Risk Management & Real Options X. Financial Options Analysis

Risk Management & Real OptionsX. Financial Options Analysis Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course 2004-05

An option is a gamble… What’s the value of this gamble? £ 5 50% £ v? 50% £ 2 © Scholtes 2004

An option is a gamble… Naïve valuation £ 5 50% £ v? 50% £ 2 v= expected return = 50%£6+50%£3=£4.50 © Scholtes 2004

Market valuation: Assumes there is a market for gambles Assume payoff determined by the same coin flip £ 6 50% Gamble in the market (stock) £ 4 Incorporates “market risk premium” £ 3 50% £ 5 New gamble 50% £ v? £ 2 50% © Scholtes 2004

First approach… Valuation principle: Since the two gambles are based on the same uncertainty, they should have the same expected returns © Scholtes 2004

Change the new gamble Adding £x to both payoffs of the new gamble should change its value to w=£v+x £ 6 50% £ 4 £ 3 50% £ 5+x Surely, w=v+x 50% £ w? £ 2+x 50% But for x=1, both gambles are the same Hence 4=v+1, i.e., v=3 is consistent with the existing gamble, not v=3 1/9! © Scholtes 2004

Second approach:Market based valuation 5 Existent gamble 1 3 2 2 Existent gamble 2 1 -1 16 New gamble v? 1 © Scholtes 2004

Second approach:Market based valuation 5 Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Existent gamble 1 3 2 2 Existent gamble 2 1 -1 16 New gamble v? 1 © Scholtes 2004

Second approach:Market based valuation 5 Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Existent gamble 1 3 2 Equations for replicating portfolio: 2 Existent gamble 2 1 -1 16 New gamble v? 1 © Scholtes 2004

Second approach:Market based valuation 5 Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Existent gamble 1 3 2 Equations for replicating portfolio: 2 Existent gamble 2 1 -1 Solution: 16 New gamble v? 1 © Scholtes 2004

Second approach:Market based valuation 5 Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Existent gamble 1 3 2 Equations for replicating portfolio: 2 Existent gamble 2 1 -1 Solution: 16 New gamble v? Price of replicating portfolio: 1 © Scholtes 2004

Second approach:Market based valuation 5 Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Existent gamble 1 3 2 Equations for replicating portfolio: 2 Existent gamble 2 1 -1 Solution: 16 New gamble v? Price of replicating portfolio: 1 Replicating portfolio has precisely the same payoffs as the new gamble Ergo: Price for the new gamble = price of replicating portfolio (v = 9) © Scholtes 2004

In-class example 5 Stock 3 2 1 Cash 1 1 1 Call option at strike price 4 v? 0 © Scholtes 2004

Why is this conceptually correct?Arbitrage and Equilibrium • Option and replicating portfolio have the SAME future payoffs, no matter how the future evolves • In equilibrium, option has a buyer AND seller • If price of the option < price of the replicating portfolio then no-one will sell the option • O/w someone buys the option, sells the replicating portfolio and pockets the difference  Risk-less profit (arbitrage) • If price of the option > price of the replicating portfolio then no-one will buy the option • O/w someone sells the option, buys the replicating portfolio and pockets the difference  Risk-less profit (arbitrage) • The only option price that is consistent with the existing market prices is the price of the replicating portfolio © Scholtes 2004

Why is this conceptually correct?Consistency • A weaker argument then no-arbitrage is that of “valuation consistency” • This argument does not require the existence of a market but replaces it by assumptions on valuations • Recall the problem: • Two chance nodes with different payoffs but the same underlying “random experiment” (“same flip of the coin”) • We have already valued one of the chance node • 1st key assumption: Linear valuation • “Constant returns to scale”: if all payoffs of a chance node are multiplied by the same factor then the value of that chance node is multiplied by that factor as well • “Adding values”: The value for the sum of two chance nodes with the same underlying random experiment is the sum of the value of the chance nodes • 2nd key assumption: “Law of one price” • If two chance nodes follow the same underlying random experiment (“same flip of coin”) and have the same payoffs then their values are the same © Scholtes 2004

Example 5 Chance Node 1 3 2 1 Chance Node 2 1 All moves are triggered by the same flip of the coin 1 1 Chance Node 3 v? 0 © Scholtes 2004

Example 0 2 1 2 5 = 2 v? Chance Node 1 3x + 3 2 1 Chance Node 2 1 All moves are triggered by the same flip of the coin 1 1 Chance Node 3 v? 0 © Scholtes 2004

Example 0 2 1 2 5 = 2 v? Chance Node 1 3x + 3 2 1 Chance Node 2 1 All moves are triggered by the same flip of the coin Only “consistent” value for v is v=(3-2)/3=1/3 1 1 Chance Node 3 v? 0 © Scholtes 2004

Option pricing in a binomialmodel: The general case Call with exercise price K uS Risky investment in stock returns u>1, d<1 S dS (1+r) Risk-free Investment r=one-period risk-free rate 1 (1+r) Cu Options contract C=? Cd All parameters are known, except for C Call option: Cd=Max{dS-K,0} Cu=Max{uS-K,0}, © Scholtes 2004

Option pricing in a binomialmodel: The general case • Invest £x in stock and £y in bank (negative amounts mean short sales and borrowing, resp.) • Equations for replicating portfolio • Solution © Scholtes 2004

Replicating the option payoffs • Price of the option is the price of the replicating portfolio C=£x+£y • After some simple algebra C=x+y becomes • Notice that 0<q<1, provided u>1+r>d (sensible assumption) • Can interpret C as an expected payoff discounted at the risk-free rate • However, q has nothing to do with the actual probability that the stock moves upwards! • Can interpret q is the “forward price” for a contract that pays Cu=1 if the stock moves up and Cd=0 o/w © Scholtes 2004

Replicating the option payoffs • Price of the option is the price of the replicating portfolio C=£x+£y • After some simple algebra C=x+y becomes • Option pricing principle:Price of the call option is its expected payoff if upwards probability was q, discounted at the risk-free rate • This is called risk-neutral pricingand q is called therisk-neutral upward probability © Scholtes 2004

Keeping track of the replicatingportfolio • It is always a good idea to keep track of the replicating portfolio if you value an option • Recall the equations for amount £x in stock and amount £y in risk-less investment (long-term government bond): • Holding £x in stock and £y in risk-less money exactly replicates the option in our model © Scholtes 2004

Multi-period models • Single-period stock price model • Given a stock price S today, the stock will move over a period t to uS (upward move) with probability p and to dS (downward move) with probability (1-p) • u and d are numbers with u>1+r>d>0 • typically d=1/u • Let us see how this model develops over time… uS p S 1-p dS © Scholtes 2004

Example of unfolding of stock price uncertainty(p=50%) © Scholtes 2004

What’s the distribution of the value after many periods? Mathematical result: Binomial model of moving up by factor u with probability p and down by factor d with probability 1-p is, for many periods, an approximation of log-normal returns, i.e., log(Sn/S0) is approximately normal with mean © Scholtes 2004

Does that make sense? • Suppose returns rt=St/St-1 of a stock over small time periods are independent and have an unknown distribution • Consider t=0,1,…,T (e.g. T=52 weeks). What is the distribution of the long run (say annual) return? • ST/S0=r1*r2*…*rT • ln(ST/S0)=ln(r1)+ln(r2)+…+ln(rT) • Therefore the central limit theorem provides an argument that long-run returns tend to be log-normally distributed, even if short-run returns are not • A random variable X is called log-normally distributed if log(X) is normally distributed © Scholtes 2004

Histogram of log-normal variable(Simulation of exp(Y), where Y is normal with mean 10%and standard deviation 40%) © Scholtes 2004

Histogram of correspondingnormal variable Y © Scholtes 2004

Estimating parameters for the lattice model • Choose base period, e.g. a year, and estimate mean n and variance s2 of the log stock price return over the base period • e.g. based on historic data • Partition base period into n periods of length t=1/n • Recall that log-return log(Sn/S0) is approximately normally distributed with • Setting n=1/t gives the equations © Scholtes 2004

Estimating parameters forthe lattice model • System • consists of two equations in three unknowns p, u, d • Can remove the degree of freedom arbitrarily, e.g. by setting d=1/u • Corresponding solution of the system is • With this choice of parameters the binomial lattice is a good approximation of normally distributed log stock price returns with mean n and volatility s © Scholtes 2004

Alternative parameter choice • System consists of two equations in three unknowns p, u, d • Can remove the degree of freedom arbitrarily, e.g. by setting p=50% • Corresponding solution of the system is • This choice of parameters incorporates the trend  in the upwards and downwards moves, as opposed to the earlier choice which incorporates trend in probability p. © Scholtes 2004

Example • Data: • Stock price is currently £62, Estimated standard deviation of logarithm of return = 20% over a year (T=1) • European call option over 2 months at strike price K=£60 • Risk-free rate is 10%, compounded monthly (r=0.1/12, t=1/12) • Conversion of this information to lattice parameters: • Risk-neutral probability: q=((1+r)-d)/(u-d)=0.559 • Notice: Risk neutral probability (and therefore the options price) are independent of the probability p of upward moves • The important parameters, u,d, only depend on the standard deviation  of the log returns (volatility), not on the mean  (trend) © Scholtes 2004

In-class example • Given annual volatility of 25%, what are u and d for a lattice with weekly periods? • Given a risk-free interest rate of 5% p.a. what is the weekly risk-free interest rate r? (assume continuous compounding, i.e., capital grows by factor exp(t5%) over a period of length t) • What is the “risk-neutral probability” q? • If the stock price today is £50 and we have a one-week call option with strike price £51, what is the value of the option in a single-period lattice (i.e. only one up or down move)? © Scholtes 2004

Corresponding decision tree Price move month 2 Price move month 1 Exercise? yes up no up yes down no yes up down no yes down no © Scholtes 2004

Corresponding decision tree Price move month 2 Price move month 1 Exercise? yes up no up yes These two nodes are identical since moving up first and then down is the same as moving down first and then up down no yes up down no yes down no © Scholtes 2004

Simplified decision tree Price move month 2 Price move month 1 Exercise? yes up no up yes down no up down yes down no We will value this decision tree using non-arbitrage valuation of chance nodes © Scholtes 2004

In-class example • Given annual volatility of 25%, what are u and d for a lattice with weekly periods? • Given a risk-free interest rate of 5% p.a. what is the weekly risk-free interest rate r? (assume continuous compounding, i.e., capital grows by factor exp(t5%) over a period of length t) • What is the “risk-neutral probability” q? • If the stock price today is £50 and we have a one-week call option with strike price £51, what is the value of the option in a single-period lattice (i.e. only one up or down move)? • Now value a 3-week call option with the same strike price © Scholtes 2004

No-arbitrage valuation versus discounted expected values • We can, conceptually, also value the decision tree with discounted expected values at the chance nodes • Assuming, in the former example, an annual expectation of log stock returns of 15%, we obtain the upward move probability (see earlier slide) • The expected monthly return on the stock is 1.42% • If we value the chance nodes by their expected value, discounted by the stock return expectation then the obtain the value £4.38 • Why is this the wrong value? © Scholtes 2004

The Black-Scholes formula • Black and Scholes have found a formula that allows you to compute the value of a European call option without the use of a lattice • C is the option price, S is today’s stock price, K is the strike price and T is the time to maturity • Make sure that n, s, r and T refer to the same base unit, i.e. if risk-free interest, means and standard deviation of log-returns are annual then T is measured in years as well • N(x) is the standard normal cumulative distribution function (N(x)=P(X<=x) where X is a standard normal (i.e. mean 0, variance 1) • Normsdist(x) in Excel © Scholtes 2004

The Black-Scholes formula • First observation: Option value • is independent of mean n of the underlying stock price • Second observation (after some calculus): Option value increases with increasing volatility s • Do you have an intuitive argument for this observation? © Scholtes 2004

In-class example • Given annual volatility of 25%, what are u and d for a lattice with weekly periods? • Given a risk-free interest rate of 5% p.a. what is the weekly risk-free interest rate r? (assume continuous compounding, i.e., capital grows by factor exp(t5%) over a period of length t) • What is the “risk-neutral probability” q? • If the stock price today is £50 and we have a one-week call option with strike price £51, what is the value of the option in a single-period lattice (i.e. only one up or down move)? • Now value a 3-week call option with the same strike price • Now calculate the value of the 3-week call option with the B-S formular © Scholtes 2004

The Black-Scholes formula • The B-S price for the option that we had valued earlier in the two-stage lattice is £ 3.82 (against £3.90 in our model) • If we use more time periods (e.g. a half-monthly or weekly lattice), then the lattice approximation of B-S becomes better and better • Mathematical result: As Dt gets smaller and smaller, the value obtained by a lattice valuation approaches the Black-Scholes value • See Luenberger, Chapter 13, for more explanations • So why do we do lattices, then? • B-S applies only to European option • European option can only be exercised at maturity • American options can be exercised at any time until they mature • More realistic for real options • American options can be priced by the lattice model! © Scholtes 2004

Risk Management & Real Options X. Financial Options Analysis