Financial Modeling. Class #06B. Financial Modeling MSS
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1 Financial Modeling Class #06B Financial Modeling MSS
2 Class Overview Equity options We will cover three methods of determining an option s price 1. Black-Scholes-Merton formula 2. Binomial trees 3. Monte Carlo simulations Financial Modeling MSS
3 Three Valuation Methods Black-Scholes-Merton: European call options and put options Assumes stock price moves a certain way called GBM GBM stands for Geometric Brownian Motion Binomial Trees Can approximate GBM using discrete steps Answers are therefore close to Black-Scholes-Merton Adds flexibility Monty Carlo Simulation Most flexible Most difficult (need to specify the correct price process) We will specify GBM in this class Financial Modeling MSS
4 General Comment In general, we will value European call options on non-dividend paying stocks The base values for the stock and option are: Price (today) P 0 $36.00 Exercise Price EX $40.00 riskfree rate (CCR) r f 4.00% Time to Maturity T 1 year Volatility (CCR) σ 24.00% Dividend Rate (CCR) ρ 0.00% Financial Modeling MSS
5 Black-Scholes-Merton Formula Financial Modeling MSS
6 Black-Scholes-Merton The value of a European call option is: V C, t = [ N( d ) P] [ N( d ) PV ( EX )] 1 2 d 1 = ln ( P ) PV ( EX ) σ T σ T + 2 d 2 = d 1 σ T N ( d ) = cumulative normal prob. function Financial Modeling MSS
7 Using CCRs How do I calculate the PV(EX) using the riskfree rate quoted as a CCR? PV ( EX ) = = = = EX e 40 e r f T Financial Modeling MSS
8 Standard Normal Distribution In Excel use the normsdist(x) function Value the following option Price (today) P 0 $36.00 Exercise Price EX $40.00 riskfree rate (CCR) r f 4.00% Time to Maturity T 1 year Volatility (CCR) σ 24.00% Dividend Rate (CCR) ρ 0.00% Financial Modeling MSS
9 Binomial Trees Financial Modeling MSS
10 Binomial Method for Option Pricing Simplify the underlying stock price process Stocks can go up or down discretely in any period Make a portfolio out of known things: stocks and bonds Portfolio should replicate the payoffs of the option Solve for the price of the portfolio today The price of a portfolio that replicates the payoffs of an option, costs the same as the option itself (no arbitrage!) Solve for how a risk-neutral investor would calculate and discount the expected payoffs of the option Use these risk-neutral probabilities Financial Modeling MSS
11 Replicating Portfolios The idea of replication is one of the key concepts in finance The concept is right up there with diversification Two securities, with exactly the same payoffs in all states of the world, must sell for the same price today, otherwise we have an arbitrage situation If we can replicate the payoffs of a difficult security, using common-place securities, then we can get the price of the complicated security by adding up the components used to make (replicate) it!!! Financial Modeling MSS
12 Binomial Method for Option Pricing Stock price tree Call option tree (EX=$40) C Payoff of Call = Max (S T EX, 0) Financial Modeling MSS
13 Replication and Option Pricing Replicating Portfolio Buy 1 share of stock at $36.00 Borrow $27.21 at 4% Riskfree Rate (owe $28.32 in one year) Net outlay is $8.79 today Payoff in one year is depends on whether stock goes up or down: If the stock goes Down Up Value of Stock Repay loan Net Payoff Payoff Structure is exactly 3.02 the call s payoff! so 3.02 C = $8.79 C = $ Financial Modeling MSS 2012
14 Replication and Option Pricing Solve for the probabilities that are consistent with: 5.76 a) The option payoffs b) The price you just solved for 2.90 c) A risk-neutral investor 0.00 C = pc + + (1 exp r f T p) C 2.90 = p ( p)0 p = Financial Modeling MSS 2012
15 Generalizing the Binomial Tree We can break the year into any number of periods Here we break the year into three periods S 0 u 2 S 0 u 3 S 0 u S 0 u 2 d S 0 S 0 u d S 0 d S 0 u d 2 S 0 d 2 S 0 d 3 Financial Modeling MSS 2012
16 Multi-Step Binomial Trees For a tree with steps, you will need t = T Steps a = exp ( r t) f u ( σ t) = exp p = a u d d d ( σ t) = exp df = exp ( r t) f Financial Modeling MSS
17 Call Option Tree Start at the far right Calculate the final option price Work your way back to the left using risk neutral probabilities C 0 Calculate as discounted expected vale Calculate as discounted expected vale Calculate as discounted expected vale Calculate as discounted expected vale Calculate as discounted expected vale max[s 0 u 3 -EX,0] max[s 0 u 2 d-ex,0] max[s 0 u d 2 -EX,0] max[s 0 d 3 -EX,0] Financial Modeling MSS 2012
18 Monte Carlo Simulations Financial Modeling MSS
19 Simulated Stock Prices Suppose you have a one-year option Divide the year into periods 12 months per year is one idea t = 1 yr 12 = Note that the monthly riskfree rate is: exp(r f t) Calculate risk-neutral parameters Drift: µ = (r f ½σ 2 ) t Volatility: σ month = σ sqrt( t ) Financial Modeling MSS
20 Simulated Stock Prices (2) Basic methodology Next month s stock price equals this month s stock price raised by the drift and moved by the shock µ P = P e e t+ 1 t σ ε month t = P e t µ + σ ε month t Financial Modeling MSS
21 Simulated Stock Prices (3) To create a random shock we need to calculate a standard normal variable: In Excel, rand() produces a uniform random number Then, norminv( rand(), 0, 1 ) approximates a standard normal random number denoted ε t or N[0,1] You can also generate standard normal random numbers using a built-in Excel function: Tools Data Analysis Random Number Generation The random movement = σ month ε t Price on date t+1 is a function of the drift and the random movement: month t P = P e µ + σ ε + t 1 t Financial Modeling MSS
22 How to use Monte Carlo Analysis Simulate stock prices for a year This is called a single draw and denoted j For a European call option, check the final stock price: S T Calculate: max( S T EX, 0) Write down the draw number j the calculated value Repeat the simulation 10,000 times (draws) Remember to write down the value for max( S T EX, 0) for each draw j Calculate the average of all 10,000 values across the draws Then discount the expected value to get today s price Financial Modeling MSS
23 Monte Carlo Analysis You can use the MC Template to keep track of results Optimizing your program is useful 1,000 draws can take over a minute 10,000 draws can take over ten minutes! Financial Modeling MSS
24 Practice Problem The base values for the stock and option are: Price (today) P 0 $36.00 Exercise Price EX $40.00 riskfree rate (CCR) r f 4.00% Time to Maturity T 1 year Volatility (CCR) σ 24.00% Dividend Rate (CCR) ρ 0.00% Steps for simulation Steps 12 You expect the stock will go up 18% next year What is the risk-neutral drift and volatility ( µ and σ month ) for: P P t+ 1 = t e µ+σ d ε t Financial Modeling MSS
25 Practice Problem (2) Go to the spreadsheet Financial Modeling MSS
26 Exotic Options Financial Modeling MSS
27 Survey Basic exotics Packages Forward starts Compounds Powers Binaries Path-dependent exotics Lock-ins Barriers Lookbacks Asian Options Multivariate exotics Exchanges FX translations Rainbows Financial Modeling MSS
28 Barrier Options Consider a knock-in call option on a European stock Also known as an Up-And-In call option Strike price of $40.00 and a knock-in barrier at $45.00 Suppose the option was purchased when the stock was at $36.00 If the option expires with the stock at $42.00, but the stock never reached the barrier level of $45.00 during the life of the option, the option expires worthless On the other hand, if the stock first rises to the $45.00 barrier, this causes the option to knock-in If the option expires with the stock at $43.00, then the options is worth $3.00 at expiration Financial Modeling MSS
29 Barrier Options (2) EX = 40.00; Knock-In = 45.00; Barrier shown by blue line EX shown by red line dec jan feb mar apr may jun jul aug sep oct nov dec Financial Modeling MSS
30 Wrap-Up and Takeaways Financial Modeling MSS
31 What Should I Know for the Final Exam? 1. You should be able to calculate the price of an option using one of three methods Black-Scholes-Merton formula Bionomial tree Monte Carlo simulation 2. You should be able to simulate a stock price MUST use the correct risk-neutral process 3. Useful reference: Options, Futures, and Other Derivatives by John C. Hull Financial Modeling MSS
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