Models FOR Financial Economic (MFE or 3/F)

1. Models FOR Financial Economic (MFE or 3/F) Review Sessions -Jasdeep Sidhu

2. About Me • I am senior, Math Major. • Actuarial Exams: • Exam P : Fall 2010 • Exam FM : Fall 2010 • Exam MFE : Spring 2011 • Exam MLC : Fall 2011 • Feel free to contact me if you have any questions: jss380@psu.edu

3. MFE Exam • Exam is a three hour multiple-choice examination usually 30 questions. • Exam MFE is offered via computer-based testing (CBT). • Uses a normal probability calculator instead of a normal probability table. • Even though its CBT you don’t get preliminary pass/fail you have to wait about 8 weeks. • Registration Deadline: March 8th 2012 for April sitting which is from April 19-25th.

4. Study Sources • A manual is must: • ASM or Actex • SOA Sample Questions (76): • http://www.beanactuary.org/exams/preliminary/exams/syllabi/MFE_SampleQS1-76.pdf • Additional practice: • Actuarial Brew: sample questions. • TIA: The Infinite Actuary. • Ron’s class on MFE.

5. Schedule • Week 1: FM derivative market review and Binomial Tree Model for determining the price of options. • Week 2: Black-Scholes Formula, Greeks, and Delta Hedging. • Week 3: Asian, Barrier, Compound, Gap, Exchange and other exotic options. • Week 4: Brownian Motion, Ito Lemma, Black-Scholes Equation, and Sharpe Ratio. • Week 5: Stochastic Integration, Binomial Tree Models for interest rates. • Week 6: Black Scholes for Bond Options, CIR, and Vasicek Models. • Week 7: Questions and review of most important concepts.

6. Objective • Not cover all the material in complete detail • Focus on topics/ concepts more common on exams. • Cover formulas and questions important to pass the exam. • Try to go over as many SOA Sample problems we can.

7. Getting Started • Call Option: An option, but not an obligation, to buy at a specified price. • Purchased Call payoff = Max[0,(Spot – Strike)] • Put Option: An option, but not an obligation, to sell at a specified price. • Purchased Put payoff = Max[0,(Strike – Spot)] • CAUTION: Buyer of put option = seller of asset & Seller of put option = buyer of asset.

8. Forward Price Pre-paid Forward Price • Price paid at time T for asset at time T. • For stocks with no dividends: • Stock with discrete dividends: • Stock with continuous dividends • Price paid now for asset at time T i.e. present value of the forward. • For stocks with no dividends: • Stock with discrete dividends: • Stock with continuous dividends

9. Put-Call Parity (very imp.) • Using Forwards: • No Dividends: • Discrete Dividends: • Continuous Dividends: • On Currency: • (note: this is a confusing. Here is a tip: in each question they will always have a dominating currency and you must always have the conversion rate as dominant currency/other currency. Also the price of call and put should be in dominant currency. Rd = rate of dominant current, Rf = rate of other currency.)

10. Put-Call Parity Contd. • Relationship between currency options: • For Bonds: C(K, T) – P(K, T) = B0 – PV0,T(Coupons) – e-rTK • For two different stocks (this will be discussed in greater detail when we talk about special types of options): C(St, Qt, T – t) – P(St, Qt, T – t) = PV[Ft,T(S)] – PV[Ft,T(Q)] Where St and Qt are two different stock prices at time t. -note another way to look at St and Qt : C(St, Qt, T – t): here St is the price of the stock you purchase and Qt is essentially the strike price.

11. Types of Options • American Options: you can exercise them any time you want. • European Options: you can only exercise at the end of maturity. • Bermudian Options: you can exercise them only at pre-determined, finite amount of time before maturity. Notes: Bermudian Options are not syllabus directly. For stocks with no dividends: American call is equal to European Call but this not always for Put options.

12. Early Exercise For American Options • Reason to exercise early for American Call: • Dividends can be received on the stock • Reasons not to exercise early • Interest can be earned on the strike price • Insurance protection against decline in value of stock • American-style call options on a nondividend-paying stock should never be exercised prior to expiration.

13. Comparing Options • Higher volatility makes the option more expensive (better chance of payoff). • Longer time to expiry always makes American Option more valuable. • Longer time to expiry always makes European options worth more if there are no dividends. • But if there are dividends, longer-dated European options may or ma not be worth more.

14. Arbitrage Inequalities (also imp.) • If K1 < K2 < K3 • Slope Here we see for call as strike price(K) increases the price of call decreases. • Convexity Here we see for call not only does the price decrease when K increases rate at which decreases also decreases.

15. Other important points for theory questions on the exam • Bounds for Option Prices: • Bounds for European options(w/o using American options):

16. Binomial Tree Approach to Price Options • One of 2 ways to price options: Binomial Tree and Black Scholes. (Relatively simple topic but very computationally intensive and this topic has many exam questions also.) • Binomial Tree: We break the time to expiry into periods. In each period price of stock can only move up or down i.e. increase or decrease with fixed probabilities and we recursively calculate the price of the option evaluating it at each possible stock price. • Risk Neutral: One does not want/ expect a reward for risk i.e. there is no risk premium. Is the risk neutral probability of moving up. u/d: Is fraction by which the stock moves up/down.

17. Standard Binomial Model – Forward Tree (Risk Neutral) • So to calculate the value for the option: • It is simply the Present value of the expected payoff. Cu = Max( 0, Vu – K) , Cd = Max( 0, Vd – K) C = PV ( P* Cu + (1-P*) Cd) • Arbitrage – Check for No-Arbitrage in a Binomial Model

18. Replicating Portfolio Method • Another way to calculating the prices other than the PV of the expected payout is the replicating portfolio method. • We replicate the option with a portfolio of number of shares of stock called Delta, and an investment in the amount of B in a risk free bond. ( B is bond, NOT borrow as in fact it is the amount we lend.) • Recall from FM: • Call Option = Lend money – Short Stock • Put Option = Borrow money – Long Stock • This will help to check if you got B correct i.e. • B is positive for Put Option. • B is negative for Call Option.

19. We get two equations: • Su * delta + FV (B) = Vu • Sd * delta + FV (B) = Vd • Here we have same payoff with our actual option and using a replicated option, amount of shares time its price and plus the FV of the amount lent equals the payoff from our option. Solving these eqs gives us generic formulas for B, delta. Solving for Option Price:

20. Now we use Real Probabilities • This is an important shift and to mark the change we are going to get rid of the asterisk (*) in superscript of probabilities (p). • Here we are using alpha instead of ‘r’. • Before we also assumed that discounting rate for option(gamma) was same as that of stock (alpha) which was same risk free rate(r). But not any more • Alpha: The stock growth rate. • Gamma: The discounting rate for Option.(this is negative for Put Option). • Delta: The rate of dividend.

21. Solving for Gamma Talk about multiple- period binomial trees.

22. Pricing Options on Futures • Assumptions: We assume that futures have same price as forwards. • The probability formula simplifies because the forward price of the future contract is the same as current price. • For Future Option has a dividend rate = risk free rate that is the formula for u/d simplifies.

23. Alternative Trees • Cox-Ross-Rubinstein Binomial model: • Lognormal Binomial model: Jarrow-Rudd • Some Important Similarities b/w Models:

24. Important Points to remember • Is the Binomial Model Realistic? No – because of these assumptions: • Volatility is constant • Large stock price movements do not suddenly occur • The periodic stock returns are independent of each other. • Delta decreases to -1 as Put Option gets further into money. Delta increases to +1 as Call Option gets further into money. • For Currency options: • risk free rate is domestic risk free rate. • Continuous dividend rate is foreign risk free rate.

25. Exam Tips: Binomial Models • For Binomial tree with a lot of branching, where they ask you to calculate the call option price, see if it is easier to calculate the put option price( if it has a lot of zero payouts) and then just use put call parity. • For American Option you have to check at each node if it is a good idea to exercise the option at that time, it can save you some time if there is no dividend as American call = European call. As European options are easier to calculate. • There are many ways we discussed here to solve for price, always use the method that you are most comfortable with. • Always remember to read the question carefully, be wary of the question says : a forward or future pricing model, if it is a call or a put option and check if it is a standard binomial model or one of the alternative model. (This is a very generic advice but something to keep in mind as I personally have made many mistakes because of it.)