Pricing Derivatives & Options
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Pricing Derivatives & Options

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Interest rate futures contracts...

Interest rate futures contracts
Options on futures
Mortgage-backed securities
Interest rate caps- and floor
Swap options
Commodity linked bonds
Zero-coupon treasury trips

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Pricing Derivatives & Options Pricing Derivatives & Options Presentation Transcript

  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options Derivatives is an investment whose value to day or at some future date is derived entirely from the value of other assets, the underlying asset. Examples are: Interest rate futures contracts Options on futures Mortgage-backed securities Interest rate caps- and floor Swap options Commodity linked bonds Zero-coupon treasury trips Major Break-through in the valuation of derivatives came with two finance professors at MIT, Black and Scholes, came out with a formula that related the price of a call option to the price of the stock to which the option applies. 1
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Examples Forwards and Futures Represents the obligation to buy (sell) a security or commodity at a pre- specified price, known as the forward price, at some future date The most important financial forward market is the inter-bank forward market for currencies, particularly dollars for yen and dollars for Euros. Swaps Is an agreement between two investors, or counterparties as they are sometimes called, to periodically exchange the cash flow of one security for the cash flow of another. The last date of exchange determines the swap maturity. Forwards, Futures and Swaps zero-cost instruments. Options Gives their buyers the right, but not the obligation, to buy (call option) or sell (sell option) an underlying security at a pre-specified price, known as the strike price. 2
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options Values of Calls and Puts at Expiration: 3
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Examples Options cont. Warrants Warrants are options, usually calls, that companies issue on their own stock. In contrast too options, which are mere bets between investors on the value of the company’s underlying stock – in which the corporation never gets involved – warrants are contracts between a corporation and an investor. Embedded Options A number of corporate securities have option like components. Ex. Convertible bonds, callable or refundable corporate bonds. Corporate equity and debt contain option like characteristics Real Assets Many real assets may be viewed as derivatives. Ex. Copper mine/Mortgage-back securities/Structured Notes 4
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Basics Two basic components: 1. Concept of Perfect tracking 2. Principle of no arbitrage A fair market price, is simply a no-arbitrage restriction between the tracking portfolio and the derivative. It is always possible to develop a portfolio consisting of the underlying asset and a risk-free asset that perfectly tracks the future cash flows of a derivative. A perfect tracking portfolio is a combination of securities that perfectly replicates the future cash flows of another investment. Note: In absence of tracking error, arbitrage exists if it costs more to buy the tracking portfolio than the derivative, or vice versa. 5
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Forward Value Ex. Obligation (forward) to buy Ekornes stock one year from now for 100 kroner! The price sells to day for 97 and Ekornes will not pay dividends in the period. One-year zero coupon bonds (par. 100) currently sell for 92. Cost Today Cash Flow one Year from today Strategy 1(forward) ? S1 − 100 Strategy 2 (tracking 97-92=+5 portfolio) S1 − 100 Strategy 2 costs 5 kroner. If strategy 1 cost > 5 kroner, arbitrage exists. Go short in strategy 1 and go long in strategy 2 (sell forward (short), buy the share and short 200 kroner par of zero coupon bonds). If Strategy 1 cost < 5 kroner, arbitrage exists. Go long in strategy 1 and go short in strategy 2 (buy forward (long), selling short the share and go long 200 kroner par of zero coupon bonds). kroner 5 is a fair value of the attractive obligation to buy Ekornes for kroner 100 one year from now. 6
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Model The derivative valuation models induce that the current price of the underlying asset determines the price of the derivative to day. Hence, a call option has a known value at expiration date of the call when the stock price to day is known. Linear functions of the underlying asset’s future payoffs: Static investment strategies (buy and hold) ex. Forward contracts Non-linear functions of the underlying asset’s future payoff: Dynamic investment strategies (continuous rebalancing) ex. Option contracts The ability to perfectly track a derivative’s payoff with a dynamic strategy requires that the following conditions are met: 1. The price of the underlying security must change smoothly; that is, it does not make large jumps. 2. It must be possible to trade both the derivative and the underlying security continuously. 3. Markets must be frictionless. 7
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Model If the price of the underlying security follows a binomial process, the investor can still perfectly track the derivative’s future cash flows. UP Give high degree of accuracy, when the binomial periods are small and numerous. Down 8
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Model Binomial Model Tracking of a Structured Bond OBX 1.500 (up state) 1.275 750 (down state) 1,10 (up state) 1 1,10 (down state) 331,75 (up state) =100,00+6,75%(100,00) +225,00 106,75 (down state) =100,00+6,75%(100,00) +0,00 9
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Model The binomial process allow perfect tracking of the value of the derivative applying a tracking portfolio consisting of the underlying asset and a risk-free bond. Two steps: Identification of the tracking portfolio (TP) and valuation of the TP. Identification: Find the perfect tracking portfolio is the major task in valuing a derivative. With binomial processes, the tracking portfolio is identified by solving two equations in two unknowns, where each equation corresponds to one of the two future nodes to which one can move (up or down). Up node: ∆Su + B (1 + rf ) = Vu Solved simultaneously yields a unique solution for ∆ and B. Down node: ∆Sd + B(1 + rf ) = Vd Valuation: The fair market value of the derivative equals the amount it costs to buy the tracking portfolio. Buying ∆ shares and B dollars of the risk free asset. Thus, V = ∆S0 + B 10
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Model Example structured bond Next period Value Today’s value Up State Down State TP portfolio ∆(1,275)+B ∆(1,500)+1,1B 0,3(750)+1,1B Derivative ? 331,75 106,75 Two equations in two unknowns The Solution : ∆ ⋅1.500 + 1,1⋅ B = 331, 75 ∆ ⋅ 750 + 1,1⋅ B = 106, 75 ∆ = 0,3 and B = -107,50 Valuation: V = ∆S0 + B V = 0,3 ⋅1.275 − 107,50 = 275 The value of the derivative equals the value of the tracking portfolio. 11
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach Based on our binomial results: Important 1: The value of the derivative in relation to the value of the underlying asset does not depend on the probabilities of up and down. Important 2: The grade of investor risk aversion was not necessary for calculation fair market values. Why not? This information is already captured by the price of the underlying asset on which we base out valuation of the derivative. Note: Once the stock price is known, risk aversion and mean return information are superfluous, not that they are irrelevant. 12
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach The risk-neutral valuation method: Step 1. Identify risk-neutral probabilities that are consistent with investors being risk neutral, given the current value of the underlying asset and its possible future values. Step 2. Multiply each risk-neutral probability by the corresponding future value for the derivative and sum the products together. Step 3. Discount the sum of the products in step 2 (the probability weighted average of the derivative’s possible future values) at the risk free rate. Fewer steps than the tracking portfolio approach. 13
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach A General formula for risk-neutral probabilities: π u + (1 − π )d = 1 + rf For our example: where rf = 10% rf − is the risk − free rate u − up node = 1,1765 u − rate of return at the up node d − down node = 0,5882 d − rate of return at the down node rearraging the Expression : rearraging the Expression : 1 + 0,1 − 0,5882 π= = 0,87 1 + rf − d 1,1765 − 0,5882 π= u−d Valuation of the derivative: 331, 75 ⋅ 0,87 + 106, 75 ⋅ (1 − 0,87) = 302,5 302,5 = 275 1.1 14
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach Risk-Neutral Probabilities and Zero-cost Forward and Future Prices Because futures contracts are one class of popularly traded financial instruments with known terms (the future price) and known market values, it is often useful to infer risk-neutral probabilities from them. To use the prices of zero-cost forwards and futures to obtain risk-neutral probabilities, it is necessary to slightly modify the risk-neutral valuation formulas. For futures, the expected cash flows at the end of the period is π ( Fu − F ) + (1 − π )( Fd − F ) /1 + rf = 0 The no-arbitrage futures price is the same as a weighted average of the expected futures prices at the end of the period, where the weights are the risk-neutral probabilities: F = π ⋅ F + (1 − π ) ⋅ F u d At the end of the period (maturity): Fu = Su and Fd = Sd. Substitution gives us: F = π ⋅ Su + (1 − π ) ⋅ S d 15
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach Using Risk-Neutral Probabilities to obtain Future Prices OBX 1.500 (up state) 1,275 750 (down state) 1,10 (up state) 1 1,10 (down state) 1.500-F (up state) 750 -F (down state) Using 0,87 and 0,13 we can derive future prices: 0.87(1.500) + 0,13(750) = 1402,50 consistent (1275 *1,1=1402,75) 16
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Wall Street Approach It is possible to rearrange to identify the risk neutral probabilities π and 1 –π from futures prices. This yields: F − Fd π= Fu − Fd and at maturity F − Sd π= Su − S d If future prices can appreciate 10% (up state) or depreciate 10% (down state), we can calculate risk neutral probabilities. In the up state, Fu=1.1F, in the down state Fd=0,9F. Applying the above equation: F − 0.9 ⋅ F 0,1F π= = = 0,5 1,1 ⋅ F − 0,9 ⋅ F 0, 2 F 17
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Multiperiod Binomial Risk neutral valuation can be applied whenever perfect tracking is possible. However, when large jumps in the value of the tracking portfolio or the derivative can occur, perfect tracking is in general not possible. The exception is a binomial price process. Numerical example in a multiperiod setting: 121 110 38=121-83 38=121 - 83 100 99 90 16=99 - 83 81 0 18
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Multiperiod Binomial Risk neutral valuation method(working backward through the tree diagram) 121 π + 99 (1-π) = 110 π = 0,5 Value of the option at node U is: 0,5 (38) + 0,5 (16) = 27 Value of the option at node D is (the same π) 0,5 (16) + 0,5 (0) = 8 The value at the derivative is 0,5 (27) + 0,5 (8) = 17,5 19
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Multiperiod Binomial Algebraic Representation of Two-Period Binomial Valuation In two steps: In one step: Node u : π V + (1 − π )Vud π V + (1 − π )Vdd V = π uu + (1 − π ) ud 1 + rf 1 + rf π Vuu + (1 − π )Vud Vu = π 2Vuu + 2π (1 − π )Vud + (1 − π ) 2 Vdd 1 + rf V= 1 + rf π Vud + (1 − π )Vdd Vd = 1 + rf To day ' s value V : π V + (1 − π )Vd V= u 1 + rf The discounted expected future value of the derivative with an expected value that is computed with risk-neutral probabilities rather than true probabilities. Note: π can vary along the nodes of the three diagram. 20
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Valuation Techniques Numerical Techniques: The techniques for valuing virtually all the new financial instruments developed by Wall Street firms consists of Numerical Methods; that is, no algebraic formula is used to compute the value of the derivative as a function of the value of the underlying security. Instead, a computer is fed a number corresponding to the price of the underlying security along with some important parameter values. The computer derives the numerical value of the derivative + sometimes, the number of shares of the underlying share in the tracking portfolio. Simulation: Generate random numbers to generate outcomes and then averages the outcomes of some variable to obtain values (together with the standard deviation). Exclude mortgages and American put options. 21
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Valuation Techniques Binomial-Like Numerical Methods: Simplification. First, the binomial method can be used to approximate many kinds of continuous distributions if the time periods are cut to extremely small intervals. One popular continuous distribution is the lognormal distribution. The natural logarithm of the return of a security is normally distributed when the price movements of the security are determined with by the lognormal distribution. Once the annualized standard deviation, σ, of the normal distribution is known, u and d are estimated as follows: 1 u = eσ T/N and d = u where T = number of years to Expiration N = number of binomial periods e = Exponential ConsTant (2.718281828) Thus T / N = square root of the number of years per binomioal period 22
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Valuation Techniques All derivative valuation procedures make use of a short term risk-free return. The most common used input for the risk free rate is LIBOR. 23
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options Summary and Conclusions The price movements of a derivative are perfectly correlated over short time intervals with the price movements of the underlying asset on which it is based. Hence, a portfolio of the underlying asset and a riskless security can be formed that perfectly tracks the future cash flows of the derivative. To prevent arbitrage, the tracking portfolio and the derivative must have the same value. 24
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Put-Call Parity Profiles: Buy a call and sell a put 25
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Put-Call Parity 26
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Corporate Securities It is possible to view equity as a call option on the assets of the firm and to view risky corporate debt as riskless debt worth PV(K) plus a short position in a put option on the asset of the firm(-p0) with a strike price of K. 1. Equity The call option characteristic of equity arises because of the limited liability of corporate equity holders. K V0 E0 = Max(Vo – Debt(K), 0) 2. Debt The put option characteristic of debt arises because of the put-call formula. Debt are assets less a call option: K S0 D0 = S0 – c0 The put-call: c0-p0=S0 – PV(Debt), substitution D0 = PV(Debt(K)) – p0 27
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Corporate Securities Hence, any characteristics of the assets of the firm that affect option values will alter the values of debt and equity (for example the variance of the asset return). The options result implies that the more debt a firm has, the less in the money is the implicit option in equity. Thus, knowing how option risk is affected by the degree to which options is in or out of money may shed light on how the mix of debt and equity affects the risk of the firm’s debt and equity securities. Finally, because stock is an option on the assets of the firm, a call option on the stock of a firm is really an option on an option, or a compound option (Geske,79). 28
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Valuation Assume that the one-period risk-free rate is constant, and that the ratio of price in the next period to price in this period is always u or d. The hypothetical probabilities that would exist in a risk-neutral world must make the expected return on the stock equal the risk free rate. The risk neutral probabilities satisfy: pu +(1-p)d=1+rf , giving the relationship 1 + rf − d π= and u−d u − 1 − rf 1−π = u−d 29
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Valuation The proper no-arbitrage call value c0, as a function of the stock price S0 becomes: π max [uS0 − K , 0] + (1 − π ) max [ dS0 − K , 0] c0 = 1 + rf and the GENERAL binomial formula : N 1 N! c0 = (1 + rf ) N ∑ j =0 j !( N − 1)! π j (1 − π ) N − j π max 0, u j d N − j S0 − K    Ex. Find the value of a three month at-the-money call option on DNB, trading at 32 kroner. Assume rf =0, u=2, d=0.5 and the number of years 3. 1 + 0 − 0,5 0,5 1 1 2 π= = = , and 1 − π = 1 − = 2 − 0,5 1,5 3 3 3 2 2 2 3 1 1  2  1  2   2   (256 − 32) + 3     (64 − 32) + 3     (0) +   (0) = 15, 41  3  3  3  3 3   3 Since the discount rate is 0, 15,41 is the value of the call option. 30
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Binomial Valuation 256 224 128 64 64 32 32 32 16 16 8 0 4 0 31
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Black&Scholes Valuation Black-Scholes Formula If a stock that pays no dividends before expiration of an option has return that is lognormally distributed, can be continuously traded in frictionless market, and has a constant variance, then, for a constant risk-free rate, the value of a European call option on that stock with a strike price of K and T years to expiration is given by c0 = S0 N (d1 ) − Ke − r ⋅T N (d1 − σ T ) where ln( S0 / PV ( K )) σ T d1 = + σ T 2 The Greek letter σ is the annualized standard deviation of the natural logarithm of the stock return, ln() represents the natural logarithm, and N(z) is the probability that a normally distributed variable with a mean of zero and variance of 1 is less than z. 32
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Black&Scholes Valuation Ex. Black-Scholes Formula The non-dividend paying stock Ekornes has a current price of 150 kroner and a volatility of 20 percent per year. What is the price of a 3 month European option with strike price of 150 when the risk-free rate is 5%: 3 −0,05⋅ ln(30 / 28e 12 ) 0, 2 3 /12 0, 0815 0, 2 ⋅ 0,5 d1 = + = + = 0,865 0, 2 3 /12 2 0, 2 ⋅ 0,5 2 c0 = 30 ⋅ N (0, 74) − 27.652 ⋅ N (0, 74 − 0, 2 ⋅ 0,5) c0 = 30 ⋅ N (0, 74) − 27.652 ⋅ N (0, 74 − 0, 2 ⋅ 0,5) c0 = 23.1105 − 20, 4326 = 2, 678 33
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Estimating Volatility Using Historical Data Note that B&S is based on the volatility of instantaneous volatility: Procedure for calculation of historical volatility: 1. Obtain historical returns for the stock the option is written on. 2. Covert returns to gross returns (1+return in decimal form). 3. Take the natural logarithm of the decimal version of the gross return. 4. Compute the unbiased sample variance of the logged return series and annualize it by multiplying it by the square root of ratio of 365 to the number of days in the return interval. 34
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Estimating Volatility The implied volatility approach Look at other options on the same security. If market values of the options exist, there is a unique implied volatility that maked the B&S model consistent with the market price of a particular option. The EXCEL-book: NewtRaphImp-vol.xls shows a particular form for implementation. Method 2: New ton Raphson technique. S 100 Price K 125 Strike Markør 0 r 12 % Interest rate Initial 1.24304 tau 0.25 Maturity sigma 0.471234 Volatility Derivative 25.5193 d(1) -0.70193 N'(d1) 0.510386 d(2) -0.93754 C (call price) 3 Mål kjøp 3 35
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: B&S Greeks The Greeks of B&S Formula Delta: The sensitivity to Stock price Changes (Gamma measures Delta changes) Vega: The sensitivity to Volatility Changes Theta: The sensitivity to Expiration Changes Rho: The sensitivity to Risk-free Interest rate Changes 36
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options: Complex Assets The Forward Price Version of the Black-Scholes Model c0 = e − r ⋅T  F0 N (d1 ) − KN (d1 − σ T )    where ln( F0 / K ) σ T d1 = + σ T 2 US $.40 ⋅1, 06 Forward Prices i) = US $0,3926 1, 08 ii ) $102 ⋅1, 06 − $4 ⋅ 1, 06 − $4 = $100, 0017 iii ) ($800 − $20) ⋅1, 06 = $826.80 iv) ($18 + $1 − $1) ⋅ (1, 06 = $19, 08 Call Option Prices 0,3926 ⋅ N (d1 ) − $0,5 ⋅ N (d1 − 0, 25) ln(0,3926 / 0,5 i ) c0 = where d1 = + 0,125 = −0,84; c0 = $0, 01 1, 06 0, 25 $100, 0017 ⋅ N (d1 ) − $100 ⋅ N (d1 − 0, 25) ln(100, 0017 /100 ii ) c0 = where d1 = + 0,125 = 0,13; c0 = $9,39 1, 06 0, 25 $826,80 ⋅ N (d1 ) − $850 ⋅ N (d1 − 0, 25) ln(826,80 / 850 iii ) c0 = where d1 = + 0,125 = 0,13; c0 = $68, 22 1, 06 0, 25 $19, 08 ⋅ N (d1 ) − $20 ⋅ N (d1 − 0, 25) ln($19, 08 / 20 iv) c0 = where d1 = + 0,125 = −0, 06; c0 = $1, 43 1, 06 0, 25 37
  • Styring og Fondsmegling Dr.oecon Per B Solibakke Pricing Derivatives and Options Summary and Conclusions Despite a few biases in the Black & Scholes option pricing formula, it appears that the formula work reasonably well when properly implemented. The Excel-book: Black&ScholesImpl.xls shows various implementations. Black-Scholes Option-Pricing Formula S 25 Current stock price X 25 Exercise price r 6.00 % Risk-free rate of interest T 0.5 Time to maturity of option (in years) Sigma 30 % Stock volatility d1 0.2475 <-- (LN(S/X)+(r+0.5*sigma^2)*T)/(sigma*SQRT(T)) d2 0.0354 <-- d1-sigma*SQRT(T) N(d1) 0.5977 <-- Uses formula NormSDist(d1) N(d2) 0.5141 <-- Uses formula NormSDist(d2) Call price 2.47 <-- S*N(d1)-X*exp(-r*T)*N(d2) Put price 1.73 <-- call price - S + X*Exp(-r*T): by Put-Call parity 1.73 <-- X*exp(-r*T)*N(-d2) - S*N(-d1): direct formula 38