Numerical Optimisation Applied to Monte Carlo Algorithms for Finance. Phillip Luong

Transcription

1 Numercal Optmsaton Appled to Monte Carlo Algorthms for Fnance Phllp Luong Supervsed by Professor Hans De Sterck, Professor Gregore Loeper, and Dr Ivan Guo Monash Unversty Vacaton Research Scholarshps are funded jontly by the Department of Educaton and Tranng and the Australan Mathematcal Scences Insttute.

2 Introducton Optons are fnancal contracts that offers the rght (but not the oblgaton) to buy (call opton) or sell (put opton) an asset at an agreed prce. Optons are often used to create safer fnancal strateges, to mnmse the rsk of losng a lot of money. In Fnancal Mathematcs, calculatng the prce of these optons quckly s mportant. There are many dfferent types of optons, dependng on the asset beng sold and the terms of the two partes. Dfferences between opton types wll nclude Strke Prce and when the opton may be exercsed. The Strke Prce refers to the agreed prce to pay for the asset. An example of a class of optons are European Optons, whch can only be exercsed at a sngle tme pont n the future (known as the Tme of Expry). In contrast, Amercan Optons may be exercsed at any tme pont between the tme of purchase and the tme of expry. In fnance, the nuances to the agreed-upon opton makes the contract tself dffcult to accurately value. In addton to the demand to quckly value the current prce of such optons, t s mperatve to create a method whch accurately models the future prce of an opton. In ths project, we wsh to fnd a method to construct a model to correctly prce the value of these optons wth the use of Monte Carlo Smulatons and Optmsaton technques. Prevously, many researchers have been able to model the prce of varous types of optons. The prce of a European opton may be calculated va the Black-Scholes Partal Dfferental Equaton (.) (Black and Scholes, 973). Ths model calculates the prce of a European opton, U, whch wll change dependng on the prce of an asset, x R+ (dollars), and the tme of purchase, t [0, T ] (years). Furthermore, we defne the Strke Prce, K, Volatlty, σ, Rsk-Free Rate, r, and Tme of Expry, T. The Volatlty s the standard devaton of the prce of the asset over tme, and the Rsk-Free Rate s the theoretcal rate of an nvestment wth zero rsk (for example, n a savngs account). The value of European optons s calculated by solvng the followng Partal Dfferental Equaton: U 2U U + σ 2 x2 2 + rx = ru. t 2 x x (.)

3 We apply the followng boundary condtons: We consder the prce of an asset cannot decrease below $0. If the prce of an asset s orgnally worth $0, we say that t s opton s worth $0 for all t [0, T ]. In cases lke ths, we would theoretcally purchase the asset, rather than purchasng the opton, snce t can only ncrease n value. So we have u(0, t) = 0 t [0, T ]. The prce of the opton wll converge towards the prce of the asset for very expensve assets (assumng the Strke Prce remans the same): u(x, t) x as x. At the tme of expry, the value of an opton s only non-zero f the asset s worth more than the Strke Prce. In ths case, the value of the opton s: u(x, T ) = (x K)+ := max(x K, 0). Solvng Equaton. analytcally gves us the followng soluton: U (x, t) = xφ(z) Ke r(t t) Φ(z σ T t)), (.2) x σ2 where z = ln + (T t) r +, K 2 σ T t and Φ(z) s the Normal Cumulatve Dstrbuton functon evaluated at z. In ths report, we shall refer to equaton (.2), as the Black-Scholes functon. 2

4 Fgure. Exact plot of the Black-Scholes PDE, wth boundary condton, u(x, T ) = (x K)+. Alternatvely, we apply the Call-Spread Boundary, a functon composed of a dfference of two European Optons wth dfferent Strke Prces (K and K2 ) Fgure.2 Exact plot of the Black-Scholes PDE (Equaton.), wth the call-spread functon as the boundary condton, u(x, T ) = (x k )+ u(x k2 )+. 3

5 The goal of ths project s to model the value of the European Call opton, and Call-Spread Functon at any tme pont t [0, T ] usng a bass of polynomals, a bass of Black-Scholes functons, and a combnaton of the two. Ths wll be completed by applyng optmsaton technques such as the least-squares method. 2 Methodology 2. Smulaton of Asset and Opton Prces Frstly, we smulated the prce of a partcular stock usng Monte Carlo Smulatons. For these smulatons, we fx the ntal stock prce (x0 = 00), rsk-free rate (r = 0), volatlty (σ = 0.3), and tme untl expry (T = 4). We wsh to use ths data to model the prce of the call opton at t = 2. We calculate the logarthm of the stock prce to ensure that the value of x remans postve at every tme step (Equaton 2.). ln(x+ ) = ln(x ) + rdt σ2 dt + σdwt 2 (2.) Here, Wt denotes a Wener Process, wth tme ncrements normally dstrbuted N (0, dt) and dt = T /# steps (years). In our smulatons, we use 00 tme steps. Next, we use the data to model the prce of U (x, t ~c) at tme, t [0, T ], and where ~c s the vector contanng all the parameters to the model. Here, we utlse the Longstaff-Schwartz method. The Longstaff-Schwartz method uses the smulated asset prces at tme, t, and the value of U (x, T ~c) to calculate the expected value of f (x; ~c) = U (x, t ) (Longstaff and Schwartz, 200). The results for a sample of these smulatons are calculated by usng equaton (2.). 4

6 Fgure 2.: Depctng the 20 Monte Carlo Smulatons of the stock prce over 4 years The frst 20 smulatons of the 00,000 Monte-Carlo smulatons. Smulatons have the parameters, x0 = 00, r = 0, v = 0.3, T = 4, σ = 0.3. The lne at t = 2 s present to show our am of the project. In ths partcular case, we attempt to model the prce of the a call opton, meanng that at tme, T, the call opton of the asset s worth $(x K)+, where K = 00. We calculate the prce of y at the tme t = 4, usng the value of the European Opton (Equaton 2.2), and the value of the call-spread functon (Equaton 2.3) at the tme of expry. y = U (x (T ), T ) = (x K)+ (2.2) y = U (x, T ) = (x (T ) K )+ (x (T ) K2 )+ (2.3) After these calculatons, we generate a set of data-ponts; where (x, y ) form a par representng the prce of the asset at tme, t, and the prce of U (x (T ), T ) respectvely. 2.2 Modellng the Prce of Opton Prces va the Least Squares Method Next, we use the data-ponts to optmse for ~c, where ~c mnmses 5

7 ss(x ; ~c) = X (y f (x (t ); ~c))2 (2.4) I va the least-squares method (Equaton 2.4). Here, I s the set of all smulated data ponts {(x, y )}, each x s the stock value at t = t, and ss(x ; ~c) s the sum of squares formula gven the data ponts. Furthermore, f (x (t ); ~c) s the bass functon that we are usng to model the prce of the opton (ether (2.6), (2.7) or (2.8)), and y s dependent on the formula we are evaluatng, whether t s the value of the call opton (2.2) or the call-spread functon (2.3). The models that we wsh to ft the model are A p-degree polynomal f (x ~c) = p X c x (2.5) =0 where ~c = [c0... cp ] are the coeffcents to each polynomal term. A sngle Black-Scholes formula (wth a constant c0 ) f (x (t ) ~c) = c0 + c bls(x, K, r, t, σ) (2.6) where ~c = [c0 c K σ] are the constant and coeffcents of the Black-Scholes functon, and the parameter of the varyng Strke Prce and Volatlty. A sum of two Black-Scholes formulae and a lnear equaton f (x ~c) = c0 + c x + c2 bls(x, K, r, t, σ ) + c3 bls(x, K2, r, t, σ2 ) (2.7) where ~c = [c0 c c2 c3 K K2 σ σ2 ] are the constant and coeffcents to the Black-Scholes functons, and the parameters to the Strke Prces and Volatltes. We fnd the optmal value of ~c, va three methods usng MATLAB s nbult lnear solvers, \ and lscov, and the non-lnear solver, fmnunc (MATLAB, 207). 6

8 2.2. The Lnear Solvers, lscov and \ The lnear solver, lscov solves the matrx equaton A~c = ~y, (2.8) where A s a m n matrx, ~c s a n vector, and ~y s a m vector. In our case, m = 06 and n = p+, where p denotes the order of the polynomal. Snce the dmensons ~c and ~y dffer, we multply AT on both sdes n order to solve for ~c. Ths s what the \ operator solves. In addton to solvng for ~c, lscov smplfes the problem by performng a QR decomposton on A frst. ~c = (AT A) AT ~y, These problems were prmarly effectve to deduce ~c when we modelled the data to f (x ~c) = Pp =0 c x. However, snce bases (2.6) and (2.7) contaned non-lnear parameters, we cannot just use lscov or \ to fnd ~c The Non-Lnear Solver, fmnunc Non-lnear solvers are necessary n ths research project to fnd parameters located n the BlackScholes functons, lke n bases (2.6) and (2.7). The nbult MATLAB non-lnear least squares solver s fmnunc, whch reles on the Quas-Newton Method or the Trust Regon Method, to fnd the optmal ~c R from an ntal guess, c~0. Whle both methods use gradent nformaton, they have dfferences n ther algorthms. As both methods are nbult solvers n MATLAB, the specfc detals to the algorthms are not fully provded here (MATLAB, 207). The Quas-Newton Method s a lne-search method whch nvolves performng a lne search to fnd the step length by estmatng a Hessan Matrx to calculate the step drecton. In MATLAB, t s possble to ether supply a gradent functon derved analytcally, or to allow t to numercally estmate the gradent at each step (MATLAB, 207). Such optons gve changes to the overall algorthm whch may affect the rate of convergence. 7

9 In contrast, the Trust Regon Method frst selects a step sze before choosng a steppng drecton. In ths case, the step-sze s dependent on the gradent, meanng t s necessary for us to supply the gradent functon. The gradent functons are all provded n Appendx A. In addton, to accelerate the non-lnear optmsaton processes, we embedded lscov nto our non-lnear solvers, mnmsng the objectve functon over the coeffcents. In ths case, the coeffcents, c0, c,... become a functon of the non-lnear parameters, K, σ, K2,.... Detals are suppled n Appendx A. We consder the Black-Scholes Problem wth bass (2.7), where c0 (K, σ) and c (K, σ) are optmally solved va ~c(k, σ) = (AT A) AT ~y, where A s the matrx (2.9) bls(x, K, σ) bls(x00,000, K, σ) (2.0) Then, the sum of squares problem s ss(c0 (K, σ), c (K, σ), K, σ) = X (y (c0 (K, σ) + c (K, σ)bls(k, σ))2 := g(k, σ). (2.) I We compared the number of teratons requred to fnd K and σ wth the Quas-newton method wthout a suppled gradent. Due to tme constrants, we were only able to mplement ths method to model a sngle Call Opton usng bass (2.6). In addton, we also ended up approxmatng the gradent functons as g(k, σ) ~c(k, σ) ss(~c, K, σ) ss(~c, K, σ) = ~c(ss(~c, K, σ)) +, (2.2) g(k, σ) ~c(k, σ) ss(~c, K, σ) ss(~c, K, σ) = ~c(ss(~c, K, σ)) +, (2.3) and where w ~ = [K σ]. 8

10 3 Computatonal Results and Dscusson 3. Polynomal Estmates of the Black-Scholes Equaton Fgure 3.: Estmates of the Black-Scholes soluton usng Polynomals (Equaton 2.5) of Increasng Order obtaned va lscov Table 3.: Coeffcents of the Polynomal Estmates obtaned va lscov Bass c0 c c2 c3 c4 Lnear N/A N/A N/A Quadratc N/A N/A Cubc Quartc N/A

11 Fgure 3.2: Comparson of the Quartc Estmates provded by Dfferent Optmsaton Methods f (x ~c) = c0 + c x + c2 x2 + c3 x3 + c4 x4 Table 3.2: Coeffcents of the Quartc Estmates va dfferent Optmsaton Methods Method c0 c c2 c3 c4 lscov \ Intal Guess Quas-Newton (no gradent gven) Quas-Newton (gradent gven) Trust Regon Fgure 3.2 shows that the lnear methods (such as lscov), are provdng better estmates than the non-lnear methods (both Quas-Newton Methods and the Trust-Regon Method). As expected, ncreasng the order of the polynomal, results n a better estmaton of the Black Scholes equaton (Fgure 3.). However, these results tend to dverge at the lower and 0

12 upper end of the doman. Interestngly, as we ncrease the order of the polynomal, the parameter values wll start to dffer (Table 3.2). There were slght (but neglgble) dfferences between the lnear methods, however, the Quas-Newton method and Trust Regon method gave very dfferent results. Snce these parameters obtaned by the non-lnear methods gve a hgher objectve value than the lnear methods, we needed to nvestgate why these methods were not workng. We suspect that ths s due to the ll-condtonng of the problem, however, further work wll be needed to verfy ths. 3.2 Estmates usng the non-lnear Bases of the Black-Scholes Equaton Fgure 3.3: Estmate of the European Opton usng a sngle Black-Scholes Formula Bass f (x ~c) = c0 + c bls(x, K, r, t, σ). Table 3.3: The Coeffcents of the European Opton usng a sngle Black-Scholes Formula Bass

13 Method c0 c K σ Expected Soluton Intal Guess Quas-Newton (no gradent gven) Quas-Newton (gradent gven) Trust Regon Wth the ntal guess, [c0, c, K, σ] = [0.5, 0.5, 75, 0.00], Fgure 3.3 show that both the Quas-Newton and Trust Regon methods both estmate the prce of a call opton accurately. However, the Quas-Newton method s hghly dependent on the ntal guess, where an ntal guess of K = 65 wll cause the algorthm to fnd ncorrect parameter values. For the non-lnear algorthms, we only allowed a maxmum of 00 teratons the QuasNewton and Trust Regon. Although the results for the trust-regon method appear not to converge, f we ncrease the number of teratons, these results appear to eventually converge. Addtonally, ncreasng number of data ponts also reduces the number of teratons requred to reach the optmal coeffcent values. Fgure 3.4: Estmates for the Call-Spread functon usng bass (2.7) f (x ~c) = c0 + c x + c2 bls(x, K, r, t, σ ) + c3 bls(x, K2, r, t, σ2 ). For ths data, we set K, K2, σ, and σ2 to 90, 0, 0.3, and 0.3 respectvely. Then usng the bass (2.7) we used fmnunc to fnd ~c = [c0, c, c2, c3, K, K2, σ, σ2 ]. The frst curve uses a good ntal guess (Table 3.4). In contrast the second curve was gven 2

14 a bad ntal guess (Table 3.4), whch estmated parameters further away than the actual value. Fgure 3.5: Estmates for the Call-Spread functon wth startng strke estmates K = 60 and K2 = 40 Table 3.4: Coeffcents of the Call-Spread functon usng bass (2.7) Intal Guess c0 c c2 c3 K K2 σ σ2 Expected Soluton Good Guess Intal Good Guess Result Bad Guess Intal 0 0 Bad Guess Result Fgure (3.5) Intal 0 Fgure (3.5) Result Ths method used the Quas-Newton Method wth an estmated gradent functon for a maxmum of 00 teratons. In all cases, the method ended before reachng the maxmum number of 00 teratons. 3

15 Fgure 3.6: Convergence of Model Parameters usng non-lnear Optmsaton vs the lscov nested non-lnear Optmsaton (2.) plotted on a sem-log scale Table 3.5: The Progresson of the Parameters for ncreasng teratons # Iteratons c0 c K σ Expected Soluton (NL) (NL) (NL) (NL) (NL) (NL) (NL) (NL) (nested NL) (nested NL) (nested NL) Note: NL refers to non-lnear 4

16 In Fgure 3.5, we only changed the ntal values of K (60) and K2 (40), to observe the change n K when usng the non-lnear methods. We notce that the value of K and K2 doesn t move too much, and nstead, s compensated by changng the values of σ and σ2. Ths could be troublesome, as t results n naccuraces for lower and greater values of stock (shown on Fgure 3.)). As such, ncreasng the number of unknown parameters results n a decrease n the performance of these methods, requrng better ntal guesses and more teratons. It s possble for us to reduce the amount of teratons by nestng the lnear optmsaton process n the non-lnear solvers, as we now explan for a call opton. In Fgure 3.6, we used the Quas-Newton Method wth no suppled gradent functon, the sngle Black-Scholes bass (f (x ~c) = c0 + c bls(x, K, r, t, σ)) and an ntal guess of [K, σ] = [75, 0.00]. Clearly, the Nested non-lnear optmsaton requred far less teratons than the orgnal method (20 opposed to 640) for all parameters. The nested method shows great promse to optmse other functons, such as the Call-Spread functon. 4 Future Drectons Further work n ths project wll be taken to look at extendng the optmsaton process to value more complex opton types such as Amercan Optons and Bermudan Optons. It would be nterestng to also consder other types of spread functons such as the Butterfly Spread Boundary Condton, Down-and-n Payoff functon and the Dscrete Down-and-out barrer put functon. These functons wll affect the amount of Black-Scholes formulae requred to accurately construct a model. 5

17 Fgure 4.: Exact Plot of the Black-Scholes PDE wth the Butterfly-Spread Boundary Condton K + K2 + ) + (x K2 )+ (4.) 2 As we sometmes experenced ssues wth the strke prce, K, and volatlty parameter, σ, U (x, T ) = (x K )+ 2(x reachng negatve values whch make no sense and break the model; t may be necessary to approach ths mnmsaton problem as a constraned problem, rather than the current unconstraned problem. In MATLAB, ths would mean changng the method to fmncon rather than fmnunc and then supplyng the necessary constrants. Alternatvely, we may substtute K = ec and σ = eθ (where c and θ are parameters that span the Real Lne) n order to ensure the two parameters reman postve. In addton, n these cases, t may be benefcal to add a penalty term to our objectve functon, to ensure that the Strke Prces of the ndependent Black-Scholes formulae reman dstant from each other. An example penalty functon s penalty,j = γ,j. K Kj (4.2) for some constant γ,j. We may also nclude other sets of bases. Longstaff and Schwartz have prevously used a set of weghted Laguerre polynomals and have suggested potentally usng other bass polynomals 6

18 to model the prce of optons (Longstaff and Schwartz, 200). Ths s somethng that can also be explored. In ths research project, we used standard least squares to create an estmate of the prce of the opton. Ths could be done by provdng a better weghtng for the more mportant regons of the curve. 7

19 Appendx A The Least Squares Gradent Functons Note here that τ = T t, and knowng that z r z x and = = 2 ln( ) + τ ( 2 ), K 2 σ Kσ τ σ τ we obtan the followng partal dervatves x bls e rτ φ(z) e rτ Φ(z σ τ ) + φ(z σ τ ), and = Kσ τ σ τ (A.) z bls z = x φ(z) Ke rτ φ(z σ τ ), (A.2) ~c(ss(~c; x)) = 2 X ( ~c(f (~c; x )) (y f (~c; x )), (A.3) I where ~c(f (~c; x )) s dependent on the bass functon we are usng. The Gradent functons of the Sum of Squares formula are an N vector (where there are N parameters) consstng of the partal dervatves of the Sum of Squares wth respect to each parameter. ss c ss c2 ~css =... ss cn For the n-degree polynomal cases (2.5), the gradent vector entres follow a general formula n X ss X j = jx (y ( c x )). cj j=0 I (A.4) If the bass functon s a cubc functon f (x ~c) = 3 X c x, =0 8

20 we would get the followng gradent functon P c2 x2 c3 x3 )) + I (y (c0 + c x + P 3 2 c I (y (c0 + c x + c2 x + c3 x )). ~c(ss(~c; x : I)) = 2 P 2c2 I x (y (c0 + c x + c2 x2 + c3 x3 )) P c3 I x (y (c0 + c x + c2 x + c3 x )) (A.5) If we consder bass (2.6): f (x ~c) = c0 + c bls(x, K, r, t, σ), we obtan the followng gradent functon (A.6) P c))) I (y f (x ~ P I bls(x, K, σ)(y f (x ~c))). ~c(ss(~c; x : I)) = 2 P bls c I (x, K, σ)(y f (x ~c)) P c I bls (x, K, σ)(y f (x ~ c )) (A.7) When we consder the Black-Scholes Problem, where c0 (K, σ) and c (K, σ) are optmally solved va ~c(k, σ) = (AT A) AT ~y, and A s the matrx (A.8) bls(x, K, σ).....,. bls(xn, K, σ) we obtan the followng gradent functon P ss I (x, K, σ)(y f (x ~c)) ss I (x, K, σ)(y f (x ~c)). ~c(ss(k, σ; x : I)) = 2 P. Consequently, the gradent functons become g(k, σ) ~c(k, σ) ss(~c, K, σ) = ~c(ss(~c, K, σ)) + and g(k, σ) ~c(k, σ) ss(~c, K, σ) = ~c(ss(~c, K, σ)) +, 9 (A.9)

21 where the Partal Dervatves of ~c obtaned by ~c(k, σ) AT (AT A) = (AT A) ( ), (A.0) AT (AT A) ~c(k, σ) = (AT A) ( ). (A.) and If we consder bass (2.7) f (x ~c) = c0 + c x + c2 bls(x, K, r, t, σ ) + c3 bls(x, K2, r, t, σ2 ) we obtan the followng gradent functon P c)) I (y f (x ~ P x (y f (x ~c)) I P bls(x, K, σ )(y f (x ~ c )) I P bls(x, K2, σ2 )(y f (x ~c)) I. ~c(ss(~c; x : I)) = 2 P bls c2 I (x, K, σ )(y f (x ~ c )) P bls c2 I (x, K2, σ2 )(y f (x ~c)) 2 P bls c3 I (x, K, σ )(y f (x ~c)) P bls c3 I 2 (x, K2, σ2 )(y f (x ~c)) 20 (A.2)

22 References Black, F. and Scholes, M. (973), The prcng of optons and corporate labltes, Journal of poltcal economy 8(3), Longstaff, F. A. and Schwartz, E. S. (200), Valung amercan optons by smulaton: a smple least-squares approach, The revew of fnancal studes 4(), MATLAB (207), verson (R207b), The MathWorks Inc., Natck, Massachusetts. 2