Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models

Transcription

1 Stochastic Processes and Stochastic Calculus - 9 Complete and Incomplete Market Models Eni Musta Università degli studi di Pisa San Miniato - 16 September 2016

2 Overview 1 Self-financing portfolio 2 Complete markets 3 Extensions of Black-Scholes 4 Incomplete market models 5 Back in discrete times

3 Self-financing portfolio Consider a market consisting in d + 1 assets with prices (S 0 t, S 1 t,... S d t ) t 0. A portfolio is self-financing if its value changes only because the asset prices change. No money is withdrawn or inserted after the initial forming of the portfolio. A portfolio strategy (Ht 0, H t ) t 0 is an (d + 1)-dim adapted process The corresponding value process is d V t = HtS i t i = Ht 0 St 0 + H t S t i=0 A portfolio is self-financing if V n = H 0 n S 0 n + H n S n (discrete time) dv t = H 0 t ds 0 t + H t ds t (continuous time)

4 Self-financing portfolio In terms of discounted prices: dṽ t = H t d S t

5 Self-financing portfolio In terms of discounted prices: dṽ t = H t d S t Proposition For any adapted process H t = (H 1 t,..., H d t ) t 0 and any initial value V 0 = x, there exists a unique adapted process (H 0 t ) t 0 such that the strategy (H 0 t, H t ) t 0 is self-financing. Proof. t Ṽ t = x + H 0 t = x + 0 t H s d S s = H t S t + H 0 t 0 H s d S s H t S t.

6 Complete markets Definition An F T -measurable random variable X is an attainable claim if there exists a self-financing portfolio worth X at time T. Definition A market is complete if every contingent claim is attainable. Theorem Assume that the market is arbitrage-free. Then, the following two statements are equivalent: the market is complete the martingale probability is unique.

7 Complete market models It is theoretically possible to perfectly hedge contingent claims. Gives a unique no-arbitrage price. Allows us to derive a simple theory of pricing and hedging. Is a rather restrictive assumption.

8 Black-Scholes model Assuming a constant volatility B-S model gives a unique no-arbitrage price of an option ds t = S t (µdt + σdb t ) The pricing formula depends only on one non-observable parameter: σ where C(t, S t ) = xn(d 1 ) Ke r(t t) N(d 2 ), d 1,2 = log(s t/k) + (r ± σ 2 /2)(T t) σ. T t In practice two methods are used to evaluate σ.

9 Black-Scholes model 1 The historical method: Since S T = S 0 exp the random variables ( ) ST log, log S 0 [ σb T ( S2T S T ) ] (µ σ2 T 2 ) ( ) SNT,..., log, S (N 1)T are independent Gaussian distributed with variance σ 2 T. Estimate σ using asset prices observed in the past. 2 The implied method: we recover σ by inversion of the Black-Scholes formula using quoted options. No explicit formulas! Numerical methods need to be used.

10 Volatility smile Options based on the same underlying but with different strike and expiration time yield different implied volatilities.

11 Time-dependent volatility models ds t = S t (µ(t)dt + σ(t)db t ) Similar formulas as in the Black-Scholes model replacing σ 2 (T t) T t σ 2 (s)ds. ( t ( ) S t = S 0 exp µ(s) σ2 (s) t ) ds + σ(s)db s Does not avoid the volatility smile!

12 Local volatility models The volatility depends on the time and the stock price: ds t = S t (µ(t, S t )dt + σ(t, S t )db t ) Note that F S t = Ft B. The market is still complete. For each X L 2 (F B T, P ), there exists a replicating portfolio V t = e r(t t) E [X F t ] = F (t, S t ), H t = F x (t, S t).

13 Need for more realistic models... In local volatility models, σ is perfectly correlated with the stock price. Empirical studies reveal that the previous models can not capture heavy tails and asymmetries present in log-returns in practice. The real market is incomplete.

14 Stochastic volatility models Model volatility as a random process driven by its own source of randomness. It is consistent with the highly variable and unpredictable nature of volatility. Let B 1 t, B 2 t be two independent Brownian motions. { ( dst = S t µt dt + σ t dbt 1 ) dσ t = α(t, σ t )dt + β(t, σ t )db 2 t

15 Stochastic volatility models Let B t := (B 1 t, B 2 t ) and F t = F B t. Girsanov theorem: motion i.e. ( B t ) t 0 H s ds is a 2-dim t P -Brownian dp ( T dp = exp H s db s 1 T ) H s 2 2 ds t t B t 1 := Bt 1 Hs 1 ds and B2 t := Bt 2 Hs 2 ds 0 0 are two independent Brownian motions w.r.t. P.

16 Stochastic volatility models If then H 1 t = µ t r σ t, ds t = S t (rdt + σ t d B 1 t ) which means that the discounted price is a P -martingale d S t = S t σ t d B 1 t. There is no restriction on the process H 2 t. Consequently, there are many probability measures under which the traded asset is a martingale.

17 Stochastic volatility models Note that F S t Ft B1. The model is not complete! Let X L 2 (Ω, F T, P ). By martingale representation theorem: T T X = X 0 + Ks 1 d B s 1 + Ks 2 d B s for some processes K 1 t, K 2 t. Hence T X = X Ks 1 T d S s + Ks 2 d B s 2 σ s Ss 0 But the second integral can not be written as an integral w.r.t. d S s.

18 Incomplete market models Under a stochastic volatility model, the market is incomplete. No unique price. More random sources than traded assets. It is not always possible to hedge a generic contingent claim. Captures more empirical characteristics.

19 Limitations Analytically less tractable. No closed form solutions for option prices. Option prices can only be calculated by simulation The practical applications of stochastic volatility models are limited.

20 Trinomial model An attempt to improve the Binomial Model (CRR)... We add a third possible state at which the stock price will not change.

21 Trinomial Model Absence of arbitrage a < R < b. Indeed, absence of arbitrage implies the existence of a probability P such that discounted prices are P -martingales. Let P (S 1 = 1 + a) = p 1 and P (S 1 = 1 + b) = p 2. Then, [ ] S 0 = E S1 = S 0(1 + a)p 1 + S 0 (1 + b)p 2 + S 0 (1 p 1 p 2 ), 1 + R 1 + R or equivalently 1 + R = (1 + a)p 1 + (1 + b)p 2 + (1 p 1 p 2 ). Hence, necessarily a < R < b.

22 Pricing in the Trinomial Model For the one-step trinomial model, the discounted price is a P -martingale if and only if where 1 + R = (1 + a)p 1 + (1 + b)p 2 + (1 p 1 p 2 ), P (S 1 = 1 + a) = p 1 and P (S 1 = 1 + b) = p 2. We have to solve one equation with two unknown quantities. No unique risk-neutral price!

23 Hedging in the Trinomial Model Consider a financial derivative on the asset S with value X t = f (S t ). At time 0, we want to construct a hedging strategy for X 1 H1 0 S1 0 + H 1 S 1 = f (S 1 ). Hence, (H1 0, H 1) must satisfy H1 0 S1 0 + H 1 S 0 (1 + a) = f (S 0 (1 + a)) H1 0 S1 0 + H 1 S 0 = f (S 0 ) H1 0 S1 0 + H 1 S 0 (1 + b) = f (S 0 (1 + b)) (1) We have to solve a system of three equations with two unknown quantities. We are unable to replicate the portfolio!