FMO6 Web: Polls:

Transcription

1 FMO6 Web: Polls: Improving numerical methods. Optimization. Dr John Armstrong King's College London November 20, 2018

2 What's coming up Improving numerical methods Time to complete survey Optimization in MATLAB

3 Improvements Improving numerical methods We'll discuss Richardson Extrapolation which can be used to improve many numerical methods Four methods of improving Monte Carlo methods: Antithetic sampling (which we've seen already) Importance sampling Control variate method. Quasi Monte Carlo

4 Improvements Richardson Extrapolation Richardson Extrapolation Suppose we have a numerical method to compute y using some function y depending on a parameter ɛ. y = lim ɛ 0 y(ɛ) Suppose moreover we know that y(ɛ) = y + Cɛ n + O(ɛ n+1 ) for some constant C. By comparing two estimates for y and taking a linear combination we can get the ɛ n term to cancel giving us a new method with faster convergence: y k (ɛ) := kn y(ɛ) y(kɛ) k n 1 = y + O(ɛ n+1 )

5 Improvements Richardson Extrapolation Proof. y(ɛ) = y + C(ɛ n ) + O(ɛ n+1 ) y(kɛ) = y + C((kɛ) n ) + O(ɛ n+1 ) Subtracting k n copies of the rst equation to one copy of the second k n y(ɛ) y(kɛ) = (k n 1)y + O(ɛ n+1 ) k n y(ɛ) y(kɛ) k n 1 = y + O(ɛ n+1 )

6 Improvements Richardson Extrapolation Example Numerical dierentiation We know three estimates for the derivative f (x) already: Name Formula Convergence f (x+ɛ) f (x) Forward O(ɛ) ɛ f (x) f (x ɛ) Backward O(ɛ) ɛ Central f c f (x+ɛ) f (x ɛ) (ɛ) := O(ɛ 2 ) 2ɛ Applying Richardson extrapolation to central estimate f c (ɛ) f r (ɛ) := (2)2 f c (ɛ) f c (2ɛ) = 4 ( ) f (x + ɛ) f (x ɛ) 1 ( ) f (x + 2ɛ) f (x 2ɛ) 3 2ɛ 3 4ɛ = f (x + 2ɛ) + 8f (x + ɛ) 8f (x ɛ) + f (x 2ɛ) 12ɛ This must have an error of O(ɛ 3 ) or better. (In fact it is O(ɛ 4 ). Exercise: Prove this using Taylor's theorem).

7 Improvements Richardson Extrapolation Example: Integration We wish to compute an integral by the trapezium rule. f : [a, b] R. Dene ɛ = (b a)/n where N is the number of steps. Compute estimate for the integral I 1 using N steps, ɛ 1 = (b a)/n. Compute estimate for the integral I 2 using 2N steps, ɛ 2 = 1 2 ɛ 1. So take k = ɛ 2 /ɛ 1 = 1 in Richardson method. 2 Trapezium rule converges at rate n = 2. New estimate is: I R = ( I 1 I 2 ) = 1 3 I I 2

8 Improvements Richardson Extrapolation MATLAB implementation of Richardson Extrapolation % Perform intergration by the trapezium rule then apply richardson % extrapolation to obtain estimates with improved convergence function richardsonestimate = integraterichardson( f, a, b, N ) h1 = (b-a)/n; h2 = (b-a)/(2*n); estimate1 = integratetrapezium( f,a,b,n); estimate2 = integratetrapezium( f,a,b,2*n); k = h2/h1; n = 2; richardsonestimate = (k^n*estimate1 - estimate2)/(k^n-1); end

9 Improvements Richardson Extrapolation Interpretation I R = 1 3 I I 2 I 2 is computed using trapezium rule with 2N steps I 2 = h 2 (f (x 0)+2f (x 1 )+2f (x 2 )+...+2f (x N 2 )+2f (x N 1 )+f (x N )) I 1 is computed using N steps I 1 = h 2 (f (x 0) +2f (x 2 )+...+2f (x N 2 ) +f (x N )) So I R is given by: I R = h 3 (f (x 0)+4f (x 1 )+2f (x 2 )+...+2f (x N 2 )+4f (x N 1 )+f (x N )) So Richardson extrapolation in this example is equivalent to Simpson's rule.

10 Improvements Richardson Extrapolation Importance of Richardson's rule Richardson's rule can be applied to many numerical methods e.g. integration methods and nite dierence methods. You can generalize it to cancel higher orders, or simply apply it more than once. Unfortunately it cannot be used for Monte Carlo pricing as the error term is not a constant multiple of ɛ 1 2

11 Improvements Antithetic Sampling Revision: Antithetic Sampling Suppose we have a Monte Carlo pricer based on drawing n normally distributed random numbers ɛ i It is often better to compute the price using a sample based on ɛ i and ɛ i rather than to use a sequence of 2n independent random variables. Theory: If X 1 and X 2 are random variables with E(X 1 ) = E(X 2 ) then ( ) X1 + X 2 E(X 1 ) = E 2 But ( ) X1 + X 2 Var = (Var(X 1) + Var(X 2 ) + 2 Cov(X 1, X 2 )) Let X 1 be estimate based on the n variables ɛ i. Let X 2 be estimate based on ɛ i. We will often have Cov(X 1, X 2 ) is

12 Improvements Antithetic Sampling MATLAB implementation of Antithetic sampling % Price a call option by antithetic sampling function [price,errorestimate] = callantithetic( K,T,... S0,r,sigma,... npaths ) logs0 = log(s0); epsilon1 = randn( npaths/2,1 ); epsilon2 = -epsilon1; logst1 = logs0 + (r-0.5*sigma^2)*t + sigma*sqrt(t)*epsilon1; logst2 = logs0 + (r-0.5*sigma^2)*t + sigma*sqrt(t)*epsilon2; ST1 = exp( logst1 ); ST2 = exp( logst2 ); discountedpayoffs1 = exp(-r*t)*max(st1-k,0); discountedpayoffs2 = exp(-r*t)*max(st2-k,0); price = mean(0.5*(discountedpayoffs1+discountedpayoffs2)); errorestimate = std(0.5*(discountedpayoffs1+discountedpayoffs2))/sqrt(npaths/2) end

13 Improvements Antithetic Sampling Antithetic Sampling Results Parameters: S0 = 100, K = 100, σ = 0.2, r = 0.14, T = 1, N = Results: Method Price Standard error estimate BlackScholes Formula Naive Monte Carlo Antithetic Sampling Conclusion: Antithetic sampling is easy to implement and often rather eective.

14 Improvements Importance Sampling Importance Sampling Monte Carlo pricing is an integration method. You can use substitution to change one integral to another integral by re-parameterizing Equivalently you can change the distribution from which you draw your samples so long as apply appropriate weights to correct for this. Monte Carlo integration is exact when the price function is constant If we can re-parameterize so the price function is nearer to being constant, we will have reduced the variance of the Monte Carlo algorithm.

15 Improvements Importance Sampling Importance Sampling Example Suppose we want to price a far out of the money knock out call option Suppose that for 99% of price paths the option will end out of the money This means that 99% of price paths in the Monte Carlo calculation will give us no information. Instead: nd a way to generate the 1% of price paths where the option ends up in the money; compute the expectation for these paths; re-weight by multiplying by 100. For simplicity, let's do this for a vanilla call option to see how it improves upon ordinary Monte Carlo.

16 Improvements Importance Sampling Calculation Generate stocks prices at time T using the formula: log(s T ) = log(s 0 ) + (r 12 ) σ2 T + σ T N 1 (u) where u is uniformly distributed on [0, 1]. Option is in the money only if log(s T ) log(k). Equivalently only if: ( ) log(k) log(s0 ) (r (1/2)σ 2 ) T u u min := N σ T So only generate values u on the interval [u min, 1], then multiply resulting expectation by 1 u min to account for the fact that we have only generated 1 u min of the possible samples. We know the other samples would have given 0 for the option payo.

17 Improvements Importance Sampling MATLAB implementation of Importance Sampling function [price,error] = callimportance( K,T,... S0,r,sigma,... npaths ) logs0 = log(s0); % Generate random numbers u on the interval [lowestu,1] lowestu = normcdf( (log(k)-logs0 - (r-0.5*sigma^2)*t)/(sigma*sqrt(t)) ); u = rand(npaths,1)*(1-lowestu)+lowestu; % Now generate stock paths using norminv( u ). lowestu was chosen % so that the lowest possible stock price obtained is K. Note that % we are only considering a certain proportion of possible stock prices logst = logs0 + (r-0.5*sigma^2)*t + sigma*sqrt(t)*norminv(u); ST = exp( logst ); discountedpayoff = exp(-r*t)*(st-k); % Since we only simulate a certain proportion of prices, the true % epectation of the final option value must be weighted by proportion proportion = 1-lowestU; price = mean(discountedpayoff)*proportion; error = std(discountedpayoff)*proportion/sqrt(npaths);

18 Improvements Importance Sampling Importance Sampling Results Parameters: S 0 = 100, K = 200, σ = 0.2, r = 0.14, T = 1, n = Note that this is far out of the money, so naive Monte Carlo will perform badly. Results: Method Price Standard Error BlackScholes Formula Naive Monte Carlo Importance Sampling Conclusions: Importance Sampling is more dicult to implement than antithetic sampling, but can produce excellent improvement for far out of the money options

19 Improvements Control Variate Method The Control Variate Method - Idea Suppose that we wish to price a Knock Out option We have an analytic formula for the price of a Call Option with the same strike. Maybe, rather than pricing a Knock Out option directly, it would be a better idea to estimate the dierence between the price of a Knock Out option and the price of the Call Option using Monte Carlo instead. Price of Knockout Option Price of Call Option + Estimate of dierence (1) Because the dierence is probably smaller than the price we're trying to estimate, the variability in a Monte Carlo estimate of the dierence is probably lower than the variablility in a Monte Carlo estimate of the price.

20 Improvements Control Variate Method Control Variate - Example that proves it can work Consider the extreme case of pricing a knock out option where the barrier is so high it will very rarely be hit. In the control variate method, we will estimate that the dierence between the call price and the knock-out option price is zero even if we use a tiny sample (e.g. a sample of one). The control variate method will converge to the exact answer immediately. The naive method will be no more accurate than pricing a call by Monte Carlo, so only converges slowly.

21 Improvements Control Variate Method The Control Variate method Suppose we have a random variable M with E(M) = µ and wish to nd µ. Suppose we have another random variable T with E(T ) = τ with τ known. Dene M = M + c(t τ). E(M ) = µ too for any c R. Our previous example was the special case when c = 1. Var(M ) = Var(M) + c 2 Var(T ) + 2c Cov(M, T ) Choose c to minimize this c = Cov(M, T ) Var(T, T ) Var(M ) = (1 ρ 2 ) Var(M) where ρ is the correlation between M and T.

22 Improvements Control Variate Method Control Variate method, worked example Let us price a Call Option by Monte Carlo We expect the price of a Call Option to be correlated with the price of the stock, so let's use the stock price as our control variate.

23 Improvements Control Variate Method function [price,errorestimate, c] = callcontrolvariate( K,T,... S0,r,sigma,... npaths,... c) % Usual pricing code logs0 = log(s0); epsilon = randn( npaths,1 ); logst = logs0 + (r-0.5*sigma^2)*t + sigma*sqrt(t)*epsilon; ST = exp( logst ); discountedpayoffs = exp(-r*t)*max(st-k,0); % Standard formula for control variate method m = discountedpayoffs; t = exp(-r*t)*st; tau = S0; covmatrix = cov(m,t); if nargin<7 c = -covmatrix(1,2)/covmatrix(2,2); end mstar = m + c*(t-tau); % Result price = mean(mstar); errorestimate = std(mstar)/sqrt(npaths);

24 Improvements Control Variate Method Control Variate Results Parameters: S0 = 100, K = 100, σ = 0.2, r = 0.14, T = 1, n = Method Result Standard Error BlackScholes Formula Results: Naive Monte Carlo Control variate Note, to compute the error I xed c and then re-ran to compute the same error as I was concerned using the same data to nd c and estimate error may lead to bias. Conclusions: The control variate technique is easy to implement. It can produce signicant improvements in the Monte Carlo price.

25 Improvements Low Discrepancy Sequences Low Discrepancy Sequences In one dimension, evenly distributed sample points give better performance than Monte Carlo sampling (i.e. the rectangle rule is faster than Monte Carlo). In high dimensions there are better sets of sample points available. Given an N and a dimension d, you can generate a low discrepancy sequence of N points in [0, 1] d so that if you wish to estimate a function f : [0, 1] d R you will get a better estimate by sampling at the points in the low discrepancy sequence than you would be Monte Carlo. Using a low discrepancy sequence rather than pseudo-random numbers is called quasi-monte Carlo. Low discrepancy sequences are also called quasi-random numbers. Quasi Monte Carlo converges faster than Monte Carlo but you have to be careful: you can only guarantee better results as N tends to innity and you want good results for low N. See Joshi More Mathematical Finance (or many other sources) for details on how to use low discrepancy sequences in practice.

26 Improvements Low Discrepancy Sequences Generating Low discrepancy sequencey in MATLAB d = 2; s = sobolset ( d ); points = net (s,1000); You can replace sobolset with haltonset for another low discrepancy sequence. Try plotting a scatter plot of the results and comparing with the halton version and genuine random numbers. Aside: this is an example of object oriented MATLAB code. We are creating a complex data object and then calling special functions to work with that kind of data object

27 Improvements Summary Summary of improvements to numerical methods With little eort you can use techniques such as Richardson extrapolation to improve the convergence of a numerical method. There are various variance reduction techniques available for Monte Carlo. If you need to speed up your Monte Carlo pricer why not try all of them at once?

28 Improvements Summary Feedback Please ll in the online feedback for the module. There are separate paper forms to rate your classes.

29 Introduction Introduction

30 Introduction Outline Modern Portfolio Theory Markowitz's theory The ecient frontier quadprog for quadratic optimization Calibrating models The jump diusion model Incomplete market models fminunc for unconstrained, nonlinear optimization Dynamic optimization Revision of delta hedging strategy The Galerkin method fmincon for constrained, nonlinear optimization

31 Introduction Strange terminology Programming is another term for optimization. Linear programming = solving linear optimization problems Quadratic programming = solving nonlinear optimization problems Linear programming is so-called because it was originally used for optimizing military questions, which was phrased as how best to conduct military programs. Operations Research, is, according to Wikipedia, a discipline that deals with the application of advanced analytical methods to help make better decisions. The name comes from military operations If you apply optimization to real world business problems, you are performing operations research.

32 Introduction Uniqueness Find x that minimizes f (x) subject to the constraint x X. If x exists it may not be unique. This is not a problem it just means that two equally good solutions can be found. When nding numerical solutions, there may be a variety of equally good solutions found When testing the correctness of your optimization, you should see if two competing methods both give the same value for f (x) rather than seeing if they give the same value for x.

33 Introduction Convexity For convex functions f on convex domains X, a local minimum is a global minimum. For this reason it is much easier to minimize a convex function numerically than a general function. Convex problems have lots of nice properties, e.g. a theory of dual problems. We won't cover this in this course.

34 Introduction Algorithms There are many optimization algorithms. We will let MATLAB choose the best algorithm for us. For large/dicult problems one may wish to choose the algorithm carefully. Dierent algorithms exist for small, medium, large and ridiculously large problems. Specialist algorithms exist for certain problems (e.g. quadratic and linear problems) Most algorithms assume f is smooth, but algorithms do exist for non-smooth, convex f. We will be interested in local algorithms that nd local minima. This is obviously not a problem for convex problems. Global algorithms are, inevitably, slower.

35 Modern Portfolio Theory Modern Portfolio Theory

36 Modern Portfolio Theory Modern Portfolio Theory Theory developed my H. Markowitz in 1950's. Not very modern! There are n assets whose returns over some xed time period T are random variables R i. R i = price of asset i at time T - current price of asset i current price of asset i The random variables R i are assumed to have a known mean µ and covariance matrix Σ. We wish to invest a xed amount P over the time period T in a xed portfolio of the assets. The portfolio is determined by choosing the weights w i to assign to each asset with w i = 1. We are allowed to short sell so the w i may take any real values.

37 Modern Portfolio Theory Optimization problem Suppose we wish to have an expected return on our portfolio of r, what portfolio should we choose to minimize the standard deviation of the portfolio? If w is the vector of weights, then the return of the portfolio has a distribution with mean: and variance: µ T w w T Σw So problem can be stated as: Minimize w T Σw Subject to the constraints i w i = 1 and µ T w = r. Σ is a positive denite matrix, µ is a vector.

38 Modern Portfolio Theory Quadratic optimization problem Minimize 1 2 x T Hx + f T x Optionally subject to constraints of the form: Ax b A x = b l x u For some matrices A and A and vectors vecb, b, l and u. We write x y if every entry of the vector x is less than or equal to the corresponding entry in y. The vectors l and u may contain the values and + respectively. To solve this problem simply use quadprog

39 Modern Portfolio Theory MATLAB quadprog For a problem as dened on the previous slide with A = Aeq and b = beq [x, fval, exitflag ]= quadprog (H,f,A,b, Aeq, beq, lb, ub, x0, options ); x is the optimal value of the vector fval is the value of 1 2 x T Hx + f T x exitflag indicates whether the optimization worked. You must check this! You can use the empty array [] where a constraint is not needed. The options parameter allows you to ne tune the optimization if you wish. It can be omitted. quadprog standard for quadratic programming

40 Modern Portfolio Theory Markowitz problem as quadratic optimization The condition i w i = 1 and µ T w = r can be rewritten A w = b where ( ) ( ) A =, b 1 = µ 1 µ 2 µ 3... µ n r Thus the Markowitz problem is just a special case of quadratic programming.

41 Modern Portfolio Theory MATLAB implementation function [ ret, variance, w ] = markowitzoptimizeret (... r, mu, sigma ) H = sigma ; n = size ( sigma,1); f = zeros (n,1); Aeq = [ ones (1, n ); mu ]; beq = [1; r ]; [w,~, exitflag ] = quadprog (H,f,[],[], Aeq, beq,[],[],[]); assert ( exitflag >0); ret = mu * w ; variance = w ' * sigma * w ; end

42 Modern Portfolio Theory Remarks If you run the code on the previous slide, MATLAB prints out a lot of messages and a warning. To switch o the messages you can customize the options To switch o the warning you use the warning command to indicate that you aren't interested in the specic warning message.

43 Modern Portfolio Theory Quieter MATLAB implementation function [ ret, variance, w ] = markowitzoptimizeret(... r, mu, sigma ) warning('off','optim:quadprog:willrundiffalg'); H = sigma; n = size(sigma,1); f = zeros(n,1); Aeq = [ ones(1,n); mu]; beq = [1; r]; options = optimset('quadprog'); options = optimset(options,'display','off'); [w,~,exitflag]=... quadprog(h,f,[],[],aeq,beq,[],[],[],options); assert(exitflag>0); ret = mu * w; variance = w' * sigma * w; end

44 Modern Portfolio Theory Creating options To create a set of options rst call optimset passing in a single parameter, the name of the optimization routine you will call. This returns an options object. options = optimset ( ' quadprog ' ); Call optimset a second time to modify the options. This time you should pass in your options object and then a set of key/value pairs that indicate what options you would like to set. We're setting the option Display to off. options = optimset ( options, ' Display ', ' off ' ); The function optimset returns the new modied options object.

45 Modern Portfolio Theory Using the options You can now pass this customized set of options object to the optimization function. [w,~, exitflag ]=... quadprog (H,f,[],[], Aeq, beq,[],[],[], options ); The reason for this procedure is to make sure that you choose the default values for all options except the ones you wish to customize.

46 Modern Portfolio Theory Available options When optimizing in MATLAB you can use options to set things like: How much information to print during the calculation (Display) The actual algorithm to use (Algorithm) How accurate the answer needs to be The maximum number of steps to perform of the algorithm before giving in etc. Consult the MATLAB documentation for details of the available options and their settings.

47 Modern Portfolio Theory Ecient frontier using all stocks 0.02 The Markowitz efficient frontier Expected return Standard deviation

48 Modern Portfolio Theory Given an expectation r, there is a corresponding minimum standard deviation. As r varies, this traces out the ecient frontier. The points represent the performance of individual stocks in the FTSE 100 They all lie inside the ecient frontier. The code to plot the ecient frontier is available from the course website.

49 Modern Portfolio Theory Ecient frontier with 10 stocks The Markowitz efficient frontier Expected return Standard deviation

50 Modern Portfolio Theory Remarks on Markowitz theory The Markowitz model only makes sense if you think standard deviation is a good measure of risk. For normal distributions, this is uncontroversial. For more complex distributions more general utility optimization makes more sense. This is why normally distributed returns are often assumed. For large markets, the assumptions are very unrealistic Precisely known drift and covariance matrix for returns Unlimited buying and selling It tends to produce unreasonable portfolios that exploit quirks in the data. Example: suppose X and Y are two highly correlated stocks with the same price. Markowitz optimization might construct a portfolio consisting of 1000 units of X and sell 999 units of Y. Despite its shortcomings, it is enormously inuential and is where the idea of portfolio optimization originated.

51 Optimizing utility Optimizing utility

52 Optimizing utility General setup Suppose that we have n assets X 1, X 2,... X n and we have a stochastic model for the price of these assets. There are various constraints on our trading for example we might only be allowed to buy or sell a certain quantity of each asset. Suppose that we have a utility function u : R R We wish to nd the trading strategy that maximizes our expected utility.

53 Optimizing utility Utility functions A utility function is a concave function. Popular examples are: power utility with risk aversion parameter η x 1 η 1 1 η η 1, x > 0 u(x) = ln(x) η = 1, x > 0 x 0 exponential utility with risk aversion parameter λ u(x) = 1 e λx Note that power utility assigns innite negative utility to losing money. This means that only trading strategies that have 0 chance of bankruptcy will ever be considered. For example, using power utility in a Markowitz type problem would eectively mean prohibiting short selling.

54 Optimizing utility Computing expected utility Computing expected utility is just a matter of computing an expectation so we can use all the techniques we have developed for computing expectations Rectangle rule, converges with rate... Trapezium rule, converges with rate... Simpson's rule, converges with rate... Monte Carlo, converges with rate... Gaussian quadrature, converges with rate... Compute the condence interval for Monte Carlo using... Try to reduce the variance of Monte Carlo using...

55 Optimizing utility One period problem A trader wishes to invest in a number of stocks that all follow multivariate geometric Brownian motion model. ds i t = S i t(µ i dt + σ i dw i t ) where each S i t is a vector of stock prices, µ i a drift vector, σ i is a volatility. The W i are Brownian motions with given correlation matrix. Equivalently d(log S i ) t = (µ i 12 (σi ) 2 ) dt + σ i dw i t We wish to choose a static portfolio of given quantities in each stock to maximize the expected utility at time T. Trading at intermediate times is not allowed.

56 Optimizing utility One period solution, sketch Use... to generate normally distributed random variables with the desired correlation matrix. Use the... method with one step to simulate log stock prices at time T. We simulate log stock prices so that... Hence compute the expected utility at time T as a function of the proportions invested in each stock using the Monte Carlo method. This will converge at a rate... Using fmincon or fminunc nd the optimal proportions that minimize the expected disutility (=minus the utility).

57 Optimizing utility Dynamic Programming Example hedging X 1 is the stock and it follows the Black Scholes model. X 2 is a call option on the stock. We know the payo at maturity and we know that we have a customer who is willing to buy one call option today at the Black Scholes price plus a small commission. We have the following constraints on our trading: We can sell up to one unit of the call option at time 0. We can't buy the call option. We can't trade in the call option at other times. We can only trade in the stock at 20 evenly spaced time points. i.e. continuous time trading is not possible. How should we trade?

58 Optimizing utility Numerical methods for dynamic programming Dynamic programming problems are very hard General purpose algorithms exist, but they are not very ecient. The basic problem is that the space of possible strategies is innite dimensional and it is hard to nd good low dimensional approximations. To resolve this problem we use the Galerkin approach.

59 Optimizing utility Galerkin method - idea Choose a nite set of candidate strategies based on intuition and experience: Strategy S 1 : delta hedging. Sell the customer the option, delta hedge at each time time point (and then pay out as required option). Strategy S 2 : no hedging. Sell the customer the option and don't bother hedging. Strategy S 3 : stop-loss hedging. Sell the customer the option, perform stop-loss at each time time point. Strategy S 4 : trade in the stock alone. Borrow money to buy the stock, wait till maturity and then sell the stock. We can form linear combinations of strategies. Suppose that we're a bank, we can instruct trader i to follow strategy i (scaled up or down by a factor of α i ) and then see what the net eect is. This is strategy α i S i.

60 Optimizing utility Galerkin method We have n strategies S n. We can calculate by Monte Carlo N possible scenarios for asset prices and compute the prot and loss of each strategy. This gives a vector x (i) of prot and loss for each strategy with rows corresponding to each scenarios. It is crucial to use the same scenarios for each strategy. Linear combinations i α is i are also valid strategies (possibly subject to some constraints on the α i ). The prot and loss of the combined strategy in each scenario is i α ix (i). Find the values of α i, subject to constraints, that minimize 1 u( N j i α i x (i) j ). The index j is running over scenarios. i is running over stratgies.

61 Optimizing utility Output of the Galerkin method The Galerkin method will give us a portfolio of strategies. This combined strategy is unlikely to be a perfectly optimal solution to the problem, but it is guaranteed to be at least as good as any individual strategy. Thus the Galerkin method allows you to improve performance by diversifying over strategies. This generalizes the notion of diversication for static trading strategies to dynamic trading strategies. Note that the method can be applied equally well to static trading strategies. This allows you to optimize static trading strategies even when Markowitz assumptions do not hold. Note: so long as the utility function is smooth and concave, this will be a smooth convex optimization problem.

62 Optimizing utility Our example We have already (in previous weeks) shown how to compute the prot and loss of strategies S 1, S 2 and S 3 given scenarios for the stock price. S 4, buying and holding stock, is trivial to evaluate. Thus we can compute the vectors x (i). The customer is only willing to buy up to one call option. We cannot sell call options. This gives constraints: α 1 0 α 2 0 α 3 0 α 1 + α 2 + α 3 1 We can write the last equation in matrix form as ( )α 1

63 Optimizing utility end function [ alpha ] = optimizeutility(... lambda,... pnlarray,... A,... b,... lb,... ub) function d = expecteddisutility( alpha ) pnl = pnlarray*alpha; u = mean( 1 - exp(-lambda*pnl)); d = -u; end nstrategies = size(pnlarray,2); alpha0 = zeros(nstrategies,1); alpha0(1)=1; options = optimset('fmincon'); options = optimset(options,... 'Display','off', 'Algorithm', 'active-set'); [alpha,~,exitflag] = alpha0, A,b,[],[],lb,ub,[],options ); assert( exitflag>0 );

64 Optimizing utility fmincon fmincon is MATLAB's function for constrained nonlinear optimization It takes similar parameters to both quadprog and fminunc. You specify the objective function by creating an appropriate MATLAB function. In this case we use the exponential utility function to compute the objective. Just like quadprog you specify constraints using various matrices and vectors such as A and b. fmincon also allows you to specify a general non linear constraint function if necessary. You should center and scale the problem well. This means that the interesting behaviour should happen near zero and the length scale should be roughly 1. (Recall that similarly you should center and scale your problem well before using the integral function.)

65 Optimizing utility Applying this to the problem We can now nd the optimal combination of strategies to use for our problem. We choose a concrete process for the price. S follows the Black Scholes model with K = 100, S = 100, T = 0.5, r = 0.03, µ = 0.2, σ = 0.2 The customer is willing to pay 1.1 times the BlackScholes predicted price for the call option. We can now generate stock price scenarios and compute the prot and loss vector x (i) for each strategy. We then call optimizeutility to nd the optimal combination of strategies.

66 Optimizing utility The optimal portfolo of strategies How much of each strategy is optimal? Delta hedge strategy Stop loss strategy No hedge strategy Invest in stock 1.2 Quantity of strategy Lambda (risk aversion)

67 Optimizing utility Interpretation of results The optimal strategy depends upon risk preferences λ. Note that our model has a very high drift µ = 0.2. This explains why investing in the stock seems like a good idea. Our model is a discrete time model. So perfect delta hedging is impossible. This is reected in the charge the customer pays on top of the BlackScholes predicted price. Delta hedging is not the optimal strategy. This is because it is risky in discrete time. Depending upon our risk aversion we may prefer no hedging to delta hedging If our risk aversion is high, we might prefer not to trade since no risk free strategy is available.

68 Optimizing utility Remarks on using Galerkin method It is likely that you would want to report the expected utility of the strategy. If you want to do this, note that the returned fval will not be an unbiased estimate of the disutility of performing our strategy. To estimate the utility correctly, you must try the portfolio on a new random sample of scenarios. In general you should always test a strategy on out-of-sample data At every time step you can re-run the Galerkin method to nd a new strategy which incorporates the information received since the previous time step.

69 Optimizing utility Generalization This idea can be generalized easily. Include transaction costs in the model. We simply need to change the computations of the prot and loss vectors to take this into account. Use a dierent model to generate stock paths. You can use any model at all, or historic data, to simulate price paths and hence compute prot and loss vectors. Add in other strategies, assets etc. Although the Galerkin method does not nd the true optimal solution to the problem, it does allow us to nd improved solutions easily. Moreover it can be applied very generally.

70 Summary Summary We have seen how quadprog, fmincon can be used to perform: Static portfolio optimization Dynamic portfolio optimization The lecture notes also discuss how to perform calibration of a pricing model to market prices using the fminunc function. References: Markowitz : "Portfolio Selection". The Journal of Finance (1952) Koivu & Pennanen: "Galerkin methods in dynamic stochastic programming". Optimization 59 (2010)