t))s(t)dw Q S (t) for every time t, then this local volatility function has to satisfy σ 2 (t, K) = EQ [γ 2 (t, ν(t))e t0 r(s)ds S(t) = K] E Q [e t r(s)ds 0 S(t) = K] = E Qt [γ 2 (t, ν(t)) S(t) = K]. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

53 Calibration By mimicking stochastic volatility models A particular case with closed form solution Consider the two-factor model with local volatility (LVHW) { ds(t) = (r(t) q)s(t)dt + σ(t, S(t))S(t)dW Q S (t), (30) dr(t) = [θ(t) αr(t)]dt + σ r dwr Q (t), (31) Calibration by mimicking a Schöbel and Zhu / Hull and White stochastic volatility model (SZHW) ds(t) = (r(t) q)s(t)dt + ν(t)s(t)dw S (t), (32) dr(t) = [θ(t) αr(t)]dt + σ r dw r (t), (33) dν(t) = k[λ ν(t)] dt + ξdw ν(t), (34) The local volatility function is given by: σ 2 (T, K) = E Q T [ν 2 (T ) S(T ) = K] = E Q T [ν 2 (T )] if we assume independence between S and ν = (E Q T [ν(t )]) 2 + Var Q T [ν(t )] Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

54 Calibration By mimicking stochastic volatility models A particular case with closed form solution Under the T -Forward measure: dν(t) = [k(λ ν(t)) ρ rνσ r b(t, T )ξ]dt + ξ dwν TF (t) T ν(t ) = ν(t)e k(t t) + k(λ t ρrνσr b(u, T )ξ k T )e k(t u) du + t ξe k(t t) dwν TF (u) where b(t, T ) = 1 α (1 e α(t t) ) so that ν(t ) conditional on F t is normally distributed with mean and variance given respectively by E Q T [ν(t ) F t] = ν(t)e k(t t) + (λ ρrνσr ξ )(1 e k(t t) ) αk + ρrνσr ξ α(α + k)) (1 e (α+k)(t t) ) Var Q T [ν(t ) F t] = ξ2 2k (1 e 2k(T t) ) Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

55 Calibration By mimicking stochastic volatility models A particular case with closed form solution σ 2 (T, K) = (E Q T [ν(t )]) 2 + Var Q T [ν(t )] ( = ν(0)e kt + (λ ρrνσr ξ )(1 e kt ) + ρrνσr ξ ) 2 αk α(α + k)) (1 e (α+k)t ) + ξ2 2k (1 e 2kT ) = σ 2 (T ) Figure: ξ = 20%, k = 50%, α = 5%, ν(0) = 10%, σ = 1%, λ = 20%, ρ rν = 1% Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

56 Numerical results Numerical results (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

57 Calibration Numerical results We have first calibrated the local volatility and the interest rate parameters to market option data per end July 2007 given in the paper A. van Haastrecht, R. Plat and A. Pelsser, Valuation of Guaranteed Annuity Options Using a Stochastic Volatility Model for Equity Prices, IME, 2010, 47(3), More precisely, a calibration of the Hull and White interest rate model to the U.S. swaptions market leads to α = and σ r = Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

58 Numerical results The equity considered in this section is the S&P500 (U.S.). van Haastrecht et al. determined the effective (10 years) correlation between the stock and the interest rates in the BSHW process by using time series analysis of the interest rates and S&P500 (U.S.) index over the period from February 2002 to July They found a correlation coefficient ρ Sr = 14.64% between the interest rates and the equity price. We took q = 0. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

59 Numerical results As van Haastrecht et al. considered GAO with a 10 year maturity, they calibrated the equity to the terminal distributions of the equity price at that time. In our case the calibration of the volatility of the equity consists in building the local volatility surface given by the equation σ 2 (T, K) = C(K,T ) + qc(k, T ) qk C(K,T ) + KP(0, T )E T K Q T [r(t )1 {S(T )>K} ] 1 2 K. 2 2 C K 2 We need the local volatility for all strikes and all maturities. As a first simple case, we have assumed that the implied volatility is constant with respect to the maturity. A plot of the complete implied volatility surface and the resulting market call option prices can be found on the next slides. Following the Monte Carlo approach, we have found the corresponding local volatility surface. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

60 Numerical results Figure: Implied volatility surface (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

61 Numerical results Figure: The resulting market call option prices (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

62 Numerical results (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

63 Numerical results (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

64 Numerical results In the table on the next slide, we compare the implied volatility (for a range of seven different strikes and a fixed maturity T = 10) with volatility given by all the three models after being calibrated. We can see that the local volatility model (LVHW) and the stochastic volatility model (SZHW) are both well calibrated since they are able to generate the Smile/Skew quite close to the implied one. However, contrarily to the SZHW model, the LVHW model is calibrated over all maturities. This will have an influence upon the hedging strategy. A plot of the Smile/Skew generated by the market and these three models is given in a figure. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

65 Numerical results Strike Mkt vol SZHW vol LVHW vol BSHW (ATM vol) % 27.50% 27.56% 25.80% % 26.60% 26.55% 25.80% % 26.20% 26.29% 25.80% % 25.80% 25.71% 25.80% % 25.40% 25.46% 25.80% % 25.00% 25.10% 25.80% % 24.40% 24.61% 25.80% Table: calibration results (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

66 Numerical results Figure: Smile/Skew generated by the market and the three models after calibration (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

67 Numerical results Guaranteed Annuity Options Guaranteed Annuity Option Pricing Now we will compare prices of GAOs obtained with our LVHW model to prices given by using the BSHW and the SZHW model computed in van Haastrecht et al.. We consider, as in van Haastrecht et al., that the policyholder is 55 years old, the retirement age is 65, leading to a maturity T of the GAO option of 10 years. The fund value at time 0, S(0) = 100. The survival rates are based on the PNMA00 table of the Continuous Mortality Investigation (CMI) for male pensioners (available at Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

68 Numerical results Guaranteed Annuity Options In the following table we show prices for the GAO given for different guaranteed annuity rates rx G for each of the considered models. The results for the SZHW model are obtained using the closed form expression derived in van Haastrecht et al. and the pricing formula for the BSHW is derived in Ballotta and Haberman (2003). The results for the LVHW model are obtained by using Monte Carlo simulations ( simulations and 500 steps). The following table and figure show the corrections induced by the LVHW and the SZHW models with respect to the BSHW model. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

69 Numerical results Guaranteed Annuity Options rx G BSHW SZHW LVHW stderr ± 95 % interval 7% % % % % % % % Table: GAO total value Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

70 Numerical results Guaranteed Annuity Options Figure: LVHW and the SZHW models corrections with respect to the BSHW model (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

71 Numerical results Guaranteed Annuity Options rx G SZHW-BSHW LVHW-BSHW LVHWinf LVHWsup -BSHW -BSHW 7% % % % % % % % Table: Corrections Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

72 Numerical results Guaranteed Annuity Options The following table presents the time value given by the difference between the option value and its intrinsic value. The volatility of an option is an important factor for this time value since this value depends on the time until maturity and the volatility of the underlying instrument s price. The following figure is a plot of the time value given by all considered models. Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

73 Numerical results Guaranteed Annuity Options rx G BSHW SZHW LVHW 7% % % % % % % % Table: GAO time value Griselda.Deelstra@ulb.ac.be (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

74 Numerical results Guaranteed Annuity Options Figure: Time value (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

75 Future research Future research (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

76 Future research Future research sensibility studies, other market studies,... hedging problem by using Likelihood ratio method hedging in local volatility versus stochastic volatility model influence for path-dependent insurance products like Ratchets (ULB) DGVFM: Local Volatility Berlin, April 29, / 77

77 Future research Thank you for your attention (ULB) DGVFM: Local Volatility Berlin, April 29, / 77