Diusion processes. Olivier Scaillet. University of Geneva and Swiss Finance Institute

Transcription

1 Diusion processes Olivier Scaillet University of Geneva and Swiss Finance Institute

2 Outline 1 Brownian motion 2 Itô integral 3 Diusion processes 4 Black-Scholes 5 Equity linked life insurance 6 Merton model

3 Introduction In the previous chapters, we considered discrete stochastic processes. Here, we consider continuous-time processes with continuous sample paths. The main example of such processes is the Brownian motion.

4 Bownian motion A Brownian motion B = {B t ; t 0} starting at B 0 = b is a process such that 1 B has independent increments 2 B t+h B t follows a Normal distribution N(0, σ 2 h) 3 the sample paths of B are continuous. The process B is called standard Brownian motion if σ 2 = 1 and B 0 = 0. It is often denoted by W and is also called Wiener process.

5 Distribution of increments The increments of a Brownian motion are stationary as the distribution of B t+h B t only depends on h. Conditionally on B s = x, we have that B t follows a Normal distribution N(x, t s), that is, F (t, y, s, x) = P[B t y B s = x] has a density function f (t, y, s, x) = F (t,y,s,x) 1 = y exp( 1 2π(t s) 2 (y x) 2 t s ).

6 Forward and backward equations The transition density f satises the following partial dierential equations: forward equation: f backward equation: f t = 1 2 f 2 y 2 s = 1 2 f 2 x 2 The forward equation is obtained by xing B s = x, whereas the backward equation is obtained by xing B t = y.

7 First passage time The rst passage time T (x) of the standard Brownian motion at point x is dened by T (x) = inf{t : B t = x} (the requirement of continuous sample paths is obvious for the denition to make sense). The random variable ( T )(x) has a density function f (t) = x exp x2, t 0. 2πt 3 2t

8 Distribution of the maximum We can also characterize the density of the maximum m t = max{b s ; 0 s t}. We note that T (m) t m t m. We could prove that the random variable m ( t has ) a density function 2 f (m) = πt exp m2, m 0. 2t The sample paths of the Brownian motion satisfy the principles of translation and reection.

9 Example: Bachelier's application One of the rst applications of the Brownian motion was proposed by Bachelier (1900). Bachelier used the Brownian motion to describe the evolution of prices on Paris stock exchange. Bachelier assumed that the innitesimal price increments dx t of a nancial asset are proportional to the increments db t of a standard Brownian motion, that is, dx t = σdb t. Starting at an initial value X 0 = x, the value of the process at time t is X t = x + σb t.

10 Example: Bachelier's application (cont'd) A major drawback of this specication is that the price has a non zero probability of getting negative. In order to solve this issue, we rather model the relative increments with respect to the prices (returns) as a standard Brownian motion, that is, dxt X t = σdb t or equivalently dx t = σx t db t. The second expression looks like a dierential equation. However, there are two diculties: 1 the variables in the equation are stochastic 2 the sample paths of B t are not dierentiable even though they are continuous.

11 Example: Bachelier's application (cont'd) The mathematical solution to this problem was found by Itô in the 40's using a new kind of integral: the stochastic integral. In particular, it allows to write X t = x + σ t 0 X sdb s where we integrate with respect to the random element B.

12 Stochastic Itô integral The stochastic integral is built in a similar way as the Riemann integral. The integral is rst dened on a class of piecewise constant processes and is then extended to a larger class by approximation. There are nonetheless two major dierences between Riemann and Itô integrals. The rst one is the convergence type: the Riemann integral converges in R whereas the Itô integral is approximated by sequences of random variables that converge in L 2, the space of square-integrable random variables (nite variance).

13 Stochastic Itô integral (cont'd) The second dierence is the following. Riemann sums approaching the integral of a function f : [0, T ] R have the form n 1 j=0 f (s j)(t j+1 t j ) with 0 = t 0 < t 1 < < t n = T and s j an arbitrary point in [t j, t j+1 ] for all j. The value of the Riemann integral does not depend on the points s j [t j, t j+1 ]. In a stochastic context, the sums have the form I (f n ) = n 1 j=0 f (s j)(w tj+1 W tj ). The limit of such approximations does depend on the choice of intermediary points s j [t j, t j+1 ]. In order to solve the ambiguity, we take s j = t j for all j. As we choose the left point of the interval, the approximations at a particular date only depend on the information known at this date, and not on future events.

14 Stochastic Itô integral (cont'd) Itô's integral is denoted by f (s)dw 0 s and is dened such that lim n E[ f (s)dw 0 s I (f n ) ] = 0. The stochastic integral has the following properties: 1 linearity: t (αf (u) + βg(u))dw 0 u = α t f (u)dw 0 u + β t g(u)dw 0 u [ 2 isometry: E ] t [ ] f (u)dw 0 u 2 t = E f (u) 0 2 du [ ] t 3 martingale property: E f (u)dw 0 u F s = s f (u)dw 0 u The stochastic integral is the core building block of the denition of diusion processes.

15 Diusion processes The stochastic integral allows us to dene integral equations of the form X t = X 0 + t µ(s, X 0 s)ds + t σ(s, X 0 s)dw s whose solution is called diusion process. We often write this equation in a compact dierential form dx t = µ(t, X t )dt + σ(t, X t )dw t called stochastic dierential equation. The term µ(t, X t ) is called drift The term σ(t, X t ) is called volatility (or diusion factor) and corresponds to the instantaneous standard deviation.

16 Applications in nance The geometric Brownian motion is used in Black-Scholes (1975) to model the evolution of stock prices: ds t = ms t dt + σs t dw t The Ornstein-Uhlenbeck process is used in Vasicek (1979) to model the evolution of the short rate r t : dr t = b(a r t )dt + sdw t The square root process is used in Cox-Ingersoll-Ross (1985) to model the evolution of the short rate r t : dr t = b(a r t )dt + s r t dw t

17 Euler discretization The Euler discretization X t+h X t = µ(t, X t )h + σ(t, X t ) hɛ t+h, ɛ t+h N(0, 1) corresponds to the discretization of dx t = µ(t, X t )dt + σ(t, X t )dw t with a time step of size h. It can be used to simulate approximate sample paths of the process by taking h small enough.

18 Itô's lemma Let X t be a diusion process such that X t = X 0 + t µ(s, X 0 s)ds + t σ(s, X 0 s)dw s, or equivalently dx t = µ(t, X t )dt + σ(t, X t )dw t. We aim at characterizing the evolution of Y t = f (t, X t ) through its stochastic dierential equation. In standard calculus, we would use f (t,xt) f (t,xt) the total dierential or chain rule dy t = dt + dx t X t. In stochastic calculus, we get an additional term σ 2 (t, X t )dt in comparison to the usual rule f (t,x t) X 2

19 Itô's lemma (cont'd) Itô's lemma gives the following relationship for Y t = f (t, X t ): dy t = f (t,xt) t dt f (t,x t) σ X 2 2 f (t,xt) (t, X t )dt + dx X t. Replacing dx t by its denition: [ ] dy t = f (t,xt) f (t,xt) + µ(t, X t X t ) f (t,x t) σ 2 X 2 2 (t, X t ) dt + f (t,x t) σ(t, X X t )dw t. The additional factor is a second order factor that is not negligible in comparison to rst order factors and is due to fast oscillations of the sample path of the process.

20 Itô's lemma (cont'd) Itô's formula ressembles a second order Taylor extension of f (t, X t ) where we replace dx t by its expression we apply the following rules: (dt) 2 = 0, dtdw t = 0, (dw t ) 2 = dt. In compact form, we get: dy = df = f t dt + f X dx ( f tt (dt) 2 + 2f Xt ) dxdt + f XX (dx )2 ) ( f XX σ2 dt = f t dt + f X (µdt + σdw ) + 1 ( 2 = f t + f X µ + 1 ) 2 f XX σ2 dt + σf X dw

21 Multidimensional Itô's lemma Let X = (X 1, X 2,...X d ) be a d-dimensional diusion process: dx t = µ t dt + σ t db t with µ = (d 1) vector, σ = (d k) matrix, and B = (B 1, B 2,...B k ) = (k 1) vector is a k-dimensional Brownian motion. Take dy t = df (t, X t ), then ( dy t = f t (t, X t ) + f x (t, X t )µ t + 1 ) 2 tr(σ tσ t f xx(t, X t )) dt +f x (t, X t )σ t db t with f t = f t, f x = row vector with partial derivatives f x i, and f xx = matrix with second partial derivatives 2 f x i x j.

22 Multidimensional Itô's lemma (cont'd) More concise form: dy t = Lf (t, X t )dt + f x (t, X t )σ t db t L is called the innitesimal generator or the Dynkin. Lf (t, X t ) = f t (t, X t ) + f x (t, X t )µ t tr(σ tσ t f xx(t, X t )) Other rule: dy t = f t (t, X t )dt + i f xi (t, X t )dxt i + 1 f xi,x 2 j dxt i dx j t i,j and product dx i t dx j t computed with the conventions dtdt = 0, dtdb i t = 0, db i t dbi t = dt and db i t dbj t = 0 if i j.

23 Integration by part (example) Let { dx 1 t = µ 1 t dt + σ1 t db t dx 2 t Show the integration by part rule ˆ t 0 = µ 2 t dt + σ2 t db t ˆ t Xs 1 dx s 2 = Xt 1 X t 2 X 1 X ˆ t Xs 2 dx s 1 0 σ 1 s σ 2 s ds and note that it diers from standard dierentiation rule d(uv) = du v + u dv and standard integration by part rule uv = du v + u dv.

24 Integration by part (example) Take Y t = Xt 1X t 2 and apply Itô's lemma ( with ) f = x 1 x Hence: f t = 0, f x = (x 2, x 1 ), f xx =. 1 0 Since d = 2 and k = 1, ( we get σ t = (σt 1 ), σt 1 ) which gives the (σ (2 2) matrix σ t σ t 1 = t ) 2 σt 1 σt 2 σt 1 σt 2 (σt 2 ) 2 ( ) σ Hence: σ t σ t 1 f xx = t σt 2 (σt 1 ) 2 and tr(σ (σt 2 ) 2 σt 1 σt 2 t σ t f xx) = 2σt 1 σt 2 and we get dy t = (X 2 t µ 1 t + X 1 t µ 2 t + σ 1 t σ 2 t )dt + (X 2 t σ 1 t + X 1 t σ 2 t )db t or d(x 1 t X 2 t ) = X 2 t dx 1 t + X 1 t dx 2 t + σ 1 t σ 2 t dt Q.E.D.

25 Black-Scholes model The Black-Scholes formula is a European option valuation formula. The Black-Scholes model considers an economy with two asset classes: a riskless asset, T-bill or zero coupon bond a risky asset, stock The value of the riskless asset M t grows at a continuously compounded constant interest rate r and is normalized such that M 0 = 1, that is, dm t = rm t dt with solution M t = e rt. The value of the risky asset S t follows a geometric Brownian motion, that is, ds t = ms t dt + ss t dw t with solution S t = S 0 exp ( (m 1 2 s 2 )t + sw t ).

26 Portfolio Let F = {F t ; t 0} be the ltration generated by the observation of the price history, F t = {S u ; 0 u t}. A portfolio is a pair α = {α t ; t 0} and β = {β t ; t 0} of F -adapted random processes. The pair (α, β) is a time-dependent portfolio containing α t units of stock and β t units of bond. The value of the portfolio at time t (α, β) is given by the value function V t = α t S t + β t M t.

27 Valuation by replication The portfolio is said to be self-nancing if dv t = α t ds t + β t dm t. It means that the changes in the value of the portfolio are only due to variations of the prices: there is no addition or withdrawal of funds. The self-nancing portfolio (α, β) replicates the payo function of a European call option if its value is equal to the value of the call at maturity, that is, V T = (S T K) +. By absence of arbitrage (AOA), we then have that the replicating portfolio value must be equal to the call price.

28 Valuation by replication (cont'd) Let us consider the value function V t of the portfolio as a function of time and the stock price V t = V (t, S t ) = α(t, S t )S t + β(t, S t )e rt. Using Itô's lemma, we get V (t,st) dv t = ds S t + [ V (t,st) t s 2 S 2 t Also, the self-nancing condition gives dv t = α(t, S t )ds t + β(t, S t )re rt dt. 2 V (t,s t) S 2 ] dt.

29 Valuation by replication (cont'd) By identication of the factors of ds t and the factors of dt and using the self-nancing constraint, we get that V (t, S t ) = α(t, S t ) [ S V (t, St ) + 1 ] t 2 s 2 V (t, S 2 St 2 t ) = β(t, S t )re rt. S 2 Multiplying the rst equation by rs t and adding it to the second equation, [ we obtain V (t, St ) + 1 ] t 2 s 2 V (t, S 2 St 2 t ) + V (t, S t) rs t S 2 S. = α(t, S t )rs t + β(t, S t )re rt which, by using α(t, S t )rs t + β(t, S t )re rt = rv (t, S t ), leads to the the pricing equation V (t, S t ) + 1 t 2 s 2 V (t, S 2 St 2 t ) + V (t, S t) rs t rv (t, S t ) = 0. S 2 S

30 Black-Scholes formula We get the Black-Scholes formula by solving this parabolic dierential equation, using the payo function of the option as a terminal condition V (T, S T ) = (S T K) + : V (t, S t ) = S t Φ (d 1 (T t, S t )) Ke r(t t) Φ (d 2 (T t, S t )) d 1 (T t, S t ) = ln(s t/k) + ( r s 2) (T t) s T t d 2 (T t, S t ) = d 1 (T t, S t ) s T t where Φ is the standard normal cumulative distribution. Depending on the derivative product and the assumptions, it is not always as easy to determine an explicit solution, however.

31 Volatility estimation The formula depends on the volatility s of the stock returns that we estimate either using historical returns (historical volatility) or using option prices and the inverse Black-Scholes formula (implied volatility).

32 Equity linked life insurance Concepts used in the Black-Scholes model can be applied to life insurance models. For usual life insurance contracts, a so-called technical interest is paid. Insurance companies usually invest part of their reserves on nancial markets. The expected return is higher than the riskless rate but there is an additional risk. Equity linked life insurances transfer part of this risk to the owner of the policy.

33 Equity linked life insurance (cont'd) The interest rate is higher than for a standard contract, but it is random. Suppose that the payo function of the life insurance depends on a reference index (e.g., the price of a nancial asset, portfolio, or market index). We consider here a contract paying the maximum between some index value and a guaranteed amount b. This approach reduces the risk for the insurer.

34 Equity linked life insurance (cont'd) Let the index value X t follow a geometric Brownian motion dx t = mx t dt + sx t dw t. The payment takes place at a random time T that is triggered if the insured dies (or in case of any other event dened in the contract such as a job loss, a divorce, a wedding). The payo at time T is max(x T, b) = X T b.

35 Types of life insurances We consider two types of contracts based on a xed maturity t 0 : Term insurance, for which the value X T b is paid at the time of death T [0, t 0 ]. Pure endowment insurance, for which the value X t0 b is paid if the insured is still alive at time t 0. In general, we combine the two types of life insurances, possibly with dierent amounts b.

36 Model Suppose that T = T a corresponds to the remaining lifetime of an a-years old insured and that it is independent to the index value X. The probability that the insured does not die in the t years to come is given by 1 F Ta (t) = P[T a > t]. Also, F Ta (t) = P[T a < t] gives the probability that the insured dies during the t years to come. The density of T a at u is denoted by f Ta (u).

37 Model (cont'd) The distribution of T a depends on the age, the gender, the country, the health, etc. There are thus two sources of risk in an equity linked life insurance: the nancial risk due to the evolution of the index the mortality risk of the insured. These two sources are modelled with X and T a and have an impact on the insurance payo.

38 Model (cont'd) Note that the payo function X Ta b can be rewritten as b + max(x Ta b, 0), which is equivalent to the payment of a xed amount b plus a payment max(x Ta b, 0). For a xed T a, we can use the Black-Scholes formula to value a payo max(x Ta b, 0) as it is equivalent to a European call with maturity T a on the index X and with strike price b.

39 Model (cont'd) For a maturity T a = u, the price today is C(u, X 0 ) = X 0 Φ (d 1 (u, X 0 )) be ru Φ (d 2 (u, X 0 )) d 1 (u, X 0 ) = ln(x /b) + ( r s 2) u s u d 2 (u, X 0 ) = d 1 (u, X 0 ) s u where Φ(x) is the standard Normal distribution CDF. In order to compute the insurance premium, we nally weight these prices according to the probability of each possible maturity.

40 Premium of pure endowment insurance The payo of the pure endowment insurance is given by X t0 b at time t 0 if the insured is still alive. The probability to be alive at time t 0 is given by P[T a > t 0 ] = 1 F Ta (t 0 ). The premium today is denoted by Π e and is given by the price 0 today of the unique random payo at time t 0, that is, the sum of the discounted xed amount b and the call price be rt 0 + C(t 0, X 0 ) multiplied by the survival probability 1 F Ta (t 0 ): Π e = 0 (be rt 0 + C(t 0, X 0 )) (1 F Ta (t 0 )).

41 Premium of term insurance For the term insurance, the payo may happen at any time between [0, t 0 ] if the insured dies. If we knew that the death would happen at time T a = u, the value today of the security would be be ru + C(u, X 0 ). The premium today is denoted by Π t and is given by the sum of 0 these values weighted by the density f Ta (u) on all possible dates u between 0 et t 0 : Π e = t (be ru + C(u, X 0 )) f Ta (u)du.

42 Merton's intertemporal model Merton's intertemporal model considers a problem of optimal consumption and portfolio allocation in continuous time. We aim at maximizing an expected utility by optimizing consumption and investment across time, and having an initial wealth w. We suppose that there are N nancial assets, whose prices are dened by dst i = m i St idt + s ist i d dw j j=1 t. The evolution of each nancial asset depends on d sources of noise W j t, j = 1,..., d.

43 Merton's intertemporal model (cont'd) We dene the price of the riskless asset by dm t = rm t dt with M 0 = 1. We may invest in N risky assets and a riskless asset, which gives the multidimensional price process X = (M, S 1,..., S N ). Let c t be the consumption level at time t and Z the nal wealth. We aim at maximizing [ ] t U(c, Z) = E u(c 0 t, t)dt + F (Z), where u and F stand for the utility of consumption and nal wealth, respectively.

44 Merton's intertemporal model (cont'd) We use nancial assets to transfer wealth from one date to the other and to nance future consumption. An investment strategy is given by the process θ = (θ 0, θ 1,..., θ N ). Such a strategy is said to nance the consumption plan c = {c t ; t [0, T ]} and the nal wealth Z if it satises N i=1 θi t S t i = w + t N 0 i=1 θi s ds s i t c 0 sds 0 pour t [0, T ] et N i=1 θi T S T i = Z. For some initial wealth w, we determine the optimal c, z et j.

45 Merton's intertemporal model(cont'd) We may obtain explicit solutions for some particular utility functions (for inst. u(w) = w α /α) using stochastic optimal control techniques. Those kind of models are used in pension funds where we invest the received premia in nancial assets in order to nance future retirement.