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Vidyasagar rocond09

1. 1. A Tutorial Introduction to Quantitative Finance M. Vidyasagar Executive Vice President Tata Consultancy Services Hyderabad, India sagar@atc.tcs.com ROCOND 2009, Haifa, Israel Full tutorial introduction to appear in Indian Academy of Sciences publication, available upon request.
2. 2. And Now for Something Completely Diﬀerent!
3. 3. Outline • • • • • • Preliminaries: Introduction of terminology, problems studied, etc. Finite market models: Basic ideas in an elementary setting. Black-Scholes theory: What is it and why is it so (in-)famous? Some simple modiﬁcations of Black-Scholes theory What went wrong? Some open problems that would interest control theorists
4. 4. Preliminaries
5. 5. Some Terminology A market consists of a ‘safe’ asset usually referred to as a ‘bond’, together with one or more ‘uncertain’ assets usually referred to as ‘stocks’. A ‘portfolio’ is a set of holdings, consisting of the bond and the stocks. We can think of it as a vector in Rn+1 where n is the number of stocks. Negative ‘holdings’ correspond to borrowing money or ‘shorting’ stocks. An ‘option’ is an instrument that gives the buyer the right, but not the obligation, to buy a stock a prespeciﬁed price called the ‘strike price’ K. A ‘European’ option can be exercised only at a speciﬁed time T . An ‘American’ option can be exercised at any time prior to a speciﬁed time T .
6. 6. European vs. American Options The value of the European option is {ST −K}+. In this case it is worthless even though St > K for some intermediate times. The American option has positive value at intermediate times but is worthless at time t = T.
7. 7. The Questions Studied Here • What is the minimum price that the seller of an option should be willing to accept? • What is the maximum price that the buyer of an option should be willing to pay? • How can the seller (or buyer) of an option ‘hedge’ (minimimze or even eliminate) his risk after having sold (or bought) the option?
8. 8. Finite Market Models
9. 9. One-Period Model Many key ideas can be illustrated via ‘one-period’ model. We have a choice of investing in a ‘safe’ bond or an ‘uncertain’ stock. B(0) = Price of the bond at time T = 0. (1 + r)B(0) at time T = 1. It increases to B(1) = S(0) = Price of the stock at time T = 0. S(1) = S(0)u with probability p, S(0)d with probability 1 − p. Assumption: d < 1 + r < u; otherwise problem is meaningless! Rewrite as d < 1 < u , where d = d/(1 + r), u = u/(1 + r).
10. 10. Options and Contingent Claims An ‘option’ gives the buyer the right, but not the obligation, to buy the stock at time T = 1 at a predetermined strike price K. Again, assume S(0)d < K < S(0)u. More generally, a ‘contingent claim’ is a random variable X such that X= Xu if S(1) = S(0)u, Xd if S(1) = S(0)d. To get an option, set X = {S(1) − K}+. Such instruments are called ‘derivatives’ because their value is ‘derived’ from that of an ‘underlying’ asset (in this case a stock). Question: How much should the seller of such a claim charge for the claim at time T = 0?
11. 11. An Incorrect Intuition View the value of the claim as a random variable. X= Xu with probability pu = p, Xd with probability pd = 1 − p. So E[X, p] = (1 + r)−1[pXu + (1 − p)Xd]. Is this the ‘right’ price for the contingent claim? NO! The seller of the claim can ‘hedge’ against future ﬂuctuations of stock price by using a part of the proceeds to buy the stock himself.
12. 12. The Replicating Portfolio Build a portfolio at time T = 0 such that its value exactly matches that of the claim at time T = 1 irrespective of stock price movement. Choose real numbers a and b (investment in stocks and bonds respectively) such that aS(0)u + bB(0)(1 + r) = Xu, aS(0)d + bB(0)(1 + r) = Xd, or in vector-matrix notation [a b] S(0)u S(0)d B(0)(1 + r) B(0)(1 + r) = [Xu Xd]. This is called a ‘replicating portfolio’. The unique solution is [a b] = [Xu Xd] S(0)u S(0)d B(0)(1 + r) B(0)(1 + r) −1 .
13. 13. Cost of the Replicating Portfolio So how much money is needed at time T = 0 to implement this replicating strategy? The answer is c = [a b] S(0) B(0) = [Xu Xd] S(0)u S(0)d B(0)(1 + r) B(0)(1 + r) = (1 + r)−1[Xu Xd] qu qd −1 S(0) B(0) , where with u = u/(1 + r), d = d/(1 + r), we have q := qu qd = u d 1 1 −1 1 1   1−d =  u −d  u −1 u −d Note that qu, qd > 0 and qu + qd = 1. So q := (qu, qd) is a probability distribution on S(1).
14. 14. Martingale Measure: First Glimpse Important point: q depends only on the returns u, d, and not on the associated probabilities p, 1 − p. Moreover, −1 S(1), q] = S(0)u 1 − d + S(0)d u − 1 = S(0). E[(1 + r) u −d u −d Under this synthetic distribution, the discounted expected value of the stock price at time T = 1 equals S(0). Hence {S(0), (1 + r)−1S(1)} is a ‘martingale’ under the synthetic probability distribution q. Thus c = (1 + r)−1[Xu Xd] qu qd is the discounted expected value of the contingent claim X under the martingale measure q.
15. 15. Arbitrage-Free Price of a Claim Theorem: The quantity c = (1 + r)−1[Xu Xd] qu qd is the unique arbitrage-free price for the contingent claim. Suppose someone is ready to pay c > c for the claim. Then the seller collects c , invests c − c in a risk-free bond, uses c to implement replicating strategy and settle claim at time T = 1, and pockets a riskfree proﬁt of (1 + r)(c − c). This is called an ‘arbitrage opportunity’. Suppose someone is ready to sell the claim for c < c. Then the buyer can make a risk-free proﬁt.
16. 16. Multiple Periods: Binomial Model The same strategy works for multiple periods; this is called the binomial model. Bond price is deterministic: Bn+1 = (1 + rn)Bn, n = 0, . . . , N − 1. Stock price can go up or down: Sn+1 = Snun or Sndn. There are 2N possible sample paths for the stock, corresponding to each h ∈ {u, d}N . For each sample path h, at time N there is a payout Xh. Claim becomes due only at the end of the time period (European contingent claim). In an ‘American’ option, the buyer chooses the time of exercising the option.
17. 17. Deﬁne dn = dn/(1 + rn), un = un/(1 + rn), 1 − dn un − 1 . qu,n = , qd,n = un − dn un − dn; Introduce a modiﬁed stochastic process Sn+1 = Snun with probability qu,n, Sndn with probability qd,n. Then E{(1 + rn)−1Sn+1|Sn, Sn−1, . . . , S0} = Sn, for n = 0, . . . , N − 1. Thus the discounted process  n−1     (1 + ri)−1Sn i=0 where the empty product is taken as one, is a martingale.
18. 18. Replicating Strategy for N Periods We already know to replicate over one period. Extend argument to N periods. Suppose j ∈ {u, d}N −1 is the set of stock price transitions up to time N − 1. So now there are only two possibilities for the ﬁnal sample path: ju and jd, and only two possible payouts: Xju and Xjd. Denote these by cju and cjd respectively. We already know how to compute a cost cj and a replicating portfolio [aj bj] to replicate this claim, namely: cj = (1 + rn−1)−1 cjuqu,n + cjdqd,n . [aj bj] = [cju cjd] Sjuj Sjdj Bj(1 + rN −1) Bj(1 + rN −1) −1 .
19. 19. Do this for each j ∈ {u, d}N −1. So if we are able to replicate each of the 2N −1 payouts cj at time T = N − 1, then we know how to replicate each of the 2N payouts at time T = N . Repeat backwards until we reach time T = 0. decreases by a factor of two at each time step. Number of payouts
20. 20. Arbitrage-Free Price for Multiple Periods Recall earlier deﬁnitions: dn un 1 − dn un − 1 dn = ,u = , qu,n = . , qd,n = 1 + rn n 1 + rn un − dn un − dn; Now deﬁne N −1 qh = qh,n, ∀h ∈ {u, d}N , n=0  −1 N −1 c0 :=  (1 + rn) Xhqh. n=0 h∈{u,d}N c0 is the expected value of the claim Xh under the synthetic distribution {qh} that makes the discounted stock price a martingale. Moreover, c0 is the unique arbitrage-free price for the claim.
21. 21. Replicating Strategy in Multiple Periods Seller of claim receives an amount c0 at time T = 0 and invests a0 in stocks and b0 in bonds, where [a0 b0] = [cu cd] S0u0 S0 d 0 B0(1 + r0) B0(1 + r0) −1 . Due to replication, at time T = 1, the portfolio is worth cu if the stock goes up, and is worth cd if the stock goes down. At time T = 1, adjust the portfolio according to [a1 b1] = [ci0u ci0d] S1u1 S1 d 1 B1(1 + r1) B1(1 + r1) where i0 = u or d as the case may be. Then repeat. −1 ,
22. 22. Important note: This strategy is self-ﬁnancing: a0S1 + b0B1 = a1S1 + b1B1 whether S1 = S0u0 or S1 = S0d0 (i.e. whether the stock goes up or down at time T = 1). This property has no analog in the one-period case. It is also replicating from that time onwards. Observe: Implementation of replicating strategy requires reallocation of resources N times, once at each time instant.
23. 23. Black-Scholes Theory
24. 24. Continuous-Time Processes: Black-Scholes Formula Take ‘limit’ at time interval goes to zero and N → ∞; binomial asset price movement becomes geometric Brownian motion: 1 µ − σ 2 t + σWt , t ∈ [0, T ], 2 where Wt is a standard Brownian motion process. µ is the ‘drift’ of the Brownian motion and σ is the volatility. St = S0 exp Bond price is deterministic: Bt = B0ert. simple option: XT = {ST − K}+. Claim is European and a What we can do: Make µ, σ, r functions of t and not constants. What we cannot do: Make σ, r stochastic! (Stochastic µ is OK.)
25. 25. Theorem (Black-Scholes 1973): price is log(S0/K ∗) 1 √ √ C0 = S0Φ + σ T 2 σ T The unique arbitrage-free option − K ∗Φ log(S0/K ∗) 1 √ √ − σ T 2 σ T , where c 1 −u2 /2 du √ e Φ(c) = 2π −∞ is the Gaussian distribution function, and K ∗ = e−rT K is the discounted strike price.
26. 26. Black-Scholes PDE Consider a general payout function erT ψ(e−rT x) to the buyer if ST = x (various exponentials discount future payouts to T = 0). Then the unique arbitrage-free price is given by C0 = f (0, S0), where f is the solution of the PDE ∂f 1 2 2 ∂ 2f + σ x = 0, ∀(t, x) ∈ (0, T ) × (0, ∞), 2 ∂t 2 ∂x with the boundary condition f (T, x) = ψ(x). No closed-form solution in general (but available if ψ(x) = (x − K ∗)+).
27. 27. Replicating Strategy in Continuous-Time Deﬁne ∗ Ct = C0 + t 0 ∗ ∗ fx(s, Ss )dSs , t ∈ (0, T ), where the integral is a stochastic integral, and deﬁne ∗ ∗ ∗ αt = fx(t, St ), βt = Ct − αtSt . Then hold αt of the stock and βt of the bond at time t. Observe: Implementation of self-ﬁnancing fully replicating strategy requires continuous trading.
28. 28. Some Simple Modiﬁcations of Black-Scholes Theory
29. 29. Extensions to Multiple Assets Binomial model extends readily to multiple assets. Black-Scholes theory extends to the case of multiple assets of the form (i) St (i) = S0 exp 1 (i) µ(i) − [σ (i)]2 t + σWt , t ∈ [0, T ], 2 (i) where Wt , i = 1, . . . , d are (possibly correlated) Brownian motions. (1) (d) Analog of Black-Scholes PDE: C0 = f (0, S0 , . . . , S0 ) where f satisﬁes a PDE. But no closed-form solution for f in general.
30. 30. American Options An ‘American’ option can be exercised at any time up to and including time T . So we need a ‘super-replicating’ strategy: The value of our portfolio must equal or exceed the value of the claim at all times. If Xt = {St − K}+, then both price and hedging strategy are same as for European claims. Very little known about pricing American options in general. Theory of ‘optimal time to exercise option’ is very deep and diﬃcult.
31. 31. Sensitivities and the ‘Greeks’ Recall C0 = f (0, S0) is the correct price for the option under BlackScholes theory. (We need not assume the claim to be a simple option!) ∂C0 ∂∆ ∂ 2C0 ∆= , ,Γ = = 2 ∂S0 ∂S0 ∂S0 ∂C0 ∂f (t, Xt) ∂f (t, Xt) ,θ = − ,ρ = . ∂σ ∂t ∂r ‘Delta-hedging’: A strategy such that ∆ = 0 – return is insensitive to initial stock price (to ﬁrst-order approximation). Vega = ν = ‘Delta-gamma-hedging’: A strategy such that ∆ = 0, Γ = 0 – return is insensitive to initial stock price (to second-order approximation).
32. 32. What Went Wrong?
33. 33. What Went Wrong? My view: The current ﬁnancial debacle owes very little to poor modeling of risks. Most signiﬁcant factors were: • Complete abdication of oversight responsibility by US government – led to over-leveraging and massive conﬂicts of interests • OTC trading of complex instruments – made ‘price discovery’ impossible ⇒ Net outstanding derivative positions: \$ 760 trillion! ⇒ U.S. Annual GDP: \$ 13 trillion, World’s: \$ 60 trillion ⇒ 90% of derivatives traded OTC – No regulation whatsoever! • ‘One-way’ rewards for traders: Heads the traders win – tails the depositors and shareholders lose • Real bonuses paid on virtual proﬁts, and so on Please read full paper for elaboration.
34. 34. Some Open Problems That Would Interest Control Theorists
35. 35. Some Open Problems • Incomplete markets: Replication is impossible • Multiple martingale measures: Which one to choose? • Computing the ‘greeks’ when there are no closed-form formulas • Alternate models for asset price movement: Why geometric Brownian motion?
36. 36. Incomplete Markets Consider the one-period model where the asset takes three, not two, possible values.   S(0)u  with probability pu, S(1) = S(0)m with probability pm,   S(0)d with probability p , d To construct a replicating strategy, we need to choose a, b such that [a b] S(0)u S(0)m S(0)d (1 + r)B(0) (1 + r)B(0) (1 + r)B(0) No solution in general – replication is impossible! = [Xu Xm Xd].
37. 37. Martingale measures Let us try to ﬁnd a synthetic measure q = (qu, qm, qd) so that E[(1 + r)−1S(1), q] = S(0). Need to choose qu, qm, qd such that qu + qm + qd = 1, quu + qmm + qdd = 1 + r. Inﬁnitely many martingale measures! Is there a relationship?
38. 38. Finite Market Case: DMW Theorem Consider d assets over N time instants, each assuming values in R. We allow inﬁnitely many values for each asset, correlations, etc. Let ˜ P denote the law of these assets (law over RdN ). Theorem (Dalang-Morton-Willinger): are equivalent: The following statements ˜ ˜ • There exists a probability measure Q on RdN that is equivalent to P ˜ such that, under Q, the process {Sn}N n=0 is a martingale. • There does not exist an arbitrage opportunity. Moreover, if either of these equivalent conditions holds, then it is pos˜ sible to choose the measure Q in such a way that the Radon-Nikodym ˜ ˜ derivative (dQ/dP ) is bounded.
39. 39. Existence of a Replicating Strategy Deﬁne ˜ ˜ V−(X) := min E[X, Q], V+(X) := max E[X, Q], ˜ Q∈M ˜ Q∈M where M is the set of martingale measures. [V−(X), V+(X)] is arbitrage-free. Then every price in Note: V−(X) is the maximum that the buyer of the claim would (normally) be willing to pay. Similarly, V+(X) is the minimum that the seller of the claim would (normally) be willing to accept. Theorem: A European contingent claim with the payout function X is replicable if and only if V−(X) = V+(X). Modulo technical conditions, replicability ≈ unique martingale measure.
40. 40. Minimum Relative Entropy Martingage Measures ˜ In incomplete markets (M is inﬁnite), choose Q to minimize the relative entropy: ˜ ˜ Q∗ = arg min H(Q ˜ Q∈M ˜ P ), where n ˜ H(Q ˜ P) = qi qi log . pi i=1 ˜ Choose martingale measure that is ‘closest’ to the real-world law P . ˜ Q∗ has a nice interpretation in terms of maximizing utility. Since M is ˜ ˜ ˜ a convex set and H(Q P ) is convex in Q, this is a convex optimization problem.
41. 41. Computing the Greeks Even for simple geometric Brownian motion models, no closed form for fair price C0 = f (0, S0) unless claim X is a simple option. So how to compute the ‘greeks’ ? One possibility: Evaluate fair price(s) using stochastic (Monte Carlo) simulation; then diﬀerentiate numerically. Absolutely fraught with numerical instabilities and sensitivities. Answer: Malliavin calculus. Originally developed for Brownian motion, now extented to arbitrary L´vy processes. e
42. 42. Alternate Models for Asset Prices Geometric L´vy processes: Assume e St = S0 exp(Lt), where {Lt} is a L´vy process. e A L´vy process is the most general process with independent incree ments. For all practical purposes, a L´vy process consists of a Wiener e process (Brownian motion) plus a countable number of ‘jumps’. Sad fact: Replication is possible only for geometric Brownian motion asset prices!
43. 43. Ergo: GBM (Geometric Brownian Motion) is the only asset price model for which there exists a unique martingale measure and the theory is very ‘pretty’. Unfortunately, real world distributions are quite far from Gaussian, in the range [µ − 10σ, µ + 10σ]. Moreover, they are also ‘skewed’ in the positive direction. A tractable theory of pricing claims with real world probabilities is still waiting to be discovered.
44. 44. Conclusions Quantitative ﬁnance has deﬁnitely made pricing and trading more ‘scientiﬁc’, but several poorly understood issues still remain: • Irrational behavior • Non-equilibrium economics (e.g. incomplete information) • Correlated increments in asset price returns • Eﬀects of alternate asset price models on option pricing and/or trading strategies. In short, what is unknown dwarfs what is known. Thank You!