Jadé Dieteren Black Scholes Model with the Radon Nikodym Derivative

Transcription

1 Jadé Dieteren Black Scholes Model with the Radon Nikodym Derivative MSc Thesis

2 Black Scholes Model with the Radon Nikodym Derivative Jadé Dieteren September 14, 2014 Abstract This research concerns valuing European options. The goal of this paper is to explain how to implement the Black Scholes model for option valuation. We calculate the expected value of the options and of the Radon Nikodym derivative, which is a random variable. Furthermore, we use that for large number of N (number of paths) in a Monte Carlo simulation, the call or put option price at time t under the risk neutral measure Q, is equal to the option price at time t under the probability measure P multiplied by the Radon Nikodym derivative. 1

3 1 Introduction This research concerns valuing European options. The goal of this paper is to explain how to implement the Black Scholes model for option valuation (Black and Scholes, 1973). We calculate the expected value of the options and of the Radon Nikodym derivative, which is a random variable. Furthermore,we use that for large number of N (number of paths) in a Monte Carlo simulation, the call or put option price at time t under the risk neutral measure Q, is equal to the option price at time t under the probability measure P multiplied by the Radon Nikodym derivative (Hammersley and Handscomb, 1965). The base of the Monte Carlo methods is the analogy between probability and volume. In measure theory the intuitive notion of probability is formalized. That is, if there is a set of outcomes then the probability of an event is defined to be its measure to that of a universe of possible outcomes. The Monte Carlo methods uses this fact in reverse. The volume of a set is calculated by interpreting the volume as a expectation. Although there are more complicated cases, this means sampling randomly from a universe of possible outcomes and taking the fraction of random draws that fall in a particular set as an estimate of the set s volume (Glasserman, 2003). By using the law of large numbers one can show that the estimate converges to the correct value as the number of draws increases. This article builds on the findings of Black and Scholes (1973). The Black Scholes model is used to simulate the paths in the Monte Carlo simulation and the analytical form of the Black Scholes model and Black model is used to test our restults. The main assumptions of the model are that the stock price follows a geometric Brownian motion and the stock price changes are lognormally distributed. However, at the time it was published the model caused a great improvement in calculating the value of options and it is still used for many option valuation problems. Glasserman (2003) extensively explains the features of the Monte Carlo methods from the perspective of financial engineering since the Monte Carlo simulation has become an essential tool in the pricing of derivative securities and in risk management. Therefore, the theory presented in the book is of great value for this paper since we focus on implementing the Black Scholes model using the Monte Carlo method. In section 2 the basics of implementing the model will be explained. The models that will be crucial for the implementation of the model are introduced. In section 3 the main results are derived with the use of the Euler discretization method and Girsanov s theorem. In section 4 we start building the model by introducing the variables that remain constant during the steps of the implementation. In section 5 we build the model with the assumption that the interest rate remains constant. In section 6 the same steps of building the model 2

4 will be discussed for a model with a term structure of interest rates. In section 7 we show an example of the model with a term structure of interest rates. As a final note, a conclusion will be drawn. 2 Basics This section starts by introducing the models that are used. Furthermore, we will explain the main results of this paper. These derivations are crucial for understanding the model. 2.1 Model Basic models we will use in the remaining of the paper. Black Scholes Model: ds = µsdt + σsdw (1) db = rbdt (2) Where S is the stock price, µ is the drift or the expected return on the stock and σ is the volatility of the stock returns. Furthermore, B is the risk free bond price and r is the risk free interest rate. Analytical Black Scholes Option Pricing Model: d 1 = ln( S K )+ 1 2 (r+σ2 )(T t) σ T t (3) d 2 = d 1 σ T t (4) C(S, t) = SN (d 1 ) Ke r(t t) N (d 2 ) (5) P (S, t) = N ( d 2 )Ke r(t t) N ( d 1 ) (6) Besides the parameters that are already specified, we have that K is the strike price of the option, T is the time to maturity in years, so t = 1, 2, 3,..., T. Then C(S, t) is the value of an European call option at time t. P (S, t) is the value of an European put option at time t. From the Black Scholes model we easily derive the Black model, once we note that F (t) = S(t) D T (t) where F (t) is the forward price of the stock and D T (t) is the discount facturing using the risk free interest rate r. The Black model will be useful for the implementation of the Monte Carlo simulation. 3

5 Black Model d 1 = F (t) ln( K )+ 1 2 σ2 T σ T d 2 = d 1 σ T C(S, t) = D T (F (t)n (d 1 ) KN (d 2 )) P (S, t) = D T (KN ( d 2 ) F (t)n ( d 1 )) (7) (8) (9) (10) For the Black model we now have the additional parameters F (t) (defined as the forward price of the stock) and D T, the discount factor. We will use the analytical Black Scholes model and the Black model to verify the results we obtain with the use of the MC-simulation. 3 General We will now derive our main results. We will build on these results during the implementation of the model. 3.1 Euler Method According to the Black Scholes model the following holds: ds = µsdt + σsdw. Let t = T M where T is the time to maturity and M is the number of time steps of t until T. This is a stochastic differential equation. A numerical way for solving this is using the Euler method (Kloeden and Platen, 1992). For the general case, consider the stochastic differential equation: dx(t) = a(t, X(t))dt + b(t, X(t))dW (t) (11) Then, according to the Euler method, we can discretize this to: X(t + t) = X(t) + a(t, X(t)) t + b(t, X(t)) W (t) where W (t) = W (t + t) W (t) (12) If we apply the above to the Black Scholes stochastic differential equation, we find that it can be approximated by: S(t + t) = S(t) + µs(t) t + σs(t) W (t) where W (t) = W (t + t) W (t) (13) Furtheremore, by construction we know that W (t) iid N (0, t) (14) Therefore, in our Monte Carlo simulation we have W (t) by tε(t) 4

6 3.2 Girsanov s Theorem To derive the Radon Nikodym derivative we apply Girsanov s Theorem, which discribes the effect of change of measure on Brownian motions (Girsanov, 1960). Let g(w (t + t)) be a function containing a stochastic process, or more particular a Brownian motion. Furthermore, let f P (t, W (t); t+ t, W (t+ t)) be the probability density function of g(w (t+ t)) under the real world probability measure P and let f Q (t, W (t); t + t, W (t + t)) be the probability density function of g(w (t + t)) under the measure Q. Then, E P [g(w (t + t))] = g(w (t + t))f P (t, W (t); t + t, W (t + t))dw(t + t) (15) and E Q [g(w (t + t))] = g(w (t + t))f Q (t, W (t); t + t, W (t + t))dw(t + t) (16) Then, we can rewrite the expectation under the measure Q in the following way: E Q [g(w (t + t))] = g(w (t + t))( f Q (t,w (t);t+ t,w (t+ t)) f P (t,w (t);t+ t,w (t+ t)) )f P (t, W (t); t + t, W (t + t))dw(t + t) (17) Note that we have multiplied the equation for E Q [g(w (t + t))] with f P (t,w (t);t+ t,w (t+ t)) f P (t,w (t);t+ t,w (t+ t)). as the Radon Nikodym derivative, ex- We define f Q (t,w (t);t+ t,w (t+ t)) f P (t,w (t);t+ t,w (t+ t)) pressed by R(t + t). That is E Q [g(w (t + t))] = g(w (t + t))r(t + t)f P (t, W (t); t + t, W (t + t))dw(t + t) (18) This result shows that we can calculate E Q [g(w (t+ t))] with the probability density function under the measure P and the Radon Nikodym derivative, which is also a stochastic process. Hence, E Q [g(w (t + t))] = E P [g(w (t + t))r(t + t)] (19) Note that R(t + t) is a function of the random variable W (t + t) and therefore the Radon Nikodym deriavtive R(t + t) is also a random variable. 5

7 3.3 Radon Nikodym Derivative We can derive R(t) in the following way for the Brownian motion dw Q + γdt = dw P As we have stated before, W (t) N (0, t). For the Brownian motions dw Q and dw P γdt we know that W (t) = W (t + t) W (t) N (0, t) W P (t + t) N (W (t), t) (20) (21) (22) W Q (t + t) N (W (t) + γ(t), t) (23) Consider W (t) = W (t + t) W (t). From the definition of the Radon Nikodym derivative we have R(t) = f Q (t, W (t); t + t, W (t + t)) f P (t, W (t); t + t, W (t + t)) = = 1 2π t exp( 1 ( W (t) γ t) 2 2 t ) 1 2π t exp( 1 ( W (t)) 2 2 t ) ( W (t) exp( 2 2 W (t)γ t+(γ t) 2 ) t ) exp( 1 ( W (t)) 2 2 t ) = exp(γ W (t) 1 2 γ2 t) (24) Since W (t) is normally distributed with mean 0 and volatility t, the likelihood ratio that we now have obtained is a lognormal random variable with drift 0 and volatility γ An extension would be to simulate paths of Brownian motions. If we have increments W (t i ) we can simulate the path for every t i = i t where i = 1, 2,..., M. If these increments W (t i ) are independent and if W (t i ) iid N we can take the product of the Radon Nikodym derivates to calculate the Radon Nikodym derivative for the path. We now also have the time dependent variable µ i. We find R(t) = f Q (t i, W (t i ); t + t i, W (t i + t i )) f P (t i, W (t i ); t i + t i, W (t i + t i )) M = exp(γ i W (t i ) 1 2 γ2 i t) i=1 M = exp( γ i W (t i ) 1 M γi 2 t) 2 i=1 i=1 (25) 6

8 The following consist of a heuristic proof. Glasserman shows the formal proof in Monte Carlo Methods in Financial Engineering (2003). If we let M this converges to T exp( γ(s)dw (s) 1 T γ(s) 2 ds) (26) Here we use that M t is equal to T. 3.4 Radon Nikodym Derivative in Black Scholes Model We will now show how to derive the Radon Nikodym derivative for the Black Scholes model that we will use in the analysis. We start with (13) derived above. Furthermore, let Let dw P dw Q + λ(t, W )dt. Then the following conditions hold W (t) = W (t + t) W (t) N (0, t) W P (t + t) N (W (t), t) W Q (t + t) N (W (t) + λ(t), t) We are now able to derive the likelihood ratio R(t + t) = dq dp Hence, we obtain R(t + t) = exp(λ W 1 2 λ2 t) (27) (28) (29) by using (24). (30) Again, in the Monte Carlo simulation we can approximate dw with tε t, where ε t N (0, 1) since dw N (0, t). Then finally, we will aproximate the Radon-Nikodym derivative in the Monte Carlo simulation by R(t + t) = exp(λ tɛ t 1 2 λ2 t) (31) 3.5 Derivation of λ Using the Black Scholes model, we can mathematically derive λ. We know that under the measure Q the expected return is the risk free interest rate r. Then ds = µsdt + σsdw P = µsdt + σs(dw Q + λdt) = (µ + λσ)sdt + σsdw Q Hence, µ + λσ = r and rewriting this yields λ = λ r σ Hence, we can rewrite the Radon-Nikodym in the following way R(t + t) = exp( µ r σ W 1 2 (µ r σ )2 t) (32) (33) (34) 7

9 3.6 Property of Radon Nikodym Derivative One of the properties of the Radon-Nikodym derivative is that E(R(t+ t)) = 1. This result follows from (30), that is E[R(t + t)] = E[exp(λ W 1 2 λ2 t)] = E[ exp(λ W ) 1 = 2 λ2 t E[exp(λ W )] exp( 1 2 λ2 t) Since the denominator only consists of constants. ] (35) Then note that E[exp(λ W )] is the moment generating function of dw, which is normally distributed with mean 0 and variance t. Hence, we can further rewrite (35) exp(λ tλ2 ) exp( 1 2 λ2 t) = exp( 1 2 tλ2 ) exp( 1 2 λ2 t) = 1 (36) 3.7 Logarithmic transformation Since we have applied the Euler-scheme, we will continue our analysis with d ln(s(t)) and d ln(r(t)) instead of S(t) and R(t). We will derive both d ln(s(t)) and d ln(r(t)) using Ito s lemma. For non-random µ and σ we have From this we know d(ln(s)) = 1 S (µsdt + µsdw ) 1 2 σ2 dt = (µ σ2 )dt + σdw 2 (37) ln(s(t)) = ln(s(0) + σw (t) + (µ σ2 (38) 2 )t) Hence, we can derive the following expression which is useful to test our model analytically S(t) = S(0) exp(σw (t) + (µ σ2 2 t) Furthermore, for small time steps t ln(s(t + t)) = ln(s(t)) + (µ σ2 2 ) t + σ tε(t) Hence we obtain S(t + t) = S(t) exp((µ 1 2 σ2 ) 2 t + σ tε(t)) (39) (40) (41) 8

10 4 Basics of the Implementation Now we can start building the model. The following variables will remain constant in every step of the implementation. We define a constant standard deviation, denoted by σ. A constant market price of (equity) risk λ, that is λ = µ r σ. Hence, intuitively λ is the excess return required for investing in stocks instead of risk free bonds per unit of volatility. Although we will let the risk free interest rate vary according to a yield curve or stochastic, we will let λ remain constant. That is, µ will vary if we change r. A stock price at t = 0, notation S(0) and a strike price K. The maturtiy time T of the option. The number of time steps M until the option matures. Then we define the length of the time steps t as t = T M. The ε t is standard normally distributed. Therefore we generate for every t = 1, 2, 3,..., T a ε t and we do this for every path n = 1, 2, 3,..., N. Every row in the matrix shows one single path and every column shows the value for ε t at a particular time t. If we have values for ε t, then we can construct values for dw for every matrix entry simply by dw = tε t. Since ε t N (0, 1) we know that dw N (0, t). At this stage we can simply check our implementation so far by calculating the mean and the variance at every time t. 5 The Model with Constant Interest Rates If r, the risk free rate, and µ, the interst rate with risk premium, are constants we now have to include a value for these constants. 5.1 S(t) under the Real World Measure P We can simulate the stock price for every path under P using S P (t + t) = S P (t) + µs P (t) t + σs P (t)dw (42) We start with our fixed S(0) and then we can generate the price at every t = 1, 2, 3,..., T the stock price for every path. Then we obtain the mean at time t by 1 N N i=1 S(t) where N is the number of paths we have generated for our Monte Carlo simulation. A small check for the model so far is comparing the mean of the Monte Carlo simulation for every time t with E P [S(t)]. 9

11 5.2 S(t) under the Risk Neutral Measure Q Now the stock price at time t is S Q (t + t) = S Q (t) + rs Q (t) t + σs Q (t)dw (43) To obtain the prices and the mean of the simulation at t = 1, 2, 3,..., T we can use the same method as stated above. Similarly to what we did in the previous section we can check our model by comparing the mean of the Monte Carlo simulation for every time t with E Q [S(t)]. 5.3 Radon Nikodym Derivative We can calculate the R(t) for every path by using R(t + t) = exp( µ r σ tεt 1 2 (µ r σ )2 t) (44) As derived before E[R(t)] = 1. Therefore, we can again test our implemented model by calculating the mean of R(t) in the Monte Carlo simulation. That is, R(t) and checking whether this mean is converging to 1 for large N. 1 N N i=1 5.4 Check of the Program with the Radon Nikodym Derivative We know that E Q [e rt X(S(T ))] = E P [e rt R(0, T )X(S(T ))] By the Monte Carlo simulation we can approximate this with (45) E Q [e rt X(S(T ))] = 1 N N e rt R(0, T ; n)x(s(t ); n) i=1 (46) where N is the number of paths in the Monte Carlo simulation. 5.5 Call Option For the call option, the option value at time T is X(t) = max(e rt [S(T ) K], 0) (47) We can test our results since max(e rt [S P (T ) K], 0)R(T ) will converge to max(e rt [S Q (T ) K], 0) for large N. Hence, we should find that 1 N N max(e rt [S P (T ) K], 0)R(T ) = 1 N i =1 N max(e rt [S Q (T ) K], 0) i=1 (48) 10

12 5.6 Put Option For a put option, we can calculate the option value the following way X(T ) = max(e rt [K S(T )], 0) (49) For large N max(e rt [K S P (T )], 0)R(T ) will converge to max(e rt [K S Q (T )], 0). Hence, in our model we should find that 1 N N max(e rt [K S P (T )], 0)R(T ) = 1 N i =1 N max(e rt [K S Q (T )], 0) i=1 (50) 5.7 Analytical Black Scholes Formula For a constant interest rate r we can easily check our obtained results under the risk neutral measure Q by comparing it to the option value that we obtain with the analytical Black Scholes model. Hence, for large N we find that for the call option 1 N N max(s Q (t) K, 0) = SN (d 1 )K exp( r(t t))n (d 2 ) i =1 (51) And for the put option 1 N N max(k S Q (t), 0) = N ( d 2 )K exp( r(t t)) N ( d 1 ) i =1 (52) 6 The Model with Term Structure of Interst Rates Now we will continue by making the model more general. Instead of a constant risk free interst rate r, we will now modify the model such that we can let our risk free interst rate change over time. 6.1 Variables We first have to implement a term structure of interst rates. Hence, for every t we enter the risk free interst rate, r(t). These are, besides the variables introduced before, the only variables that we have to enter. After we have the term structure of interest rates, we can use this to obtain other variables that will be useful for implementing the model. The discount factor for every t. That is, D(t) = exp( r(t)t t). 11

13 Next we can calculate the forward rates at time t: F (t) = S0 D(t). The discount factor at time t, so forward price at time t divided by the forward price at time t + t. That is F (t) D( t) = F (t + t) = S 0 D(t) S 0 D(t+ t) D(t + t) = D(t) exp( r(t)(t + t)) = exp( r(t)t) = exp( r(t) t) (53) For every step t we can now calculate the forward discount factor. From these discount factors, we can also calculate the forward interst rates, denoted by r ft (t). We calculate this the following way 1 D( t) t = ln( D(t) D(t+ t)) t = ln(exp(r(t) t)) t = r ft (54) Finally, we can calculate the drift term at time t, denoted by µ(t) and the forward drift term, denoted by µ ft (t). Note that we now have a µ that is time dependent and therefore we will obtain a term structure for µ. Since we have a constant λ and a constant σ and we have already calculated the values for r(t) we can calculate the values for µ(t) in the following way λ = Rewriting this expression yields µ(t) r(t) σ µ(t) = λσ + r(t) (55) (56) With our forward interst rates we can also calculate the forward µ(t), denoted by µ ft (t) µ ft (t) = λσ + r ft (t) (57) 12

14 6.2 S(t) under the Real World Measure P Now we can generate the stock value S P (t). Recal that by the use of the Euler method we obtained (13). However, instead of a constant µ we now have µ rf (t) which we have calculated for every t = 1, 2, 3,..., N. Hence, we will implement S P (t + t) = S P (t) + µ ft (t)s P (t) t + σs P (t) tε t (58) We can check the model by comparing the mean under probability measure P with E P [S(t + t)] = exp(µ ft (t) t)s(t) (59) Hence, for large N we should find that 1 N N i=1 S P(t) converges to E P [S(t)]. 6.3 S(t) under the Risk Neutral Measure Q Similar to the previous section we can generate S Q (t) with the forward risk free interest rate r ft (t), which we have calculated for every t = 1, 2, 3,..., N. That is S Q (t + t) = S Q (t) + r ft (t)s Q (t) t + σs Q (t) tε t (60) We can check the model by comparing the mean under probability measure Q with E Q [S(t)] = F (t) = S 0 D(t) (61) So again, for large N we should find that 1 N N i=1 S Q(t) converges to E Q [S(t)]. 6.4 Radon Nikodym Derivative Now we continue with calculating the Radon Nikodym derivative. We can calculate the R(t) for every path using R(t + t) = exp(λ tε t 1 2 λ2 t) (62) The condition that E[R(t)] = 1 still holds. Therefore, we can again test our implemented model by calculating the mean of R(t) in the Monte Carlo 1 N simulation. That is, N i=1 R(t) and checking whether this mean is convering to 1 for large N. 6.5 Check of the Program with Radon Nikodym Derivative Similarly to section 5.4 we can test the model with the Radon Nikodym derivative. Again, we know that E Q [e rt X(S(T ))] = E P [e rt R(0, t)x(s(t))] (63) 13

15 6.6 Call Option For the call option the value at time t is calculated by (47). Then similarly to section 5.5 we can test our results by (48). 6.7 Put Option For a put option we can calculate the value according to (49). We can test our model by (50). 6.8 Analytical Black Formula To test the model that we have implemented so far, we can apply Black s analytical formula introduced in section 2.1. At every t = 1, 2, 3,..., T we should find that the option value obtained by performing the Monte Carlo simulation will converge to the option value accoreding to the analytical Black formula if N. That is for a call option 1 N N max(s Q (t) K, 0) = D T (F (t)n (d 1 ) KN (d 2 )) i =1 (64) For the put option 1 N N max(s Q (t) K, 0) = D T (KN ( d 2 ) F (t)n ( d 1 )) i =1 (65) 7 An Example with a Term Structure of Interest Rates We will give an example of the program that is implemented. We will mainly use the same structure as in section 6. First we have to define the variables that need to be chosen. Then we let the program run and we find the option value with the use of the Monte Carlo simulation. We will also check the program by multiplying the stock value under the P measure with the Radon Nikodym derivative to obtain the stock value under the Q measure. We do Monte Carlo simulations. 7.1 Variables The following variables need to be entered λ 0.25 M 20 t 0.25 S(0) 100 K

16 Since we have defined T = M t, the maturity time T of the option is 5. In this example we will have a term structure of interest rates as stated in the following table. The term structure needs to be entered. We have a strictly increasing term structure of interest rates. The rate of increasing is not in line with reality. However, this makes the example more useful. We have included the discount factors, the time dependent µ ft and the forward interest rate that we will use later. t r(t) D(t) µ ft (t) r ft (t) Next we need a matrix of ε(t) in order to approximate W (t). There are 21 columns since we include time zero in the paths. Hence, first we generate the matrix with the ε(t) by using the inverse of the cumulative normal distribution with random probability (the probability where we want to evaluate the inverse function), mean (µ) zero and standard deviation (σ) 1. Then we generate a second matrix where we multiply every path (row) of the matrix with t. Note that the mean at every time t should be approximately zero. 7.2 S(t) under the Real World Measure P By using (58) we now calculate the stock value under the real world measure P and check whether the mean of the Monte Carlo simulation converges to (59). 15

17 Our findings are listed in the following table. t mean S P (t + t) E P [S(t + t)] z-score S(t) under the Risk Neutral Measure Q Here one can use (60) to simulate the paths and (61) to test the program. We have listed our findings in the following table. 16

18 t mean S Q (t + t) E Q [S(t + t)] z-score Check of the Program with Radon Nikodym Derivative Next we calculate the Radon Nikodym derivative using (62). Furthermore, we calculate the option value under the P measure and we multiply this with the Radon Nikodym derivative to obtain the risk neutral value (see formula (63)). We can compare this with the option value calculated using the analytical Black Scholes formula. This is listed in the following table. These findings are for call options, but the same reasoning holds for put options. 17

19 t Option value using MC Option value using analytical Black Scholes z-score This example shows that with the method using the Radon Nikodym derivative and the analytical Black Scholes formula we find approximately the same option value. As mentioned before we have simulations. In this case the absolute value z-score is at most Conclusion We will now give a summary of the results in this paper. We have first introduced the Black Scholes model and the Analytical Black Scholes model and Black model. We continued with more theory that we needed to implement the model. The main finding was the Radon Nikodym derivative that is crucial in this paper. Then we continued with building the program, firstly with a constant interest rate and secondly with a term structure of interest rates. We discussed several steps of the implementation and introduced various possibilities to test whether the implementation is correct. Finally, we give an example of the program with a term structure of interest rates. As mentioned in the previous section we have a z-score of at most 0.5. We could decrease the z-score by decreasing t (so increasing the number of steps m). Furthermore we could simulate more paths in the Monte Carlo simulation. 18

20 However, with this program we find approximately the same option values using the risk neutral measure, the real world measure and the Radon Nikodym derivative and the analytical Black Scholes model. Hence, this program is a good tool for finding the option value of an European option. An extension of this paper is to include a stochastic interest rate. That is, the Black Scholes Hull White model. The one factor Hull White Model (Hull and White, 1990) can be incorporated in the Black Scholes world to obtain a model with stochastic interest rates. One can still apply the theory of the Radon Nikodym derivative, but more variables should be introduced. 19

21 9 References Black, F., and Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81: Girsanov, I. V. (1960) On Transforming a Certain Class of Stochastic Processes by Absolutely Continuous Substution of Measures. Theory of Probability and its Applications, Vol. 5, No 3: pp Glasserman, P. (2003). Monte Carlo Methods in Financial Engineering, Springer- Verlag, New York. Monte Carlo Methods, Hal- Hammersley, J. M., Handscomb, D. C. (1965). sted Press, New York. Hull, J., and White, A. (1990). Pricing interest-rate-derivative securities, Review of Financial Studies 3: Kloeden, P. E., and Platen, E. (1992). Numerical Solution of Stochastic Differential Equations, Springer. 20