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

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Presentazione sul tema: "American Options with Monte Carlo Tommaso Gabbriellini Siena, 20 Maggio 2011."— Transcript della presentazione:

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

2 - 2 - Black&Scholes Very basic recap The Black&Scholes model assumes a market in which the tradable assets are: - A risky asset, whose evolution is driven by a geometric brownian motion - the money market account, whose evolution is deterministic

3 - 3 - Black&Scholes (2) Valuing a derivative contract A derivative can be perfectly replicated by means of a self-financing dynamic portfolio whose value exactly matches all of the derivative flows in every state of the world. This approach shows that the values of the derivative (and of the portofolio) solves the following PDE where the terminal condition at T is the derivatives payoff.

4 - 4 - Black&Scholes (3) There exists a very important result: the Feynman-Kac theorem. It mathematically states the equivalence between the solution of this PDE and an expectation value. If f(t 0,S(t 0 )) solves the B&S PDE, then it is also solution of i.e. its the expected value of the discounted payoff in a probability measure where the evolution of the asset is This probability measure is the Risk Neutral Measure

5 - 5 - Black&Scholes and numerical methods Since there exist such an equivalence, we can compute option prices by means of two numerical methods PDE:finite difference (explici, implicit, crank-nicholson) suitable for optimal exercise derivatives Integration Quadrature methods Monte Carlo Methods suitable for path dependent options

6 - 6 - European Option Lets recall the European Call/Put option: Its a derivative contract in which the holder of the option has the right to buy/sell the asset at expiry at a fixed price (the strike). The price at time t can be computed as where the expectation is taken in the risk neutral probability.

7 - 7 - American Put Option The american version of the put option gives the holder the right to exercise it any time before the expiration date. Will there be cases in which it is convenient to early exercise the option? Yes. Heres a case. Imagine you bought an american put and at t 1 the stock drops to zero, with no chance to ever going back to a strictly positive value (like in the Black&Scholes model)

8 - 8 - American Put Option (2) K = 10 t1t1 T The holder, at t 1, 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 t 1, the holder gets K 2. The option is exercised later, suppose at maturity, the value is Its convenient to exercise at t 1 !

9 - 9 - American Call Option What about american call option? Will there be cases in which it is convenient to early exercise the option? Well, it depends on dividends. Imagine you bought an american call and at t 1 the stock goes so high that the probability to finish out of the money at expiry is negligible (S >> K)

10 American Call Option (2) No dividends t1t1 T K = 10 The holder, at t 1, 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 t 1, the holder gets S(t 1 ) - K 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) Its better to wait!

11 t1t1 T K = 10 The holder, at t 1, 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 t 1, the holder gets S(t 1 ) - K 2. The option is exercised later, suppose at maturity, the value is approximately (remember the assumption) American Call Option (3) With dividends It might be better to exercise With dividends things are different. As in the previous example, but now the stock pays a dividend yield q:

12 Bermudan Put Option The bermudan option is similar to an american option, except that it can be early exercised once only on a specific set of dates t1t1 t2t2 t3t3 t4t4 t5t5 T In the graph Put at strike K, maturity 6 years, and each year you can choose whether to exercise or wait.

13 Bermudan Put Option. A simple example Lets consider a simple example: a put option which can be exercised early only once. t1t1 T

14 Valuation of the simple example Can we price this product by means of a Monte Carlo? Yes, lets see how. Lets implement a MC which actually simulates, besides the evolution of the market, what an investor holding this option would do (clearly an investor who lives in the risk neutral world). In the following example we will assume the following data S(t) = 100, K = 100, r = 5%, = 20%, t1 = 1y, T = 2y

15 Valuation of the simple example 1. We simulate that 1y has passed, computing the new value of the asset and the new value of the money market account 2. At this point, I (the investor) could exercise. How do I know if its convenient? In case of exercise I know exactly the payoff Im getting. In case I continue, I know that it is the same of having a European put option.

16 Valuation of the simple example (2) In mathematical terms I have the payoff in t 1 is So the premium of the option is the average of this discounted payoff calculated in each iteration of the monte carlo procedure Where P(t 1, T, S(t 1 ), K) is the price of a put (which I can compute analytically!) In the jargon of american products, P is referred to the continuation value, i.e. the value of holding the option instead of early exercising it.

17 Some more considerations I could have priced this product because I have an analytical pricing formula for the put. What if I didnt have it? Brute force solution: for each realization of S(t 1 ) I run another Monte Carlo to price the put. This method (called Nested Monte Carlo) is very time consuming. For this very simple case its time of execution grows with N 2 … which becomes prohibitive when you deal with more than one exercise date!

18 Introducing Longstaff-Schwarz A finer solution For each realization of S(t 1 ) I go on with the following step simulating S(T) t0t1T For each path compute at time t 1 the discounted payoff given the value S(T) i.e. t0t1Tdisc.payoff

19 Introducing Longstaff-Schwarz (2) Plot the discounted payoff P i versus S i (t 1 ) (as an example, by means of the scatter plot in excel) t1disc.payoff

20 Introducing Longstaff-Schwarz (3) On this plot, add the analytical price of the put as a function of S i (t 1 ) t1disc.payoffput

21 Introducing Longstaff-Schwarz (4) The analytical price of the put is a curve which kinds of interpolate the cloud of monte carlo points. Observation. Today the price can be computed by means of an average on all discounted payoff (i.e. the barycentre of the cloud made of discounted payoffs) Maybe…. The future value of an option can be seen as the problem of finding the curve that best fits the cloud of dicounted payoffs (up to the date of interest)???

22 Introducing Longstaff-Schwarz (5) Below theres a curve found by means of a linear regression on a polynomial of 4° order.

23 Introducing Longstaff-Schwarz (6) We now have a pricing formula for the put to be used in my MC: The formula is obviously fast: the cost of this algorithm is performing the best fit Please note that I could have used any form for my curve (non only a polynomial). This method has the advantage that it can be solved as a linear regression, which is fast.

24 Longstaff-Schwarz algorithm Lets consider now a generic bermudan option Heres the Longstaff-Schwarz algoritm 1.Generate the MC trajectories of the underlying up to maturity 2.Compute the payoff at maturity and discount it to the previous exercise date 3.Regress the last column as a function of the previous one, compute the continuation value for each path and calculate what you would get from exercise Continuation value 20.1 Exercise

25 Longstaff-Schwarz algorithm (2) 4.Compare those two numbers. In this particular path the payoff in case of exercise is greater than the continuation value. Exercise it and go to next step and discount the payoff. 5.As in step 3, compute the continuation value and the payoff in case of exercise 6.Now the continuation value is greater. Dont exercise: the payoff value is replaced with the discounted adjacent number (more on this in next slide) Continuation value 20.1 Exercise Continuation value 14.5 Exercise Continuation value 14.5 Exercise Continuation value 14.5 Exercise

26 Longstaff-Schwarz algorithm (3) Theoretically we should have done this This is correct, but it is generally less accurate because the continuation value provided by the interpolating function is accurate only in a region close to the exercise boundary. Thats why it is used the previous step. Continuation value Exercise Continuation value 14.5 Exercise Ok, iterate till you get the price!

27 What is the Longstaff-Schwarz algorithm Recall that pricing a derivative means solving a backward partial differential equation i.e. starting from the payoff, and proceeding backward in time, you compute at each time and for each value of S the option value. Did I say option value? Well, I could have said continuation value Therefore I can naturally price american/bermudan products

28 What is the Longstaff-Schwarz algorithm (2) Longstaff-Schwarz method is thus a way to introduce backward evaluation in a Monte Carlo approach (which is naturally forward looking)

29 Il presente documento è distribuito da MPS Capital Services Banca per le Imprese S.p.A. a mezzo posta e/o in forma elettronica, esclusivamente ad investitori istituzionali ovvero ad operatori qualificati, così come definiti nellart. 31 del Regolamento Consob n° del 1° luglio 1998 e successive modifiche ed integrazioni. MPS Capital Services Banca per le Imprese S.p.A. è una società appartenente al Gruppo MPS ed un intermediario autorizzato ai sensi di legge. Il Documento é destinato esclusivamente allutilizzo ed alla consultazione da parte della clientela di MPS Capital Services Banca per le Imprese S.p.A. e viene diffuso per mera finalità informativa ed illustrativa; esso non intende sostituire in alcun modo le autonome e personali valutazioni che i singoli destinatari del Documento sono tenuti a svolgere prima della conclusione di qualsiasi operazione per conto proprio o in qualità di mandatari. Le informazioni e le opinioni contenute nel presente Documento si basano su fonti ritenute affidabili ed elaborate in buona fede, tuttavia né MPS Capital Services Banca per le Imprese S.p.A. né altra società appartenente al Gruppo MPS rilasciano alcuna dichiarazione o garanzia, espressa o implicita, relativamente allaccuratezza, completezza e correttezza delle stesse. Le opinioni, previsioni o stime contenute nel presente Documento sono formulate con esclusivo riferimento alla data di redazione dello stesso, e non vi è alcuna garanzia che risultati o qualsiasi altro evento futuro saranno coerenti con le opinioni, previsioni o stime qui contenute. Tutte le opinioni espresse nel presente documento sono soggette a modifica senza preavviso. Qualsiasi riferimento diretto ed indiretto ad emittenti o titoli non é, né deve essere inteso, quale offerta di vendita o acquisto di strumenti finanziari di qualsiasi tipo. MPS Capital Services Banca per le Imprese S.p.A. e nessuna delle società del Gruppo MPS, né alcuno dei loro amministratori, rappresentanti, funzionari, quadri o dipendenti, può essere ritenuta responsabile per eventuali perdite determinate dallutilizzo del presente Documento. MPS Capital Services Banca per le Imprese S.p.A. e le società del Gruppo MPS, gli amministratori e/o rappresentanti e/o le rispettive persone ad essi strettamente legate, possono avere rapporti di natura bancaria e finanziaria con eventuali emittenti qui citati ovvero avere interessi specifici con riferimento a società, strumenti finanziari o operazioni collegate al presente Documento. Per esempio MPS Capital Services Banca per le Imprese S.p.A. e le società del Gruppo MPS possono svolgere attività dinvestimento e dintermediazione, avere rapporti partecipativi diretti ed indiretti con emittenti qui menzionati e prestare ad essi servizi di consulenza; inoltre, con particolare riferimento agli strumenti finanziari eventualmente citati, esse possono altresì svolgere attività di prestito-titoli, sostenerne la liquidità con attività di market making su mercati regolamentati o sistemi di scambi organizzati. MPS Capital Services Banca per le Imprese S.p.A. potrebbe strutturare titoli ed operazioni con rendimenti collegati a parametri e strumenti finanziari qui menzionati. Si specifica che lelenco dei potenziali conflitti dinteresse indicati può non esaurire il complesso dei conflitti stessi. Per quanto non riprodotto nelle presenti Avvertenze, si fa espresso rinvio a quanto riportato nel sito internet ed alle relative condizioni del servizio. Procedendo alla lettura di questo documento, si accettano automaticamente le limitazioni e le avvertenze precedentemente riportate.


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