Numerical methods for derivatives

1. Numerical methods for derivatives Hans Dewachter

2. Schedule of classes 1-3 • Class 1: introduction into Wiener processes and their discrete approximations • Class 2 : Portfolio strategies • Class 3 : excercise!

3. Wiener Processes • Introduction: the class of Wiener processes • Asset price dynamics and the no-arbitrage principle • Practical approaches to price derivatives • Binomial trees • Monte Carlo simulation • Numerical procedures for solving partial differential equations

4. Wiener processes • Why study the Wiener process? • Answer: Most standard derivative pricing models start from the assumption that the underlying sources of risk are well approximated by Wiener processes. • Implication: Prices will be accurate only to the extent that reality conforms to these processes

5. Wiener processes • Notation: • Wiener process W(t) or z(t) • Change in the Wiener process dW(t) or dz(t) • Construction of Wiener process • Probabilities under specific probability measure • 2 probabilities (up and down movement per unit time dt) • Step size(s) (up movement and down movement per unit time dt) • The limit for dt going to zero is the Wiener process.

6. Wiener processes • The limit for dt going to zero is the Wiener process iff: • Under the probability measure (defined by the respective up and down probability) • E(dW) = 0 dt puSu + (1-pu) Sd = 0 • E(dW(t2) = 1 dt pu(Su)2+ (1-pu) (Sd)2= 1

7. Constructing Wiener Processes • Interest: find the time path of a Wiener process over the period 0 to T with observations at discrete time values: W(t) for t=1, 2, …, T. • Strategy: • Define a basis steplength dt. • Define the initial value for W(t=0) = 0 • Simulate the step process for given stepsize and probabilities (1/dt) times • Collect the values for the Wiener process at the points (i/dt), i=1,2,….,T.

8. Constructing Wiener Processes • Construction of Wiener processes using excel

9. Properties of Wiener process • Wiener process has continuous sample paths (no jumps in continuous time) • The Wiener process differences are normally distributed! • W(t’) – W(t) is distributed as N(0, t’-t) • Expected change = 0 • Variance = t’-t • Standard deviation = sqrt(t’-t)

10. Generalized Wiener Process • Generalized Wiener allows for non-zero mean and any variance of changes • dX(t) = a.dt + b.dW(t) • E(X(t’) – X(t) ) = a.(t’-t) • Var(X(t’)-X(t)) = b2.(t’-t) • Example: excel file

11. Ito Processes ! • Ito processes allow us to have first order Markov system: state dependent mean and variances! • dX(t) = A(X,t).dt + B(X,t).dW(t). • Where A(X,t) and B(X,t) are functions of X and t. • Ito processes are used as a good approximation of financial series!

12. A simple but important Ito process: GBM • GBM: Geometric Brownian Motion • dX(t) = A(X,t).dt + B(X,t).dW(t). • A(X,t) = a.X(t) • B(X,t) = b X(t) • dX(t) = a.X(t).dt + b.X(t).dW(t). • Thus a dynamic equation for instantaneous returns! • dX(t)/X(t) = a. dt + b. dW(t) !

13. Recap on Wieners • Wiener processes are mathematical constructs in continuous time that: • Allow us to model risks (in terms of discrete changes) by normal random variables • Can be used in the construction of Ito processes • Are easily simulated by means of random up and down movements • But: imply continuous sample paths.

14. Risk Neutral Valuation Techniques • A risk neutral world • The risk premium and change of probability measure • Pricing derivatives

15. A risk neutral world • Suppose agents (speculators) are risk neutral. Describe the dynamics of financial assets according to drift (trend) and variance. • Drift: all financial assets would have the same expected return: the riskless rate. • Variance is not constrained in this world

16. Risk Neutral Valuation • In a risk neutral world : all financial assets would have the same expected return: the riskless rate. • All assets: implies also any type of derivative asset. • Pricing strategy for derivatives: • Create a risk neutral world (change of measure) • Solve for the derivative price dynamics • Ito-calculus • Simulation techniques

17. Creating the Risk Neutral world: change the probability measure • For every Wiener process and for any dynamics of dX, • dX(t) = A(X,t).dt + B(X,t).dW(t) • There exists a set of probabilities q, such that under this measure (denoted by *) the dynamics are given by: • dX*(t) = r.X(t).dt + B(X,t).dW*(t). • (Called Girsanov’s theorem).

18. Creating the Risk Neutral world: change the probability measure • dX(t) = A(X,t).dt + B(X,t).dW(t), A(X,t) = aX, B(X,t) = X q = ( exp(rdt) – d ) / ( u –d ) u = exp( sqrt (variance. dt) ) d=exp(-sqrt (variance.dt) ) • Effects of using this probability measure q is: • Variance of X and X* are exactly the same • The mean change of X* is changed to the riskless interest rate r

19. Solving for the expected derivative dynamics • No arbitrage condition states that expected return per instant time must equal the riskless rate. • Let C(X,t) denote the price of the derivative security written on asset with price X • E(dC(X,t)) = r(t).C(X,t) dt

20. Solving for the current price of the derivative • How to solve for C(X,t), given that E(dC(X,t)) = r(t).C(X,t) dt • Three methods • Binomial tree models • Monte Carlo simulations • Solving partial differential equations

21. Solving by means of binomial trees • At maturity, T, payoff is known in each possible state of the world i.e. C(X,T) is known • Approximate the time unit dt, normalize this to 1 . • Work backwards from T to T-dt to find the value C(X,T-dt), using the no-arbitrage condition: E( C(X,T) – C(X,T-dt) ; T-dt) = r(T-dt). C(X,T-dt). dt • Use the risk neutral measure to compute the expectation • Repeat the steps until you reach the value C(X,t), to find the current value of the derivative!

22. Example of binomial tree: European call option • Assumptions (situation) • Underlying follows Geometric Brownian Motion with variance of 48% a year and a = 12% a year. • interest rates are constant at 6% a year • Call is two months before expiration • Strike price = 100 • Current price = 100.

23. Example ctd • Practical steps to be taken: • Define the unit of time dt • Generate the tree of the underlying • Specify the final pay-off C(X,T) at expiration • Define the risk neutral probabilities • Work backwards to find the current value

24. Step 1: Define the unit of time dt • We choose (for instance) dt = 1 month. • Therefore under the normal probability measure: • Expected change per month = 1% • Variance = 4% per month. • So from the GBM assumption (after renormalizing dt=1): • X(t+1)-X(t) = 0.01 X(t) + sqrt(0.04) X(t) dW • dW(t) = 1 with probability p • dW(t) = -1 with probability (1-p)

25. Changing probability measure • Expected return must be riskless interest rate • X(t)exp(rdt)=pXu(t+dt) + (1-p)Xd(t+dt) • exp(rdt)=pXu(t+dt) /X(t)+ (1-p)Xd(t+dt)/X(t) • exp(rdt)=p.u+ (1-p).d • u = Xu(t+dt) /X(t): percentage increase in X • d = Xd(t+dt)/X(t): percentage decrease in X

26. Changing probability measure • Variance must remain unaffected • p.u2 + (1-p).d2 – exp(rdt) 2 = observed variance • u = Xu(t+dt) /X(t): percentage increase in X • d = Xd(t+dt)/X(t): percentage decrease in X • Imposing symmetry • u=1/d

27. Changing probability measure • Solution of problem is then • p = ( exp(r.dt) – d ) / (u – d ) • u=exp(sqrt(variance.dt)) • d=exp(-sqrt(variance.dt)) • In our example • u = exp(.20) = 1.2214 • d = 1/u = 1/1.2214 = 0.8187 • p = ( exp(0.005) - .8187 )/ (1.2214 - .8187) = 0.4626 0.4626

28. Tree for two months with base time unit one month

29. Terminal option value

30. Step 3: working backward through the tree E( C(X,T) – C(X,T-dt) ; T-dt) = r(T-dt). C(X,T-dt). dt E(C(X,T)) = (1 + r(T-dt).dt). C(X,T-dt) Using (1 + r(T-dt).dt) = exp(r(T-dt).dt) C(X,T-dt) = exp(-r(T-dt).dt)E( C(X,T) ) E( C(X,T) ) = p Cu(X,T) + (1-p).Cd(X,T)

32. Binomial pricing method • Binomial pricing method reconstructs in steps the Wiener process. • Then calculate the risk neutral probability • Construct X under the risk neutral probability (tree for X) • Work backwards through the tree for C, using the terminal value for C, and the risk neutral probability • This method can be used for any type of univariate derivative! Most general. • If stepsizes are taken small enough, then the method will be accurate

33. Monte Carlo simulation • Easy to use method that is easily extended to multivariate derivatives, i.e. derivatives that depend on more than a single underlying. • Instead of stepwise building the tree, one uses the known distributional characteristics of the underlying process. • Works as an approximation (less accurate than other methods.

34. Monte Carlo simulation • Intuition: • Suppose that the world consists of risk neutral agents, what is the price of an asset? • Price= E ( exp(-rT) V(T,X) ) • V(T,X) = derivative payoff function • r is the interest rate • E: the expectation in a risk neutral world

35. Monte Carlo simulation • Procedure: • Simulate a number of paths under the risk neutral measure • Calculate the derivative pay-off • Compute the average derivative pay-off (take the arithmetic average • Discount this average back by multiplying by exp(-rT).

36. Monte Carlo simulation • Simulation in risk neutral world using GBM: • real world: dX(t) = a.X(t).dt + b.X(t).dW(t). • Risk neutral world: • dX(t) = r.X(t).dt + b.X(t).dW*(t). • Solution: • X(T) = X(0) exp( ( r – b2/2)T + b. (W*(T)-W*(0) ) ) • W*(T) – W*(0) is normal with mean zero and variance T • W*(T) – W*(0) = sqrt (T).e, with e standard normal. • X(T) = X(0) exp( ( r – b2/2)T + b. sqrt (T).e )

37. Monte Carlo simulation • Step 2 simulation : • Take a large number of standard normal variables e(i), i=1,…, M. • For each e(i) compute the derivative price V() • Example • European call: V= max(0, X-F) • Simulation of derivative proceeds is then • V(e(i)) = max(0, X(e(i)) – F) for i=1,…, M. • With X(T) = X(0) exp( ( r – b2/2)T + b. sqrt (T).e(i) )

38. Monte Carlo simulation • Step 3: computing the price : • Take the average value over all V(X(e(i)),T) • Discount this average back to the present by multiplying by exp(-rT). • Result: Price (V) = E ( exp(-rT).V(X,T) ) • Example in excel.