Option Pricing with Delayed Information Mostafa Mousavi University of California Santa Barbara Joint work with: Tomoyuki Ichiba CFMAR 10th Anniversary Conference May 19, 2017 Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 1 / 19
Introduction Types of Delays Risk Minimizing Hedging Strategies with Delayed Information: Schweizer (1994) presents the general case of restricted information. Some other works in this direction are Frey (2000), Mania et al. (2008), Kohlmann and Xiong (2007) and Ceci et al. (2017). Absence of Arbitrage: Kabanov and Stricker (2006) provides an absence of arbitrage condition in discrete time models with delayed information. Kardaras (2013) studies market viability in scenarios that the agent has delayed or limited information. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 2 / 19
Discrete Time Model N-period Binomial tree model of Cox et al. (1979). Probability Space (Ω, F, P) Ω = {0, 1} N F: Borel σ-algebra P: Probability measure under which coordinate maps Z k, k = 1,..., N are independent Bernoulli random variables. Filtration {F k, k = 0,..., N}, where F k = σ(z 1,..., Z k ). Risky asset price S k S k := S 0 u I k d k I k, I k = The discounted risky asset price S k k Z l, k = 1,..., N, l=1 S k := e rk S k, k = 1,, N. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 3 / 19
Delayed Filtration G k := F k H, k = H,..., N, G k := F 0, k = 0,..., H 1. = { k, k = 0,..., N 1} A G : Hedging strategy based on the delayed information. A G : Set of {G k }-adapted stochastic processes s.t k = 0, k = 0,..., H 1. P 0 u 4 P 0 u 3 P 0 u 2 P 0 u 3 d P 0 u P 0 u 2 d P 0 P 0 ud P 0 u 2 d 2 P 0 d P 0 ud 2 P 0 d 2 P 0 ud 3 P 0 d 3 P 0 d 4 time 1 time 2 time 3 time 4 Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 4 / 19
Portfolio Value Process V H (x 0, )(ω) := x 0 e rh + H S H (ω), V 0 (x 0, )(ω) := e rh V H (x 0, )(ω) = x 0 + H S H (ω), V k (x 0, )(ω) := e r(h k) V H (x 0, ) (ω), k = 0,..., H 1, e rk x 0 + k 1 l=h ) S l (ω) ( (l 1) H l +S k (ω) (k 1) H, k = H,..., N. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 5 / 19
Absence of Arbitrage Definition (Arbitrage) An arbitrage opportunity is the strategy (x 0, ) (R, A G ) such that Theorem max 0(x 0, ) (ω)} ω Ω = 0, P(V N (x 0, ) 0) = 1, P(V N (x 0, ) > 0) > 0. There does not exists any arbitrage opportunity in our discrete time model, in the domain of (R, A G ) strategies. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 6 / 19
Super-Replication Price and Value Process Super-Replication Price: { π(ϕ) = inf max V 0 (x 0, )(ω) = x 0 + S } H H (ω) (x 0, ) Γ ω Ω where Γ = {(x 0, ) R A G : V N (x 0, ) ϕ P a.s.} Super-Replication Value Process: V k (S k H, S k ) := V k (x 0, ), k = 0,..., N, Where π(ϕ) = max ω Ω {V 0(x 0, )(ω)} Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 7 / 19
An N-period Model with H = N 1 periods of Delay Theorem For a European-style contingent claim with payoff ϕ := Φ(S N ) for some convex function Φ( ) in the N-period binomial model with H = N 1 periods of delay, the super-replication price is ( π(ϕ) = max x0 + e rh H S 0 u H, x0 + e rh H S 0 d H), where the corresponding strategy (x 0, ) is given by j 0, j = 0, 1,..., H 1, H = N 1 = Φ(S 0u N ) Φ(S 0 d N ) S 0 (u N d N ) x 0 = e rn un Φ(S 0 d N ) d N Φ(S 0 u N ) u N d N. and Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 8 / 19
An N-period Model with H = N 1 periods of Delay Geometrical representation when N = 2 and H = 1. ϕ(s 2 ) ϕ Optimal Line V 1 (S 0, S 1 = S 0 u) V 1 (S 0, S 1 = S 0 d) x 0 S 0 d 2 S 0 d S 0 S 0 ud S 0 u S 0 u 2 S 2 Figure : Super-replication Strategy in a 2-period Binomial Model with a 1-period Delay. The optimal line characterizes the super-replication strategy. The slope of it is 1 and its intercept is x 0. The super-replication price is π(ϕ) = max {V 1 (S 0, S 1 = S 0 d), V 1 (S 0, S 1 = S 0 u)}. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 9 / 19
An N-period Model with H = N 1 periods of Delay The super replication price π(ϕ) π(ϕ) = max ω Ω {V 0(S 0, S H ) (ω)} = max j {0,...,H} e rn E Q j [ϕ(s N )] = max j {0,H} e rn E Q j (ϕ(s N )) ( ) = e rn max p u ϕ(s 0 u N ) + q u ϕ(s 0 d N ), p d ϕ(s 0 u N ) + q d ϕ(s 0 d N ), Notation: (p u, q u ) = (p H, q H ) and (p d, q d ) = (p 0, q 0 ). Here {Q j } H j=0 are probability measures on (Ω, F) defined by Q j (I N = N) := p j = 1 Q j (I N = 0) = 1 q j, p j := uj d H j e r(n H) d H+1 u H+1 d H+1, j = 0,..., H. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 10 / 19
An N-period Model with H periods of Delay: Dynamic Programming Approach Theorem For a European-style contingent claim with payoff ϕ := Φ(S N ) for some convex function Φ( ) in the N-period binomial model with H N 1 periods of delay, the payoff function for all the intermediary (H + 1)-period subtrees are convex with respect to the corresponding risky asset prices. Key Recursive Equation (For k = H,..., N 2): V k (S k H, S k ) = H e [p r j V k+1 ( S k H u, S k H u H+1 ) j=0 ] +q j V k+1 ( S k H d, S k H d H+1 ) 1 {Sk =S k H u j d }, H j Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 11 / 19
An N-period Model with H periods of Delay: Explicit Approach Define measure Q on (Ω, F) s.t. the risky asset price S k satisfies S k := S 0 u I k d k I k, I k = k Z l, k = 1,..., N, where {Z l, l = 0,..., N} is a Markov chain with transition matrix ( ) qd p Q = d on {0, 1} for l = 1,..., N H 1. q u p u Besides, for l = N H,..., N l=1 Q (Z N = = Z N H = 1 Z N H 1 = 1) = p u, Q (Z N = = Z N H = 1 Z N H 1 = 1) = q u, Q (Z N = = Z N H = 1 Z N H 1 = 0) = p d, Q (Z N = = Z N H = 1 Z N H 1 = 0) = q d. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 12 / 19
An N-period Model with H periods of Delay: Explicit Approach Theorem For a European-style contingent claim with a convex payoff function φ(s N ) L (Ω, F, P) in an N-period Binomial model with H periods of delay, the super hedging price is { } π(ϕ) = e rn max E Q (ϕ (S N ) Z 0 = 1), E Q (ϕ (S N ) Z 0 = 0). I k = k l=1 Z l and S k := S 0 u I k d k I k are NOT Markov processes! Under measure Q, probability of a downward move preceded by a downward move is higher than the probability of a downward move preceded by an upward move. So, variance of S k is higher under this measure than the initial measure P. if delay H = 0, the transition matrix would have identical rows. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 13 / 19
Continuous Time Asymptotics Fix T > 0. µ n = µt δ 2 n, σ n = σ T δ n, u n = exp (µ n + σ n ), d n = exp (µ n σ n ), r n = rt δ 2 n, H n = HT δ 2 n. Where the order δ n = 1 / n, as in Donsker s theorem. Sm n ( )] m = S 0 exp [mµ n + σ n 2 Z i m, m = 0,..., n, i=1 S (n) t := S nt /T n, 0 t T, Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 14 / 19
Continuous Time Asymptotics Theorem The sequence of processes (S (n) ) n N converges in distribution to the process (S t ) 0 t T with dynamics ds t = rs t dt + σs t dw t, 0 t T. Where (W t ) 0 t T is a Brownian motion, and we have the enlarged volatility σ = 2H + 1σ. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 15 / 19
Exaggerated Volatility Smile Figure : Volatility smile for the Call and Put options in the binomial model with and without delayed information (H n = 1 100 year 2.52 days and 0 day respectively). The parameters are σ = 0.1, T = 1, r = 0, S 0 = 40, and n = 100 Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 16 / 19
Exaggerated Volatility Smile Figure : Volatility smile for the Call and Put options in the binomial model with delayed information (H n = 1 250,000 year 30 seconds). The parameters are σ = 0.1, T = 1, r = 0, S 0 = 40, and n = 250, 000 Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 17 / 19
Thank you! Questions? Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 18 / 19
References Ceci, C., Colaneri, K., and Cretarola, A. (2017). The föllmer schweizer decomposition under incomplete information. Stochastics, pages 1 35. Cox, J. C., Ross, S. A., and Rubinstein, M. (1979). Option pricing: A simplified approach. Journal of financial Economics, 7(3):229 263. Frey, R. (2000). Risk minimization with incomplete information in a model for high-frequency data. Mathematical Finance, 10(2):215 225. Kabanov, Y. and Stricker, C. (2006). The dalang morton willinger theorem under delayed and restricted information. In In Memoriam Paul-André Meyer, pages 209 213. Springer. Kardaras, C. (2013). Generalized supermartingale deflators under limited information. Mathematical Finance, 23(1):186 197. Kohlmann, M. and Xiong, D. (2007). The mean-variance hedging of a defaultable option with partial information. Stochastic analysis and applications, 25(4):869 893. Mania, M., Tevzadze, R., and Toronjadze, T. (2008). Mean-variance hedging under partial information. SIAM Journal on Control and Optimization, 47(5):2381 2409. Schweizer, M. (1994). Risk-minimizing hedging strategies under restricted information. Mathematical Finance, 4(4):327 342. Mostafa Mousavi (UCSB) Option Pricing with Delayed Information May 19, 2017 19 / 19