What is Financial Mathematics? 1 Introduction • Financial Mathematics is a collection of mathematical techniques that find applications in finance, e.g. – Asset pricing: derivative securities. – Hedging and risk management – Portfolio optimization – Structured products • There are two main approaches: – Partial Differential Equations – Probability and Stochastic Processes 2 Short History of Financial Mathematics • 1900: Bachelier uses Brownian motion as underlying process to derive option prices. • 1973: Black and Scholes publish their PDE-based option pricing formula. • 1980: Harrison and Kreps introduce the martingale approach into mathematical finance. • Financial Mathematics has been established as a separate academic discipline only since the late eighties, with a number of dedicated journals. 3 Structure of this talk • Preliminary notions: Time value of money, financial securities, options. • Arbitrage and risk–neutral valuation via a one–period, two–state toy model. • Modelling stock price behaviour • Naive stochastic calculus • PDE approach to finance • Martingale approach to finance • Numerical methods • Current Research 4 Preliminary Notions Discounting and Financial Instruments • Finance may be defined as the study of how people allocate scarce resources over time. • The outcomes of financial decisions (costs and benefits) are – spread over time – not generally known with certainty ahead of time, i.e. subject to an element of risk • Decision makers must therefore – be able to compare the values of cashflows at different dates – take a probabilistic view 5 Discounting • The time value of money: R1.00 in the hand today is worth more than the expectation of receiving R1.00 at some future date. • Thus borrowing isn’t free: the borrower pays a premium to induce the lender to part with his/her money. This premium is the interest. • We shall make the simplifying assumptions that – There is only one interest rate: All investors can borrow and lend at this (riskless) rate. – The interest rate is constant over time. – The same rate applies for all maturities. 6 • Let r denote the continuously compounded interest rate, so that one unit of currency deposited in a (riskless) bank account grows to erT units in time T . • Thus an amount X at time T is the same as Xe−rT now. • Discounting allows us to compare amounts of money at different times. 7 Returns • The return on an investment S is defined by S R = ln T S0 i.e. ST = S0eRT The random variable R is essentially the “interest” obtained on the investment, and may be negative. • Investors attempt to maximize their expected return. Fundamental relationship in finance: • E[Return] = f (Risk) where f is an increasing function. 8 Securities • Securities are contracts for future delivery of goods or money, e.g. shares, bonds and derivatives. • One distinguishes between underlying (primary) and derivative (secondary) instruments. • Examples of underlying instruments are shares, bonds, currencies, interest rates, and indices. • A derivative (or contingent claim) is a financial instruments whose value is derived from an underlying asset. • Examples of derivatives are forward contracts, futures, options, swaps and bonds. 9 • There are two main reasons for using derivatives: Hedging and Speculation. • Thus derivatives are essentially tools for transferring risk, and will allow one to diminish or increase one’s exposure to uncertain events. • An option gives the holder the right, but not the obligation to buy or sell an asset. • A European call option gives the holder the right to buy an asset S (the underlying) for an agreed amount K (the strike price) on a specified future date T (maturity). 10 • Thus the payoff at expiry is max{S(T ) − K, 0} • Since the payoff can never be negative, but is sometimes positive, options aren’t free. The premium paid for the option is related to the risk (“probability”) that the share price is greater than the strike at expiry. 11 Risk-Neutral Valuation • Consider a toy model with just two trading dates t = 0 and t = T , and just two financial assets – A risk–free bank account A paying a constant simple rate r = 10% over the interval [0, T ]. – A risky stock S. Today’s stock price is S0 = 10. • At time T , there are only two possible states of the world, UP and DOWN. 12 • We model this using the tuple (Ω, P, F , T, F, (At, St)t∈T) • Here Ω = {Up, Down}, and P is a probability measure on Ω. 13 • Consider a European call option on S with strike price K = 11 and maturity T . At maturity the call option has the following possible values: • How would we find “the” fair price C0 for this contract at t = 0? 14 • Two possibilities come to mind: – METHOD I. Calculate the expected value of the future payoff, and discount this to the present. Thus 1 [P(UP) · 11 + P(Down) · 0] 1.1 = 10 · P(UP) C0 = ∗ PROBLEM: How do we determine the measure P? If we consider both states equally likely, the value of the call option will be C(0) = 5 – METHOD II. The price of the option will be determined by the market, in particular by supply and demand. 15 16 • The correct price can be determined by an arbitrage argument, as follows: • Consider a portfolio θ = (θ0, θ1) containing an amount θ0 in the bank and a quantity θ1 shares. The initial value of the portfolio is V0(θ) = θ0 + 10θ1. • We want to ensure that the portfolio has the same value as the call option in all states of the world at expiry. UP DOWN VT (θ) = 1.1θ0 + 22θ1 = 11 VT (θ) = 1.1θ0 + 5.5θ1 = 0 i.e. 10 2 θ= − , 3 3 2 shares, and buy • Thus if you borrow 10 3 3 the resulting portfolio has the same cashflows at maturity as the call option. 17 • To exclude arbitrage, the initial value of the option must be the same as the initial value of the portfolio, i.e. 10 2 10 C(0) = − + 10 · = 3 3 3 • Arbitrage is the possibility of making a profit without the possibility of making a loss. • In the preceding example, if the option costs less than the portfolio, then – Short the portfolio; – Use the proceeds to buy the option; – And put the remainder in the bank. 18 • Note that the option price using discounted expected values was 5, which is higher than 10/3. How can this be? • If we insist on using the probability measure P, then the share itself is priced “incorrectly”. – Its value ought to have been 1 1 1 S0 = (22) + (5.5) = 12.5 1.1 2 2 – but the real price is S0 = 10. 19 • This reflects the fact that investors are risk averse. In order to take on the risk of the share, investors require a risk premium Rp: 1 10 = S0 = EP[ST ] 1 + r + Rp 1 1 1 = [ · 22 + · 5.5] 1.1 + Rp 2 2 • Suppose that we now change the probability measure to a new measure Q under which investors are risk–neutral, i.e. under which they do not require a risk premium. • In this world, the current value of the share is its discounted expected value. 1 10 = [Q(UP) · 22 + (1 − Q(UP)) · 5.5] 1.1 1, which implies that Q(UP) = 3 and Q(DOWN) = 2 3. 20 • If we price the option using the discounted expected value under the risk–neutral measure Q, we get 1 1 2 10 C(0) = ( · 11 + · 0) = 1.1 3 3 3 • and this is CORRECT!!! • Principle of Risk–Neutral Valuation: – The t = 0–value of an option is its discounted expected value. – However, the expectation is taken under a risk–neutral probability measure, which we can calculate. – And not under the “real–world” probability measure, which we can never know. 21 Modelling Stock Prices • Any model of stock price behaviour must be stochastic, i.e. incorporate the random nature of price behaviour. The simplest such models are random walks. • Partition the interval [0, T ] into subintervals of length ∆t 0 = t0 ≤ t1 ≤ · · · ≤ tN = T N = T ∆t • Let Xtn , n = 1, 2, . . . N be a family of random variables, and let S0 be the stock price at t = 0. We might (naively) attempt to model the stock price process by Stn+1 = Stn + Xtn+1 22 • Thus St = S0 + t X Xu u=1 • The intuition behind this is that the price at time t + ∆t equals the price at time t plus a “random shock”, modelled by Xt. • We also assume that these shocks are independent. • Efficient Markets Hypothesis: Stock price processes are Markov processes. 23 • Fact: If Xn are independent random variables, then X var( n Xn) = X n var(Xn) • Thus if the Xn are independent, identically distributed, then the variance of the sum is proportional to the number of terms. • So the variance of the stock price in our naive random walk model is proportional to the elapsed time. 24 • We attempt to build a continuous–time model of stock price behaviour over an interval [0, T ]. As a first approximation, we use Bernoulli shocks every unit time, i.e. we let ( Xt = +∆S with probability 0.5 −∆S with probability 0.5 • Note that Var(Xt) = ∆S 2 Var(ST ) = N X Var(Xtn ) n=1 = N ∆S 2 ∆S 2 = T ∆t 25 • How large should the jumps in stock price be? To ensure that Var(ST ) goes to neither 0 nor ∞ as ∆t → 0, we must have √ ∆S = o( ∆t) • Note that for differentiable functions f (t), we have ∆f ≈ f 0(t)∆t, i.e. ∆f = o(∆t) • This shows that St cannot be differentiable! 26 • To build a continuous version of our model, we use the Central Limit Theorem: If Xn is a largish family of iid random variP ables, then n Xn is approximately normally distributed. • Thus: After a largish number of shocks, the stock price in our naive random walk model will be approximately normally distributed. • We seek a continuous-time version of the random walk — a stochastic process that is changing because of random shocks at every instant in time. 27 Brownian motion • Brownian motion is a continuous–time stochastic process Bt, t ≥ 0 with the following properties: (1) Each change Bt − Bs = (Bs+h − Bs ) + (Bs+2h − Bs+h ) + · · · + (Bt − Bt−h ) is normally distributed with mean 0 and variance t − s. (2) Each change Bt − Bs is independent of all the previous values Bu, u ≤ s. (3) Each sample path Bt, t ≥ 0 is (a.s.) continuous, and has B0 = 0. • Brownian motion is a martingale: EsBt = Bs s≤t where Es denotes the expectation at time s. 28 GBM • For stock prices, the Brownian motion model is inadequate. We expect the change in price to be proportional to the current price. • A better model for share prices is given by the stochastic differential equation dSt = µSt dt + σSt dBt • Here µ is the drift, i.e. the rate at which the share price increases in the absence of risk. The differential dBt models the randomness (risk), and the parameter σ, known as the volatility, models how sensitive the share price is to these random events. • This share price process is called a geometric Brownian motion. 29 Value process • Consider a market with a share St whose price process is a GBM dSt = µSt dt + σSt dBt • Let the risk–free interest rate be r, i.e. the risk–free bank account At satisfies the DE dAt = rAt dt At is the riskless asset. It has drift r and zero volatility. • Given a dynamic portfolio θt = (θt0, θt1), the value process Vt(θ) is defined by Vt(θ) = θt0 At + θt1 St • It satisfies the SDE dVt = θt0 dAt + θt1 dSt = (rθt0 At + µθt1 St) dt + θ1 σSt dBt 30 • The value of the portfolio at time T is therefore Z T VT (θ) = V0 (θ) + Z0T + [rθt0 At + µθt1 St] dt θ1 σSt dBt 0 • We now see that we need to be able to evaluate integrals of the form Z T f (t, ω) dBt(ω) 0 • The obvious method would be to regard the above as a Riemann–Stieltjes (or Lebesgue–Stieltjes) integral. 31 Stochastic Calculus Naive Approach • Let f (x) be a differentiable function on an interval [a, b]. Partition this interval: a = x0 < x1 < x2 < xn = b where xi+1 − xi = ∆x • Then by Taylor series expansion, we get f (xi+1 ) − f (xi) = f 0 (xi)∆x + + 1 00 f (xi)(∆x)2 2! 1 000 f (xi)(∆x)3 + terms involving ∆x4 , ∆x5 , . . . 3! • Thus f (b) − f (a) = n−1 X i=0 n−1 X [f (xi+1 ) − f (xi)] n−1 1 X 00 0 = f (xi)∆x + f (xi)(∆x)2 + . . . 2 i=0 i=0 32 • As ∆x → 0, we get f (b) − f (a) = lim X ∆x→0 f 0 (xi)∆x i X 1 + lim f 00 (xi)(∆x)2 + . . . 2 ∆x→0 i Z b Z b 1 = f 00 (x) (dx)2 + . . . f 0 (x) dx + 2 a a • In ordinary calculus, only the first term counts (by the Fundamental Theorem of Calculus), and the other terms are zero. • This is because the quadratic variation of any “ordinary” function is zero, i.e. lim X (∆g)2 = 0 ∆x→0 for any “ordinary” function g. 33 • But Brownian motion is different: Consider ∆B = Bt+∆t − Bt. This is a normally distributed random variable with E[∆B] = 0 and variance var(∆B) = ∆t. • Consider next the random variable (∆B)2. This has E[(∆B)2] = var[∆B] = ∆t var[(∆B)2 ] = E[(∆B)4 ] − (∆t)2 = 2(∆t)2 << ∆t • Thus the variance of (∆B)2 is ≈ 0, i.e. though ∆B is a random variable, (∆B)2 is a constant (!! I promise that this can be made precise.) • It follows that lim ∆t→0 X 2 E(∆B) = lim ∆t→0 X ∆t = T where T is the total elapsed time. Thus the quadratic variation of Brownian motion is non–zero. 34 • Also lim ∆t→0 X 4 E(∆B) = 2 lim X (∆t)2 = 0 ∆t→0 because g(t) = t is an “ordinary” function, with quadratic variation zero. • Hence we cannot ignore the second–order term 1 2 Z b f 00 (x) (dx)2 a in the case that x = B. • But we can ignore all higher–order terms. • We thus have the following rules for stochastic calculus: (dBt)2 = dt dBt · dt = (dt)2 = 0 35 • Suppose that f (t, x) is a C 1,2–function, and let Xt = f (t, Bt). Applying these rules to a second order Taylor series, we obtain: Theorem: (Ito’s Formula) dXt = 1 ∂ 2f ∂f + ∂t 2 ∂B 2 ! dt + ∂f dBt ∂B • Ordinary calculus shows that for a function f (t, x) we have df = ∂f ∂f dt + dx ∂t ∂x • In stochastic calculus, we get another term, due to the non–zero quadratic variation of Brownian motion. 36 • Since Brownian motion has non-zero quadratic variation, Brownian sample paths are (a.s.) of unbounded variation. • This means that in general the Ito stochasR tic integral 0T f dBt cannot be interpreted as a Riemann–Stieltjes integral. • Nevertheless, the stochastic integral can be defined with semimartingale integrators (using an approximation in a L2– space, rather than an (almost) pointwise limit). • Fact: The Ito integral Z Mt = t f (u, Bu ) dBu 0 is a (local) martingale, i.e. Z t Z f dBu = Es 0 s f dBu 0 37 Stock price process parameters • Let’s have another look at volatility. The GBM model for stock prices is dSt = µSt dt + σSt dBt Thus dS 2 E = σ 2 dt S and thus σ 2 dt is the variance of the return of the stock over a small period dt. • It follows that σ is the standard deviation of the annual return of the stock S. • This can be measured from market data. 38 • Can we also measure the drift µ? No. • So the correct, real-world dynamics of a share price are unknowable: We can get the volatility, but not the drift. • Amazingly, we don’t care!! 39 Black-Scholes Model PDE Approach • Consider again market with a share St whose price process satisfies the SDE dSt = µSt dt + σSt dBt • Let the risk–free interest rate be r, and let At be the riskless bank account, with dynamics dAt = rAt dt • Let V (t, St) be European–style derivative whose value depends on both the share price and time. Consider a portfolio Π which contains 1 derivative, and n shares, i.e. its value is Πt = Vt + nSt 40 • A small amount of time dt later, the share price has changed. The value of the portfolio changes by dΠt = dVt + n dSt • By Ito’s Formula, ∂V ∂V 1 ∂ 2V 2 dVt = dt + dS + dS ∂S 2 ∂S 2 ∂t ∂V ∂V ∂V 1 2 2 ∂V = dt + σS + µS + σ S dBt ∂t ∂S 2 ∂S 2 ∂S • Hence ∂V ∂V 1 ∂V dΠt = + µS + σ 2 S 2 2 + nµS ∂t ∂S 2 ∂S ∂V + n dBt + σS ∂S dt 41 • Now if we take n = − ∂V ∂S (i.e. the portfolio is short − ∂V ∂S shares), then the portfolio is unaffected by a random change in the stock price: dΠt = ∂V 1 ∂V + σ2S 2 2 ∂t 2 ∂S dt (1) • Thus, for a brief instant, the portfolio is risk–free. By a no–arbitrage argument, it must earn the same return as the risk–free bank account, i.e. ∂V dΠt = rΠt dt = r V − S ∂S dt (2) 42 • Equating (1) and (2), we get ∂V 1 2 2 ∂ 2V ∂V + σ S + rS − rV = 0 2 ∂t 2 ∂S ∂S • This is the famous Black–Scholes PDE. It is a second–order parabolic PDE, i.e. essentially a heat equation. • It is now clear why we don’t care about the drift µ of the underlying asset S: It does not appear in the BS PDE!! 43 Black–Scholes Model Risk–Neutral Approach • Since we don’t care about the drift rate µ of an underlying asset, we may as well simplify our asset price dynamics by assuming that all assets have the same drift. • The riskless asset (bank account) has drift r, which we can actually see. We thus assume that all assets have the same return, namely the risk–free rate r. • Mathematically, this corresponds to a change of measure — from a real world, unknowable probability measure P to a knowable, risk–neutral measure Q. In the risk–neutral world, the dynamics of S are dSt = rSt dt + σS dBt Thus we change the drift of the asset from µ to r. 44 • Mathematically, this is accomplished using the Cameron–Girsanov Theorem: Let dYt = µ(t, ω) dt + θ(t, ω) dBt be an Ito process in a filtered probability space (Ω, Ft, P) Suppose that there exists processes u(t, ω) and ν(t, ω) such that θ·u=ν−µ Define a process M by 1 t ||u||2 ds) Mt = exp( u dBs − 2 0 0 and define a measure Q by Z t Z dQ = MT dP Then under Q, Y –dynamics are ˆt dYt = ν(t, ω) dt + θ(t, ω) dB ˆt is a Q–Brownian motion. where B • Amazingly, a change of measure changes only the drift and not the volatility. 45 • We can calculate option prices in the risk– neutral world, because the asset price dynamics/distributions are known. • But: Prices in the real– and risk–neutral world are the same! It is just probabilities that are changed. Fundamental Theorem of Asset Pricing: There are no arbitrage opportunities • if and only if there exists a risk–neutral measure. 46 PDE = Risk–Neutral • Consider a European call option C on a share S with strike K and maturity T . The volatility of the underlying share S is σ and the risk–free rate is r. • We must solve the following BVP: ∂V ∂V 1 2 2 ∂ 2V + rS + σ S − rV = 0 ∂t 2 ∂S 2 ∂S V (T ) = Φ(ST ) = max{ST − K, 0} • Theorem: In the risk–neutral world, the Vt discounted value process A = e−rtVt is t a martingale: −rT e Z VT = V0 + T e−rtσSt dBt 0 47 • It follows that the expected value of e−rtVt at any time is its current value, and thus the value of the call option with strike K and maturity T is given by V0 = E0 [e−rT VT ] = e−rT E0 [max{ST − K, 0}] • In the same way, the value of any European– style contingent claim V with maturity T and payoff (boundary condition) V (T, ST ) = Φ(ST ) is simply V0 = e−rT E[Φ(ST )] • This is a version of the Feynman–Kac Theorem, which gives the solution to a large class of parabolic PDE’s as an expectation of a diffusion (here with loss of mass, represented by discounting). 48 Computational Toolbox • Numerical integration • Optimization techniques • Finite difference methods • Lattice/tree methods • Monte Carlo and quasi–Monte Carlo methods • Statistical techniques: Principal component analysis, factor analysis, maximum entropy 49 • Time series analysis • Numerical solution of SDE’s • Dynamic programming • Stochastic control theory 50 Current Research • Altenatives to Black–Scholes – Stochastic volatility models. – Jump diffusions. – Levy processes. • Interest–rate modelling. • Pricing in incomplete markets. • Pricing/measuring/hedging credit risk. 51 • Capital adequacy based risk measures • Real options • Differential game theory • Entropy–based option pricing • Viability theory • Non-standard finance 52

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