1. JUMP DIFFUSION MODELS Karina Mignone Option Pricing under Jump Diffusion

3. Previous knowledge  Diffusion: A process that has diffusion has random motion causing random distribution of data.  Option: In finance, an option is a contract between a buyer and a seller that gives the buyer of the option the right, but not the obligation, to buy or to sell a specified asset on or before the option's expiration time, at an agreed price (the strike price). This is often done in the stock market.

4.  Call option: an option to buy.  Put option: an option to sell.  European option: An option that can be exercised only at expiry date.  American option: An option that can be exercised at any date between the issue date and the expiry date.  Stochastic process: a random process; non-deterministic.  Stochastic drift: The average change over time in a random process.

5. Brownian motion  A zero-mean continuous-time stochastic process with independent increments (also known as a Wiener process).*  It is the scaling limit of a random walk in one dimension  According to the model the returns on a certain stock in successive, equal periods of time are independent and equally distributed. *definition from www.mathworks.com

6.  According to the geometric Brownian motion model the future price of financial stocks has a lognormal probability distribution and their future value can therefore be estimated with a certain level of confidence.  Brownian motion however assumes a constant expected rate of return and volatility and does not consider discontinuity

7. Background  First model Bachelier’s Gaussian model for stock returns  60 years later Osborne proposed the use of Brownian motion to ensure price levels did not achieve negative values which as in Bachelier’s model  Then, Brownian motion was known as the only process with stationary and independent increments that has continuous paths

8.  Brownian motion is a main tool for financial modeling however fits data poorly  It underestimates the likelihood of a large movement in the underlying  Instead need a model that conforms to empirical data: a stochastic model that is skewed (to the left) and leptokurtic (more peaked at its mean and with greater probability mass in its tails)

9. Different Option Pricing methods  Press proposed a model where the natural logarithm of the stock price is assumed to follow a distribution that is a Poisson mixture of normal distributions and a Brownian motion, in the following way  Where Y 1 …Y k is a sequence of mutually independent random variables normally distributed, N t is a Poisson process and W t is Brownian motion

10.  Black-Scholes model (extension of Brownian motion) [The model develops partial differential equations whose solution, the Black–Scholes formula. Close approximation to real observed prices, however assumes the stock price follows a geometric Brownian motion with constant drift and volatility.]

11.  Cox, Ross and Rubinstein binomial model [the model uses a "discrete-time" (lattice based) model of the varying price over time of the underlying. Binomial model provides a discrete time approximation to the continuous process underlying the Black-Scholes model]

12.  Merton (1976) and Tucker and Pond (1988) provide a more thorough discussion of mixed-jump processes. Mixed-jump processes are formed by combining a continuous diffusion process and a discrete-jump process and may capture local and nonlocal asset price dynamics.

13.  Merton derived an option pricing formula as the underlying stock returns are generated by a mixture of both continuous and the jump processes.  The two basic building blocks of every jump- diffusion model are the Brownian motion (the diffusion part) and the Poisson process (the jump part).

14. Jump Diffusion  Assume the underlying follows Brownian motion, plus jumps governed by Poisson distribution  (The size of the jump can be given by a distribution of our choice)  Poisson process is used for modeling systematic jumps caused by surprise effects  [The arrival times follow a Poisson distribution]  =mean arrival rate of event during time interval dt.

15. Event occurring Event will not occur

16.  2 important properties of Poisson process  1. Probability of at least one event occurring in is  2. Probability of two or more events occurring in is o( ). In other words, we do not see 2 or more events happening at the same time.

17. Stochastic differential equation for an option with jumps  From Black-Scholes Model : Represents the jump

18.  Another way of writing it:  if no jump occurs  if one jump occurs

19.  The jump size follows a log-normal distribution, where m is the average jump size, v is the volatility of jump size and N(0, 1) is the standard normal distribution.

20.  From a risk management perspective, jumps allow to quantify and take into account the risk of strong stock price movements over short time intervals, which appears non-existent in the simple diffusion framework.

21.  It can be shown that for all derivatives with convex payoff (which includes regular call and put options) the price always increases when jumps are present—regardless of the average jump direction.  This increase in price can be interpreted as compensation for the extra risk taken by the option writer due to the presence of jumps, since this risk cannot be eliminated by hedging.

22. Example 1  when there are no jumps, the jump diffusion model reduces to the Black– Scholes model, in which returns follow a normal distribution

23. Example 2  the effect of jumps can be observed clearly by "turning down" the volatility of the diffusive component to zero

24. Summary  for the purpose of studying option pricing, a jump diffusion model, in which the price of the underlying asset is modelled by two parts:  a continuous part driven by Brownian motion, and  a jump part following a Poisson process  Represents a good model for option pricing.

25. Any questions?

26. References  http://www.hoadley.net/options/bs.htm, date accessed 22/5/11 http://www.hoadley.net/options/bs.htm  http://www.few.vu.nl/en/Images/werkstuk-dmouj_tcm39-91341.pdf, date accessed 22/5/11 http://www.few.vu.nl/en/Images/werkstuk-dmouj_tcm39-91341.pdf  http://what-when-how.com/finance/jump-diffusion-model-finance/, date accessed 19/5/11 http://what-when-how.com/finance/jump-diffusion-model-finance/  http://www.datasimfinancial.com/UserFiles/articles/PIDE.pdf, date accessed 12/5/11 http://www.datasimfinancial.com/UserFiles/articles/PIDE.pdf  http://www.math.nyu.edu/~benartzi/Slides5.2.pdf, date accessed 24/5/11 http://www.math.nyu.edu/~benartzi/Slides5.2.pdf  http://janroman.dhis.org/stud/AFI%20Jump%20diffusionPPT.pdf, date accessed 22/5/11 http://janroman.dhis.org/stud/AFI%20Jump%20diffusionPPT.pdf  http://www.ems.bbk.ac.uk/for_students/msc_finance/TOF2ctpt_emec043p/slides4.pdf, date accessed 24/5/11 http://www.ems.bbk.ac.uk/for_students/msc_finance/TOF2ctpt_emec043p/slides4.pdf  http://demonstrations.wolfram.com/DistributionOfReturnsFromMertonsJumpDiffusionModel/, date accessed 24/5/11 http://demonstrations.wolfram.com/DistributionOfReturnsFromMertonsJumpDiffusionModel/  http://docs.google.com/viewer?a=v&q=cache:yOkPsHM3WggJ:www.ems.bbk.ac.uk/for_stud ents/msc_finance/TOF2ctpt_emec043p/lecture4.pdf+jump+diffusion+lecture&hl=en&gl=au &pid=bl&srcid=ADGEESg0c1YzS7Eb9cecGB- mEdSGYIDnhP8MuVGFrgY0VH45loi95rhG5ESBXw1rC5cTyze4nW3ww13cEUaUXgrOYqNlvJZ X6ogQXPettJsfTu5BIZ6Fy9hV2ot3fnSaWiE5ap8udYDU&sig=AHIEtbQQK9V8I7GFyAdwk6Sh xwMjJq6yBQ, date accessed 25/5/11  http://www.google.com.au/search?hl=en&q=Brownian+motion&tbs=dfn:1&tbo=u&sa=X&ei =MrzdTbP1GMvnrAff9Nj2CQ&ved=0CB0QkQ4&biw=1131&bih=687, 26/5/11 http://www.google.com.au/search?hl=en&q=Brownian+motion&tbs=dfn:1&tbo=u&sa=X&ei =MrzdTbP1GMvnrAff9Nj2CQ&ved=0CB0QkQ4&biw=1131&bih=687

27. Demonstration  http://demonstrations.wolfram.com/DistributionOfR eturnsFromMertonsJumpDiffusionModel/ http://demonstrations.wolfram.com/DistributionOfR eturnsFromMertonsJumpDiffusionModel/  Download from here  http://demonstrations.wolfram.com/MertonsJumpDif fusionModel/ http://demonstrations.wolfram.com/MertonsJumpDif fusionModel/

28.  If v(x,t) is smooth enough, then V(S(t),t) is also a jump diffusion, providing a partial integrodifferential equation (PIDE):