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transaction costs preclude perfect replication of contingent claims and introduce a tradeoff between the risk and return to hedging . a standard approach to dealing with this tradeoff
is to use utility functions for evaluating hedging errors ( hodges and neuberger 1989 ) and the utility indifference principle for pricing options ( davis et al .
in contrast to the elegance of black - scholes delta hedging , these models typically lack explicit solutions and require powerful numerical methods to obtain case - by - case approximations of optimal hedges .    in some situations the particular structure of the optimal hedging strategy
is known , which facilitates numerical analysis .
one example is the black - scholes model with proportional transaction costs , where the hedging strategy of a trader with cara utility function is defined by a no - trade region ( davis et al .
when the hedger s stock holdings are outside this region , she carries out a trade that brings the stock position onto the nearest boundary point .
numerical methods that approximate value functions have proved successful in this model , see , e.g. , monoyios ( 2004 ) .
a different approach is presented in zakamouline ( 2006 ) who derives analytic approximations of the two boundaries of the no - trade region .
motivated by asymptotic results in whalley and wilmott ( 1997 ) , his approach combines a clever guess of a parametric functional form with the estimation of coefficients as functions of model parameters .
these estimations , however , require a numerical approximation of the optimal hedging strategy .
in this paper we propose an alternative approach to derive explicit hedging strategies for general nonlinear transaction costs .
the approach is based on genetic programming which offers several advantages over classical numerical approaches .
first , by construction it is constrained to output closed - form expressions that are functions of the model parameters .
second , it delivers efficient results at low cost by solving the two tasks of finding an optimal hedging strategy and a closed - form representation for it in one integrated process .
third , no assumption is made on the functional form of the approximation .
genetic programming therefore avoids the disadvantage of ex post calibration methods and , at the same time , offers a technique that deals with cases in which there is no a priori information on the structure of the optimal solution .
the closed - form approximations of hedging strategies derived with this heuristic method can be directly tested in different market situations to verify their efficiency before their integration into an automated trading system .
the hard work of obtaining these hedging strategies is left to the computer .
once this is done , hedging is without sweat .    in the example considered in this paper
we determine closed - form expressions for the hedging strategy of an option writer with cara utility function under proportional and quadratic transaction costs .
the price of the underlying follows a geometric brownian motion .
solutions to this problem with a large range of parameter values are derived at various requirements on the simplicity of hedging strategies , ranging from full - fledged computer programs to short analytic functions .
parameters include volatility , interest rate , strike price of the european call , and transaction cost parameters .
information available to the hedger consists of stock price , black - scholes delta and gamma , time to maturity , and current portfolio holdings , in addition to the parameter values which are random but constant until maturity of an option .
our method produces better results than zakamouline ( 2006 ) when transaction costs are proportional . under
quadratic transaction costs we find simple near - optimal linear trading strategies that consist of a trade intensity and a no - trade reference solution .
the no - trade path can be interpreted as a modified black - scholes delta and the trade intensity as a modified gamma . in summary
, the paper demonstrates the practical use of genetic programming in deriving hedges under nonlinear transaction costs .
we consider the hedging problem faced by a risk - averse writer of a european call option .
the option writer can trade in the underlying whose price follows a geometric brownian motion @xmath0 with constant drift @xmath1 and volatility @xmath2 .
she also has access to a money account that pays interest at constant rate @xmath3 .
we set @xmath4 to obtain the risk - neutral measure .
the option writer can trade only at dates @xmath5 with @xmath6 , and @xmath7 the expiry date of the option .
trading incurs transaction costs .
we assume that the total transaction cost of buying or selling @xmath8 shares at the current price @xmath9 is given by @xmath10 with @xmath11 and @xmath12 .
it is deducted from the trader s money account at the time a transaction takes place .
the transaction cost has two components : ( a ) a proportional cost @xmath13 which arises in markets with a bid - ask spread of size @xmath14 and mid point given by the stock price @xmath9 ; and ( b ) a nonlinear cost which arises in markets when a trade can ` walk up ' an order book , i.e. , obtains worse prices the larger the trade .
lcl +   + risk aversion & @xmath15 & 0.5 + proportional cost & @xmath13 & 10bp - 200bp + quadratic cost & @xmath16 & 5bp - 100bp + volatility & @xmath2 & uniformly drawn from @xmath17 $ ] + interest rate & @xmath3 & uniformly drawn from @xmath18 $ ] + strike price & @xmath19 & uniformly drawn from @xmath20 $ ] + & & where the spot price @xmath21 + maturity & @xmath22 & 3 months + hedge frequency & @xmath23 & @xmath24 years ( one day ) +   +   +    trading strategies are real - valued functions of the form @xmath25 where @xmath26 is the number of shares bought or sold at time @xmath27 , and @xmath28 is a parameter vector which is constant over the lifetime of the option , cf .
table  [ table : parameters ] .
the function @xmath29 can depend on @xmath30 , as well as time @xmath27 , stock price @xmath31 , the black - scholes greeks @xmath32 and @xmath33 , and the number of shares @xmath34 held after trading in period @xmath35 . in our application , this function will be represented by a computer program .
share holdings and money market account evolve as @xmath36 with initial portfolio @xmath37 .
we make the usual assumption that there are no transaction costs at maturity . the terminal wealth of an option writer who sold one european call option with strike price @xmath19 at time @xmath38
is therefore given by @xmath39^+.\ ] ]    the option writer s objective is to maximize expected utility @xmath40 $ ] from terminal wealth , where the expectation is taken over the probability measure defined by on the space of sample paths .
given two hedging strategies @xmath41 and
@xmath42 , we measure their relative performance for an option @xmath43 by the difference in certainty equivalents @xmath44\right ) - u^{-1}\left({\mathbb{e}}[u(w_{\phi_{\theta}''}(t))]\right)\ ] ] where @xmath45 $ ] is the expected utility of terminal wealth when hedging strategy @xmath29 is applied to option @xmath30 .
@xmath46 is the monetary gain from using strategy @xmath41 instead of @xmath42 to hedge option @xmath30 , and @xmath41 performs better than @xmath42 , the larger is @xmath46 .
we will assume that the option writer has cara utility @xmath47 for this class of utility functions , @xmath48 is independent of initial money holdings which therefore can be set to zero for convenience .
it also allows to determine option prices through the indifference principle : the option premium is determined by the amount of money that when paid to the option writer gives the same utility as not writing the option and forgoing the premium . if the initial endowment is @xmath49 ( which gives utility -1 if no option is written ) , then the price is @xmath50 \right ) = ( 1/\gamma ) \exp(- r t ) \
, \ln \left ( { \mathbb{e}}\exp(-\gamma w_{\phi_{\theta}}(t ) ) \right)\ ] ] for a hedger with strategy @xmath51 .
this indifference price will typically depend on the initial share endowment through its impact on ( a ) the trading strategy of the option writer and ( b ) the opportunity cost which is the utility of an investor with no position in the option .
solving the hedging problem described in section  [ sec : model ] is a straightforward application of genetic programming ( koza 1992 ) .
we use a parallel steady - state algorithm with tournament selection , as illustrated in table  [ table : gpalgo ] .
ll +   + worker processes & master process +   +    randomly generate 250 programs + for iteration 1, ... ,250 : + randomly generate 20,000 options + do at least 125 tournaments : + select and evaluate 4 random programs + rank them by performance + replace worst 2 by copies of best 2 + cross and mutate the copies + end do + send best program to master + exchange good programs with neighbors + end for    &    randomly generate 100,000 options + for iteration 1, ... ,250 : + wait for all workers to do at least 125 + tournaments + tell workers to send programs and start + the next iteration + receive programs from workers + evaluate programs + save best program and report results + end for     +   +    the algorithm uses some 100 - 500 autonomous sub - populations of hedging strategies deployed on separate worker processors arranged in a circle topology . from time to time , the worker processes send their best program to a master process ; exchange good candidate solutions with their neighbors ; and generate a new set of test options . the master process uses a separate large fixed set of options to identify the current globally best strategy .
a byte code program representation is used for genetic recombination ( crossover and mutation ) , and a machine code representation ( nordin 1997 ) is used for fast computation of hedging decisions .
a simple built - in compiler translates byte code to machine code .
our software also contains a byte code disassembler whose output can be processed by a c compiler , or analyzed with maxima or similar tools for symbolic math manipulation .
we first apply the genetic programming ( gp ) method to the model with proportional transaction costs ( with @xmath52 and @xmath53 ) .
two different treatments will be applied .
trading strategies are first evolved using the gp approach without any information about the structure of the optimal hedge strategies . we choose to ignore the knowledge that under proportional transaction costs the optimal strategy is defined by a no - trade region because under other cost structures such a priori information is typically not available . in a second treatment , we derive a simple analytic closed - form expression to approximate the no - trade region .      the structure of the hedging strategies is given by .
strategies are represented as computer programs whose operators include arithmetic expressions as well as _ min _ , _ max _ , conditional assignments , and forward jumps ( conditional and unconditional ) .
the output from a program is interpreted as a decision of how much stock to buy or sell given the current combination of option parameters and variables .
the programs can be used as is in an automated trading application , although they may be difficult to analyze due to the occurrence of conditional jumps and assignments .
[ cols= " < , > " , ]     figure  [ fig : tradingqc ] provides an illustration of our results for an example option and stock price path . as in figure
[ fig : tradingpc ] , the left panel depicts trade volumes for each day and each stock position in @xmath54 $ ] for a typical trading strategy evolved with the gp algorithm , but in the this case with quadratic transaction costs .
the left panel shows that the trade volume is approximately a linear function of the stock position for most stock positions and most periods in time .
intersecting the graph with the horizontal plane at zero on the vertical axis , one obtains a curve through combinations of time and stock positions where there is no trade , and such that for other nearby points on the graph , the trade volume on the corresponding day is a constant fraction of the difference between the corresponding no - trade position and stock position .
these observations suggest that linear trading strategies will provide good performance under quadratic transaction costs .
the right - hand panel of figure  [ fig : tradingqc ] shows the projection of the no - trade curve ( grey ) along with the gp hedge ( bold black ) , and the delta hedge ( dotted ) .
also shown is the trade intensity ( black ) , defined as trade volume divided by the absolute difference between the current stock position and the no - trade stock position .
the latter closely tracks the black - scholes delta .
the hedging strategy adjusts the stock position towards the no - trade stock position , but this adjustment is quite slow , as the trade intensity is typically less than 20% .
we therefore observe large deviations between actual stock holdings and the no - trade curve .
closer to maturity the trade intensity is more volatile but then declines quickly ( resembling the widening of the no - trade region under proportional costs ) .
this general pattern of sluggish adjustment leads to substantial savings in transaction costs compared to the black - scholes delta .      to explore the performance of linear hedging strategies under quadratic costs , we apply the gp algorithm to trading strategies of the form @xmath55\ ] ] where @xmath56 is the no - trade stock position and
@xmath57 is the trading intensity , and both are analytic expressions .
the functions are constrained to be independent of stock holdings but are allowed to depend on time , greeks , and the parameter vector @xmath30 which are suppressed in .    as in the proportional cost case , making use of information about the ( approximate ) structure of the optimal hedge leads to strategies that are slightly better than those obtained without such information . in the current case , the mean improvement in performance amounts to 0.3 cents per option .
this improvement is of the same order of magnitude as in the proportional cost case , where the a priori information was known to be correct .
we therefore conjecture that linear strategies of the form ( [ eq : stratqc ] ) are near - optimal for hedgers with cara utility functions and quadratic transaction costs when the initial stock position is not too far from its desired level .
the following approximation was obtained by means of the same simplification approach as in section  [ sec : prop_notrade ] .
it produces hedging decisions that perform as well as the best strategy in its class . for a given date @xmath58 , the no - trade stock position @xmath59 and the trading intensity @xmath60 are given by @xmath61 respectively , where @xmath62 , as before .
the paper demonstrates the merits of a gp approach to solving optimal hedging problems under transaction costs .
this heuristic method delivers approximations that are analytic functions of various parameters describing the option contract , the dynamic of the underlying , and transaction costs .
these explicit approximations to optimal hedging strategies can easily be tested and integrated into an automated trading system .
other estimation methods , such as zakamouline s ( 2006 ) , derive closed - form approximations using ad hoc specifications of the functional form and an exact ( numerical ) solution for their calibration .
in contrast to such approaches , ours is both simpler and more general , as it requires neither explicit solutions nor assumptions about functional forms .
preliminary results indicate that our gp method also works well for exotic options , other nonlinear transaction costs and stochastic volatility models but that is beyond the scope of this short note .
= 2em = -2em
hedging in the presence of transaction costs leads to complex optimization problems . 
 these problems typically lack closed - form solutions , and their implementation relies on numerical methods that provide hedging strategies for specific parameter values . in this paper 
 we use a genetic programming algorithm to derive explicit formulas for near - optimal hedging strategies under nonlinear transaction costs . 
 the strategies are valid over a large range of parameter values and require no information about the structure of the optimal hedging strategy .    _ keywords : _ hedging , transaction costs , closed - form approximations , genetic programming .    
 _ jel classification : _ g13 .