Introduction to Arbitrage-Free Pricing: Fundamental Theorems

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1 Introduction to Arbitrage-Free Pricing: Fundamental Theorems Dmitry Kramkov Carnegie Mellon University Workshop on Interdisciplinary Mathematics, Penn State, May 8-10, / 24

2 Outline Financial market Pricing = Replication Black and Scholes formula Fundamental theorems Martingale Representation Summary 2 / 24

3 ISDA Market Survey Notional amounts outstanding at year-end, all surveyed contracts, 1987-present Notional amounts in billions of US dollars Year-end outstandings for interest rate swaps Year-end outstandings for currency swaps Year-end outstandings for interest rate options Total IR and currency outstandings Total credit default swap outstandings Total equity derivative outstandings 1987 $ $ $ , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , , by International Swaps and Derivatives Association, Inc. Brief excerpts may be reproduced provided the source is stated.

4 Financial security Financial Security = Cash Flow Example (Interest Rate Swap) To owner today t 1 t 2 t 3 t 4 t 5 t 6 From owner Pricing problem: compute fair value of the security today. 3 / 24

5 Classification of financial securities We classify all financial securities into 2 groups: 1. Traded securities: the price is given by the market. Financial model = All traded securities 2. Non-traded securities: the price has to be computed. Remark This black-and-white classification is quite idealistic. Real life securities are usually gray. In this lecture we shall deal with Arbitrage-Free Pricing. 4 / 24

6 Arbitrage-free price Inputs: 1. Financial model (collection of all traded securities) 2. A non-traded security. Arbitrage strategy (intuitive definition): 1. start with zero capital (nothing) 2. end with positive and non zero wealth (something) Assumption The financial model is arbitrage free. Definition An amount p is called an arbitrage-free price if, given an opportunity to trade the non-traded security at p, one is not able to construct an arbitrage strategy. 5 / 24

7 Replication Cash flow of non-traded security: today t 1 t 2 t 3 t 4 t 5 t 6 Replicating strategy: 1. starts with some initial capital X 0 2. generates exactly the same cash flow in the future today t 1 t 2 t 3 t 4 t 5 t 6 X 0 6 / 24

8 Methodology of arbitrage-free pricing Theorem An arbitrage-free price p is unique if and only if there is a replicating strategy. In this case, p = X 0, where X 0 is the initial capital of a replicating strategy. Main Principle: (Unique) Arbitrage-Free Pricing = Replication 7 / 24

9 Problem on two calls Problem Consider two stocks: A and B. Assume that A: 95% $120 B: 5% $120 $100 $100 5% $80 95% $80 Consider call options on A and B with the same strike K = $100. Assume that T = 1 and r = 5%. Compute the difference C A C B of their arbitrage-free prices. 8 / 24

10 Pricing in Black and Scholes model There are two traded assets: a savings account and a stock. We assume that the interest rate is zero: r = 0. The price of the stock: ds t = S t (µdt + σdw t ). Here W = (W t ) t 0 is a Wiener process and µ R: drift σ > 0: volatility Problem (Black and Scholes, 1973) Compute arbitrage-free price V 0 of European put option with maturity T and payoff Ψ = max(k S T, 0). 9 / 24

11 Replication in Black and Scholes model Basic principle: Replicating strategy: 1. has wealth evolution: Pricing = Replication X t = X 0 + t 0 u ds u, where X 0 is the initial capital and t is the number of shares at time t; 2. generates exactly the same payoff as the option: X T (ω) = Ψ(ω) = max(k S T (ω), 0), P-a.s.. Two standard methods: direct (PDE) and dual (martingales). 10 / 24

12 PDE method Since X T = f (S T ) we look for replicating strategy in the form: X t = v(s t, t) for some deterministic v = v(s, t). By Ito s formula, dx t = v s (S t, t)ds t + (v t (S t, t) σ2 S 2 t v ss (S t, t))dt. But, (since X is a wealth process) dx t = t ds t. Hence, v = v(s, t) solves PDE: { vt (s, t) σ2 s 2 v ss (s, t) = 0 v(s, T ) = max(k s, 0) 11 / 24

13 Martingale method Observation: replication problem is defined almost surely and, hence, is invariant with respect to an equivalent change of probability measure. Convenient choice: martingale measure Q for S. We have ds t = S t σdw Q t, where W Q is a Brownian motion under Q. Replication strategy: (by Martingale Representation Theorem) X t = X 0 + t Risk-neutral valuation: (no replication!) 0 ds = E Q [Ψ F t ]. V 0 = E Q [Ψ]. 12 / 24

14 Martingale method The computation of hedging delta is conveniently done with Clark-Ocone formula: σs t t = E Q [D Q t [ψ] F t ], where D Q is the Malliavin derivative under Q. For example, for European put D Q t [max(k S T, 0)] = 1 {ST <K}D Q t [S T ] = 1 {ST <K}σS T, resulting in t = 1 S t E Q [1 {ST <K}S T ] F t ] = Q[S T < K F t ], where d Q dq = S T S / 24

15 Arbitrage-free pricing: general financial model There are d + 1 traded or liquid assets: 1. a savings account with zero interest rate. 2. d stocks. The price process S of the stocks is a semimartingale on (Ω, F, (F t ) 0 t T, P). Question Is the model arbitrage-free? Question Is the model complete? In other words, does it allow replication of any non-traded derivative? 14 / 24

16 Fundamental Theorems of Asset Pricing Let Q denote the family of martingale measures for S, that is, Q = {Q P : S is a local martingale under Q} Theorem (1st FTAP) Absence of arbitrage Q. Theorem (2nd FTAP) Completeness Q = / 24

17 Risk-Neutral Valuation Consider a European option with payoff Ψ at maturity T. The formula V 0 = E Q [Ψ], where Q Q is called Risk-Neutral Valuation. Arbitrage-free models: Unique Arbitrage-Free Pricing = Replication Complete models: (no replication!) Arbitrage-Free Pricing = Risk-Neutral Valuation 16 / 24

18 Free Lunch with Vanishing Risk For 1st FTAP to hold true the following definition of arbitrage is needed (Delbaen and Schachermayer, Math Ann, 1994): 1. There is a set A Ω with P[A] > For any ɛ > 0 there is a strategy X such that 2.1 X is admissible, that is, for some constant c > 0, X c. 2.2 X 0 ɛ (start with almost nothing) 2.3 X T 1 A (end with something) 17 / 24

19 Verification of the absence of arbitrage Assume that F t = Ft S (the information is generated by S). Then without loss in generality (Ω, F, (F t ) 0 t T, P) is a canonical probability space of continuous functions ω = ω(t) on [0, T ] and S t (ω) = ω(t). Suppose t S t = S 0 + µ t du + Wt P, 0 where µ t = µ((s u ) u t, t) and W P is a P-Brownian motion. Problem Find (necessary and sufficient) conditions on µ = (µ t ) for the absence of arbitrage (No FLVR). 18 / 24

20 Solution Levi s theorem = that the only possible martingale measure Q is such that W Q t = S t S 0, is a Q-Brownian motion. Then by 1st FTAP No FLVR P Q. One can show (easy!) that P Q T 0 µ 2 t dt < P + Q a.s.. 19 / 24

21 Martingale Representation (MR) (Ω, F T, F = (F t ) t [0,T ], P): a complete filtered probability space. Q: an equivalent probability measure. S = (S j t ): J-dimensional martingale under Q. Martingale representation (MR= MR(S)): every martingale M = (M t ) under Q admits an integral representation with respect to S, that is, t M t = M 0 + H u ds u, t [0, T ], 0 for some predictable S-integrable process H = (H j t). Completeness in Mathematical Finance. Jacod s Theorem (2nd FTAP): MR holds iff Q is the only martingale measure for S. 20 / 24

22 Brownian framework Hereafter, we assume that P = Q and that there is a d-dimensional Brownian motion W such that F t = F W t, t [0, T ]. Ito s MR(W ) : every martingale M = (M t ) admits an integral representation t M t = M 0 + H u dw u, t [0, T ], 0 for some adapted process H = (H t ) such that T 0 H 2 t dt <. 21 / 24

23 Forward setup Input: volatility process Σ = (Σ t ) taking values in J d-matrices. Stocks prices S = (S j ) are a J-dimensional Ito s martingale: S t = S 0 + t 0 Σ u dw u By Ito s MR(W ) we have that MR(S) holds if and only if for every H there is K such that HdW = KdS = KΣdW which, by linear algebra, is the case if and only if rank(σ) = d, (dp dt a.s.). Very easy to verify! 22 / 24

24 Backward setup Input: terminal values ψ = (ψ j ) for S: S t = E[ψ F t ], t [0, T ]. This setup arises, e.g, when liquid securities are derivatives: forwards, futures, options; also, in equilibrium. Ito s MR(W ) = the existence of Σ = (Σ t ) such that t S t = S 0 + Σ u dw u 0 Conditions on ψ so that MR(S) holds rank(σ) = d? 23 / 24

25 Summary (FTAP1): Model is arbitrage-free There is a martingale measure Q for prices S. (FTAP2): Model is complete There is only one martingale measure Q for S. MR property holds for S under Q. Easy to verify in forward setup. Non-trivial research topic in backward framework. Unique Arbitrage-Free Pricing = Replication. Replication is hard. In complete models we can avoid replication and compute prices as expectations E Q [Ψ]. 24 / 24