Introduction to Financial Derivatives

Transcription

1 Introuction to Financial Derivatives Week of December n, 3 he Greeks an Wrap-Up Where we are Previously Moeling the Stochastic Process for Derivative Analysis (Chapter 3, OFOD) Black-Scholes-Merton Moel (Chapter 4, OFOD) Black-Scholes-Merton Moel for Options (continue) (Chapter 4, 6-7, OFOD).. Where we are his Week (Chapter 8, OFOD) Heging & he Greeks Course Wrap-up an Final Review Last Day of Classes: December 4 th Final Exam: Dec 7 th ; 9:am Noon, Merg Assignment For Week of December n Rea: Hull Chapter 8 (he Greeks) Problems (Due December 3 r ) Chapter 6 (5, 7e): 4, ; 3 Chapter 7 (6, 7e): 7, 8, 5;.3.4

2 Plan Look-back briefly at Black-Scholes-Merton Differential Equation an Risk-Neutral Valuation Black-Scholes Formula Non-Divien paying Stock For Divien paying stock Applications of Black-Scholes-Merton heory Derivatives of a Stock Paying a Divien Yiel (& Inex Options) Derivatives on Currency exchange (FX) Derivatives on Futures (an Commoities) he Greeks.5 Previously Black-Scholes-Merton Differential Equation Derive from the Stock Price process an Ito s lemma, knowing the erivative is a function of the Stock Price process Form riskless portfolio: short one erivative & long stock Gives the ifferential equation f f f rs S rƒ t S S With bounary conitions, it s solution escribes the erivative Risk-Neutral Valuation applies as the BSM Diff Eq equation is inepenent of any variable affecte by risk preference only S, t, σ, an r no expecte return, µ Principle of Risk-Neutral Valuation Assume the expecte return of the unerlying asset is r, i.e. µ = r Calculate the expecte payoff from the erivative Discount the expecte payoff at the risk-free rate, r.6 Valuing a Forwar Contract with Risk-Neutral Valuation Consier a long forwar contract on a non-ivien paying stock, S, that matures at with elivery price K he payoff at maturity is S K Denoting the value of the forwar contract at time zero by f, means that f e Eˆ S K, iscounting the expecte payoff in a risk neutral worl Expecte return on the unerlying asset is r so E ˆ( S ) r Se Present value of expecte payoff is ˆ f e ES Ke r e Se Ke S Ke.7 he Black-Scholes Formulas Black-Scholes Formulas for the Present Value of a European Call, c (Put, p ) with expiration an strike K on a non-ivien paying stock with price S cs N ( ) Ke N ( ) p K e N( ) S N( ) where an ln( S / K) ( r / ) ln( S / K) ( r / ).8

3 Implie Volatility Diviens he implie volatility of an option is the volatility for which the Black-Scholes price equals the market price here is a one-to-one corresponence between prices an implie volatilities European options on ivien-paying stocks are value by substituting the stock price less the present value of iviens into Black-Scholes Only iviens with ex-ivien ates uring life of option shoul be inclue he ivien shoul be the reuction in the stock price expecte often only 8% of eclare amount because of tax reasons raers an brokers often quote implie volatilities rather than ollar prices.9. Black s Approximation for Dealing with Diviens in American Call Options Set the American price equal to the maximum of two European prices:. he st European price is for an option maturing at the same time as the American option. he n European price is for an option maturing just before the final ex-ivien ate. Option Results for a Stock paying a known Divien Yiel So, it is the same probability istribution for S at time when he stock starts at price S an provies a ivien yiel q, or he stock starts at price S e -q an pays no ivien We may raw the following conclusion as a rule: When valuing a European option of term on a stock paying a known ivien yiel q, reuce the current stock price from S to S e -q an then value the option as though the stock ha no ivien, but were starting with the price S e -q. 3

4 Option Results for a Stock paying a known Divien Yiel Extening Black-Scholes formulas to a stock paying a ivien yiel q (replace S by S e -q ) q c Se N( ) Ke N( ) q p Ke N( ) S e N( ) where since ln( S / K) ( r q / ) ln( S / K) ( r q / ) q Se q ln ln Se ln K ln S K ln K ln e S ln q K q.3 Derivation of the Black-Scholes Differential Equation w/divien Yiel As equation for W oes not involve z, the portfolio Π is riskless uring time t, an must return the risk-free rate W r t Substituting for W an Π gives f f f f ( S qs ) t r( f S) t t S S S Which results in the Black-Scholes-Merton ifferential equation for a stock with ivien yiel as f ( r q) S f S f rf t S S.4 Risk-Neutral Valuation reux for Stock paying a Divien Yiel he concept of risk-neutral valuation is the single most important result for erivative analysis he Black-Scholes equation is again foun inepenent of all variables affecte by risk preference Only variables are S, t, σ, an r no expecte return, µ he solution to the ifferential equation is therefore the same in a risk-free worl as it is in the real worl his leas to the principle of Risk-Neutral Valuation Assume the expecte total return of the unerlying asset is r he iviens provie a return of q, the stock price growth rate is r-q When the growth rate is r-q, the expecte stock price at is S e (r-q) ( ) hat is: ES ˆ( ) r q Se Calculate the expecte payoff from the erivative Discount the expecte payoff at the risk-free rate, r.5 he Binomial Moel Applicable to a stock paying a ivien yiel, as covere earlier in Chapter o match stock price volatility, set t t u e ; e Risk-neutral probability of an up move is chosen so the expecte return is r-q over a time step of t an So a p ; a e u With the erivative value psu ( p) S Se ( rq) t rt f e [ pf ( p) f ] u S ƒ ( r q) t S u ƒ u S ƒ.6 4

5 European-Style Currency Options European-Style Currency Options Foreign currency is an asset that provies a ivien yiel of r f Use the formula for an option on a stock paying a ivien yiel with S, the current exchange rate, an q = r ƒ Black-Scholes gives where f ( ) ( ) f cs e N Ke N p Ke N( ) S e N( ) ln( / ) ( f / ) S K r r S K r r ln( / ) ( f / ).7 Alternatively, we can simplify the Black-Scholes formulas by using the forwar exchange rate, F, for maturity : ( r rf ) F S e Now Black-Scholes gives where ce [ F N( ) KN( )] pe [ KN( ) F N( )] ln( F / K) / ln( F / K) /.8 Futures Option Valuation from the Binomial Approach Form the portfolio: short one erivative an long futures he value at the en of one time perio is F u ( Fu F) fu ( F F) f ƒ u when ( fu f)/( Fu F ) F he value of the PF toay is ƒ F r r [( Fu F) fu] e [( F F) f] e f () f ƒ as the long future has no value at inception Substituting an simplifying gives f e [ pfu ( p) f] where p = ( )/(u ) as asserte back in Chapter 9.9 Futures Prices Drift in a Risk- Neutral Worl Define F t as the futures price at time t If we enter into a futures contract toay its value is zero After a short increment, t, it provies a payoff F t F If r is the t risk-free rate at time, risk-neutral valuation rt gives e E ˆ[ F ] t F, as the contract has no value where Ê enotes expectations in the risk-neutral worl hus we have Eˆ( F t ) F similarly for EF ˆ( ˆ t) F t, EF ( 3t) F t,... so concatenating these results Eˆ( F ) F for any he rift of the futures price in a risk-neutral worl is zero From the stock price equation with ivien yiel q equal to r S ( r q) St Sz F Fz q r. 5

6 Valuing European Futures Options Summary of Key Results Black-Scholes formula for an option on a stock paying a ivien yiel q Set S = current futures price, F Set q = risk-free rate, r, ensuring the rift of F is Results in ce FN( ) KN( ) p e KN( ) F N( ) where ln( F / K) / ln( F / K) /. We can treat stock inices, currencies, an futures like a stock paying a ivien yiel of q For stock inices, q = average ivien yiel on the inex over the option life For currencies, q = r ƒ For futures, q = r. Heging an Option Position Example A Dealer must price an manage exposure to instruments it unerwrites An option market maker has choices in how to o that epening on the specific option Coul use an exchange-trae, close substitute he traing area coul hege where the most efficient approach uses risk neutral principles hat is the iea that a long position in the unerlying can be use to create a riskless portfolio A bank has sol for $3, a European call option on, shares of a nonivien paying stock S = 49, K = 5, r = 5%, = %, = weeks, = 3% he Black-Scholes value of the option is $4, How oes the bank hege its risk to lock in the $6, profit?.4.5 6

7 Delta (Unerlying Price) & Delta Heging Delta () is the rate of change of the option price with respect to the unerlying A elta hege involves taking a position in size of - of the unerlying netting a neutral position B Option price A Delta (Unerlying Price) & Delta Heging From the Black-Scholes formula we can fin a close form expression for the elta of a Euro-style Call csn ( ) Ke N ( ) S where ln r K c By efinition N ( ) S N ( ) Ke N ( ) S S S x where N ( ) N( xx ) an N( x) e Since N( ) N( ) N( ) S S S Slope = an N( ) N( ) N( ) Ke / S hen Stock price N ( ) KN( ) e / SKe N( )/ S N ( ).6.7 Black-Scholes Example (Again) A bank has sol for $3, a European call option on, shares of a nonivien paying stock S = 49, K = 5, r = 5%, = %, = weeks, = 3% he Black-Scholes value of the option is $4, How oes the bank hege its risk to lock in a $6, profit?.8 Delta (Unerlying Price) & Delta Heging A Simulation (8.) Week Stock Price Delta Shares Purchase Cost ($) Cum Cost w/ Interest ($) ,, , (6,4) 45, ,, 57.5., (38.),5.3,54.5 Interest Cost ($) , ,

8 Delta (Unerlying Price) & Delta Heging A Simulation (8.) Delta (Unerlying Price) & Delta Heging A Simulation (8.3) able 8. able 8.3 PV $63,3 = $58,85.3 PV $56,6 = $5,7.3 Using Futures for Delta Heging he futures price for a contract on a non-ivien paying stock is F =S e r So as the price of the stock changes by S the futures price changes by Se r he elta of a futures contract is e r ue to aily mark-to-market Contrast to the elta of a forwar contract which is For a futures on an asset paying a ivien yiel we can similarly see that the elta is e (r-q) he elta of a futures contract is e (r-q) times the elta of a spot contract of the asset, H F = e (r-q) H A he position require in futures for elta heging is therefore e -(r-q) times the position require in the corresponing spot contract.33 heta (ime) heta () of a erivative (or portfolio of erivatives) is the rate of change of the value with respect to the passage of time he theta of a call or put is usually negative. his means that, if time passes with the price of the unerlying asset an its volatility remaining the same, the value of the option eclines We can evelop a close form for theta of a Euro Call as we i for elta; from the Black-Scholes equation Use a slightly more general form to explicitly affirm toay where csn ( ) Ke N ( ) ( t) S ln r ( t) t t K.34 8

9 heta (ime) heta is the change in value ue to a change in the passage of time as before c so since Finally c SN( ) rke N( ) Ke N( ) t t t SN Ke N Ke t N rke t N Ke N t ( t) rke N( ) SN( ) t t t; t t t t t ( t) ( t) ( t) ( ) ( ) ( t) ( ( t) ( t) ) ( ) ( ) c ( t) rke N( ) SN( ) t t heta (ime) For a European Call Gamma (Unerlying Price) Gamma () is the rate of change of elta () with respect to a change in the price of the unerlying asset Gamma is greatest for options that are close to the money Again we can evelop a close form for a Euro-Call c N( ) N( ) N( ) S S S S S S ln r ( t) t K Form for gamma is ominate by the form of the normal ensity function Maximize aroun the strike price for a given t Gamma Aresses Delta Heging Errors Cause By Curvature C'' C' C Call price S S' Stock price

10 Interpretation of Gamma For a elta neutral portfolio, t + ½S Gamma (Unerlying Price) S S Positive Gamma Negative Gamma.39.4 Vega (Volatility) Vega (Volatility) Vega () is the rate of change of the value of a erivatives portfolio with respect to volatility Vega tens to be greatest for options that are close to the money Again, a close form for a Euro-Call csn ( ) Ke N ( ) S ln r K c S N( ) Ke N( ) S N( ) Ke N( ) S N( ) S N( ) SN( )[ ] SN( ) is ominate by the normal istribution aroun the strike.4.4

11 Relationship Between Delta, Gamma, an heta he price of a single erivative on a non-ivien paying stock must satisfy the Black-Scholes ifferential equation A portfolio Π of such erivatives must also satisfy that ifferential equation rs S r t S S When we use the risk notation we have ; ; t S S rs S r For a elta hege portfolio (Δ=) so S r When theta is large an positive, gamma is large an negative (& vice versa).43 Managing Delta, Gamma, & Vega can be change by taking a position in the unerlying o ajust & it is necessary to take a position in an option or other erivative.44 Gamma Suppose a elta-neutral PF has gamma an a trae option has gamma the total gamma of the PF with option is Where / is the number of options to make the PF gamma-neutral Since this is likely to change the elta of the PF with the option, the unerlying asset position (now with options) will nee rebalancing Gamma - Example Suppose a elta-neutral PF has gamma -3, An the elta-gamma of a trae call option is.6 an.5, respectively he PF can be mae gamma-neutral with a (long) position of / 3 /.5, options he elta of the PF is now, x.6 =,4 herefore,,4 of the unerlying asset must be (aitionally) sol to keep the total PF elta-neutral as well as gamma-neutral

12 Vega Vega () is the rate of change of the value of a erivatives portfolio with respect to volatility P Vega tens to be greatest for options that are close to the money 3.47 Vega As with gamma, a elta-neutral PF can be mae (both gamma an) Vega neutral A elta-neutral PF has gamma-vega of -5, an -8,, respectively Suppose there are trae options as shown DELA GAMMA VEGA Portfolio. -5, -8, Option.6.5. Option.5.8. If, are the quantities of Options &, then for a gamma-vega neutral PF we must have 3.48 Vega Simultaneously For gamma-neutral: 5,.5.8 an For Vega-neutral: 8,.. So 4 & 6, But now the elta of the PF after aing these options is 4.6 6,.5 3, 4 So to restore elta neutrality to the combine PF with Options &, we nee to sell 3,4 units of the asset Gamma an Vega Limits In practice a traer responsible for all traing involving a particular asset must keep Gamma an Vega within limits set by risk management As well as observe Delta constraints

13 Rho (Interest Rate) Rho (Interest Rate) Rho is the rate of change of the value of a erivative with respect to the interest rate For currency options there are rhos For the Euro-Call, a close form results as csn ( ) Ke N ( ) S ln r K c S N ( ) Ke N ( ) Ke N ( ) r r r S N( ) Ke N( ) Ke N( ) r r S N( ) S N( ) Ke N( ) Ke N( ) r r.5 Consier a call on a non-ivien-paying stock with price 49, strike 5, risk-free rate 5%, = weeks (.3846), an volatility % Rho is Ke N( ) 8.9 So a % increase in risk-free rate from 5% to 6% increases the value of the option by approximately.89 From.4 to.5.5 Heging in Practice raers usually ensure that their portfolios are eltaneutral at least once a ay Whenever the opportunity arises, they improve gamma an vega As portfolio becomes larger heging becomes less expensive.53 3