Trading the randomness  Designing an optimal trading strategy under a drifted random walk price model


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1 Tradig the radomess  Desigig a optimal tradig strategy uder a drifted radom walk price model Yuao Wu Math 20 Project Paper Professor Zachary Hamaker Abstract: I this paper the author iteds to explore the applicatio of probability mathematics i tradig fiacial markets. To achieve that, the author will build a drifted radom walk model to simulate the price movemets of stocks, ad the based o a set of assumptios about trasactio cost ad tradig advatage, establish a tradig system. The author the will examie the expected payoff from differet tradig strategies uder this system ad coclude a optimal tradig strategy i this model. The appedix will the provide a Matlab program to simulate this model ad test all coclusios i this paper. Cotet Table Part I. Backgroud ad itroductio 2 Part II. Itroducig the variables ad the assumptios 3 Part III. A simple case, o drift (µ 0 = 0) 5 Part IV. Full model with the drift µ 0 ad the edge α 6 Part V. A review o assumptios ad limitatios of the model 8 Appedix. tradigradom.m 9
2 2 Part I. Backgroud ad itroductio Modelig stock price with some kid of radom walk process is ot a ew idea. It was most famously oted by Priceto Ecoomist Burto Malkiel s book A Radom Walk Dow Wall Street, i which Malkiel argued that movemets of stock price behave radomly ad therefore the game of tradig stocks is essetially a martigale 2. However, some other ecoomists ad may market participats believe otherwise. They argue that stock price does ot behave completely like a radom walk, ad have suggested other price models. Oe price model developed by Adrew W. Lo ad Archie Craig MacKilay is the basis of my price model i this paper. It ca be expressed as: X t = µ + X t where X t is the price of the stock at time t µ is the tred force drift ε t is the radom disturbace term 3. To uderstad the above formula, oe ca thik of stock price movemet i a treded market. Each discrete movemet is radom (represeted by ε t ), but there is also a tred force (represeted by µ) that drives the price towards certai directio. Θ (Chart.) For example, as the chart (Chart.) above shows, each price movemet is radom (as there are ups ad dows i a radom sequece); however, as the blue arrow shows, overall there is a directio (or tred force) that price oscillates aroud. This tred directio could be drive by fudametal factors (e.g. i the example of a stock, the performace ad quality of the compay) or mometum factors (e.g. if may people are buyig the same stock, the stock price is drive up, which i tur attracts Burto G. Malkiel, A radom walk dow Wall Street : icludig a lifecycle guide to persoal ivestig. New York, Norto: C Martigale: a model of a fair game where o kowledge of past evets ca help to predict future wiigs. Martigale (probability theory), Wikipedia. 3 Radom Walk Hypothesis, Wikipedia.
3 3 more buyers ad drives price further up a classic explaatio of fiacial bubbles). Therefore, a skilled trader who uderstads those factors well could profit by guessig correctly this tred directio, i other words, predictig the correct µ. Part II. Itroducig the variables ad the assumptios To start with, I will defie the variables i this model mathematically. ) Stock price X t The variable X t represets the stock price at time t. It is the exact price at which the trader ca buy or sell at. Agai, i this model: X t = µ + X t. As i this model we care oly about the relative price movemets, we could set X 0 = 0. Hece, we ca t t t t derive that X = X 0 + µ t = µ t. We will further adjust the formula for this variable below i this model to icorporate some other elemets. 2) The time t I this model time t is a discrete iteger from 0 owards, ad therefore, the price movemets is also a discrete process. 3) The drift µ I this model the drift µ will be of a costat size µ 0 represetig the part of price movemets due to the tred force i a uit time. As show i Chart., oe could easily derive the relatioship betwee µ 0 ad the tred lie agle Θ: µ 0 = taθ. However, the directio of the tred could go either way with equal probability. More details o how to model the sig of µ will be discussed below. 4) The radom disturbace term ε t As said, i this paper, the stock price will be modeled with a simple radom walk with a drift, so the radom disturbace term ε t will behave as a simple radom walk variable with a costat step size ε 0, so its probability distributio fuctio is: ε t = ε 0 P(ε t ) = 2 2 ε t = ε 0 Besides parameters i the origial price model (X t, t, µ, ε), I will also itroduce several ew variables ito this price model below. 5) The tred legth A tred i fiacial markets does ot last forever. It dies out after certai time as the fudametal factors behid it are fully reflected i the price or as the mometum eds. Hece, is used to represet how log (i the same uit of t) the tred lasts.
4 4 6) Trader s edge α As the paper discussed above, if the price treds towards a certai directio (drift µ 0), the a skilled trader might be able to gai a edge i tradig by havig a good guess about the directio of the drift. I other words, if the trader ca guess correctly whether µ is positive or egative for more tha half of the time, the she might be able to develop a system that provides a positive expected retur. I this model, we will assig a value α to represet this trader s edge. The trader will guess correctly the value of µ for a probability of α; i the other sceario with a probability of α, I assume that the trader still guess the correct absolute value of µ but with the opposite sig (hece µ). It is easy to see that, by symmetry, if µ is positive ad price is tredig up, but the trader guesses wrog ad decides to shortsell (essetially bettig the price goes dow), it is equivalet to the trader buys whe the price is tredig dow (with a egative µ). Therefore, we could coveietly covert all scearios ito oe that the trader always buys whe eterig a positio, but with µ beig positive whe the trader guesses correctly ad µ beig egative whe the trader guesses wrog. Therefore, give a fixed size of drift µ 0, the drift µ i this case could be modeled with a probability distributio fuctio: P(µ) = 2 + α µ = µ 0 2 α µ = µ 0 7) Profit Limit L ad Stop Loss S To protect their capital, traders ofte set a Stop Loss (S) that whe price goes agaist their pla ad the loss exceeds this poit, they will exit their positio with a loss. As stated above, i this model we coverted all sceario ito oe that trader buys whe eterig a positio. Hece whe X t <= X eter  S, the trader takes i a loss of S ad exits the positio. While a Profit Limit (T) is exactly the opposite, it defies a gai that the trader is satisfied with ad beyod which is uwillig to take further risk. Hece, i this model it represets that whe X t >= X eter + T, the trader takes i a gai of T ad exits the positio. Now it is importat to ote that, if the trader eters the positio at X eter ad exits at X exit it is essetially the same as eterig at 0 ad exits at X exit  X eter. exit X t = µ + X t, so X exit X eter = µ(exit eter ). As ε t are all idepedet ad have the same distributio fuctio, exit eter E( ε t ), so E(X exit X eter ) = µ(exit eter ) + E( t= 0 t= eter+ exit E( ε t )is o differet from t= eter+ exit eter ε t ). I other words, we could reset X eter =X 0 =0 ad X exit = X exiteter (of course with ew radom values for X t ) ad maitai E(X exit X eter ). Hece, i this model, we would simply reset X t to 0 after eterig/reeterig a trade, ad exit the trade whe X t <= S (for a loss) or X t >= L (for a gai). Hece, we ca get a ew formula for X t : t= 0
5 5 X t = t t µ t t eter t eter where teter is the eter time of the curret trade. 8) Trasactio cost C As tradig stocks ivolve a trasactio cost (or ofte called a commissio fee ), this variable C deotes a fixed trasactio cost for each trade. 9) Outcome of the trade O i The Outcome of the ith trade is deoted as O i. This could be a gai or loss. As itroduced above, if the price moves over the Profit Limit L or the Stop Loss S, the trader will take a gai of L or a loss of S respectively. Aother case that the trader will exit the trader ad take a gai/loss i this model is whe the tred arrives to a ed (t=), the trader will have to exit her positio at X i order to prevet the risk of a sharp reversal (a big price chage i the uwated directio). These described rules alog with the trasactio cost C gives us: L C X t >= L O i = S C X t <= S X C t = 0) Total profit of the strategy P This variable P accouts for the sum of gais ad losses of all trades durig the legth of the trade. Its value is the best measuremet of how successful a tradig strategy is. It is give by: P = O + O 2 + O O is the total umber of trades doe withi time. Part III. A simple case, o drift (µ 0 = 0) We start with a special case of the model: µ 0 = 0, the µ = 0 ad X t = X t. I this case, stock price is ideed a radom walk. This situatio happes whe there is a flat market ad price oly oscillates horizotally ad does ot seem to move cosistetly towards either directio (as i Chart 3.). (Chart 3.)
6 6 I this case, it is clear that the trader s edge α becomes irrelevat. Uder this assumptio, ituitively it is very questioable that the trader could develop a tradig system with a positive expected value without a edge ad with a trasactio cost. I fact, we ca proof that without a trasactio cost, o matter how the trader sets her Profit Limit L ad Stop Loss S, all her trades will have a expected value of 0. Coclusio 3.: If µ 0 =0 ad C=0, the for ay i, E(O i ) = 0 regardless of S, L, ad. E(O i ) = E(X exit ) = t= 0 E(X t ) P(t = exit) E(X t ) = E(X t ) + E(ε t ) = E(X t ) =...= E(X 0 ) = X 0 = 0 For t (0,) Hece, E(O i ) = 0. It is oteworthy to poit out that S ad L oly affects P(t = exit) but ot E(X t ), which ituitively explais this coclusio. With a positive C, the it is easy to see E(O i ) = 0 C = C. From here, it is also easy to see that E(P) = E( O i ) = E(O i ) = C = C (Coclusio 3.2). This coclusio ca also be verified by ruig the Matlab program tradigradom.m with the parameter drift set to 0 (e.g. try ruig tradigradom(00, 0, 0.4, 0, 0, 00, 3, 00000) ). As this coclusio suggests, i a flat market, the trader is always expected to lose o matter how she trades; ad the more she trades, the more she is expected to lose. Thus, i such markets, traders are better off ot tradig at all. Part IV. Full model with the drift µ 0 ad the edge α Now to complete the model, let us itroduce the drift variable µ 0 ad the trader s edge variable α, ad the model become much more complicated to solve. Agai, we try to start with a simpler exceptio. If the trader forgets to set her Profit Limit ad Stop Loss, or sets them to be too far from the origial price X 0 so that either of them could be effectively reached withi the tred legth, the what would happe? Mathematically, this problem could be expressed mathematically as: if Max(X 0, X, X ) < L ad Mi(X 0, X, X ) > S, what is E(P)? I such a case, a trade will ot be exited o a Profit Limit or o a Stop Loss. The oly way left for a trade to be exited i this case is whe t arrives at the tred legth ad the stock is sold at X. Therefore, the whole strategy will cosist of a sigle trade with a outcome O = X C. Thus, E(P) = E(O ) = E(X C) = E(X ) C, sice E(X ) = E( µ t ) = E( µ t ) + E( ε t ) = E(µ t ) + E(ε t ),
7 7 ad E(µ t ) = ( 2 + α)µ 0 + ( 2 α)( µ 0) = 2αµ 0, E(ε t ) = 2 ε ( ε 0) = 0, we get E(X ) = E(µ t ) + E(ε t ) = 2αµ = 2αµ 0. Hece, we arrive that E(P) = E(X ) C = 2αµ 0 C. (Coclusio 4.) This formula illustrates the tradeoff of the strategy: how much a trader expected to gai from the size of the tred (µ 0 ) ad her probability advatage to beefit from that tred (2α), versus her trasactio cost (C). We ca verify this coclusio by ruig the Matlab program tradigradom.m with the appropriate parameters (e.g. try ruig tradigradom(00, 0.6, 0.4, 0., 0000, 0000, 3, 00000) ). Now fially, we could examie the effect of Profit Limit (L) ad Stop Loss (S). Let us first assume that whe a trader is stopped out, she immediately eters the positio agai. What would the trader s expected profit be? l s E(P) = E( O i ) = E(O i ) = (L C) + ( S C) + (X C) P r = l(l C) + s( S C) + (X C) P r = (ll ss + P r X ) (l + s + P r )C (Coclusio 4.2) where l deotes the umber of trades that reaches L, s deotes the oes that eds at S, ad P r deotes the probability that there will be a residual trade (the last trade exited i betwee L ad S whe the tred eds, it could be avoided whe the payoff X C is clearly egative. To calculate l, s, ad P r is beyod the scope of this paper (iterested readers are welcome to explore these values with tradigradom.m). However, with oe more reasoable assumptio, we could approximate E(P) above. That assumptio is, µ 0 ad ε 0 are both small. This is a reasoable assumptio i modelig stock market as stock prices chage i very small time itervals (ofte a small fractio of secod), ad therefore each icremetal chage (the combied result of µ t ad ε t ) is also very small. I accout of that, let us examie what happes whe the trader exits her positio at time t due to either a Profit Limit or Stop Loss: If trader exits her positio at t due to a Profit Limit L, the we kow X t < L ad X t >= L, so X t L = X t + µ t L = µ t + (X t L) <= µ t <= µ 0 + ε 0. Similarly, we kow whe a trader exits at t due to a Stop Loss S, the X t > S ad X t <= S ( S) X t = S (X t + µ t ) = (µ t ) + ( S X t ) <= (µ t ) <= µ 0 + ε 0. Therefore, we see that if we replace S or L with X t (the price at exit time t), the error is bouded by ±(µ 0 +ε 0 ), i other words, very small. (Coclusio 4.3) Also, we will use P r = i this approximatio, as we will have a very high probability to ed up with a residual (small or big). Hece, we ca approximate the expected profit E(P) by: E(P) = E( O i ) = E(O i ) E(X ith exit C) = E(X ith exit ) C ith.exit ith.exit = E( µ t ) C = E( µ t ) C = E(X ) C ith.eter ith.eter
8 8 = 2αµ 0 C. (Coclusio 4.4) Note that this is clearly worse off tha the result we get from the o Profit Limit or Stop Loss strategy (Coclusio 4.) as it oly icrease tradig cost but does ot improve tradig gais. Hece, we arrive our fial coclusio: the theoretical optimal tradig strategy i this model is to: predict the directio of the tred, buy (or shortsell) the stock i the begiig accordig to the predicted tred directio, ad exit the positio whe the tred eds. This simple strategy gives the best expected profit of: E(P) = 2αµ 0 C. (Coclusio 4.5) Part V. A review o assumptios ad limitatios of the model The fial coclusio about optimal strategy (Coclusio 4.5) seems a little surprisig. Those who are more familiar with stock tradig will ofte hear the importace of Stop Losses ad Profit Limits. To uderstad this discrepacy, it is importat to realize that all coclusios are arrived uder all the assumptios of this model discussed above. Some key assumptios I would like to reemphasize are: ) The magitude of the drift µ 0, the disturbace term ε 0, ad the trader s edge α are all costats. Hece, the trader caot lear from her gais or losses i this model, which elimiates a importat fuctio of Stop Losses ad Profit Limits i real tradig. 2) The trader trades cotiuously i this model. Immediately after the trader exits a positio, she eters agai. This assimilates more with Program/Algorithmic tradig, while real huma traders ofte take reflective breaks or looks for optimal eter poits durig their tradig. 3) The drifts µ t ad the disturbace ε t are small. This is probably the most importat assumptio made (also probably the most doubtful oe) to achieve Coclusio 4.5. Because of the small icremet price chage, Stop Loss s fuctio of protectig traders from sharp price movemets agaist them is miimized. While i real world tradig, a trader who sets a stop loss could be protected from a large differece betwee X exit ad S. These assumptios reveal some limitatios of this model. However, this paper ad the coclusios still serve reasoably well as a exploratory effort towards fidig a optimal tradig strategy.
9 9 Appedix: tradigradom.m fuctio avg_profit = tradigradom(tred_legth, drift, disturbace, edge, stop, limit, cost, ru_time) profit = 0; for r = :ru_time x = 0; radd = rad(); if (radd < (.5 + edge)) directio = ; else directio = ; ed; t = 0; tradig = ; while (t < tred_legth) tradig = ; rade = radi(2)*23; x = x + directio * drift + rade * disturbace; if (x <= stop) profit = profit stop cost; tradig = 0; ed; if (x >= limit) profit = profit + limit cost; tradig = 0; ed; t = t + ; ed; if (tradig == ) profit = profit + x  cost; ed; ed; avg_profit = profit / ru_time; ed
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