Option Pricing with Aggregation of Physical Models and Nonparametric Learning

Option Pricing with Aggregation of Physical Models and Nonparametric Learning Jianqing Fan Princeton University With Loriano Mancini http://www.princeton.edu/ jqfan May 16, 2007 0

Outline Option pricing using regression approaches Parametrical guided nonparametric estimate of SPD Combine powers of physical models and empirical methods. Expedience in computation and implementation.

Outline Option pricing using regression approaches Parametrical guided nonparametric estimate of SPD Combine powers of physical models and empirical methods. Expedience in computation and implementation. Nonparametric estimation of implied volatility: SemiParametric estimate of SPD.

Outline Option pricing using regression approaches Parametrical guided nonparametric estimate of SPD Combine powers of physical models and empirical methods. Expedience in computation and implementation. Nonparametric estimation of implied volatility: SemiParametric estimate of SPD. Ad hoc Black-Scholes methods. Empirical Studies. 1

Valuation of contingency claims Call option: right to buy an asset for a certain price K at (European) or before (American) expiration time T. Payoff of a call option Payoff of a put option A portfolio of options Pay-off: ψ(s T ) = (S T K) + 0 50 100 150 0 20 40 60 80 100 10 0 10 20 30 40 Exotic options: 1000 1100 1200 1300 1000 1100 1200 1300 1000 1100 1200 1300 (a) Asian options, look-back options and barrier (b) (c) options. (Hull, Duffie, Steele, Bingham & Kiesel) 2

Valuation of European options Pricing of Derivatives: the discounted value of the expected pay-off in the risk neutral world: exp( T 0 r s ds)e Q (Ψ(S T ) S 0 = s 0 )

Valuation of European options Pricing of Derivatives: the discounted value of the expected pay-off in the risk neutral world: exp( T 0 r s ds)e Q (Ψ(S T ) S 0 = s 0 ) depends on option characters: strike price (payoff function); time to maturity T ; the initial stock value s 0 ; risk-free rate r t r (constant in this talk). depends on the conditional density of S T given S 0 = s 0. 3

State Price Density the density of the value of an asset under the risk-neutral world [Cox and Ross (1976)]. transition density of S T given S 0 = s 0 under the equivalent martingale Q (Harrison and Kreps, 1979).

State Price Density the density of the value of an asset under the risk-neutral world [Cox and Ross (1976)]. transition density of S T given S 0 = s 0 under the equivalent martingale Q (Harrison and Kreps, 1979). directly related to the pricing of financial derivatives. independent of pay-off function and useful for evaluating illiquid derivatives from liquid derivatives 4

Black-Scholes formula Black-Scholes model: ds t /S t = rdt + σdw t so that log S t log S 0 = (r σ 2 /2)t + σw t N((r σ 2 /2)t, σ 2 t).

Black-Scholes formula Black-Scholes model: ds t /S t = rdt + σdw t so that log S t log S 0 = (r σ 2 /2)t + σw t N((r σ 2 /2)t, σ 2 t). State-Price density: log-normal: f LN (x; σ). Black-scholes: C = e rτ K (x K)f LN(x; σ)dx = C BS (σ).

Black-Scholes formula Black-Scholes model: ds t /S t = rdt + σdw t so that log S t log S 0 = (r σ 2 /2)t + σw t N((r σ 2 /2)t, σ 2 t). State-Price density: log-normal: f LN (x; σ). Black-scholes: C = e rτ K (x K)f LN(x; σ)dx = C BS (σ). Implied volatility: σ IV = C 1 BS (c obs ). IV smile: Evidence of inadequacy of BS. 5

Remarks Moneyness: m = K/F, F = e τ(r δ) S t. normalization account for interest and dividend rate and spot price.

Remarks Moneyness: m = K/F, F = e τ(r δ) S t. normalization account for interest and dividend rate and spot price. Time to maturity: Options with limited maturities are traded at each given day. Fix time to maturity approximately. Fit individualized volatility curve or state price density. Mitigate the chance of modeling error, but use less data points. 6

Ad hoc Black-Scholes benchmark Data: Given the time to maturity τ = T t 0, {(m t,i, σ IV t,i ) : i = 1,, N t, t = t 0 1,, t 0 d}. Implementation d = 5. Quadratic fit: Dumas, Fleming and Whaley (98) σ IV t,i = a 0 + a 1 m t,i + a 2 m 2 t,i + ε t,i Pricing: Ĉ(m) = C BS (q(m)), q(m) = â 0 + â 1 m + â 2 m 2.

Ad hoc Black-Scholes benchmark Data: Given the time to maturity τ = T t 0, {(m t,i, σ IV t,i ) : i = 1,, N t, t = t 0 1,, t 0 d}. Implementation d = 5. Quadratic fit: Dumas, Fleming and Whaley (98) σ IV t,i = a 0 + a 1 m t,i + a 2 m 2 t,i + ε t,i Pricing: Ĉ(m) = C BS (q(m)), q(m) = â 0 + â 1 m + â 2 m 2. Benchmark: routinely used in industry. Outperform (DFW, 98) the models by Derman and Kani (94), Dupire (94), Rubinstein (94). GARCH-type option pricing models (Heston and Nandi 00). 7

Semiparametric Black-Scholes Fitted IV: Dec 27 31, 04 with maturities 169 173 days. 0.5 0.45 Observed Impl. Vol. Semip BS Ad Hoc BS 0.4 0.35 Implied volatility 0.3 0.25 0.2 0.15 0.1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Moneyness = Strike/Futures Local linear: Nonparametric fit the IV function and use Black- Scholes formula. Related to Aït-Sahalia and Lo (98). 8

Estimation of State Price Density European call option: C = exp( rτ) K (x K)f (x)dx SPD: f (K) = exp(rτ) 2 C K 2 (Breeden and Litzenberger 78).

Estimation of State Price Density European call option: C = exp( rτ) K (x K)f (x)dx SPD: f (K) = exp(rτ) 2 C K 2 (Breeden and Litzenberger 78). Price of option: C(S, K, T, r, δ) can be estimated statistically: (Aït-Sahalia and Lo 98) parametric and nonparametric methods. C i = C(S i, K i, T i, r i, δ i ) + ε i.

Estimation of State Price Density European call option: C = exp( rτ) K (x K)f (x)dx SPD: f (K) = exp(rτ) 2 C K 2 (Breeden and Litzenberger 78). Price of option: C(S, K, T, r, δ) can be estimated statistically: (Aït-Sahalia and Lo 98) parametric and nonparametric methods. C i = C(S i, K i, T i, r i, δ i ) + ε i. Role of stochastic models: Produce a parametric form of C( ; θ) based on ideal physical models. e.g. Black-Scholes or jump diffusion (SV) models in risk-neutral world that we never live. Bakshi et al. 1997 9

Observations Notation: F (x) SP Distribution; F (x) = 1 F (x) survivor function. Note: e rτ C t = K (y K)f (y)dy = K F (y) dy = F t,τ m t F (u) du needs only state-price distribution; easier to estimate than SPD.

Observations Notation: F (x) SP Distribution; F (x) = 1 F (x) survivor function. Note: e rτ C t = K (y K)f (y)dy = K F (y) dy = F t,τ m t F (u) du needs only state-price distribution; 3 2.5 C T,1 / (X 2 X 1 ) C T,2 / (X 2 X 1 ) (C T,1 C T,2 ) / (X 2 X 1 ) easier to estimate than SPD. 2 1.5 I FT,0 > (X 1 + X 2 ) / 2 Portfolio: Long with strike K 1 ; short with strike K 2, 1 0.5 0 0.5 both with (K 2 K 1 ) 1 shares. 1 900 950 1000 1050 1100 1150 1200 F T,0 State price distribution can be inferred from the portfolio. 10

Relation with regression Data: Let Y i = e rτ C i C i+1 m i+1 m i for fixed τ. Then, Y i F ( m i ) + O{(m i+1 m i ) 2 }, m = m i+1 + m i. 2 If BS, then F (m) = F LN (m) = Φ( log m+σ2 /2 σ ).

Relation with regression Data: Let Y i = e rτ C i C i+1 m i+1 m i for fixed τ. Then, Y i F ( m i ) + O{(m i+1 m i ) 2 }, m = m i+1 + m i. 2 If BS, then F (m) = F LN (m) = Φ( log m+σ2 /2 σ ). Direct NP Rreg: Smooth the data {( m t,i, Y t,i )} over d = 5 days. Survivor function 1 0.9 0.8 0.7 0.6 0.5 0.4 Observed Y FM NP 0.3 0.2 0.1 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Moneyness = Strike/Futures 11

Pricing Errors 3 2.5 FM Ad Hoc BS Semip BS NP 2 1.5 Pricing error 1 0.5 0 0.5 1 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Moneyness = Strike/Futures In sample: On 12/29/04, DNP = $1.04; AHBS = $1.10, SBS = $.35, ACE =$.21 12

Nonparametric learning of pricing errors hard to model pattern of pricing errors from day-to-day; can be combined with model-based method.

Nonparametric learning of pricing errors hard to model pattern of pricing errors from day-to-day; can be combined with model-based method. NP fit of pricing errors: γ t adjusts for time-to-maturity and changing volatility = 1 Ỹ t,i Y t,i F LN ( m t,i ; γ t q( m t,i ))

Nonparametric learning of pricing errors hard to model pattern of pricing errors from day-to-day; can be combined with model-based method. NP fit of pricing errors: γ t adjusts for time-to-maturity and changing volatility = 1 Ỹ t,i Y t,i F LN ( m t,i ; γ t q( m t,i )) = F c ( m t,i ) + ε t,i ACE(Automatic Error Correction): ˆ F (m) = FLN (m; γ t q(m)) + ˆF c (m). 13

ACE pricing Pricing: C t = e rτ F t,τ m t FLN (u; γ t q(u))du + e rτ F t,τ m t F t,c (u)du. Applicability: Any other stochastic models, empirically correcting pricing errors. 1 0.9 0.8 τ = 24 days τ = 52 τ = 80 τ = 171 0.7 Survivor function 0.6 0.5 0.4 0.3 0.2 0.1 0 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 Moneyness = Strike/Futures 14

Statistical properties Smaller MSE, when model-based estimate gives a good first order approximation. Theorem: Under suitable conditions, we have nh{ ˆ F (m) F0 (m) 1 F c 2 (m)h2 u 2 K(u) du o(h 2 )} w N(0, σ 2 K 2 (u) du/g(m)) where F c (m) = F 0 (m) F (m; θ 0 ) and g(m) is the density of m.

Statistical properties Smaller MSE, when model-based estimate gives a good first order approximation. Theorem: Under suitable conditions, we have nh{ ˆ F (m) F0 (m) 1 F c 2 (m)h2 u 2 K(u) du o(h 2 )} w N(0, σ 2 K 2 (u) du/g(m)) where F c (m) = F 0 (m) F (m; θ 0 ) and g(m) is the density of m. Bias of Direct NR: 1 2 F (m)h 2 u 2 K(u)du. much smaller when F (m; θ 0 ) is close to F (m). 15

Adequacy of a pricing model H 0 : F t (m) = F t (m; θ) H 1 : F t (m) F t (m; θ). This is equivalent to testing H 0 : F t,c (m) = 0.

Adequacy of a pricing model H 0 : F t (m) = F t (m; θ) H 1 : F t (m) F t (m; θ). This is equivalent to testing H 0 : F t,c (m) = 0. RSS: RSS 0 = t 0 +d Nt t=t 0 d i=1 Ỹ t,i 2 I(a m t,i b) =1 RSS 1 = t 0 +d Nt t=t 0 d i=1 (Ỹt,i ˆ Ft,c (m t,i )) 2 I(a m t,i b). GLR test (Fan et. al. 2001): T n = n a,b 2 log(rss 0/RSS 1 ), where n a,b = t 0 +d Nt t=t 0 d i=1 I(a m t,i b).

Adequacy of a pricing model H 0 : F t (m) = F t (m; θ) H 1 : F t (m) F t (m; θ). This is equivalent to testing H 0 : F t,c (m) = 0. RSS: RSS 0 = t 0 +d Nt t=t 0 d i=1 Ỹ t,i 2 I(a m t,i b) =1 RSS 1 = t 0 +d Nt t=t 0 d i=1 (Ỹt,i ˆ Ft,c (m t,i )) 2 I(a m t,i b). GLR test (Fan et. al. 2001): T n = n a,b 2 log(rss 0/RSS 1 ), where n a,b = t 0 +d Nt t=t 0 d i=1 I(a m t,i b). Null distribution: r K T n a χ 2 an. 16

Importance of ACE Null hypothesis: H 0 : F t,c (m) = 0 No correction needed. Empirical testing: Daily over 3-years with P-values < 0.001. Correction is needed every day. 0.09 R 0 0.08 R 1 0.07 0.06 0.05 Residual 0.04 0.03 0.02 0.01 0 0.01 0.4 0.5 0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 Moneyness = Strike/Futures 17

Empirical studies SP500 from 1/2/02-12/31/04 (OptionMetrics), 101K satisfying selection criteria (maturity 20-240 days; IV 70%; price 1/8). Challenges: temporal mismatch (Fleming, Ostdiek, Whaley, 96); illiquid trading (DIM); dividend rates.

Empirical studies SP500 from 1/2/02-12/31/04 (OptionMetrics), 101K satisfying selection criteria (maturity 20-240 days; IV 70%; price 1/8). Challenges: temporal mismatch (Fleming, Ostdiek, Whaley, 96); illiquid trading (DIM); dividend rates. Resolution: Use daily closing prices. Use the put-call parity to infer Future price and call option price: C t + Ke rt,ττ = P t + F t,τ e r t,ττ Empirical findings based on separately calls or puts are similar (Bakshi, Cao, and Chen,, 97; Dumas, Fleming and Whaley, 98). 18

Maturity Less than 60 60 to 160 More than 160 mean St. Dev. mean St. Dev. mean St. Dev. DITM Call price $ 323.22 89.16 342.00 108.12 354.06 117.30 ( <.8) σ bs % 43.91 9.70 35.67 7.82 30.57 5.40 Observations 6,280 9,115 4,314 ITM Call price $ 129.49 41.32 141.22 38.74 153.38 36.11 (.8.94) σ bs % 26.99 6.84 24.79 5.44 23.19 4.21 Observations 7,969 8,609 3,840 ATM Call price $ 30.61 18.60 45.47 19.45 64.27 19.20 (.94 1.04) σ bs % 18.06 5.82 19.16 5.01 19.36 4.08 Observations 10,832 8,058 3,028 OTM Call price $ 3.03 4.02 7.88 7.76 17.28 11.63 (1.04 1.20) σ bs % 18.86 5.79 17.20 4.74 17.13 4.04 Observations 7,395 9,175 4,302 DOTM Call price $ 0.28 0.16 0.51 0.78 1.39 2.15 ( > 1.20) σ bs % 37.38 11.55 25.91 8.40 20.32 4.72 Observations 4,561 8,775 4,783 19

In-sample performance Absolute pricing errors (MSE) Relative Pricing errors (MSE) 20

In-sample absolute errors Less than 60 60 to 160 More than 160 bias RMSE bias RMSE bias RMSE DITM ACE 0.05 0.49 0.03 0.46 0.05 0.54 (< 0.8) Semip-BS 0.01 0.09 0.00 0.17 0.02 0.27 Ad Hoc BS 0.06 0.18 0.18 0.43 0.33 0.60 ITM ACE 0.09 0.43 0.14 0.42 0.11 0.42 (.8.94) Semip-BS 0.01 0.35 0.04 0.51 0.07 0.68 Ad Hoc BS 0.55 0.80 0.98 1.27 0.94 1.28 ATM ACE 0.01 0.39 0.01 0.38 0.01 0.37 (.94 1.04) Semip-BS 0.11 0.66 0.16 0.79 0.17 0.87 Ad Hoc BS 0.79 1.58 0.36 1.46 0.21 1.38 OTM ACE 0.13 0.28 0.24 0.36 0.27 0.42 (1.04 1.2) Semip-BS 0.11 0.41 0.14 0.53 0.19 0.71 Ad Hoc BS 0.89 1.30 1.17 1.53 0.93 1.47 DOTM ACE 0.04 0.07 0.00 0.08 0.05 0.13 (> 1.2) Semip-BS 0.00 0.07 0.02 0.12 0.01 0.21 Ad Hoc BS 0.06 0.20 0.12 0.38 0.16 0.36 21

In-sammple relative errors Less than 60 60 to 160 More than 160 bias RMSE bias RMSE bias RMSE DITM ACE 0.02 0.18 0.01 0.16 0.01 0.18 (< 0.8) Semip-BS 0.00 0.04 0.00 0.07 0.00 0.11 Ad Hoc BS 0.03 0.08 0.08 0.19 0.13 0.25 ITM ACE 0.09 0.38 0.12 0.33 0.08 0.29 (.8.94) Semip-BS 0.01 0.37 0.04 0.46 0.05 0.52 Ad Hoc BS 0.50 0.80 0.77 1.05 0.64 0.94 ATM ACE 0.77 2.68 0.19 1.23 0.09 0.74 (.94 1.04) Semip-BS 1.03 4.36 0.57 2.38 0.33 1.50 Ad Hoc BS 7.72 16.57 2.17 5.83 0.74 2.77 OTM ACE 4.50 13.98 4.54 8.91 2.43 4.26 (1.04 1.20) Ad Hoc BS 54.27 88.78 34.62 58.15 11.03 21.27 GARCH 26.95 69.01 22.79 37.97 4.59 17.05 DOTM ACE 11.30 20.25 4.30 17.17 0.14 11.87 (> 1.2) Semip-BS 1.71 18.40 4.61 17.18 2.37 13.34 Ad Hoc BS 25.68 60.52 6.37 57.95 13.24 40.43 22

In-sample Performance comparisons 4 150 Semip BS FM Semip BS FM 3 100 2 50 Pricing error Pricing error 1 0 0 1 2 50 3 4 0.4 0.6 0.8 1 1.2 1.4 1.6 Moneyness = Strike/Futures 1.8 2 Absolute Errors 2.2 2.4 100 0.4 0.6 0.8 1 1.2 1.4 1.6 Moneyness = Strike/Futures 1.8 2 2.2 2.4 Relative Errors 23

Out-sample performance Pricing options of next Wednesdays Absolute pricing errors (MSE) Relative Pricing errors (MSE) 24

Out-sample: Absolute error Less than 60 60 to 160 More than 160 bias RMSE bias RMSE bias RMSE DITM ACE 0.10 0.40 0.08 0.52 0.18 0.67 Semip-BS 0.08 0.25 0.09 0.55 0.02 0.87 Ad Hoc BS 0.11 0.31 0.27 0.72 0.37 1.02 GARCH 0.30 0.75 0.87 2.05 1.37 4.35 ITM ACE 0.18 0.75 0.43 1.15 0.40 1.40 Semip-BS 0.12 1.02 0.08 1.56 0.04 2.17 Ad Hoc BS 0.59 1.25 1.09 1.93 0.98 2.34 GARCH 0.59 1.27 1.62 2.94 2.40 5.40 ATM ACE 0.08 1.22 0.04 1.50 0.13 1.67 Semip-BS 0.32 1.83 0.32 2.33 0.17 2.87 Ad Hoc BS 0.83 2.31 0.41 2.53 0.21 3.05 GARCH 0.74 1.86 0.21 2.59 0.52 4.57 OTM ACE 0.30 0.75 0.39 1.11 0.48 1.25 Semip-BS 0.06 1.10 0.06 1.58 0.12 2.10 Ad Hoc BS 0.59 1.52 1.04 2.07 0.84 2.59 25

Out-sample Relative errors Less than 60 60 to 160 More than 160 bias RMSE bias RMSE bias RMSE DITM ACE 0.03 0.16 0.04 0.22 0.07 0.27 Semip-BS 0.03 0.12 0.04 0.25 0.01 0.36 Ad Hoc BS 0.05 0.15 0.12 0.32 0.15 0.42 GARCH 0.12 0.31 0.34 0.78 0.56 1.58 ITM ACE 0.20 0.78 0.35 1.00 0.27 1.03 Semip-BS 0.11 1.06 0.05 1.37 0.02 1.62 Ad Hoc BS 0.54 1.28 0.85 1.63 0.67 1.73 GARCH 0.48 1.19 1.19 2.26 1.55 3.71 ATM ACE 3.22 12.15 0.49 4.34 0.03 2.82 Semip-BS 4.03 15.18 1.42 6.63 0.53 4.89 Ad Hoc BS 10.12 25.92 2.66 8.58 0.89 5.41 GARCH 6.01 18.01 0.17 6.62 0.40 7.12 OTM ACE 13.59 41.59 10.55 25.29 5.44 11.20 Semip-BS 2.08 40.64 6.44 27.27 3.57 15.55 26 Ad Hoc BS 44.14 97.03 35.93 71.04 12.41 28.20

Out-sample Performance comparisons 8 300 Semip BS FM Semip BS FM 6 250 4 200 150 Pricing error Pricing error 2 0 100 50 2 0 4 50 6 8 100 0.4 0.6 0.8 1 1.2 1.4 1.6 Moneyness = Strike/Futures 1.8 2 Absolute Errors 2.2 2.4 150 0.4 0.6 0.8 1 1.2 1.4 1.6 Moneyness = Strike/Futures 1.8 2 2.2 2.4 Relative Errors 27

Summary Propose a semiparametric method to combine theoretical (model-based) and empirical pricing (statistical) methods. Pricing errors are learned statistically and corrected. It is powerful for option pricing and estimating SPD. Proposed methods price far more accurately than the Ad hoc Black-Scholes.

Summary Propose a semiparametric method to combine theoretical (model-based) and empirical pricing (statistical) methods. Pricing errors are learned statistically and corrected. It is powerful for option pricing and estimating SPD. Proposed methods price far more accurately than the Ad hoc Black-Scholes. Semiparametric techniques are much easier to implement than existing methods (SV, JDSV, GARCH) with much faster computational savings. Can be combined with any existing model-based methods. 28

Thank You 29