Stochastic Finance - Arbitrage theory

Transcription

1 Department of Probability and Mathematical Statistics Faculty of Mathematics and Physics Charles University Prague October 8th, 2007

2 Contents Introduction

3 Introduction Hans Föllmer, Alexander Schied Stochastic Finance - An Introduction in Discrete Time Chapter 1 - Arbitrage theory

4 Introduction one-period model of financial market, trading at time t = 0 finite number of primary assets, their initial prices at time t = 0 known, future prices at time t = 1 random variables arbitrage - trading opportunities which yield a profit without any downside risk characterization of financial market with absence of such arbitrage - martingale measures, contingent claims, perfect hedge

5 One-period model financial market with d + 1 assets (equities, bonds, commodities,...) priced at initial time t = 0 and at the final time t = 1 we assume that the i th asset is available at time 0 for a price π i 0, collection π = (π 0, π 1,..., π d ) R d+1 + is called a price system prices at time 1 non-negative random variables S 0, S 1,..., S d on probability space (Ω, F, P)

6 One-period model riskless investment possibility in bonds: by assuming π 0 = 1 and S r, where r > 1 (enough, more natural would be r 0) notation S = (S 0, S 1,..., S d ) = (S 0, S), π = (π 0, π), ξ = (ξ 0, ξ 1,..., ξ d ) = (ξ 0, ξ) R d+1, where ξ i represents the number of shares of the i th asset, which an investor choose at t = 0

7 One-period model price for buying the portfolio at time t = 1 π ξ = d π i ξ i i=0 at time t = 1 the portfolio will have the value ξ S(ω) = d ξ i S i (ω) = ξ 0 (1 + r) + ξ S(ω) i=0

8 Loan and short sales ξ 0 < 0 corresponds to taking out a loan, at t = 0 we receive ξ 0 and pay back (1 + r) ξ 0 at t = 1 ξ i < 0 for i 1 means short sale of the i th asset, received amount π i ξ i can be used for buying quantities ξ j 0, j i oh the other assets price of the portfolio ξ is given by π ξ = 0

9 Definition arbitrage theory (page 5) A portfolio ξ R d+1 is called an arbitrage opportunity if π ξ 0 but ξ S 0 P-a.s. and P[ξ S > 0] > 0. with positive probability a positive profit without exposure to any downside risk market inefficiency, certain assets are not priced well absence of arbitrage implies that S i vanishes P-a.s. once π i = 0, hence with no loss in generality we assume π i 0 for i = 1,..., d.

10 Lemma 1.3. (page 5) The following statements are equivalent 1. The market model admits an arbitrage opportunity 2. There is a vector ξ R d such that ξ S (1 + r)ξ π P-a.s. and P[ξ S > (1 + r)ξ π] > 0 no arbitrage opportunity = investment in risky assets which yields with positive probability a better result than investing the same amount in the risk-free asset must be open to some downside risk

11 Definition martingale measure (page 6) characterization of arbitrage-free market models which do not admit any arbitrage opportunities Definition 1.4. A probability measure P is called a risk-neutral measure, or a martingale measure, if [ ] S π i = E i, i = 0, 1,..., d. (1) 1 + r price π i is identified as the expectation of the discounted payoff under the measure P pricing formula (1) does not take into account any risk aversion, therefore measure P is called risk-neutral

12 Fundamental theorem of asset pricing (page 6) let us define the set of risk-neutral measures which are equivalent to P P := {P P is a risk-neutral measure with P P} Theorem 1.6. A market model is arbitrage-free P. In this case, there exists a P P which has a bounded density dp /dp. Remark: in an infinite market model of tradable assets S 0, S 1, S 2,... is implication = no longer true

13 Discounted net gains (page 7) in the proof is used random vector Y = (Y 1,..., Y d ) of discounted net gains Y i := Si 1 + r πi, i = 0, 1,..., d. (2) with this notation Lemma 1.3. implies that arbitrage-free model is equivalent to the condition for ξ R d : ξ Y 0 P-a.s. = ξ Y = 0 P-a.s. (3) since Y i is bounded from below by π i, P is a risk-neutral measure if and only if E [Y ] = 0. discounted asset prices and time value of money S i, i = 0, 1,..., d. (4) 1 + r

14 Attainable payoff and its return (page 10) let V := { ξ S ξ R d+1} denote the linear space of all payoffs which can be generated by some portfolio, the set V will be called attainable payoff the portfolio that generates V V is not unique therefore it is reasonable to define the price of V V as π(v ) := π ξ if V = π S. (5) Definition Suppose an arbitrage-free market model and V V an attainable payoff such that π(v ) 0. Then the return of V is defined by R(V ) := V π(v ). π(v )

15 Expected return of V in arbitrage-free model (page 11) Proposition Suppose arbitrage-free market model and let V V be an attainable payoff such that π(v ) Under any risk-neutral measure P, the expected return equals E [R(V )] = r. 2. Under any measure Q P such that E Q [ S ] < ( ) dp E Q [R(V )] = r cov Q dq, R(V ), where P is an arbitrary risk-neutral measure in P and cov Q denotes the covariance with respect to Q.

16 Derivative securities (page 13) in real financial markets not only the primary assets are traded securities whose payoff depends in non-linear way on the primary assets S 0, S 1,..., S d, and sometimes other factors derivative securities, options, contingent claims Example forward contract: one agent agrees to sell another agent an asset S i at time 1 for a delivery price K at time 0, such contract corresponds to random payoff C fw = S i K

17 Call and put options (page 13-14) Example call/put option: the owner has the option to buy/sell the i th asset at time 1 for a fixed strike price K C call = (S i K) +, C put = (K S i ) +, C call C put = S i K relation between the price of call and put options: put-call parity π(c call ) = π(c put ) + π i K 1 + r. (6)

18 Reverse convertible bond (page 15) Example reverse convertible bond: pays interest r which is higher than interest r of riskless bond at maturity t = 1 the issuer may convert the bond into a predetermined number of shares of asset S i instead of paying the nominal value in cash equals purchase of standard bond and the sale of a put option suppose that 1 is the price of a reverse convertible bond at t = 0, nominal value at maturity is 1 + r, and that it can be converted into x shares of the i th asset conversion will happen if S i < K := (1 + r)/x and the payoff the reverse convertible fond is 1 + r x(k S i ) +

19 Contingent claims (page 15) Definition A contingent claim is a random variable C on the underlying probability space (Ω, F, P) such that 0 C < P-a.s. contingent claim C is a derivative of S 0,..., S d if it is measurable with respect to the σ-field σ(s 0,..., S d ) generated by the assets, i.e., if C = f (S 0,..., S d ) for a measurable function on R d+1 it is a contract which is sold at t = 0 and which pays a random amount C(ω) 0 at time 1 security with negative terminal value can be reduced to combination of a non-negative contingent claim and a short position in some of S 0,..., S d

20 Arbitrage-free price of a contingent claim (page 16) so far, we have fixed prices π i of standard assets S 0,..., S d for a contingent claim C it is not clear, what the correct price should be our goal: identify possible prices which do not generate arbitrage in the market trading C at time 0 for a price π C corresponds to introducing a new asset π d+1 := π C and S d+1 := C. (7) Definition 1.22.: A real number π C 0 is called an arbitrage-free price of a contingent claim C if the market model extended according to (7) is arbitrage-free.

21 The set of arbitrage-free prices (page 16) the set of all arbitrage-free prices for C is denoted Π(C), the respective lower and upper bounds of Π(C) are π (C) := inf Π(C) and π (C) := sup Π(C) Theorem 1.23.: Suppose that the set P of equivalent risk-neutral measures for the original market model is non-empty. Then {[ ] } Π(C) = E C P P such that E [C] <. (8) 1 + r Moreover, the lower [ and ] upper bounds are given by π (C) = inf E C P P 1+r and π (C) = sup P P [ E C 1+r ].

22 Superhedging duality (page 18) the definition of arbitrage bound π (C) can be restated as: { π (C) = inf m R ξ R d with m + ξ Y C } 1 + r P-a.s. π (C) is the smallest amount of capital which, if invested risk-free, yields a superhedge (superrepliacation) of C in the sense that (m π ξ) + ξ S 1 + r C 1 + r P-a.s. in this view is π (C) in Theorem often called a superhedging duality

23 Lower and upper bounds of call and put option (page 18) consider arbitrage-free market model, and let C call = (S i K) + be a call option [ ] for any P P is C call S i so that π i E C call 1+r from Jensen s inequality, we obtain the lower bound: [E C call ] [ S (E i 1 + r 1 + r ] K ) + ( = π i 1 + r K ) r thus, we have following universal bound for any arbitrage-free market model: ( π i K ) + π (C call ) π (C call ) π i. (9) 1 + r

24 Lower and upper bounds of call and put option (page 18) for a put option C put = (K S i ) +, we could obtain in similar way ( ) K r πi π (C put ) π (C put ) K 1 + r. (10) in many situations, the universal bounds (9) and (10) are in fact attained

25 Attainable contingent claim and replicating portfolio (page 20) Definition A contingent claim C is called attainable (replicable, redundant), if there exists a portfolio ξ R d+1 such that C = ξ S = ξ 0 (1 + r) + ξ S P-a.s. Such strategy ξ is called a replicating portfolio for C. if we can replicate a given contingent claim C by some portfolio ξ, then the problem of determining a price for C has simple solution, the price of C should be equal to the cost ξ π of its replication

26 Unique arbitrage-free price (page 20) Theorem Suppose that the market model is arbitrage-free and that C is a contingent claim. 1. If C is attainable, then the set Π(C) of arbitrage-free prices for C consists of the single element ξ π, where ξ is any replicating portfolio for C. 2. If C is not attainable, then either π (C) = +, or π (C) < π (C) and Π(C) = (π (C), π (C)).

27 Example (page 21) Example We consider a call option C call = (S i K) + traded in a arbitrage-free market model. if the risk-free return r 0 and if C call is not deterministic, then Jensen s inequality yields ( (π i K) + π i K ) + [ (S < E i K) + ] for all P P. 1 + r 1 + r time value of a call option: the value of the right to buy the i th asset at t = 0 is strictly less than any arbitrage-free price for C call

28 Complete arbitrage-free market model (page 22) Definition An arbitrage-free market model is called complete if every contingent claim is attainable. In every market model following inclusion holds for each P P: V = {ξ S ξ R d+1 } L 1 (Ω, ω(s 1,..., S d ), P ) L 0 (Ω, F, P ) = L 0 (Ω, F, P) if the market is complete, these inclusions are in fact equalities linear space V is finite dimensional and the model can be reduced to a finite number of relevant scenarios - atoms of (Ω, F, P)

29 Characterization of a complete arbitrage-free market model (page 23) Proposition For any p [0, ], the dimension of the linear space is given by dim L p (Ω, F, P) = sup{n N partition A 1,..., A n of Ω with A i F and P[A i ] > 0}. Moreover, n := dim L p (Ω, F, P) < if and only if there exists a partition of Ω into n atoms of (Ω, F, P). we will use this result in following theorem

30 Characterization of a complete arbitrage-free market model (page 23) Theorem An arbitrage-free market model is complete if and only if there exists exactly one risk-neutral probability measure, i.e., if P = 1. In this case, dim L 0 (Ω, F, P) d + 1. application of this theorem in Example 1.33., pages 23-25