Option Pricing. Chapter 12 - Local volatility models - Stefan Ankirchner. University of Bonn. last update: 13th January 2014

Transcription

1 Option Pricing Chapter 12 - Local volatility models - Stefan Ankirchner University of Bonn last update: 13th January 2014 Stefan Ankirchner Option Pricing 1

2 Agenda The volatility surface Local volatility Dupire s formula Practical note: how to calibrate the loc. volatility function Further reading: Chapter 1 in J. Gatheral: The volatility surface, B. Dupire: Pricing with a smile. Risk 7, Stefan Ankirchner Option Pricing 2

3 Implied vola Implied vola varies with different strikes different maturities Recall the volatility smile: Stefan Ankirchner Option Pricing 3

4 Volatility surface Volatility surface = graph of the the implied volatility as a function of t2m and moneyness Data: Exxon Mobile call prices, CBO on 5/12/2010. Stefan Ankirchner Option Pricing 4

5 Local volatility and transition densities Proof of Dupire s formula Definition of local volatility One way to make model prices consistent with market prices for Plain Vanilla Options is to assume that the volatility is a function of time and the underlying price: Definition: function. We will see that ds t = rs t dt + S t σ(t, S t )dw Q t. (1) the mapping (t, x) σ(t, x) is called local volatility European call / put prices uniquely determine the local volatility function, our pricing PDEs for American and exotic options remain almost the same: only the constant vola has to be replaced with the local volatility function. Stefan Ankirchner Option Pricing 5

6 Local volatility and transition densities Proof of Dupire s formula Local volatility is determined by European Call Prices Suppose that the market call prices C(T, K) are known for all possible expiration dates T > 0 and strike prices K 0. Then the local volatility function is given by C σ(t, K) = 2 T + rk C K K 2 2 C K 2 (2) Equation (2) is called Dupire s formula. For the proof we use the transition density of the price process S t. Stefan Ankirchner Option Pricing 6

7 Local volatility and transition densities Proof of Dupire s formula Transition probabilities Let φ(0, x; T, y) be the probability density of S T at y conditional to S 0 = x. This means that for any nice B R P 0,x (S T B) = φ(0, x; T, y)dy. Definition: S t. B φ(0, x; T, y) is called transition density of the process Notation: In the following we fix the initial condition (0, x) and simply write φ(t, y) = φ(0, x; T, y). Stefan Ankirchner Option Pricing 7

8 Local volatility and transition densities Proof of Dupire s formula Kolmogorov forward equation Theorem (Kolmogorov forward equation) Suppose that S t satisfies ds t = rs t dt + S t σ(t, S t )dw Q t, S 0 = x. Then the transistion density φ(t, y) of S t satisfies the PDE φ(t, y) = T y (ryϕ(t, y)) y 2 ( y 2 σ 2 (T, y)φ(t, y) ). Remark: The Kolmogorov forward equation is sometimes also called Fokker-Planck equation. Stefan Ankirchner Option Pricing 8

9 Local volatility and transition densities Proof of Dupire s formula Option prices The price of a call option expiring at T and struck at K satisfies C(T, K) = e rt E Q [ (S T K) +] = e rt (y K) + φ(t, y)dy R = e rt (y K)φ(T, y)dy. K Stefan Ankirchner Option Pricing 9

10 Local volatility and transition densities Proof of Dupire s formula Partial derivatives Observe that C(T, K) = e rt (y K)φ(T, y)dy K C(T, K) = rc(t, K)+e rt (y K) φ(t, y)dy (3) T K T C(T, K) = e rt φ(t, y)dy (4) K K 2 K 2 C(T, K) = e rt φ(t, K) (5) Proof. Stefan Ankirchner Option Pricing 10

11 Local volatility and transition densities Proof of Dupire s formula Proof of Dupire s formula With the Kolmogorov forward equation we get C(T, K) T = rc(t, K) + e rt (y K) φ(t, y)dy K T [ = rc(t, K) e rt (y K) (ryϕ(t, y)) K y 1 2 ( y 2 2 y 2 σ 2 (T, y)φ(t, y) ) ] dy (6) Stefan Ankirchner Option Pricing 11

12 Local volatility and transition densities Proof of Dupire s formula Proof of Dupire s formula cont d Suppose that A1) lim y (y K) ( y y 2 σ 2 (T, y)φ(t, y) ) = 0, ( A2) lim y y 2 σ 2 (T, y)φ(t, y) ) = 0, A3) lim y (y K)ryφ(T, y) = 0. Then we can show Dupire s formula: σ(t, K) = Proof. 2 C T +rk C K K 2 2 C K 2 Stefan Ankirchner Option Pricing 12

13 Local volatility for pricing Question: How to price American or exotic options that are not actively traded? Derive the local volatility function from standard European options, use the local vola function in the pricing PDE for the American or exotic option considered, and solve the PDE numerically. Stefan Ankirchner Option Pricing 13

14 Example: Pricing D&O call with a local volatility model Assume that ds t = rs t dt + S t σ(t, S t )dw Q t, S 0 = x. and that σ(t, x) has been calibrated from market prices of standard calls or puts. Let v(t, x) be the time t D&O call value under the assumption that it has not been knocked out before t and that S t = x. Then v(t, x) satisfies the Black-Scholes PDE v t (t, x) + rxv x (t, x) σ2 (t, x)x 2 v xx (t, x) rv(t, x) = 0, with boundary conditions v(t, L) = 0, 0 t T, v(t, x) = (x K) +, x > L. Stefan Ankirchner Option Pricing 14

15 time-space parameterization of the local vola function Local vola from BS implied vola How to find the local vola function? Dupire s formula requires market prices for all strikes and maturities! In reality we have market prices for only a finite number of Plain Vanilla Calls. Idea: Parameterize the local vola function and choose the parameters such that the model prices for European calls & puts are as close as possible to the real market prices. Stefan Ankirchner Option Pricing 15

16 time-space parameterization of the local vola function Local vola from BS implied vola A simple time-space parameterization A simple parameterization of the local vola function is given by Let σ(t, y) = a 0 + a 1 y + a 2 y 2 + a 3 t + a 4 t 2 + a 5 ty T 1 T m be the maturities of traded calls. K i,1,..., K i,n(i) strikes of traded calls expiring at T i, 1 i m. C(T i, K i,j ) = market price of the traded call with expiration T i and strike K i,j. Stefan Ankirchner Option Pricing 16

17 time-space parameterization of the local vola function Local vola from BS implied vola A simple time-space parameterization cont d Assume that the local vola function satisfies σ(t, y) = a 0 + a 1y + a 2y 2 + a 3t + a 4t 2 + a 5ty. For a given set of parameters (a 0,..., a 5 ) one can calculate the model call prices Ĉ i,j = e rt i K i,j (y K i,j )φ(0, x; T i, y)dy. for all 1 i m and 1 j n(i). The quadratic distance to the market prices is given by error(a 0, a 1, a 2, a 3, a 4, a 5 ) = n(i) m (C(T i, K i,j ) Ĉi,j) 2 i=1 j=1 Stefan Ankirchner Option Pricing 17

18 time-space parameterization of the local vola function Local vola from BS implied vola A simple time-space parameterization cont d With a search algorithm one can find (a0,..., a 5 ) such that error(a 0,..., a 5) min error(a 0,..., a 5 ). (a 0,...,a 5 ) The local volatility function is then approximately given by σ(t, y) = a 0 + a 1y + a 2y 2 + a 3t + a 4t 2 + a 5ty. Stefan Ankirchner Option Pricing 18

19 time-space parameterization of the local vola function Local vola from BS implied vola BS implied vola and local volatility Notation: σ BS (T, K) = BS implied vola for a call with strike K and exp. date T Note that the market call price C(T, K) satisfies C(T, K) = BS call(s 0, K, T, σ BS (T, K), r). We will show that the local volatility function σ(t, K) is uniquely determined by the BS implied volatility function σ BS (T, K). Therefore: instead of parameterizing and calibrating the local vola function directly, one can proceed as follows: Parameterize the BS implied vola function Calibrate the BS implied vola function to market prices Calculate the local vola function from the calibrated BS implied vola function. Stefan Ankirchner Option Pricing 19

20 time-space parameterization of the local vola function Local vola from BS implied vola Linking local vola and BS implied vola Some new variables: ( ) log moneyness y = log K e rt S 0 implied total variance w(t, y) = T σ 2 BS (T, ert S 0 e y ) Lemma The local volatility function satisfies = σ 2 (T, e rt S 0 e y ) (7) 1 y w w y (T, y) w 2 w T (T, y) (T, y) + 1 y 2 4 ( w + y 2 w )( w y )2 (T, y) Stefan Ankirchner Option Pricing 20

21 time-space parameterization of the local vola function Local vola from BS implied vola Steps in the proof of Equation (7) a) First write the BS call price as a function of log moneyness and implied total variance w C BS (y, w) := BS call(s 0, e rt S 0 e y, T, T, r). Observe that [ C BS (y, w) = S 0 Φ( y w + w 2 ) ey Φ( y ] w w 2 ) The partial derivatives of C BS satisfy 2 C BS (y, w) y 2 2 C BS (y, w) w 2 = 2 C BS (y, w) y w = BS C (y, w) = y Stefan Ankirchner Option Pricing 21

22 time-space parameterization of the local vola function Local vola from BS implied vola Steps in the proof of Equation (7) b) Write the market call price as a function of T and log moneyness: Dupire s formula implies C(T, y) := C(T, e rt S 0 e y ). 1 2 σ2 (T, e rt S 0 e y )( 2 C y 2 C y )(T, y) = C T (T, y). (8) Proof of (8). Stefan Ankirchner Option Pricing 22

23 time-space parameterization of the local vola function Local vola from BS implied vola Steps in the proof of Equation (7) c) Observe that by definition we have C(T, y) = C BS (y, w(y, T )). With this one can rewrite Dupire s formula in terms of C BS and then derive (7). Proof. Stefan Ankirchner Option Pricing 23

24 Objections to local volatility models Local volatility models are criticized because: bad statistical properties: the local volatility functions change considerably over time. E.g. parameters usually change from one week to the next bad prediction properties See Dumas, Fleming, Whaley. Implied Volatility Functions: Empirical Tests. Journal of Finance, The local volatility model predicts the smile/skew to move in the opposite direction as the underlying; in reality, both move in the same direction. See Hagan, Kumar, Lesniewski, Woodward. Managing Smile Risk. Wilmott Magazine, hedging based on volatility models is inconsistent ad hoc model; no economic explanation of the local volatility function Stefan Ankirchner Option Pricing 24