25857 Interest Rate Modelling

Transcription

1 25857 UTS Business School University of Technology Sydney Chapter 20. Change of Numeraire May 15, /36

2 Chapter 20. Change of Numeraire 1 The Radon-Nikodym Derivative 2 Option Pricing under Stochastic Interest Rates 3 2/36

3 Consider some traded asset S (e.g. stock) whose price follows; ds = μ s Sdt+σ s SdW where W is generated by the measure P. Let U, V be the prices of derivative instruments dependent on S, so that U = U(S,t) and V = V(S,t). For instance U and V could be options of different maturity or, U could be an option and V could be a bond. 3/36

4 From the discussion of Section 8.4, ds = (r +λσ s )Sdt+σ s SdW, du = (r +λσ u )Udt+σ u UdW, dv = (r +λσ v )Vdt+σ v VdW, where λ is the market price of risk associated ( with the uncertainty W and the expressions for σ u = S U σs U S) and ( σ v = S V σs V S) are obtained from application of Ito s lemma. 4/36

5 Applying Girsanov s theorem ds = rsdt+σ s Sd W, du = rudt+σ u Ud W, (1) dv = rvdt+σ v Vd W, (2) where W(t) = W(t)+ t 0 λ(τ)dτ, (3) is a Wiener process under the equivalent risk-neutral P. 5/36

6 If we let r(t) denote the (possibly stochastic) risk-free rate of interest then we can form the money market account ( t ) A(t) = exp r(s)ds, so that 0 da = radt. Applying the results of Section 6.6 concerning the s.d.e. followed by the quotient of two diffusions, we can show that ( ) ( ) U U d = σ u d W, (4) A A ( ) ( ) V V d = σ v d W. (5) A A 6/36

7 Equations (4) and (5) have zero drifts, it follows that the relative prices U/A, V/A are martingales under P. Thus, [ ] U t UT, (6) A t = E t V t A t = E t A T [ VT A T ], (7) where E t denotes the expectation under P conditional on information at time t. Equations (6) and (7) yield U t = E t [exp ( ( V t = E t [exp T t T t ] r(s)ds )U T, (8) ] r(s)ds )V T. (9) 7/36

8 Here r is possibly stochastic we leave it under the E t. In equations (8) and (9) the term exp( T t r(s)) acts as a stochastic discount factor. For a particular realisation of the interest rate process (under P) it discounts back to time t the realised values of U T and V T. The operation E t then basically averages over all such expected payoffs. In deriving (8), (9) we have used the money market account as the numeraire. 8/36

9 r r (i) (s) t T s e T t r(i) (s)ds U T U T Figure 1: Discounting the value of U T from T back to t for the particular ith realisation r (i) (s) of the interest rate process. The expectation E t in (8) averages over many such paths. 9/36

10 However it is possible to use other instruments as numeraire. For example, in pricing U we may use V as the numeraire (e.g. V could be a bond price with the same maturity as U), then we would consider the relative price Y = U V. We continue to use the arbitrage free dynamics for U and V, viz. eqns. (1), (2). Using again the result of Section 6.6 dy = (σ v σ u )σ v Ydt+(σ u σ v )Yd W. (10) If we introduce the new process W (t) = W(t) t 0 σ v (s)ds, (11) 10/36

11 then by Girsanov s theorem, we can obtain an equivalent measure P under which W (t) will be a Wiener process. It follows that under P, Y is a martingale since dy = (σ u σ v )YdW. (12) Thus Y t = E t (Y T), or, on using the definition of Y U t = V t E t [ UT V T ]. (13) Eqn. (13) may provide a more convenient pricing relationship, especially if V is the price of a bond that matures at the same time as the instrument U. 11/36

12 We note from (3) and (11) the relation between the processes W(t) and W (t), viz. W (t) = W(t)+ t 0 λ(s)ds t 0 σ v (s)ds. (14) In Table we represent the relation between the three measures P, P and P under which the financial market dynamics may be considered. We also show the relation between the related processes W(t), W(t) and W (t), in particular under which measures they are and are not Wiener processes. 12/36

13 P P P WIENER NOT WIENER NOT WIENER W E(dW) = 0 E(dW) = λdt = 0 E (dw) = (σ v λ)dt = 0 NOT WIENER WIENER NOT WIENER W E(d W) = λdt = 0 E(d W) = 0 E (d W) = σ vdt = 0 NOT WIENER NOT WIENER WIENER W E(dW ) = (σ v λ)dt = 0 E(dW ) = σ vdt = 0 E (dw ) = 0 Table 1.1: Summarising the relation between the processes W(t), W(t) and W (t) and the measures P, P and P. Here W(t) = W(t)+ t 0 λ(s)ds, W (t) = W(t) t 0 σ v(s)ds and so W (t) = W(t)+ t 0 λ(s)ds t 0 σ v(s)ds. 13/36

14 The Radon-Nikodym Derivative The Radon-Nikodym Derivative Comparing (6) and (13) we obtain a result which allows us to change numeraire, viz. [ ] [ ] U t = V t E UT UT t = A t Et. V T A T Rearranging we obtain [ ] E Vt t U T V T [ ] = E At t U T. (15) A T 14/36

15 The Radon-Nikodym Derivative The Radon-Nikodym Derivative It is also sometimes useful to obtain the Radon-Nikodym derivative which underlies the change of numeraire result in eqn. (15). Some formal manipulations allow us to reexpress (15) as E t [ UT V T ] = E t [ At V T A T V t U T V T ]. (16) Introducing a notation similar to one employed in Section 8.2 we set ξ(t,t) = A tv T A T V t, (17) so that (16) may be written E t [ UT V T ] = E t [ ξ(t,t) U T V T ]. (18) 15/36

16 The Radon-Nikodym Derivative The Radon-Nikodym Derivative The quantity ξ(t, T) is thus the Radon-Nikodym derivative which allows us to switch from calculating expectations under P to calculating expectations under P, i.e. dp = ξ(t,t)d P. (19) Consider the interval (0,T) we note from (17) that since A 0 = 1. ξ(0,t) = A 0V T A T V 0 = V T A T V 0, (20) 16/36

17 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates An application of the change of measure results is the Black-Scholes model under stochastic interest rates in Chapter 19. Consider the no-arbitrage condition (19.14) μ α σ = μ f α σ fz = φ, (21) where φ is the market price of risk associated with the stock price noise, z. Substituting (21) into the s.d.e.s (19.4) for S and (19.7) for f; ds = (α+φσ)dt+σdz, S (22) df f = (α+φσ f z )dt+σ fz dz +σ fv dv, (23) 17/36

18 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates which together with the bond pricing eqn. dp P = αdt+δdv, (24) give the arbitrage free financial market dynamics under P. By defining new processes z and v equations (22) and (23) may be written ds = αdt+σd z, S (25) df f = αdt+σ f z d z +σ fv d v. (26) 18/36

19 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates Clearly by introducing the stochastic money market account ( t ) A(t) = exp α(s)ds, S/A and f/a would become martingales under the measure P i.e. [ ] f t = E At t f T. (27) A T o 19/36

20 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates However we could also allow the bond price P to be the numeraire. Using P to denote the probability measure when P is the numeraire; [ ] [ ] f t = E At t f T = E Pt t f T, A T P T i.e., f t P t = E t [f T ], (28) since P T = 1 (recall the bond matures when the option does). 20/36

21 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates In order to operationalise (28), we need to make explicit the stochastic price dynamics from which P and hence E t can be calculated. The dynamics are those for the s.d.e. for S/P derived from (24) and (25) by an application of Ito s lemma; d(s/p) S/P = (δ2 ρσδ)dt+σd z δd v. (29) Under P a geometric Brownian motion with instantaneous variance V 2 = (σ 2 +δ 2 2ρσδ). 21/36

22 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates Since under P E[σd z δd v] = 0, and var[σd z δd v] = E[(σd z δd v) 2 ] = V 2 dt, We can define under P a new Wiener process w such that σd z δd v = Vd w. 22/36

23 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates Eqn. (29) can then be written d(s/p) S/P = (δ2 ρσδ)dt+vd w. (30) Introducing the new Wiener process t w (t) = w(t)+ 0 δ 2 ρσδ ds V 23/36

24 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates We can again apply Girsanov s theorem to write (30) as d(s/p) S/P = Vdw. (31) It is the s.d.e. (31) which generates paths under the measure P. The associated Kolmogorov p.d.e. is where X = S/P. 1 2 V 2 2 p X 2 + p = 0, (32) t 24/36

25 Option Pricing under Stochastic Interest Rates Option Pricing under Stochastic Interest Rates Hence applying the Feyman-Kac formula to the expectation on the right-hand side of (28), we see that f/p is given by the solution of subject to 1 2 V 2 2 (f/p) X 2 + (f/p) = 0, t f(x,t) = max[0,x E]. We are in fact dealing with the Black-Scholes model with r = 0, and will thus again recover the solution given by eqn. (??). 25/36

26 Previously the change of numeraire idea in an economy with just one source of risk. Suppose we have (n + 1)-risky assets whose dynamics are driven by ds i S i = μ i dt+ n s ij dw j (t) (i = 0,1,...,n), (33) j=0 where the W j (t) are independent Wiener processes. 26/36

27 We consider here the case when all assets are traded. The framework we develop is easily extended to allow the case in which some of the S i may not be traded. Let S denote the vector of risky asset prices (S 0,S 1,...,S n ). Let f k (S,t)(k = 1,2,...,m) denote a derivative instrument written on the vector of processes S. 27/36

28 From Section 10.4 there exists a unique risk-neutral measure P under which the dynamic for the S i and f k can be written ds i S i = (r q i )dt+ df k f k = rdt+ n s ij d W j (t), (i = 0,1,...,n), (34) j=0 n σ kj d W j (t) (k = 1,2,...,m), (35) j=0 where the W j (t) are the Wiener processes under P. In equation (34) the q i is the dividend yield on S i, and in (35) the σ kj are the volatility factors whose calculation is outlined in Section /36

29 The σ kj are given by f k σ kj = n s lj S l Δ kl, l=1 where Δ kl = f k. S l Defining prices ( in terms of the money market account ) t A t = exp 0 r(s)ds we would of course obtain f k (S,t) = E [ ] t e T t r(s)ds f k (S(T),T) (36) Here the distribution P under which E t is calculated is generated by the (n+1) processes (34). 29/36

30 Suppose instead we use S 0 (t) as the numeraire (it is always possible to relabel the assets so that S 0 is the new numeraire). Thus we are interested in the processes and Z i (t) = S i(t), (i = 1,...,n), (37) S 0 (t) Y k (t) = f k(t) (k = 1,...,m). (38) S 0 (t) 30/36

31 Applying the result (6.83) for the stochastic differential of the quotient of two diffusions the dynamics for the Z i and Y k processes become dz i Z i = and dy k Y k = (q 0 q i ) q 0 n s 0j (s ij s 0j ) dt+ j=0 m s 0j (σ kj s 0j ) dt+ j=1 n (s ij s 0j )d W j, j=0 (39) m (σ kj s 0j )d W j. j=1 (40) 31/36

32 The dividend yields prevent us from obtaining martingales under the new measure problem easily overcome by defining the new processes Zi (t) = Z i(t)e (q 0 q i )t, (41) Yk k(t)e q0t, (42) Their dynamics are easily calculated as and dz i Z i dy k Y k n n = s 0j (s ij s 0j )dt+ (s ij s 0j )d W j, (43) j=1 j=1 m m = s 0j (σ kj s 0j )dt+ (σ kj s 0j )d W j. (44) j=1 j=1 32/36

33 By Girsanov s theorem we can find a new measure P under which the processes W j (t) = W j (t) t 0 s 0j (τ)dτ (45) are Wiener processes. Under this new measure the dynamics for the Zi and Yk become and dz i Z i dy k Y k = = n (s ij s 0j )dwj. (46) j=1 m (σ kj s 0j )dwj. (47) j=1 33/36

34 The last equation indicates that the Yk are martingales under P, so that Y k (t) = E t[yk (T)]. (48) By use of (42) this expression becomes Y k (t) = E t [e q 0(T t) Y k (T)]. In terms of the original variable f k (t) we have [ f k (t) = S 0 (t)e t e q 0(T t) f ] k(t) S 0 (T). (49) 34/36

35 Suppose for example that [ f k (T) = max 0, when the α j are constants. Thus [ f k (T) S 0 (T) = max 0,α 0 + [ = max 0,α 0 + n j=0 n j=1 n j=1 ] α j S j (T) ] α j Z j (T) ] α j e (q 0 q i )T Zj(T) 35/36

36 Equation (49) then becomes f k (t) = S 0 (t)e t [ [ e qt qit max 0,α 0 e (q i q 0 )T + n α j Zj ]]. (T) j=1 The process under which E t system (46). is calculated is given by the 36/36