MFE Study Guide

1、mfe study guide (fall 2007)notes from mcdonalds derivative marketswritten by colby schaeffer introductionthe material in this quick study guide has been done to the best of my knowledge. some topics are only covered briefly (delta-hedging and caps/floors) while other topics have been omitted (equity

2、 linked annuities and compound options w/one discrete dividend). all exam tips are marked in red! this the 1st edition of my mfe study guide, and it may be updated in the future with better content. comments and questions may be directed via pm to colby2152 on the actuarial outpost.2nd edition notes

3、: jraven and fractl helped me tweak some little errors in the study guide. an entirely revamped spring 08 edition will be out by march 2008 with more on brownian motion, a better understanding but less detailed look at the greeks, an easier view of convexity, and perpetual options will be removed.ac

4、knowledgements: day yi, abraham weishus, bill cross and ao member “jraven”chapter 9 - parity and other option relationshipsoption exercise style· american: any time· european: end of maturityvalue of otherwise identical options: european < americanput-call paritygeneral formula: call(k,

5、 t) put(k, t) = pv(fo,t k)k: strike pricet: exercise timeput-call parity usually fails for american-style options.currency: c(k, t) p(k, t) = x0e-rt ke-rtstock: c(k, t) p(k, t) = s0 pv0,t(div) e-rtkbond: c(k, t) p(k, t) = b0 pv0,t(coupons) e-rtkdifferent assets: c(st, qt, t t) p(st, qt, t t) = pvft,

6、t(s) - ft,t(q)x0: current exchange rate denominated as $/r: euro-denominated interest rater: american or implied interest ratediv: stream of dividends paid on stockearly exercise for american options· american-style call options on a nondividend-paying stock should never be exercised prior to e

7、xpiration.· early exercise is not optimal if: c(st, k, t t) > st k· when exercising calls just prior to a dividend, early exercise is not optimal at any time where: k - pvt,t(k) > pvt,t(div)arbitrage inequalities (for both american & european)there is no free lunch!k1 < k2 <

8、; k30 c(k1) c(k2) k2 k10 p(k2) p(k1) k2 k1*if options are european, then the difference in option premiums must be less than the present value of the difference in strikespremiums decline at a decreasing rate as we consider calls with progressively higher strike prices. premiums also decline for put

9、s but when the strike price monotonically decreases.convexity of option price w.r.t. strike pricec(k3) - c(k2)/(k3 - k2) < c(k2) - c(k1)/(k2 - k1) p(k2) p(k1)/(k2 k1) p(k3) p(k2)/(k3 k2)option trendsamerican options become more valuable as time to expiration increases, but the value of european o

10、ptions may go up or down.as the strike price increases for calls or decreases for puts, the options become less valuable with their price decreasing at a decreasing rate.with dividends, longer term european options may be less valuable than shorter term european options.chapters 10, 11 - binomial pr

11、icing· assumes that the stock price can change to either an upper value or to a lower valueif the observed option price differs from its theoretical price, arbitrage is possibleu e(r )h d: delta, the number of shares to replicate the option payoff: dividend rate: volatility, the standard deviat

12、ion of the rate of return on stockat the prices sh = su, sd, a replicating portfolio will satisfy:(d ´ su ´ edh ) + (b ´ erh) = cu(d ´ sd ´ edh ) + (b ´ erh) = cdrisk neutral probability (of increase in stock)suppose that the continuously compounded expected return on t

13、he stock is a and that the stock does not pay dividends, then the true probability of an up move is:p = (e(a )h d) / (u d)multiple periods: work with future values and compute option prices retrospectively*take step-by-step answers to six digits!american options for an american call, the value of th

14、e option at a node is given bycall value = maxs k *, erh(p* value(up) + (1 - p*)value(down)*switch to k s for puts the valuation of american options proceeds as follows: at each node, we check for early exercise. if the value of the option is greater when exercised, we assign that value to the node.

15、 otherwise, we assign the value of the option unexercised. early exercise: receive dividends, advance payment of strike (interest), and lose insurancekr > stcall goes downput goes upkr < stcall goes upput goes downpricing options on other assetsstock index similar to nondividend-paying stocksc

16、urrency replace stock price with currency exchange rate and dividend rate with foreign risk-free rate: commodities replace dividend rate with lease ratebonds volatility decreases over time and interest rates are variableforwards forwards arent risk-freestocks paying discrete dividendsthe dividend is

17、 taken off the first node. the tree does not completely recombine after a discrete dividend unless it is a percentage of the stock. another solution is to use:schroders methodf = s pv(div)alternative treescox ross-rubinsteinlognormalmcdonaldassumes s = sudchapter 12 black-scholes modelthe black-scho

18、les formula is a limiting case of the binomial formula for the price of a european option.contrary to other forms of the d1 equation that you will see, this is the only one that you need to know. currency and futures options replace variables of this equation, but it remains the same.where n(x) is t

19、he cumulative normal distribution functionassumptions/properties· returns on stock are normally distributed and independent over time· volatility and risk-free rate are both known and constant· future dividends are known· there are no transaction costs/taxescurrency optionsreplac

20、e stock price with currency exchange rate and the dividend rate with foreign risk-free rate known as garman-kohlhagen modelfuturesreplace stock price with forward price and the dividend rate with risk-free rate. option greeksformulas that express the change in the option price when an input to the f

21、ormula changes, taking all other inputs as fixed., delta: option price change w.r.t stock price changedelta is the only greek that you are expected to computecall = e-t n(d1)put = -e-t n(-d1)call = put + e-t *delta of a stock is always equal to 1, gamma: measures convexity or change in delta, always

22、 > 0vega: tests if volatility is sufficient, always > 0, theta: option price change w.r.t. time to maturity change, usually < 0, rho: sensitivity to risk free rate, psi: sensitivity to the dividend ratethe greek measure of a portfolio is the sum of the greeks of the individual portfolio com

23、ponentselasticity· tells us the risk of the option relative to the stock in %termsfor a call 1, while for a put 0risk premium: r = ( r) sharpe ratio: perpetual optionseach x value is the present value of 1 when a stock of value s rises or falls to price h, where the value is (s/h)xh1 = lower va

24、lue of x, and h2 = higher value of xcallsvalue: maximum h: puts just change (h k) to (k h) and change h1 to h2chapter 13 delta hedgingmarket makers want stable portfolios, so they use delta hedging as a method of controlling risk.overnight profitdelta-gamma-theta approximation*delta and delta-gamma

25、approximations are contained within the formulawhere: black-scholes equationthis is different than the black-scholes formula that was used for pricing options. rather, this equation is a function of the greeks, stock price, volatility, and risk-free rate.chapter 14 exotic optionsasian options based

26、on the arithmetic/geometric average of underlying asset/strike price *useful for hedging currency exchange, variable annuities, and reducing volatilitygeometric(s) < arithmetic(s)barrier options payoff depends if price of asset reaches a barrier level· payoff and option premium is less valua

27、ble than those of standard options1. knock out option goes out of existence if price reaches barrier2. knock in option “comes into play” if price reaches barrier3. rebate fixed payment if asset price reaches barrierknock-in + knock-out = standard optioncompound option option whose underlying asset i

28、s another option that expires latercompound option parityx: strike price of compound optiont1: expiry of compound optiont2: expiry of underlying optioncallonoption putonoption = gap options option with trigger k2 (price that option must be exercised) and strike price k1 that differ, election is not

29、optimal use k1 for put-call parityexchange options lets you receive an asset in exchange for another at time t, pays off only if the option asset outperforms the asset it is being exchanged forvolatility depends on both assets: chapter 20 brownian motion & itos lemma introduction of terms1) stoc

30、hastic process is a random process that is also a function of time.2) brownian motion is a continuous stochastic process3) diffusion process is brownian motion where uncertainty increases over time4) martingale is a stochastic process for which ez(t2) = z(t1) if t2 > t1arithmetic brownian motionz

31、(0) = 0, z(t + s) z(t) n(0, s), z(t) is continuous: volatility or variance factor: drift factorornstein-uhlembeck processvariation of arithmetic brownian motiongeometric brownian motionitos lemmamultiplication tabledtdzdt00dz0dtsharpe ratio= “expected return per unit risk”if and thenchapter 24 inter

32、est rate modelsa stochastic interest-rate model that assumes a flat yield curve cannot be arbitrage-free.arithmetic: (similar to arithmetic brownian motion)problems:· r < 0 is possible· drift is positive, so r can goto infinity· volatility is independent of interest raterendleman-b

33、arter model: (similar to geometric brownian motion)problems:· drift sends the interest rate to infinityvasicek: problems:· volatility is independent: sharpe ratio: yield to maturity on infinitely lived bondthese are fairly extensive formulas to memorize, but there is no way around it. skip

34、 it if short on time or space in your head.cox-ingersoll-ross (cir) model: it's important to know that the sharpe ratio for cir isn't a constant; it depends on the short-term risk-free rate r according to where is a constant. cir yield to maturity formulas are too much for an soa exam which

35、says a lot.model doesnt have problems like the other models have “mean reversion” prevents interest rate from going to infinity.interest rate models allow us to generate stochastic yield curves, but the models themselves may be too restrictive.binomial interest rate modelall that is needed is a tree

36、 with short rates and p*. unless given, assume p* = 0.5.value bonds and options just like we did before, but it must be discounted at each node due to varying interest rates.types of options1) calls/puts american/european2) caps/floorsa. strike rateb. notional amountc. frequency of paymentd. length

37、of contractcaps & floors control risk and promote parity. simply, caps are like calls and floors are like puts. caplet values are equal to the difference in the strike rate and given interest rate at a node multiplied by the notional amount.black-derman-toy modelbdt model is actually not that difficult. it is an