American Options with Monte Carlo

1. American Options with Monte Carlo Tommaso Gabbriellini Siena, 20 Maggio 2011

2. Black&Scholes • Verybasicrecap • The Black&Scholes modelassumes a market in which the tradableassets are: • A risky asset, whoseevolutionisdrivenby a geometricbrownianmotion • the money market account, whoseevolutionisdeterministic

3. Black&Scholes (2) • Valuing a derivative contract • A derivative can beperfectlyreplicatedbymeansof a self-financingdynamic portfolio whosevalueexactlymatchesallof the derivative flows in every state of the world. Thisapproachshowsthat the valuesof the derivative (and of the portofolio) solves the following PDE • where the terminal conditionat T is the derivative’s payoff.

4. Black&Scholes (3) • Thereexists a veryimportantresult: the Feynman-Kactheorem. • Itmathematicallystates the equivalencebetween the solutionofthis PDE and anexpectationvalue. • If f(t0,S(t0)) solves the B&S PDE, thenitisalsosolutionof • i.e. it’s the expectedvalueof the discountedpayoff in a probabilitymeasurewhere the evolutionof the asset is • Thisprobabilitymeasureis the RiskNeutralMeasure

5. Black&Scholes and numericalmethods • Sincethereexistsuchanequivalence, we can computeoptionpricesbymeansoftwonumericalmethods PDE: finite difference (explici, implicit, crank-nicholson) suitableforoptimalexercisederivatives Quadrature methods Integration Monte Carlo Methods suitableforpathdependentoptions

6. EuropeanOption Let’s recall the EuropeanCall/Put option: It’s a derivative contract in which the holderof the optionhas the right tobuy/sell the asset at expiry at a fixed price (the strike). The price at time t can becomputedas where the expectationistaken in the riskneutralprobability.

7. American Put Option The american versionof the put optiongives the holder the right toexerciseitanytimebefore the expiration date. Will therebecases in whichitisconvenientto “early” exercise the option? Yes. Here’s a case. Imagineyouboughtanamerican put and at t1 the stock dropsto zero, with no chance toevergoing back to a strictly positive value (like in the Black&Scholes model)

8. American Put Option (2) The holder, at t1, wonders if it is worth exercising the option. Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets K 2. The option is exercised later, suppose at maturity, the value is K = 10 T t1 It’s convenient to exercise at t1!

9. American Call Option Whatabout american call option? Will therebecases in whichitisconvenientto “early” exercise the option? Well, itdepends on dividends. Imagineyouboughtan american call and at t1 the stock goes so high that the probabilitytofinish out of the money at expiryisnegligible (S >> K)

10. American Call Option (2) No dividends The holder, at t1, wonders if it is worth exercising the option. Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets S(t1) - K 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) K = 10 t1 T It’s better to wait!

11. American Call Option (3) Withdividends With dividends things are different. As in the previous example, but now the stock pays a dividend yield q: The holder, at t1, wonders if it is worth exercising the option. Which is the optimal strategy? I have to compare the values of the possibilities 1. The option is exercised at t1, the holder gets S(t1) - K 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) K = 10 t1 T It might be better to exercise

12. Bermudan Put Option The bermudanoptionissimilartoan american option, exceptthatit can beearlyexercised once only on a specific set ofdates In the graph Put at strike K, maturity 6 years, and eachyearyou can choosewhethertoexercise or wait. t5 t2 t1 t3 T t4

13. Bermudan Put Option. A simpleexample Let’s consider a simpleexample: a put optionwhich can beexercisedearlyonly once. t1 T

14. Valuationof the simpleexample Can we price thisproductbymeansof a Monte Carlo? Yes, let’s seehow. Let’s implement a MC whichactuallysimulates, besides the evolutionof the market, whataninvestor holding thisoptionwould do (clearlyaninvestorwholives in the riskneutral world). In the followingexamplewewill assume the following data S(t) = 100, K = 100, r = 5%, s = 20%, t1 = 1y, T = 2y

15. Valuationof the simpleexample 1. We simulate that 1y haspassed, computing the newvalueof the asset and the newvalueof the money market account 2. At thispoint, I (the investor) couldexercise. How do I knowifit’s convenient? In case ofexercise I knowexactly the payoff I’m getting. In case I continue, I knowthatitis the sameofhaving a European put option.

16. Valuationof the simpleexample (2) In mathematicalterms I have the payoff in t1is Where P(t1, T, S(t1), K) is the price of a put (which I can computeanalytically!) In the jargonof american products, Pisreferredto the continuationvalue, i.e. the valueof holding the optioninsteadofearlyexercisingit. So the premium of the optionis the averageofthisdiscountedpayoffcalculated in eachiterationof the monte carlo procedure

17. Some more considerations I couldhavepricedthisproductbecause I haveananalyticalpricing formula for the put. Whatif I didn’t haveit? Brute forcesolution: foreachrealizationof S(t1) I runanother Monte Carlo to price the put. Thismethod (calledNested Monte Carlo) isverytimeconsuming. Forthisverysimple case it’s timeofexecutiongrowswith N2… whichbecomesprohibitivewhenyou deal with more thanoneexercise date!

18. IntroducingLongstaff-Schwarz A finersolution ForeachrealizationofS(t1) I go on with the followingstepsimulating S(T) Foreachpathcompute at timet1 the discountedpayoffgiven the value S(T) i.e.

19. IntroducingLongstaff-Schwarz (2) Plot the discountedpayoff Pi versus Si(t1) (asanexample, bymeansof the scatter plot in excel)

20. IntroducingLongstaff-Schwarz (3) On this plot, add the analytical price of the put as a functionofSi(t1)

21. IntroducingLongstaff-Schwarz (4) The analytical price of the put is a curve whichkindsof interpolate the cloudof monte carlopoints. Observation. Today the price can becomputedbymeansofanaverage on alldiscountedpayoff (i.e. the barycentreof the cloudmadeofdiscountedpayoffs) Maybe…. The future valueofanoption can beseenas the problemoffinding the curve that best fits the cloudofdicountedpayoffs (up to the date of interest)???

22. IntroducingLongstaff-Schwarz (5) Belowthere’s a curve foundbymeansof a linearregression on a polynomialof 4° order.

23. IntroducingLongstaff-Schwarz (6) Wenowhave a pricing formula for the put tobeused in my MC: The formula isobviously fast: the costofthisalgorithmisperforming the best fit Please note that I couldhaveusedanyformformy curve (non only a polynomial). Thismethodhas the advantagethatit can besolvedas a linearregression, whichis fast.

24. Longstaff-Schwarzalgorithm • Let’s considernow a genericbermudanoption • Here’s the Longstaff-Schwarzalgoritm • Generate the MC trajectoriesof the underlying up tomaturity • Compute the payoff at maturity and discount itto the previousexercise date • Regress the last columnas a functionof the previousone, compute the continuationvalueforeachpath and calculatewhatyouwouldgetfromexercise.

25. Longstaff-Schwarzalgorithm (2) • Compare thosetwonumbers. In thisparticularpath the payoff in case ofexerciseisgreaterthan the continuationvalue. Exerciseit and go tonextstep and discount the payoff. • As in step 3, compute the continuationvalue and the payoff in case ofexercise • Now the continuationvalueisgreater. Don’t exercise: the payoffvalueisreplacedwith the discountedadjacentnumber (more on this in next slide)

26. Longstaff-Schwarzalgorithm (3) Theoreticallyweshouldhavedonethis Thisiscorrect, butitisgenerallyless accurate because the continuationvalueprovidedby the interpolatingfunctionis accurate only in a regioncloseto the exerciseboundary. That’s whyitisused the previousstep. 7. Ok, iterate tillyouget the price!

27. Whatis the Longstaff-Schwarzalgorithm Recallthatpricing a derivative meanssolving a backwardpartialdifferentialequation i.e. startingfrom the payoff, and proceedingbackward in time, youcompute at eachtime and foreachvalueof S the optionvalue. Did I sayoptionvalue? Well, I couldhavesaidcontinuationvalue Therefore I can naturally price american/bermudanproducts

28. Whatis the Longstaff-Schwarzalgorithm (2) Longstaff-Schwarzmethodisthus a way to introduce backwardevaluation in a Monte Carlo approach(whichisnaturallyforwardlooking)

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