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Risk Management & Real Options X. Financial Options Analysis Stefan Scholtes Judge Institute of Management University of Cambridge MPhil Course

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

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

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

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2 September 2004 © Scholtes 2004Page 5 First approach… Valuation principle: Since the two gambles are based on the same uncertainty, they should have the same expected returns

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2 September 2004 © Scholtes 2004Page 6 Change the new gamble £ 4 £ 6 £ 3 50% £ w? £ 2+x £ 5+x 50% Adding £x to both payoffs of the new gamble should change its value to w=£v+x 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! Surely, w=v+x

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2 September 2004 © Scholtes 2004Page 7 Second approach: Market based valuation 3 1 v? Existent gamble 1 Existent gamble 2 Newgamble

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2 September 2004 © Scholtes 2004Page 8 Second approach: Market based valuation 3 1 v? Existent gamble 1 Existent gamble 2 Newgamble Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty)

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2 September 2004 © Scholtes 2004Page 9 Second approach: Market based valuation 3 1 v? Equations for replicating portfolio: 16 Existent gamble 1 Existent gamble 2 Newgamble Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty)

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2 September 2004 © Scholtes 2004Page 10 Second approach: Market based valuation 3 1 v? Equations for replicating portfolio: Solution: 16 Existent gamble 1 Existent gamble 2 Newgamble Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty)

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2 September 2004 © Scholtes 2004Page 11 Second approach: Market based valuation 3 1 v? Equations for replicating portfolio: Solution: Price of replicating portfolio: 16 Existent gamble 1 Existent gamble 2 Newgamble Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty)

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2 September 2004 © Scholtes 2004Page 12 Second approach: Market based valuation 3 1 v? Equations for replicating portfolio: Solution: Price of replicating portfolio: 16 Existent gamble 1 Existent gamble 2 Newgamble Up/down movements for all gambles are determined by the same flip of the coin (underlying fundamental uncertainty) Replicating portfolio has precisely the same payoffs as the new gamble Ergo: Price for the new gamble = price of replicating portfolio (v = 9)

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2 September 2004 © Scholtes 2004Page 13 In-class example 3 1 v? Stock Cash Call option at strike price 4

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2 September 2004 © Scholtes 2004Page 14 In-class example 3 1 v? Equations for replicating portfolio: Solution: Price: 1 Stock Cash Call option at strike price 4 Borrowing $2/3 from the bank and investing $1/3 in the stock gives PRECISELY the same payoff distribution as the call option

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2 September 2004 © Scholtes 2004Page 15 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

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2 September 2004 © Scholtes 2004Page 16 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

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2 September 2004 © Scholtes 2004Page 17 Example 3 1 v? Chance Node 1 Chance Node 2 Chance Node 3 All moves are triggered by the same flip of the coin

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2 September 2004 © Scholtes 2004Page 18 Example 3 1 v? Chance Node 1 Chance Node 2 Chance Node 3 = v? 1 0 3x All moves are triggered by the same flip of the coin

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2 September 2004 © Scholtes 2004Page 19 Example 3 1 v? Chance Node 1 Chance Node 2 Chance Node 3 = v? 1 0 3x All moves are triggered by the same flip of the coin Only “consistent” value for v is v=(3-2)/3=1/3

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2 September 2004 © Scholtes 2004Page 20 Option pricing in a binomial model: The general case S uS dS Risky investment in stock returns u>1, d<1 1 (1+r) Risk-free Investment r=one-period risk-free rate CuCu CdCd Options contract C=? Call with exercise price K All parameters are known, except for C Call option: C u =Max{uS-K,0}, C d =Max{dS-K,0}

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2 September 2004 © Scholtes 2004Page 21 Option pricing in a binomial model: The general case Invest £x in stock and £y in bank (negative amounts mean short sales and borrowing, resp.) Equations for replicating portfolio Solution

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2 September 2004 © Scholtes 2004Page 22 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 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

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2 September 2004 © Scholtes 2004Page 23 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 pricing and q is called the risk-neutral upward probability

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2 September 2004 © Scholtes 2004Page 24 Keeping track of the replicating portfolio 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

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2 September 2004 © Scholtes 2004Page 25 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… S uS dS p 1-p

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2 September 2004 © Scholtes 2004Page 26 Example of unfolding of stock price uncertainty (p=50%)

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2 September 2004 © Scholtes 2004Page 27 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(S n /S 0 ) is approximately normal with mean

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2 September 2004 © Scholtes 2004Page 28 Does that make sense? Suppose returns r t =S t /S t-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? S T /S 0 =r 1 *r 2 *…*r T ln(S T /S 0 )=ln(r 1 )+ln(r 2 )+…+ln(r T ) 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

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2 September 2004 © Scholtes 2004Page 29 Histogram of log-normal variable (Simulation of exp(Y), where Y is normal with mean 10% and standard deviation 40%)

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2 September 2004 © Scholtes 2004Page 30 Histogram of corresponding normal variable Y

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2 September 2004 © Scholtes 2004Page 31 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

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2 September 2004 © Scholtes 2004Page 32 Estimating parameters for the 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

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2 September 2004 © Scholtes 2004Page 33 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.

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2 September 2004 © Scholtes 2004Page 34 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)

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2 September 2004 © Scholtes 2004Page 35 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)?

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2 September 2004 © Scholtes 2004Page 36 2-period stock price

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2 September 2004 © Scholtes 2004Page 37 Corresponding decision tree Price move month 1 Price move month 2 Exercise? up up up down down down yes yes yes yes no no no no

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2 September 2004 © Scholtes 2004Page 38 Corresponding decision tree Price move month 1 Price move month 2 Exercise? up up up down down down yes yes yes yes no no no no These two nodes are identical since moving up first and then down is the same as moving down first and then up

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2 September 2004 © Scholtes 2004Page 39 Simplified decision tree Price move month 1 Price move month 2 Exercise? up up up down down down yes yes yes no no no We will value this decision tree using non-arbitrage valuation of chance nodes

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2 September 2004 © Scholtes 2004Page 40 Valuation of the final decision nodes Max(stock-strike, 0) Strike price = £60

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2 September 2004 © Scholtes 2004Page 41 Valuation of the final decision nodes Existing gamble in the market New gamble

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2 September 2004 © Scholtes 2004Page 42 Non-arbitrage valuation of month 1 chance nodes

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2 September 2004 © Scholtes 2004Page 43 Non-arbitrage valuation of month 1 chance nodes Value x+y of replicating portfolio

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2 September 2004 © Scholtes 2004Page 44 Non-arbitrage valuation of second chance node in month 1 Corresponding risky gamble in the market

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2 September 2004 © Scholtes 2004Page 45 Non-arbitrage valuation of today’s chance node Optionvalue Corresponding risky gamble In the market

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2 September 2004 © Scholtes 2004Page 46 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

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2 September 2004 © Scholtes 2004Page 47 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?

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2 September 2004 © Scholtes 2004Page 48 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

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2 September 2004 © Scholtes 2004Page 49 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?

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2 September 2004 © Scholtes 2004Page 50 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

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2 September 2004 © Scholtes 2004Page 51 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!

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2 September 2004 © Scholtes 2004Page 52 Decision tree for American option Move in Month 1 Exercise? Move in Month 2 Exercise? yes yes no no

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2 September 2004 © Scholtes 2004Page 53 Decision tree for American option Move in Month 1 Exercise? Move in Month 2 Exercise? Can be merged as before yes yes no no

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2 September 2004 © Scholtes 2004Page 54 Simplified decision tree Move in Month 1 Exercise? Move in Month 2 Exercise? We will value this decision tree using non-arbitrage valuation of chance nodes yes yes no no

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2 September 2004 © Scholtes 2004Page 55 American call option valuation: Valuing final decision nodes Max(stock-strike, 0) Strike price = £60 Final period: exercise only if stock price > strike price

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2 September 2004 © Scholtes 2004Page 56 American call option valuation: Simultaneous valuation of month 1 chance node and decision node Is the value from exercising the option larger than the value from holding it?

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2 September 2004 © Scholtes 2004Page 57 American call option valuation: Simultaneous valuation of month 1 chance node and decision node Is the value from exercising the option larger than the value from holding it?

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2 September 2004 © Scholtes 2004Page 58 American call option valuation: Simultaneous valuation of month 1 chance node and decision node Is the value from exercising the option larger than the value from holding it?

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2 September 2004 © Scholtes 2004Page 59 American call option valuation: Simultaneous valuation of month 1 chance node and decision node Maximum of the value of holding the option and the value from exercising it

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2 September 2004 © Scholtes 2004Page 60 Roll-back to the root of the tree Max(holding,exercising)

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2 September 2004 © Scholtes 2004Page 61 Aside One can show mathematically that an American call option on a non- dividend-paying stock is never exercised before maturity Therefore the additional flexibility of being able to exercise before maturity has no value American calls on non-dividend-paying stocks have the same value as European calls on the same stock This is not the case for put options

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2 September 2004 © Scholtes 2004Page 62 American Put Option Strike price £63 Max(strike-stock,0)

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2 September 2004 © Scholtes 2004Page 63 American Put Option Strike price £63 Max(strike-stock,0) Max(exercise value,holding value)

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2 September 2004 © Scholtes 2004Page 64 American Put Option Strike price £63 Max(strike-stock,0) Max(exercise value,holding value)

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2 September 2004 © Scholtes 2004Page 65 American Put Option Strike price £63 Max(strike-stock,0) Max(exercise value,holding value) PrematureExercise

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2 September 2004 © Scholtes 2004Page 66 Summary of option valuation Set up the lattice for the underlying stock price Calculate the option price at maturity Value the nodes of the tree successively backwards, using Value of a node for European option = holding value Value of a node for American option = max(holding value, exercise value) Value of the origin of the tree is the current value of the option

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2 September 2004 © Scholtes 2004Page 67 Optimal decision Notice that for American options you do not only get the value of the option but also an optimal contingency plan when to exercise the option For real options, the contingency plan is much more important than the precise value, since real options are typically not traded in a market

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2 September 2004 © Scholtes 2004Page 68 Group work Carry out a binomial lattice valuation for an American call option over 5 months with annual volatility 30%, risk free interest 5% p.a., strike price $35 and current stock price $30 Compute the optimal contingency plan for the exercise decision Compute the initial replicating portfolio and the re-balancing strategy Compare with the value you obtain from the Black-Scholes formula

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