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1 Financial Options

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2 Topics in Chapter 1. What is a financial option? 2. Options Terminology 3. Profit and Loss Diagrams 4. Where and How Options Trade? 5. The Binomial Model 6. Black-Scholes Option Pricing Model

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3 1. What is a financial option? An option is a contract which gives its holder the right, but not the obligation, to buy (or sell) an asset at some predetermined price within a specified period of time. It does not obligate its owner to take any action. It merely gives the owner the right to buy or sell an asset.

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4 2.Option Terminology A call option gives you the right to buy within a specified time period at a specified price The owner of the option pays a cash premium to the option seller in exchange for the right to buy

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5 2. Option Terminology A put option gives you the right to sell within a specified time period at a specified price It is not necessary to own the asset before acquiring the right to sell it

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6 2. Option Terminology Strike (or exercise) price: The price stated in the option contract at which the security can be bought or sold. Expiration date: The last date the option can be exercised.

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7 2. Option Terminology Exercise value: The value of a call option if it were exercised today = Max[0, Current stock price - Strike price] Note: The exercise value is zero if the stock price is less than the strike price. Option price: The market price of the option contract.

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8 The price of an option has two components: Intrinsic value: For a call option equals the stock price minus the striking price For a put option equals the striking price minus the stock price Time value equals the option premium minus the intrinsic value 2. Option Terminology

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9 3. Profit and Loss Diagrams For the Disney JUN Call buyer: -$0.25 $22.50 $0 Maximum loss Breakeven Point = $22.75 Maximum profit is unlimited

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10 3. Profit and Loss Diagrams For the Disney JUN Call writer: $0.25 $22.50 $0 Maximum profit Breakeven Point = $22.75 Maximum loss is unlimited

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11 3. Profit and Loss Diagrams For the Disney JUN Put buyer: -$1.05 $22.50 $0 Maximum loss Breakeven Point = $21.45 Maximum profit = $21.45

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12 3. Profit and Loss Diagrams Consider the following data: Strike price = $25. Stock PriceCall Option Price $25$

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13 3. Profit and Loss Diagrams Exercise Value of Option Price of stock (a) Strike price (b) Exercise value of option (a)–(b) $25.00 $

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14 3. Profit and Loss Diagrams Market Price of Option Price of stock (a) Strike price (b) Exer. val. (c) Mkt. Price of opt. (d) $25.00 $0.00 $

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15 Time Value of Option Price of stock (a) Strike price (b) Exer. Val. (c) Mkt. P of opt. (d) Time value (d) – (c) $25.00 $0.00 $

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Stock Price Option value Market price Exercise value Call Time Value Diagram

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17 4. Where and How Options Trade?

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18 4. Where and How Options Trade? Options trade on four principal exchanges: Chicago Board Options Exchange (CBOE) American Stock Exchange (AMEX) Philadelphia Stock Exchange Pacific Stock Exchange

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19 4. Where and How Options Trade? AMEX and Philadelphia Stock Exchange options trade via the specialist system All orders to buy or sell a particular security pass through a single individual (the specialist) The specialist: Keeps an order book with standing orders from investors and maintains the market in a fair and orderly fashion Executes trades close to the current market price if no buyer or seller is available

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20 4. Where and How Options Trade? CBOE and Pacific Stock Exchange options trade via the marketmaker system Competing marketmakers trade in a specific location on the exchange floor near the order book official Marketmakers compete against one another for the public’s business

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21 4. Where and How Options Trade? Any given option has two prices at any given time: The bid price is the highest price anyone is willing to pay for a particular option The asked price is the lowest price at which anyone is willing to sell a particular option

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4. Where and How Options Trade?

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5. The Binomial Model

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The value of the replicating portfolio at time T, with stock price S T, is Δ S T + (1+r) T *B where Δ is the # of share s and B is cash invested At the prices S T = S o u and S T = S o d, a replicating portfolio will satisfy ( Δ * S o u ) + (B * (1+r) T )= Cu ( Δ * S o d ) + (B * (1+r) T )= Cd Solving for Δ and B Δ = (Cu - Cd) / (S o (u –d) ) B = (Cu - ( Δ * S o u) )/ (1+r) T and substitute in C = Δ S o + B to get the option price 5. The Binomial Model

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25 5. The Binomial Model Stock assumptions: Current price: S o = $27 In next 6 months, stock can either Go up by factor of u = 1.41 Go down by factor of 0.71 Call option assumptions Expires in t = 6 months = 0.5 years Exercise price: X = $25 Risk-free rate: 6% a year a day ( r = 0.06/365 a day)

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26 5. The Binomial Model Current stock price S o = $27 Ending "up" price = fu = S o u = $38.07 Option payoff: Cu = MAX[0, fu −X] = $13.07 Ending “down" price = = fd = S o d = $19.17 Option payoff: Cd = MAX[0, fd −X] = $0.00 u = 1.41 d = 0.71 X = $25

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27 5. Riskless Portfolio’s Payoffs at Call’s Expiration: $13.26 Current stock price S o = $27 Ending "up" price = fu = $38.07 Ending "up" stock value = Δ *fu = $26.33 Option payoff: Cu = MAX[0, fu −X] = $13.07 Portfolio's net payoff = Δ *fu - Cu = $13.26 Ending “down" stock price = fd = $19.17 Ending “down" stock value = Δ *fd = $13.26 Option payoff: Cd = MAX[0, fd −X] = $0.00 Portfolio's net payoff = Δ *fd - Cd = $13.26 u = 1.41 d = 0.71 X = $25 Δ =

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28 5. Create portfolio by writing 1 option and buying Δ shares of stock. Portfolio payoffs: Stock is up: Δ * S o u − Cu Stock is down: Δ * S o d − Cd

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29 5. The Hedge Portfolio with a Riskless Payoff Set payoffs for up and down equal, solve for number of shares: Δ = ( Cu − Cd ) / (S o ( u − d )) B = (Cu − ( Δ * S o u) )/ (1+r) T = - (Portfolio's net payoff )/ (1+r) T In our example: Δ = ($13.07 − $0) / $27(1.41 − 0.71) = B = - $13.26/( / 365 ) 365*0.5 = -$12.87

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30 5. The Hedge Portfolio with a Riskless Payoff Option Price C = Δ S o + B = ($27) + (-$12.87) = $18.67 − $12.87 = $5.80 If the call option’s price is not the same as the cost of the replicating portfolio, then there will be an opportunity for arbitrage. Option Premium C + (Commission)

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31 5. Arbitrage Example Suppose the option sells for $6. You can write option, receiving $6. Create replicating portfolio for $5.80, netting $6.00 −$5.80 = $0.20. Arbitrage: You invested none of your own money. You have no risk (the replicating portfolio’s payoffs exactly equal the payoffs you will owe because you wrote the option. You have cash ($0.20) in your pocket.

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32 5. Arbitrage and Equilibrium Prices If you could make a sure arbitrage profit, you would want to repeat it (and so would other investors). With so many trying to write (sell) options, the extra “supply” would drive the option’s price down until it reached $5.80 and there were no more arbitrage profits available. The opposite would occur if the option sold for less than $5.80.

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33 5. Multi-Period Binomial Pricing If you divided time into smaller periods and allowed the stock price to go up or down each period, you would have a more reasonable outcome of possible stock prices when the option expires. This type of problem can be solved with a binomial lattice. As time periods get smaller, the binomial option price converges to the Black-Scholes price, which we discuss in later slides.

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6. Black-Scholes Fischer Black and Myron Scholes published the derivation and equation of the Black-Scholes Option Pricing Model in 1973 under several assumptions. Bottom line: options of traded stocks are fundamentally priced

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35 6. Black-Scholes Option Pricing Model The stock underlying the call option provides no dividends during the call option’s life. There are no transactions costs for the sale/purchase of either the stock or the option. Risk-free rate, r RF, is known and constant during the option’s life. (More...)

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36 6. Black-Scholes Option Pricing Model Security buyers may borrow any fraction of the purchase price at the short-term risk-free rate. No penalty for short selling and sellers receive immediately full cash proceeds at today’s price. Call option can be exercised only on its expiration date. Security trading takes place in continuous time, and stock prices move randomly in continuous time.

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6. Black Scholes Equation ★ Integration of statistical and mathematical models ★ For example in the standard Black-Scholes model, the stock price evolves as dS = μ (t)Sdt + σ (t)SdWt. where μ is the drift parameter and σ is the implied volatility ★ To sample a path following this distribution from time 0 to T, we divide the time interval into M units of length δ t, and approximate the Brownian motion over the interval dt by a single normal variable of mean 0 and variance δ t. ★ The price f of any derivative (or option) of the stock S is a solution of the following partial-differential equation:

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38 6. Black-Scholes Option Pricing Model The Black-Scholes OPM:

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39 6. Black-Scholes Option Pricing Model Variable definitions: C = theoretical call premium S = current stock price t = time in years until option expiration K = option striking price R = risk-free interest rate

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40 6. Black-Scholes Option Pricing Model Variable definitions (cont’d): = standard deviation of stock returns N(x) = probability that a value less than “x” will occur in a standard normal distribution ln = natural logarithm e = base of natural logarithm (2.7183)

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41 6. Black-Scholes Option Pricing Model Example Stock ABC currently trades for $30. A call option on ABC stock has a striking price of $25 and expires in three months. The current risk-free rate is 5%, and ABC stock has a standard deviation of According to the Black-Scholes OPM, but should be the call option premium for this option?

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42 6. Black-Scholes Option Pricing Model Example (cont’d) Solution: We must first determine d 1 and d 2 :

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43 6. Black-Scholes Option Pricing Model Example (cont’d) Solution (cont’d):

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44 6. What is the value of the following call option according to the OPM? Assume: N = P = $27 X = $25 r RF = 6% t = 0.5 years σ = 0.49

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45 d 1 = {ln($27/$25) + [( /2)](0.5)} ÷ {(0.49)(0.7071)} d 1 = d 2 = (0.49)(0.7071) d 2 = First, find d 1 and d 2.

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46 6. Second, find N(d 1 ) and N(d 2 ) N(d 1 ) = N(0.4819) = N(d 2 ) = N(0.1355) = Note: Values obtained from Excel using NORMSDIST function. For example: N(d 1 ) = NORMSDIST(0.4819)

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47 6. Third, find value of option. V C = $27(0.6851) - $25e -(0.06)(0.5) (0.5539) = $ $25( )(0.6327) = $5.06

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48 What impact do the following parameters have on a call option’s value? Current stock price: Call option value increases as the current stock price increases. Strike price: As the exercise price increases, a call option’s value decreases.

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49 6. Impact on Call Value Option period: As the expiration date is lengthened, a call option’s value increases (more chance of becoming in the money.) Risk-free rate: Call option’s value tends to increase as r RF increases (reduces the PV of the exercise price). Stock return variance: Option value increases with variance of the underlying stock (more chance of becoming in the money).

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6. Negative Affects Myron Scholes was a partner of Long-Term Capital Management (LTCM) established in LTCM would use models and experienced investing knowledge to pick differences in short and long securities, profiting from the small margin. In 1998 the firm had $5B in equity and $125B in liability August 17, 1998 Russia devalues the rouble and suspended $13.5B of it’s Treasury Bonds, causing LTCM’s equity to decrease drastically, ever increasing it’s leverage ratio In order to prevent a stock market crash due to the outflux of investors and banks from LTCM’s funds the Federal Reserve organized a $3.5 B rescue effort to take over LTCM’s management Had LTCM lost leverage, all banks and creditors would pull out of other firms – including Salomon Brothers, Merrill Lynch.

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★ In the field of mathematical finance, many problems, for instance the problem of finding the arbitrage-free value of a particular derivative, boil down to the computation of a particular integral. ★ When the number of dimensions (or degrees of freedom) in the problem is large, PDE’s and numerical integrals become intractable, and in these cases Monte Carlo methods often give better results. ★ Monte Carlo methods converge to the solution more quickly than numerical integration methods, require less memory, have less data dependencies and are easier to program. ★ The idea is to use the result of Central Limit Theorem to allow us to generate a random set of samples as a valid representation of the previous value of the stock. “The sum of large number of independent and identically distributed random variables will be approximately normal.” 7. Monte Carlo method

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St = S 0 e (a – 0.5 σ 2)t + σ √tZ If V(S t,t) is the option payoff at time t, then the time-0 Monte Carlo price V(S 0,0) is where ST 1, …, ST n are n randomly drawn time-T stock prices

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7. Option Pricing Sensitivities The Greeks

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8. Parallel Computing

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drift = mu*delt; sigma_sqrt_delt = sigma*sqrt(delt); S_old = zeros(N_sim,1); S_new = zeros(N_sim,1); S_old(1:N_sim,1) = S_init; for i=1:N % timestep loop % now, for each timestep, generate info for % all simulations S_new(:,1) = S_old(:,1) +... S_old(:,1).*( drift + sigma_sqrt_delt*randn(N_sim,1) ); S_new(:,1) = max(0.0, S_new(:,1) ); % check to make sure that S_new cannot be < 0 S_old(:,1) = S_new(:,1); % % end of generation of all data for all simulations % for this timestep end % timestep loop 8. MATLAB program for Monte Carlo Peter Forsyth 2008

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function [Pmean, width] = Asian(S, K, r, q, v, T, nn, nSimulations, CallPut) dt = T/nn; Drift = (r - q - v ^ 2 / 2) * dt; vSqrdt = v * sqrt(dt); pathSt = zeros(nSimulations,nn); Epsilon = randn(nSimulations,nn); St = S*ones(nSimulations,1); % for each time step for j = 1:nn; St = St.* exp(Drift + vSqrdt * Epsilon(:,j)); pathSt(:,j)=St; end SS = cumsum(pathSt,2); Pvals = exp(-r*T) * max(CallPut * (SS(:,nn)/nn - K), 0); % Pvals dimension: nSimulations x 1 Pmean = mean(Pvals); width = 1.96*std(Pvals)/sqrt(nSimulations); Elapsed time is seconds. price = MATLAB program for Asian Options 2009 Financial Derivatives Proposal 2008/mfin7017/Chapter_2.ppt

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8. Monte Carlo Parallelism function MC (S, r, T, sigma, nSamples, nP) dataRand[nSamples] = rand(); start = idP*p/nSamples; end = (idP+1)*p/nSamples; for each idP mean_value[idP] = findMC(S,E,r,T,sigma,dataRand[start,end)); end reduce(mean_value[idP]); endfunction ★ Iterations are independent - Embarrassingly parallel ★ Parallel Random Number Generation: Master Processor ★ SPRNG (Scalable Pseudorandom Number Generation) library

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8. Binomial Parallelism function Bin (S, E, r, T, sigma, nTime ) … for step = nTime:-1:1 for i=1:nStep+1 value(i) = (pUp*value(i+1)+pDown*value(i))*exp(-r*dt); price(i) = price(i+1)*exp(-sigma*sqrt(dt)); value(i) = max(value(i), price(i)-E) endfunction

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8. Binomial Parallelism

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8. Finite Difference Parallelism Explicit Finite Differences ★ Careful: dt and dS are not independent ★ Use Ghost points to minimize the communication ★ Communication comes from one side only

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Standard options ★ Call, put ★ European, American Exotic options (non standard) ★ More complex payoff (ex: Asian) ★ Exercise opportunities (ex: Bermudian) 8. Types of options 2009 Financial Derivatives Proposal

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Bibliography FIR 7155: Global Financial Management (Spring 2013) https://umdrive.memphis.edu/cjiang/www/teaching/fir7155/powerpoint/FM12%20Ch%2008%20Show.ppt https://umdrive.memphis.edu/cjiang/www/teaching/fir7155/powerpoint/FM12%20Ch%2008%20Show.ppt Grid Enablement of Scientific Applications 2009/Presentations/FinancialDerivatives_ ppt 2009/Presentations/FinancialDerivatives_ ppt “ASX – Australian Stock Exchange.” Tuesday, March 7,2006.http://www.asx.com.au/ “Black-Scholes.” Tuesday, March 7, 2006.http://en.wikipedia.org/wiki/Black-Scholes “Case Study – LTCM.” Tuesday, March 7, 2006.http://www.erisk.com/Learning/CaseStudies/ref_case_ltcm.asp Shah, Ajaj. “Black, Merton and Scholes:Their work and its consequences.” Tuesday, March 7, “YHOO: Options for YAHOO INC.” Peter Forsyth, “An Introduction to Computational Finance Without Agonizing Pain” Guangwu Liu, L. Jeff Hong, "Pathwise Estimation of The Greeks of Financial Options” John Hull, “Options, Futures and Other Derivatives” Kun-Lung Wu and Philip S. Yu, “Efficient Query Monitoring Using Adaptive Multiple Key Hashing” Denis Belomestny, Christian Bender, John Schoenmakers, “True upper bounds for Bermudan products via non-nested Monte Carlo” Desmond J. Higham, “ An Introduction to Financial Option Valuation” Alexandros V. Gerbessiotis, “Architecture independent parallel binomial tree option price valuations” Bernt Arne Odegaard, “Financial Numerical Recipes in C++” Paul Wilmott, “Introduces Quantitative Finance”

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