Risk Neutral Modelling Exercises

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1 Risk Neutral Modelling Exercises Geneviève Gauthier Exercise.. Assume that the rice evolution of a given asset satis es dx t = t X t dt + X t dw t where t = ( + sin (t)) and W = fw t : t g is a (; F; P) Brownian motion. a) Show that the SDE admits a unique strong solution. b) The riskless asset as a time t instantaneous return at time of r t = r ( + sin (t)). Find the SDE of the risky asset rice in the risk neutral framework. What allows you to retend that the risk neutral measure exists? c) What is this risk neutral measure? Exress it in function of P. Exercise.. The instantaneous interest rate fr t : t g of the riskless asset satis es the ordinary di erential equation d r t = c ( r t ) dt; which imlies that r t = + (r ) e ct : The riskless asset rice evolution fst : t g satis es St = ex r u du : a) Show that the rice evolution satis es ds t = r t S t dt: b) The risky asset rice evolution fs t : t g is characterized by

2 ds t = S t dt + S t dw t where fw t : t g is a (; F; P) Brownian motion. Show how to change the robability measure such that we get the risky asset rice dynamics under the risk neutral measure. Exercise.3. The rices of the assets are characterize with : ds (t) = S (t) dt + S (t) dw (t) ; ds (t) = S (t) dt + S (t) dw (t) + S (t) dw (t) where W and W are indeendent (; F; ff t : t g ; P) Brownian motions. The risk free interest rate r is assumed to be constant. ff t : t g is the ltration generated by the Brownian motions, with the usual regularity conditions. Find the risk neutral market model. Exercise.4. The value in American dollar of the yen satis es du (t) = U U (t) dt + U U (t) dw U (t) : The value in Canadian dollars of the yen evolves like dc (t) = C C (t) dt + C C (t) dw C (t) : The value in Canadian dollars of an American dollar is d (t) = (t) dt + (t) dw (t) : W U ; W C and W are (; F; ff t : t g ; P ) Brownian motions such that for all t > Moreover, Corr P [W U (t) ; W C (t)] = UC, Corr P [W U (t) ; W (t)] = U and Corr P [W (t) ; W C (t)] = C : da (t) = r U A (t) dt; A () = db (t) = r C B (t) dt, B () = dd (t) = r J D (t) dt; D () = reresent the evolutions of the riskless asset in the United-States, Canada and Jaan, resectively. More recisely, A (t) is in American dollars, B (t) is exressed in Canadian dollars and D (t) is in yens.

3 Questions. a) What are the conditions so that there is no arbitrage oortunity? b) Is the market model comlete? c) What are the evolutions of the exchange rates in the risk neutral framework? Interret your results Exercise.5. Assume that W, W c and W f are indeendent (; F; ff t : t T g ; P) motions. Let eb t W t + Wt f, t T and bb t W t + c W t, t T: n o a) Show that bbt : t T is a (ff t g ; P) Brownian motion. h b) Comute Corr P bbt ; B e i t for all < t T: Brownian The rice evolution of two risky assets is and dx t = X X t dt + X X t d e B t = X X t dt + X X t dw t + X X t d f W t dy t = Y Y t dt + Y Y t d b B t = Y Y t dt + Y Y t dw t + Y Y t d c W t : We can interret W as the common chock, whereas W f and W c are the idiosyncratic chocks of X and Y resectively. c) Determine the model in the risk neutral environment, assuming that the instantaneous riskless interest rate is r: d) Is this market model comlete? Justify. e) Can you rice the contract that romises the di erence C = X T Y T between to asset? Justify all stes: Interret your results. 3

4 Solutions Exercise. a) We need to verify that. Let K = jj + jj : (i) jb (x; t) b (y; t)j + j (x; t) (y; t)j K jx yj ; 8t (ii) jb (x; t)j + j (x; t)j K ( + jxj) ; 8t (iii) E [X] < : jb (x; t) b (y; t)j + j (x; t) (y; t)j = j ( + sin (t)) x ( + sin (t)) yj + jx yj = jj ( + sin (t)) jx yj + jj jx yj jj jx yj + jj jx yj = ( jj + jj) jx yj K jx yj jb (x; t)j + j (x; t)j = j ( + sin (t)) xj + jxj jj jxj + jxj K jxj : If E [X ] < ; then the three conditions are satis ed. b) dx t = t X t dt + X t dw t = r t X t dt + ( t r t ) X t dt + X t dw t = r t X t dt + X t d W t + s r s ds : Let s = s r s ; s and note that the function s! s is continuous, which imlies that the rocess f s : s g 4

5 is redictable. Since s r s s ds = ds ( + sin (s)) r ( + sin (s)) = ds = r (4 4 cos T cos T sin T + 3T ) < ; R i then E hex P T t dt <. We can aly the Cameron-Martin-Girsanov theorem :there is a martingale measure Q on (; F) such that W f n o = fwt : t [; T ] de ne as fw t = W t + s ds; t : is a (; F; Q) Brownian motion. Therefore, dx t = r t X t dt + X t d f W t : c) According to the Cameron-Martin-Girsanov theorem, the Radon-Nikodym derivative is Z dq T Z dp = ex T t dw t t dt " R # T r ( + sin t) dw = ex t (4 4 cos T cos T sin T + 3T ) = ex " r R r r T W T r cos T sin t dw t cos T sin T + 3T 4 4 # : Consequently, Q (A) = E P "ex " r R r r T W T sin t dw t cos T cos T sin T + 3T 4 4 # A # : 5

6 Exercise. a) Let Y t = R t r u du. Y is an Itô rocess (dy t = K t dt + H t dw t )for which K t = r t and H t = since Note that jk s j ds = Let f (t; y) = e y. Itô s lemma, b) jr s j ds = + (r ) e cs ds + (r + ) e cs ds = T + (r + ) dy t = r t dt and hy i t ds t = df (t; Y t (t; Y t) dt = f (t; Y t ) dy t = S t r t dt H s ds = : e ct c < : = f, f (t; Y t ) = S t and f (; Y ) = = S. (t; Y t) dy t (t; Y t) d hy i t Let t = rt ds t = S t dt + S t dw t. We need to verify that r t = r t S t dt + S t dt + S t dw t r u = r t S t dt + S t d W t + du Z T E ex P t dt < 6

7 is verify to use Girsanov theorem. Z T E ex P t dt = E P " = E P " ex rt dt!# ( + (r ) e ct ) dt!# ex (r ) e ct dt! = ex ex < C + C e ct + C 3 e ct dt 3 Exercise.4 3. The main ideas To show that there is no arbitrage oortunity, we need to nd a risk neutral measure under which the discounted rice rocess of tradable assets are martingale. What are these tradable assets? In the ersective of a local investor, these are the three riskless assets, all exressed in Canadian dollars. The value in Canadian dollars of the American riskless asset A (t) = (t) A (t) satis es da (t) = (t) da (t) + A (t) d (t) + d ha; i (t) = (r U + ) A (t) dt + A (t) dw (t) : Similarly, the value in Canadian dollars of the Jaanese riskless asset D (t) = C (t) D (t) is characterize with dd (t) = C (t) dd (t) + D (t) dc (t) + d hd; Ci (t) = (r J + C ) D (t) dt + C D (t) dw C (t) : But the value in Canadian dollars of the American riskless asset is also A (t) = C(t) A (t). U(t) Indeed, A (t) is the value in yen of the American bank account and C (t) A (t) is the U(t) U(t) 7

8 value in Canadian dollars of A (t) which is exressed in yens. U(t) Using Itô s lemma, da C (t) C (t) (t) = A (t) du (t) + A (t) dc (t) + da (t) U (t) U (t) U (t) + C (t) U 3 (t) A (t) d hui (t) A (t) d hc; Ui (t) U (t) = r U + C U + U C U CU A (t) dt + C A (t) dw C (t) U A (t) dw U (t) : Similarly, the value in Canadian dollars of the Jaanese bank account satis es D (t) = (t) U (t) D (t). Itô s lemma gives dd (t) = (r J + U + + U U ) D (t) dt + U D (t) dw U (t) + D (t) dw (t) : U(t) Finally, B (t) is the value in yen of the Canadian bank account; B (t) is exressed C(t) C(t) in American dollars and B (t) = (t)u(t) B (t) is reverted back to Canadian dollars. Itô s C(t) lemma gives db (t) = (t) U (t) B (t) dc (t) + U (t) (t) (t) U (t) B (t) d (t) + B (t) du (t) + C (t) C(t) C(t) C(t) + (t) U (t) B (t) d hci (t) C 3 (t) (t) U B (t) d hc; Ui (t) C (t) (t) B (t) d hc; i (t) + B (t) d hu; i (t) C (t) C(t) = C + + U + r C + C C U CU C C + U U B (t) dt C B (t) dw C (t) + U B (t) dw U (t) + B (t) dw (t) : db (t) Since the value in Canadian dollars of the American bank account has two di erent exressions,, A (t) and A (t), these two rocesses must be the same Therefore, da (t) = (r U + ) A (t) dt + A (t) dw (t) da (t) = r U + C U + U C U CU A (t) dt + C A (t) dw C (t) U A (t) dw U (t) : r U + = r U + C U + U C U CU () W (t) = C W C (t) U W U (t) : () 8

9 Similarly, the value in Canadian dollars of the Jaanese bank account have two exressions: dd (t) = (r J + C ) D (t) dt + C D (t) dw C (t) dd (t) = (r J + U + + U U ) D (t) dt + U D (t) dw U (t) + D (t) dw (t) Therefore These four restrictions may be rewritten as From the two rst ones, we conclude that From the last one, r J + C = r J + U + + U U C W C (t) = U W U (t) + W (t) : = C U + U C U CU ; = C U U U ; W (t) = C W C (t) U W U (t) : U = C CU U : U = Corr [W U (t) ; W (t)] = Corr W U (t) ; CW C (t) = C Corr [W U (t) ; W C (t)] = C CU U = C CU ( C CU U ) = U which do not brings new information. Therefore U W U (t) U Corr [W U (t) ; W U (t)] U = C CU U U = C U U : 9

10 The second ste consist in nding the risk neutral measures. The Choleski decomosition allows to exress the deendent Brownian motions as a linear combination of indeendent Brownian motions: W (t) = a B (t) + a B (t) + a 3 B 3 (t) W U (t) = a B (t) + a B (t) + a 3 B 3 (t) W C (t) = a 3 B (t) + a 3 B (t) + a 33 B 3 (t) : Let and where eb i (t) = B i (t) + i (s) ds; i = ; ; 3 fw (t) = a e B (t) + a e B (t) + a 3 e B3 (t) = W (t) + fw U (t) = a B e (t) + a B e (t) + a 3B3 e (t) = W U (t) + fw C (t) = a 3B e (t) + a 3B e (t) + a 33B3 e (t) = W C (t) + (s) ds U (s) ds C (s) ds (t) = a (t) + a (t) + a 3 3 (t) U (t) = a (t) + a (t) + a 3 3 (t) C (t) = a 3 (t) + a 3 (t) + a 33 3 (t) : Therefore da (t) = (r U + ) A (t) dt + A (t) dw f (t) da (t) = r U + C U + U C U CU C C + U U A (t) dt + C A (t) dw f C (t) U A (t) dw f U (t) dd (t) = (r J + C C C ) D (t) dt + C D (t) dw f C (t) dd (t) = (r J + U + + U U U U ) D (t) dt + U D (t) dw f U (t) + D (t) dw f (t) db C + (t) = + U + r C + C C U CU C C + U U + C C U U C B (t) dw f C (t) + U B (t) dw f U (t) + B (t) dw f (t) : B (t) dt

11 To risk neutralize the system, the drift arameter must be the Canadian risk free rate: r U + = r C r U + C U + U C U CU C C + U U = r C r J + C C C = r C (r J + U + + U U U U ) = r C C + + U + r C + C C U CU C C + U U = r + C C U U C : On matrix form, the rst three exressions become 3 3 r U + r C 4 U C 5 4 U r U + C U + U C U CU r C C C r J + C r C 3 5 = : The solution is 4 U C 3 5 = 4 (r U + r C ) U (r J r U + U U + C U CU ) C (r J + C r C ) 3 5 : The rice of risk associated to the W s; ( ; U ; C ) ; have to exist. For that, ; U and C have to be strictly ositive. Since ; U ; C are constants, ; ; 3 are also constants: The Novikov condition is satis ed. We can aly Girsanov theorem to nd Q under which eb ; B e ; B e 3 are indeendent Q Brownian motions. Consequently W f ; W f U ; W f C are correlated Q Brownian motions having the same correlation structure as the P Brownian motions W ; W U ; W C : Since the rice of risk ; U ; C are uniquely determined, the risk neutral measure is unique and the market model is comlete. To rule out arbitrage oortunities, the rices of risk ; U ; C are substituted in the fourth and fth equations. Relacing ; U and C in the fourth equation leads to Similarly, working with the fth equation, U + U C CU = : Therefore = C C U CU C C + U U + U = C ( C U CU C ) + U ( U + {z U C CU ): } = C U CU C = : (3)

12 For the martingale measure to exist, we need that the yennamerican dollar exchange rate volatility satis es U = C CU U and U = C C CU : The SDE under Q are du (t) = r U r J + U C U CU U (t) dt + U U (t) dw f U (t) = (r U r J U U ) U (t) dt + U U (t) dw f U (t) ; dc (t) = (r C r J ) C (t) dt + C C (t) dw f C (t) ; d (t) = (r C r U ) (t) dt + (t) dw f (t) : 4 Exercise.5 4. Question a) Recall that First bb t W t + c W t : bb W + c W = : Second, for all t < t < ::: < t n, the increments bb t b Bt ; b Bt b Bt ; :::; b B tn b Btn are indeendent since for r < s t < u, h Cov bbs Br b ; B b i u Bt b = Cov h (W s W r ) + cws Wr c ; (W u W t ) + i cwu Wt c = Cov [ (W s W r ) ; (W u W t )] +Cov h (W s W r ) ; i cwu Wt c h i +Cov cws Wr c ; (W u W t ) h +Cov cws Wr c ; i cwu Wt c =

13 The indeendence roerty will be established once we will have rove that B b t Bs b is Gaussian. Third, because a linear combination of a multivariate Gaussian random variables is Gaussian, B b t is Gaussian. Moreover, h i E P bbt = E P [W t ] + h i E P cwt = and ar P h bbt i = ar P [W t ] + ar P h cwt i = t + t = t: Finally, the ath of W and c W being continuous, the ones of b B are also continuous. 4.. Question b) = Corr P h bbt ; e B t i Cov P h bbt ; e B t i rar P h bbt i r ar P h ebt i = t Cov P h W t + c W t ; W t + f Wt i t t Cov P [W t ; W t ] + h Cov P W t ; W f i t + h i Cov P cwt ; W t + h Cov P cwt ; W f i t = CovP [W t ; W t ] t = : A 4.. Question c) We have dx t = X X t dt + X X t dw t + X X t d f W t and dy t = Y Y t dt + Y Y t dw t + Y Y t d c W t : 3

14 Let t ex ( rt) ; t T: t is the discounted factor at t. This rocess is deterministic and satis es d t = r t dt: The evolution of the discounted rice rocesses satis es d t X t = X r X t X e t t X t dt + X t X t d W t + s ds + X t X t d and d t Y t = Y r Y t Y b t t Y t dt + Y t Y t d W t + s ds + Y t Y t d We nd ; e and b such that the drift terms are nil: X r X t X e t = Y r Y t Y b t = fw t + e s ds cw t + b s ds : We need to solve X X Y t e t b t A X Y r r = : The solution exists but it is not unique. Need to satisfy the Novikov condition to aly Girsanov theorem. For b t R, d t X t = X t X t dwt + X t X t dw f t and d t Y t = Y t Y t dw t + Y t Y t d c W t where W, f W and c W are ff t g ; Q b Brownian motions. We have 4..3 Question d) dx t = rx t dt + X X t dw t + X X t d f W t and dy t = ry t dt + Y Y t dw t + Y Y t d c W t : In nitely many martingale measures, the market is incomlete. 4

15 4..4 Question e) The contract is accessible since it su ces to hold share of the rst asset and the second one. share of E Q [ex( rt )C] = E Q [ex( rt ) (X T Y T )] = E Q B T X T E Q B T Y T = E Q B X E Q B Y martingale roerty = X Y : 5