II. Pricing Derivative Products. Risk-free portfolios. 6. Partial Differential Equations Method
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1 II. Pricing Derivative Products Risk-free portfolios 6. Partial Differential Equations Method Pricing financial derivatives with partial differential equations is based on the elimination of all randomness (risk) by utilizing the no arbitrage assumption and replicating portfolio (hedge). The idea follows the schedule T = time to maturity F = derivative contract S t = security price F T = F (S T,T) At expiration F T is known. (1) S SDE F noarbitrage PDE solution Example 6.1: Call option with exercise price X (2) F T = max(s T X, 0). At general timepoints, t, F (S t,t) is unknown. 1 2
2 The dynamics of ds implies (via Itô) the dynamics of df. Remark 6.1: df and ds are increments depending on the same innovation process, and can be eliminated with a replicating portfolio (if exists)! In order to find the PDE, referred in equation (1), construct portfolio P t (3) P t = θ 1 F (S t,t)+θ 2 S t, where θ 1 and θ 2 are amounts invested in F and S. Assuming θ 1 and θ 2 are constants. Using Itô (4) dp = P t P P dt + df + F S ds P 2 F 2(dF ) P 2 S 2 (ds)2 + 2 P df ds. F S 3 4
3 Now (5) so (6) P t =0, 2 P F 2 =0, P F = θ 1, 2 P S 2 =0, dp = θ 1 df + θ 2 ds. P S = θ 2 2 P S F =0 Remark 6.2: Generally θ 1 and θ 2 depend on time, too. Let (7) ds = a(s t,t) dt + σ(s t,t)dw t, Using Itô (8) df =(F t σ2 F ss )dt + F s ds, where F(S, t) (9) F t =, t (10) and (11) F s = F(S, t), S F ss = 2 F (S, t) S 2. Thus, dp = θ 1 df + θ 2 ds becomes (12) dp = θ 1 (F t σ2 F ss ) dt +(θ 1 F s + θ 2 )ds, where σ 2 = σ 2 (S t,t). 5 6
4 Defining θ 1 = 1 and θ 2 = F s,weget Equating the formulas for dp we get (13) dp = F t dt σ2 F ss dt, (16) rpdt=(f t σ2 F ss ) dt, i.e., portfolio (hedge portfolio) (14) becomes risk-free! P = F F s S No arbitrage implies that the return of P must be (15) dp = rp dt, i.e., (17) rp = F t σ2 F ss, or using P = θ 1 F + θ 2 S with θ 1 = 1 and θ 2 = F s, (18) where σ = σ(s t,t). r(f F s S)=F t σ2 F ss, where r is the risk-free rate. It is assumed that S t does not pay dividends (if it does then, e.g., δ units per time unit then dp = rp dt δdt). Note that this is an approximation, because θ 2 = F s actually depends on S t, which makes all what follows only approximating. 7 8
5 Arranging the terms gives a standard partial differential equation (19) to be solved. rf + rf s S + F t σ2 F ss =0 Remark 6.3: The asset price S is here a general diffusion process, given by (7). In terms of the geometric Brownian motion (20) ds = μsdt + σsdw, i.e., with a(s, t) =μs and σ(s, t) =σs, (19) becomes (21) rf + rf s S + F t σ2 S 2 F ss =0, the famous (Nobel price winning) Black-Scholes PDE. The solution depends on the boundary conditions. For example with the boundary conditions of the European option the solution leads to the Black-Scholes equation, where as American option leads to a different solution (no closed form solution is known). Examples of boundary conditions are 0 t T, S t 0, and F (S, T )=G(S, T ), where G is a known function. 9 10
6 Example 6.1: IfF is an European call option G(S, T )= max(s K, 0), where K is the exercise price, and (22) ds t = μs t dt + σs t dw t is the geometric Brownian motion (GBM), then (19) becomes Remark 6.4: The major problem in this approach is solving the PDEs under sppropriate boundary conditions. (23) rf + F t + rf s S F ssσ 2 S 2 =0 and the solution of the PDE (technical steps skipped here) gives the famous Black-Scholes call option formula (24) F (S, t) =S t N(d 1 ) Ke rτ N(d 2 ), where d 1 = log(s t/k)+(r σ2 )τ (25) σ, τ (26) d 2 = d 1 σ τ, (27) N(d i )= 1 2π di e 1 2 x2 dx are the values of the standard normal cdf at d i, i = 1, 2, and τ = T t the time to maturity
7 Extensions of the basic model Random dividends Constant dividends S t pays constant dividends δ per time unit, so that the return of the hedge portfolio (14) from price appreciation of the stock and dividends. That is the return of the hedge portfolio is (28) dp + δdt= rpdt and the PDE becomes (29) rf + rf s S + δ + F t F ssσ 2 =0. Thus, the constant dividends cause no extra problems. Suppose the dividends process is (30) dd t = a dt + σ dw t, where Wt is a standard Wiener process. The derivative price is then (31) with F = F (S t,d t,t) df = F t dt + F s ds + F d dd F ss(ds) F dd(dd) 2 (32) and (33) = F t dt + F s ds + F d dd σ2 F ss dt σ 2 F dd dt dp = θ 1 ds + θ 2 [F t dt + F s ds + F d dd F ssσ 2 dt F ddσ 2 dt ]
8 There exist no choice of the portfolio weights θ 1 and θ 2 that would make the portfolio nonstochastic. Thus the PDE approach does not work directly here. More structure is needed! One justified choice could be that the dividend process and the stock price process share the same random innovations. I.e., W is the same in ds and dd. Then dp = θ 1 [F t dt + F s (adt+ σdw) (34) +F d (a dt + σ dw )+ 1 2 F ssσ 2 dt F ddσ 2 dt + F sd σσ dt ] +θ 2 (adt+ σdw) Note that ds dd = σσ dt
9 Then selecting θ 1 = 1 and θ 2 = σf s σ F (35) d, σ dp becomes nonstochastic, and PDE approach applies. Exotic Options There are lots of different kinds of options, like lookback options, Asian, Knock-In, Knockout, ladder, multiasset, etc. In each of these the PDE is basically the same, but boundary conditions change, which lead to different solutions of the PDE
10 Solving PDEs in practice Consider again the PDE in the B-S framework (S t is GBM) (36) rf + F + rf s S F ssσ 2 S 2 =0, S 0, 0 t T. Closed form solution, like the B-S call price, cannot often be found, or the solution is difficult. Numerical solutions gives values for F (S, t) for (discrete) combinations (S i,t j ) of the stock price S and time variable t. Numerical Methods To solve the PDE numerically, one approximates the PDE with finite increments ΔS and Δt. Partitions for range of S and t are needed: 1. Select grid size for ΔS and Δt. 2. Select an appropriate range for S, S min S S max. 3. Determine the boundary conditions. 4. Determine the value of F (S, t) at the grid points
11 Let (37) and (38) df = ds = μs dt + σs dw (F t + 12 S2 σ 2 F ss ) dt + σs dw. So that the partial differential equation (19) can be written as (Black-Scholes PDE) (39) F t + rs F S σ2 S 2 2 F S 2 = rf. Suppose that the life of the option is T. Divide the interval [0,T] into n subintervals of equal length, Δt = T/n. In the same manner divide the reasonable range of the stock price into m subintervals of equal length, ΔS = S max /m. Then we have a grid (t j,s i ) where t j = jδt and S i = iδs, i =0, 1,...,m, j =1,...,n, i.e., time is running as (40) 0, Δt, 2Δt,...,(n 1)Δt, T and S as (41) 0, ΔS, 2ΔS,...,(m 1)ΔS, S max
12 Using numerical methods the idea is to replace the partial derivatives with the discrete approximation at points F ij = F (S i,t j ). For the first order partial derivatives the choices are: Backward difference: F S F i,j F i 1,j (42). ΔS Forward difference: F S F i+1,j F i,j (43). ΔS Central difference: F S F i+1,j F i 1,j (44), 2ΔS which is the average of the previous two. The preferred method is the central difference for the partial derivative w.r.t S, and forward or backward difference for the partial derivative w.r.t t (the reason becomes apparent below). The second derivative is approximated by (45) 2 F S Δ(ΔF ) 2 (ΔS) = F i+1,j 2F i,j + F i 1,j. 2 (ΔS) 2 Substituting these in (39) with S i = iδs (46) F i,j+1 F i,j Δt + riδs F i+1,j F i 1,j 2ΔS σ2 (i ΔS) 2 F i+1,j 2F i,j +F i 1,j (ΔS) 2 = rf i,j for i =1, 2,...,m 1 and j =0, 1,...,n
13 Rearranging terms and noting that the ΔS terms cancel out, we get (47) where a i F i 1,j + b i F i,j + c i F i+1,j = F i,j+1 a i = 1 2 i ( r iσ 2) Δt b i = 1+ ( r + i 2 σ 2) Δt c i = 1 2 i ( r + iσ 2) Δt To solve the problem there are several possibilities. One is the explicit difference method (and its variations) and the other is the implicit difference method. The former one is simpler, but unfortunately fairly unstable, whereas the latter is robust but computationally more expensive
14 The Implicit Finite Difference Method To illustrate the method, consider an American put on a non-dividend paying stock. The value of the put at time T (= t n ) is max(k S T, 0), where S T is the stock price at time T. Thus (48) F i,n = F (iδs, t n ) = max(k iδs, 0), i =0, 1,...,m. The value of the put, when stock price is zero, is K. So (49) F 0,j = K, j =0, 1,...,n. When S, F 0. Thus we approximate (50) F m,j =0, j =0, 1,...,n. Equations (48) (50) define the values of the put option along the three edges S = 0, S = S max, and t = T of the grid (t i,s j ), i =0, 1,...,m, j =0, 1,...,n
15 To fill the remaining nodes of the grid, we start from the points T Δt with j = n 1, where equation (47) is (51) a i F i 1,n 1 + b i F i,n 1 + c i F i+1,n 1 = F i,n for i =1, 2,...,m 1. The right-hand sides of (51) are known from equation (48). Furthermore, from (49) and (50) (52) F 0,n 1 = K F m,n 1 = 0. Equations (51) are m 1 simultaneous equations with m 1 unknowns: Given the solutions for F i,n 1,ifF i,n 1 <K iδs, then early exercise at time T Δt is optimal and F i,n 1 is set equal to K iδs. The nodes with T iδt, i =2, 3,...,m,are handled in a similar way, eventually giving F 0,1,F 0,2,...,F 0,n 1, one of which is the option price of interest. Remark 6.5: Solving F i,n 1 does not involve matrix inversion. The first equation in (51) can be used to express F 2,n 1 in terms of F 1,n 1, the second equation can be used to express F 3,n 1 in terms of F 1,n 1 and so on. The final equation provides a value for F 1,n 1, which can then be used to determine the other F i,n 1. (53) F 1,n 1,F 2,n 1,...,F m 1,n 1 to be solved
16 The relationship of the neighboring option prices in the implicit method is Explicit Finite Difference Method F i+1,j F i,j F i 1,j F i,j+1 To simplify the calculations of the implicit method, let us assume that the F/ S and 2 F/ S 2 are the same at grid points (i, j) and (i, j + 1), so that Figure 6.1: Relationship of the derivative price at time t +Δt to three values of derivative at time t in the implicit method. Thus in the implicit method the value at time t depends directly on its two neighbors, and hence indirectly (implicitly) on all option values at that time step. The advantages of the implicit method are that it is robust, and it always converges to the solution of the differential equation as ΔS and Δt approach to zero. (54) and (55) with (56) F S F i+1,j+1 F i 1,j+1 2ΔS 2 F S 2 F i+1,j+1 2F i,j+1 +F i 1,j+1 (ΔS) 2, F i,j = a i F i 1,j+1 + b i F i,j+1 + c i F i+1,j+1 a i = i Δt ( r σ 2 i ) 2(1 + r Δt) b i = 1 ( 1 σ 2 i 2 Δt ) 1+rΔt c i = i Δt ( r + σ 2 i ) 2(1 + r Δt) The disadvantage is that it is pretty tedious to program
17 Alternatively an explicit finite difference method can be obtained by taking backward approximation to F/ t in (47) and solving for F i,j 1. Example 6.2: Consider an American put option with T =0.5 years, i.e., six months, K =50,r = 10% p.a., and ds t = μs t dt + σs t dw t where σ = 40% p.a. Then we know F (S T,T) = max(50 S T, 0) and F (S t,t) 0asS t gets large. F (0,t)=50 for all t
18 Exercise 50 sigma 40% p.a r_f 10% p.a Dleta S 5 = S_max/ m, m = 20 Finite difference method Delta_t = 0.5 years /n, n = 12 Time to maturity (months) Stock Price Black-Scholes for European put Time to maturity (months) Stock Price Several improvements, like Crank-Nicolson, which is a kind of mixture of the implicit and explicit methods. A useful textbook approach to numerical methods can be found from Wilmott, P. (2000). Derivatives: The Theory and Practice of Financial Engineering. Wiley. (Or Paul Wilmott on Quantitative Finance, 2000, 2006 (2nd ed), Wiley). Table 6.1: Explicit finite difference method approximation to Black-Scholes American put option
19 Control variate technique A method called control variate technique can be utilized to improve accuracy of the numerical approximation. Suppose F 1 and F 2 are two similar type options approximated with finite difference method. Denote the approximations by F 1 and F 2. Suppose further that an accurate solution is for F 2. Then an improved approximation of F 1 is obtained as (57) F 1 = F 1 F 2 + F 2. The idea here is that the approximation errors of F 1 and F 2 are similar. An estimate for the error is F 2 F2. 38
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