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International Financial Markets and the FirmLognormal Option Pricing Model page 8-1 Ch. 8. Pricing European Options: The Lognormal Model 1. Assumptions of the Continuous-time Option Pricing Model 2. A Discrete-Time Derivation of the Continuous-Time Model 2.1. Step 1: Computing the Expected Value of a Call Option 2.1.1. A Discrete-Scale Example 2.1.1.The Expected Call Value when the Spot Rate is Lognormal 2.2. Step 2: Correcting the Call’s Expected Expiration Value for Risk 2.3. Step 3: Discounting the Risk-adjusted Expiration value of the Call at the Risk-free Rate. 2.4. Standard Notational Convention for the Continuous- time Call Pricing Model. 3. How to Use the Continuous-Time Option Valuation Formula 3.1. A Numerical Example 3.2. How to Use the Formula for Delta-hedging 4. Related Option Pricing Models 4.1. The Value of European Put Option 4.2. The Value of European Options on a Futures Contract 4.3. The Value of European Currency Options with Stochastic Interest Rates 5. Conclusions Appendix A: Derivation of the Expected Expiration Value of the Call Option Appendix B: Stochastic Calculus and the Black-Scholes Differential Equation P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-2 • Binomial model: time is discrete, S is from a discrete scale. ∆Cn+1 ∆Ct+∆t hedge ratio = ∆S = ∆S n+1 t+∆t • Black-Scholes-Merton: time is continuous, S is lognormal (i.e. from a continuous scale). ∂Ct+dt ∂Ct hedge ratio = ∂S = ∂S t+dt t • Samuelson-Rubinstein-Brennan: time is discrete, S is lognormal. Links • The BSM and SRB models yield the same formula for European options. The bimonial model converges to this formula. • The binomial and BSM approach can be used for more complicated options, like American options • binomial: stepwise, using Cn,j = q Cn+1,j+1 + (1-q) Cn+1,j 1+r • BSM: numerical solution of a partial differential equation P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-3 1. Assumptions of the Continuous-time Option Pricing Model 1. The process for the exchange rate is continuous. 2. The value of the option is a continuous and twice differentiable function of the underlying process S. [Thus: (1) over a short time interval the changes in the exchange rate will be small, and (2) the effect of a small change in the spot rate on the call price is always well-defined. Thus, hedging works.] contract value value forward contract exposure line = tangency line in S C S S–dS S S+dS 3. Trading is continuous. [You can adjuste the hedge all the time the option price is always correct.] P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-4 4. The distribution of the percentage changes in the exchange is lognormal—or the continuously compounded change in the spot rate is normally distributed. 5. The risk-free rate(s), and the variance of the ("continuously compounded") percentage changes in the spot rate are constant over the option's life. [4 and 5 correspond to the assumption in the binomial model that the process is multiplicative and that u, d, and (1+r), (1+r*) are constant over time. ] P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-5 2. A Discrete-Time Derivation of the Continuous-Time Model Samuelson [1967], Rubinstein [1976], Brennan [1979]: ˜ 1. Compute the expected value of the option at maturity, E t(C T). 2. Correct this expected value for risk. That is, compute ˜ ˜ ˜ ˜ CEQt(C T) from Et(C T), by replacing Et(S T) with CEQt(S T) = Ft,T. 3. Discount the risk-adjusted expected future value at the risk- free rate to determine the call’s value at time zero, Ct. That is, CEQt(CT)˜ (1) Ct = 1+r t,T P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-6 2.1. Step 1: Computing the Expected Value of a Call Option 2.1.1. A Discrete-Scale Example Consider a call with X = 43. ˜ S T may be 38 39 40 41 42 43 44 45 46 47 with probt,T 0 .05 .10 .15 .20 .20 .15 .10 .05 0 ˜ then C T = 0 0 0 0 0 0 1 2 3 4 with probt,T 0 .05 .10 .15 .20 .20 .15 .10 .05 0 ˜ Et(C T) = 0 + 0 + ... + (1 0.15) + (2 0.1) + (3 0.05) + (4 0) = 0.5 = (43 – 43) 0.20 + (44 – 43) 0.15 + (45 – 43) 0.10 + (46 – 43) 0.05 = [(43 0.20) + (44 0.15) + (45 0.1) + (46 0.05)] – 43 [.20 + 0.15 + 0.1 + 0.05] P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-7 (5) ˜ Et(C T) = ST Probt,T(ST) –X ST=X Probt,T(ST) ST=X = [Sum A] – X [Sum B] partial mean X prob of ending in the money P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-8 2.1.1. The Expected Call Value when the Spot Rate is Lognormal ˜ ˜ Denote Et(lnS T) by µt,T, and sdt(lnS T) by t,T. Then ∞ ˜ (7) Et(C T) = ST f(ST; µt,T,t,T)dST – X X ∞ f(S ; µ , )dS T t,T t,T T X = [Integral A] – X [Integral B] partial mean – X prob of ending in the money [après maintes péripéties:] (8) ˜ = Et(S T) N(d' ) 1 – X N(d' ) 2 where N(d'i ) denotes the cumulative standard normal probability: P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-9 n(z) N(d) z d 0 ˜ Et(ST) 1 2 ln X + 2 t,T (9) d' = , d' = d'1 – t,T . 1 t,T 2 P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-10 2.2. Step 2: Correcting the Call’s Expected Expiration Value for Risk In the BSM or binomial logic, and also in the SRB model, risk correction means replacing Et(ST) by Ft,T: (10) ˜ CEQt(C T) = Ft,T N(d1) – X N(d2) Ft,T 1 ln X + 2 t,T (11a) d1 = t,T (11b) d2 = d1 – t,T 2.3. Step 3: Discounting the Risk-adjusted Expiration value of the Call at the Risk-free Rate. CEQt(CT) ˜ (12) Ct = 1 + r t,T Ft,T X =+r 1 t,T N(d1) – 1 + r N(d2) t,T St X (13) = 1 + r * N(d1) – 1 + r N(d2) t,T t,T P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-11 2.4. Standard Notational Convention for the Continuous- time Call Pricing Model. The convention in the literature and among practitioners is to quote all data on an annualized basis. • The p.a. variance is typically denoted by the (non-subscripted) symbol 2. Thus, 2 2 t,T = (T-t), • The riskfree rate is typically a continuously compounded, p.a. interest rate, denoted by the (non-subscripted) symbol r (HC) and r* or r' (FC). Thus, (1 + rt,T) = er (T-t) and (1 + rt,T ) = er* (T-t) * Example life is 201 days T–t = 201/365 = 0.55 years volatility 14.14% p.a. variance = 0.14142 = 0.02 p.a. 2 t,T = .55 .02 = .011 r (p.a., cc): 9.7347% 1 + rt,T = e0.097347 0.55 = 1.055 r* (p.a., cc): 5.9031% 1 + rt,T = e0.059031 0.55 = 1.033 * P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-12 Thus: 1+rt,T Ft,T = St * 1+r = St e(r-r*) (T-t) t,T Ft,T St ln X = ln X + (r–r*) (T–t). (14) Ct = St e–r* (T–t) N(d1) – X e–r (T–t) N(d2) 1 ln(St/X) + (r–r*)(T–t) + 2 2(T–t) d1 = (T–t) d2 = d1 – (T–t) P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-13 3. How to Use the Continuous-Time Option Valuation Formula 3.1. A Numerical Example data: • S = USD/DEM 0.45, or 45 cents, and X = USD/DEM 0.43, or 43 cents. We use exchange rates expressed in cents, and keep in mind that the Black- Scholes-Merton formula then yields an option premium that is likewise expressed in cents. • The option expires in 210 days. Thus, (T-t) = 201/365 = 0.55 years • The volatility is 14.14%. This is a per annum figure, so the effective variance equals 2(T–t) = .55 .02 = .011, and the effective standard deviation is r(T–t) = 0.1414 0.55 = 0.10488. • The riskfree simple interest rates are 10% on USD, and 6% on DEM. Thus, 1 + rt,T = 1.055, and 1 + rt,T = 1.033. * P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-14 Thus: 1+rt,T 1.055 Ft,T = St * 1+r t ,T = 45 1.033 = 45.958 (cents) ln(F/X) = ln(45.958/43) = ln(1.0688) = .066536 1 0.066536 + 2 0.011 d1 = 0.10488 = 0.686824 d2 = d1 – 0.10488 = 0.583098. N(d1) = 0.753935 , N(d2) = 0.720086 St 45 1 + r*t ,T = 1.033 = 43.5624 (cents) X 43 X e –r (T–t) = 1 + r = 1.055 = 40.7583 (cents) t,T Ct = (43.5624 0.753935) – (40.7583 0.720086) = 3.48432 (cents) P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-15 3.2. How to Use the Formula for Delta-hedging When valuing the call, we have in fact computed the value of a portfolio containing a certain amount of DEM and USD T-bills: • The second term in the solution of Ct corresponds to a purchase of –N(d2) = –0.7196 USD T-bills, each having a face value of 43 cents. So the amount invested in domestic T-bills is N(d1) .7196 –43 1+r * = –43 1.055 = –29.33 UScents , t,T That is, one takes out a riskfree loan of 29.33. • The first term corresponds to buying N(d1) = 0.7538 DEM T- bills. Since each DEM T-bill costs DEM 1/1.033, the amount of DEM required in order to buy these T-bills is N(d1) .7538 1+rt,T * = 1.033 = DEM 0.73 N(d1) ∆spot = 1+r * = number of foreign currency units t,T to be bought spot and invested until T N(d1) ∆frwd = 1+r = number of currency units to be bought t,T forward (for delivery at T) P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-16 4. Related Option Pricing Models In this section we describe how the currency call pricing model given in (13) or (14) is related to other option valuation models in the finance literature. 4.1. The Value of European Put Option We can find the value of a European put by starting from the Put-Call-Parity Theorem and the substituting the valuation formula for the call: X St (16) Pt = 1+r – 1+rt,T * + Ct t,T X St St = 1+r + 1+rt,T * + 1+rt,T (N(d1)) * – t,T X 1+rt,T (N(d2)) X St = 1+r (1–N(d2)) – 1+rt,T (1–N(d1)) * t,T X St = 1+r N(–d2) – 1+r * N(–d1) t,T t,T X St = 1+r N(e1) – 1+r * N(e2) t,T t,T P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-17 = X e–r(T–t) N(e1) – S e–r*(T–t) N(e2) P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-18 4.2. The Value of European Options on a Futures Contract When interest rates are known, forward and futures prices are identical; and even when interest rates are uncertain we can ignore the difference between both prices as trivial. Thus, we set ft,T = Ft,T. • If the futures contract and the option expire at the same time (T), then the option on the forward contract has the same payoff as the option on the cash. Thus, it must have the same present value, too. We use (11) and (12), which is our option price expressed in terms of Ft,T rather than St: (17) Value of a European call on a futures contract expiring at T: ft,T X = 1+r N(d1) – 1+r N(d2) t,T t,T 1 ln(ft,T/X) + 2 t,T2 where d1 = and d2 = d1 – t,T t,T • If the futures contract expires at T 2>T, use ft,T2 instead of ft,T, but do not change T in the variance: use t,T2, or 2(T-t)). P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-19 4.3. The Value of European Currency Options with Stochastic Interest Rates Merton uses B(t,T) to denote 1/(1+r t,T) and B*(t,T) to denote 1/(1+r* ,T), and then shows that the model still holds when bond t prices are not fully predictable—as long as the bond's day-by-day variance is known in advance. [Popular assumption: the day-by-day variance is proportional to the bond's duration.] Then (Merton (1973); Grabbe (1983)): (19) C = S B*(t,T) N(d1) – X B(t,T) N(d2) 1 1 ln F/X ± 2 T,t ln F/X ± 2 2 (T–t) di = t,T = T–t 2 = annualized average variance of dlnF • we should use the average variance of the forward rate (for delivery at T) rather than the variance of the spot rate. • This variance typically higher because domestic and foreign interest rates are uncertain and imperfectly correlated. • Grabbe’s model therefore prices options higher than the Garman-Kohlhagen model. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-20 5. Warnings 1. The model assumes continuous rebalancing. In practice rebalancing occurs far less frequently. Thus, a linear hedge will not mimic the call price as perfectly as one would like. Hint: use Gamma-hedging: ∂C ∂C 1 ∂2C ˜ ∆C – ∂t ∆t + ∂S ∆S + 2 2 (∆S) 2 ∂S ∂C Form a portfolio of FC and FC-options that has the same delta ∂t and the ∂2C same gamma as the option to be hedged/replicated. ∂S2 2. Sudden jumps in the exchange rate, and changes in its volatility, are not taken into account by the model. Thus, delta-hedging will not protect us against jumps in the exchange rate, or changes in volatility. 3. The assumption that the variance of the log exchange rate, 2 t,T , is proportional to the horizon T–t may be inappropriate in the long run. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-21 4. Interest rates are not constant. This is important for long- term options. Hint: use the Merton-Grabbe model with the higher . 5. Exchange rate changes have distributions that are fat-tailed; that is, the probability of extreme events is somewhat higher than the lognormal model predicts. As a result, options should probably be priced higher than what the lognormal model predicts, because options thrive on uncertainty. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-22 Appendix A: Integrals A & B A.1. The Probability of Exercising (Integral B) ˜ The probability that S T ≥ X can be read off immediately from the standard normal tables: ˜ lnST – µt,T lnX – µt,T ˜ (A.2)(S T ≥ X) ( t,T ≥ t,T ) lnX – µt,T ˜ Prob(S T ≥ X) = prob(z ≥ ˜ t,T ) µt,T – lnX (A3) = prob(z ≤ ˜ ) t,T µt,T – lnX (A.4) N(d'2) , d'2 = t,T In short, we have evaluated Integral B in (7) as follows: ∞ (A.5) X ˜ f(ST; µt,T, t,T) dST = X Probt (S T ≥ X) X = X N(d'2) P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-23 ˜ A.2. The Partial Mean of S T ˜ Lemma: Assume S is lognormal with mean and stdev . Then 1 S f(S; µ, ) = exp{µ + 2 2} f(S; µ+2, ) . ˜ Proof: The density for a lognormal variate S , we have 1 lnS – µ 2 f(S; µ, ) = k exp{– 2 [ ]} 1 where k = . Thus, the integrand is 2 1 lnS – µ 2 (A.6) S f(S; µ, ) = exp{lnS} k exp{– 2 [ ]} 1 lnS – µ 2 = k exp{lnS – 2 [ ]} P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-24 Rearrange the argument of the exponential and "complete the square": 1 (lnS – µ)2 lnS – 2 2 1 –2 lnS 2 + [(lnS)2 – 2µlnS + µ2] = – 2 2 1 –2 lnS (µ+2) + (lnS)2 + µ2 (A.7) = – 2 2 1 –2 lnS (µ+2) + (lnS)2 + µ2 + [2µ2 + 4] – [2µ2 + 4] – 2 2 1 (lnS)2 – 2 lnS (µ+2) + (µ2 + 2µ2 + 4) – 2µ2 – 4 = – 2 2 1 [lnS – (µ + 2)]2 – 2 µ 2 – 4 = – 2 2 1 lnS – (µ + 2) 2 1 (A.8) = – 2 [ ] + µ + 2 2 P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-25 Thus 1 1 lnS – (µ + 2) 2 S f(S; µ, ) = exp{µ + 2 } k exp{– 2 2 [ ]} 1 (A.9) = exp{µ + 2 2} f(S; µ+2, ) where f(S; µ+2, ) is the density of a lognormally distributed variable with a shifted mean: has been replaced by µ+2. ˜ Corollary 1: The expected value of a lognormal variate S , is 1 ˜ ˜ ˜ E(S ) = exp{µ + 2 2} where = E(lnS ) and 2 = var(lnS ). Proof: ∞ (A.10) ˜ E(S ) S f(S; µ, ) dS 0 ∞ 1 = exp{µ + 2 2} f(S; µ+2, ) dS 0 1 = exp{µ + 2 2} QED P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-26 ˜ Corollary 2: The partial mean of S from S=X to S= is given (µ+2) – lnX ˜ by Et(S T) N(d'1), where d'1 = . Proof. We apply the Lemma, and then Corollary 1. S f(S; µ, ) dS (A.11) X 1 = exp{µ + 2 2} f(S; µ+2, ) dS ) X ˜ = E(S ) f(S; µ+2, ) dS X Thus: (A.12) ˜ Part A = Et(S T) N(d' ) 1 where (µ+2) – lnX (A.13) d' = ; and d' = d' – . QED. 1 2 1 P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-27 A.3. The Link Between the Probabilities and the Expected Spot Rate Combining our results of Sections A.1 and A.2, we obtain (A.14) ˜ ˜ Et(C T) = Et(S T) N(d'1) – X N(d'2) Using the first Corollary of Section A.2, we can write the d’1 and ˜ d’2-factors as explicit functions of Et(S T): µ + 2 – lnX (A.15) d'1 = 1 1 µ + 2 2 + 2 2 – lnX = 1 ˜ ln(E(S)) + 2 2 – lnX = ˜ E(S) 1 ln X + 2 2 = ˜ E(S) 1 ln X – 2 2 (A.16) d'2 = d'1 – = P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-28 Appendix B: Stochastic Calculus and the Black-Scholes-Merton Differential Equation B.1. Ito Processes Consider an additive, continuous normal random walk in continuous time. As the process is continuous, we can select any observation frequency we like. • Annual observations: x t+1 = xt + a + t,t+1 ˜ ˜ x = the level of the random walk variable a = the annual growth or "drift" = the standard deviation of the annual growth, and t,t+1 N(0,1). ˜ ˜ P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-29 • N-annual observations: x t+N = xt + a N + ( t,t+1 + t+1,t+2 +... + t+(N–1),t+N) ˜ ˜ ˜ ˜ = xt + a N + t,t+1 + t+1,t+2 +... + t+(N–1),t+N ˜ ˜ ˜ N N = xt + a N + N t,t+N ˜ t,t+1 + t+1,t+2 +... + t+(N–1),t+N ˜ ˜ ˜ where t,t+N = ˜ N ˜ N(0,1) • Quarterly observations: can we set N = 1/4 in the above? ?1 1 x t+1/4 – xt = 4 a+ 4 ˜ ˜ Yes, because this is consistent with the annual observations: a • a quarterly mean of 4 implies that the mean of the sum of a four quarterly changes will be 4 4 a = the annual expected change. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-30 2 • a quarterly variance of 4 implies that the variance of the 2 sum of four quarterly changes will be equal to 4 4 2, the annual variance. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-31 • Continuous observations: set N = dt (B1) dx = a dt + dt , a standard normal ˜ ˜ = a dt + dz where dz dt (the standard Wiener Process) . ˜ Example Think of dt as one day, that is, 1/365 year. For each of the next 365 days, you intend to make a random drawing , compute dz = 1/365 = 0.0523, and cumulate these figures. What can you say about the sum after 365 days? The sum of 365 such drawings is a standard normal: 365 var(∑i=1 dzt+i/365) = 365 var(dz) = 365 (1/365) = 1. • • A process with continuously changing drift and variance: Example: geometric random walk Assume that, at each instant, the mean and standard deviation of dS are proportional to the price level S; that is, a t = St and t = St, where and are constants. dS = [S dt + [S dz P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-32 dS S = dt + dz . • P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-33 B.2. Ito's Lemma ∂f ∂f Let f = f(x,y). Then df ≠ ∂x dx + ∂y dy if {dz, dy} are random. B.2.1. Ito’s Lemma Applied to a Function of One Ito Process Consider a univariate function f = f(x). 1 f(x1) – f(x0) = fx (x1 – x0) + 2 fxx (x1 – x0)2 1 + 6 fxxx (x1 – x0)3 + ... 1 ∆f = fx ∆x + 2 fxx (∆x)2 + 1 fxxx (∆x)3 + ... 6 1 df = fx dx + 2 fxx (dx)2 + 1 fxxx (dx)3 + ... 6 • Deterministic math: if dx is infinitesimally small, then (dx)2, (dx)3, etc. are negligible relative to dx itself. So df = fx dx. • If dx = a dt + dz, then (dx)2 = (a dt + dz) (a dt + dz) = a2 (dt)2 + 2 (a dt) ( dz) + ( dz)2 = a2 (dt)2 + 2 (a dt) ( dt) + ( dt)2 ˜ ˜ = a2 (dt)2 + 2 a (dt)3/2 + 2 2 dt ˜ ˜ P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-34 Time being deterministic, any power larger than unity of dt can be ignored. Thus, (dx)2 = 2 2 dt ˜ 1 1 (B.2) df = fx dx + 2 fxx (dx)2 = fx dx + 2 fxx 2 2 dt ˜ New term 2 2 dt? ˜ • emerges whenever there is randomness and non-linearity. • its mean equals 2 dt because E( 2) = 1, being a standard ˜ ˜ normal. • its variance equals var( 2) 4 (dt)2, but this is negligible. ˜ Thus: (dx)2 = 2 2 dt = E(2 2 dt) = 2 dt ˜ ˜ and 1 (B.3) df = fx dx + 2 fxx 2 dt Example of (B.3) in the geometric case dx Suppose x = dt + dz or dx = x ( dt + dz). 1 (B.4) df = fx dx + 2 fxx 2 dt 1 = fx x ( dt + dz) + 2 fxx x2 2 dt . • P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-35 Example: the geometric case implies lognormality of x dx Let f(x)=ln(x), and x = dt + dz. Then 1 1 fx = x and fxx = – x2 . 1 Thus dln(x) = fx dx + 2 fxx 2 dt 1 = fx x ( dt + dz)+ 2 fxx x2 2 dt 1 1 1 = x x ( dt + dz) – 2 x2 x2 2 dt 1 = ( – 2 2) dt + dz We conclude: 1 • The p.a. mean = – 2 2 of the continuously compounded return, d lnx, is systematically lower than the mean of the p.a. simple return dx/x, which is . • Over short intervals, the random component of the continuously compounded return is indistinguishable from the random component in the simple return; • Over any small interval dt, dln(x) is normal. So over any finite interval ∆t, ∆ln(x) will be normal, because it is the P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-36 "sum" of many normal instantaneous changes. Thus, x itself is lognormal. • P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-37 B.2.2. Ito’s Lemma Applied to a Function of Two Lognormal Processes Let f = f(x,y) where dx = ax dt + x dzx , dy = ay dt + y dzy . (B.5) df = fx dx + fy dy 1 +2 {fxx (dx)2 + 2 fxy dx dy + fyy (dy)2} + ... As before: (dx)2 = x2 dt and (dy)2 = y2 dt = instantaneous variances. Analogously, dx dy is the instantaneous covariance: dx dy = (ax dt + x dzx) (ay dt + y dzy) = (x dzx) (y dzy) = (x x dt ) (y y dt ) ˜ ˜ = x y x y dt ˜ ˜ • E( x y) , the correlation coefficient. Thus, E(dx dy) = x ˜ ˜ y dt, the instantaneous covariance between dx and dy. • Obviously, the variance of x y x y dt around its mean will ˜ ˜ again be of order (dt)2, implying that we can set x y x y dt ˜ ˜ equal to its mean, x y dt. (B.6) df = fx dx + fy dy P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-38 1 +2 {fxx x2 + 2 fxy x y + fyy y2} dt P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-39 Example: stock returns in domestic and foreign currency, with lognormals Consider the price process f = VS, where V is a foreign- currency stock price and S is an exchange rate, and where dV dS V = V dt + V dzV and S = S dt + S dzS d VS What is VS , the return on the stock measured in HC? Ito's Lemma: (B.7) df = fv V (v + dzv) + fs S (s + dzs) 1 + 2 {fvv V2 v2 + 2 fvs V S v s + fss S2 s2} dt . We have fV = S, fS = V, fSS = 0, fSV = 1, fVV = 0: 1 dVS = S dV + V dS + 2 {0 + 2 VS SV V S + 0} dt, d VS dV dS (B.8) VS = V + S + SV V S dt . dV Special case: return on the foreign riskfree asset ( V = r*dt): d VS dS VS = r* dt + S (cross-term is too small) . P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-40 Example: returns on a portfolio Consider the price process f = nxX + nyY, where X is the price of one asset, Y is the price of another asset, and nx and ny are the number of shares of each asset held. dX dY df = nx dX + ny dY = nx X X + nYY Y df nXX dX nYY dY f = f X + f Y (No 2nd-order terms because f is linear in X and Y.) B.2.3. Ito’s Lemma Applied to a Function of an Ito Process and Time Consider f=f(S,t), where t is calendar time. 1 (B.9) df = ft dt + fS dS + 2 fSS S2 dt Example: the Lognormal Case If dS = S ( dt + dz), then the change in f equals 1 (B.10) df = ft dt + fS S ( dt + dz) + 2 fSS S2 S2 dt P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-41 B.3. The Black-Scholes-Merton Differential Equation B.3.1. Derivation of the Differential Equation If C = C(S, t) where S is lognormal, then 1 (B.10) dC = Ct dt + CS dS + 2 CSS S2 S2 dt Hedge this by buying N units of forex worth (initially) NS. Change in the value of the hedge: dH = N (dS + S r*dt) Set N = –CS, and add this to the call. • Initial value of the portfolio: p = C + (–CS) S. • Change of p over an instant dt: (B.11) dp = dC – CS (dS + S r*dt) 1 = [Ctdt + CSdS + 2 CSS S2 S2 dt] – CS(dS + Sr*dt) 1 = [Ct – CS S r*+ 2 CSS S2 S2] dt P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-42 • No arbitrage: dp (B.12) p = r dt or dp = p r dt 1 [Ct – CS S r*+ 2 CSS S2 S2] dt = [C – CS S] r dt 1 (B.13) C r = Ct + CS S (r–r*) + 2 CSS S2 S2 This is the fundamental pde for any contingent claim C that pays out no income stream between t and T (i.e., there are only capital gains), and depends on S. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-43 B.3.2. Interpretation as a Risk-Adjusted version of the Expected Capital gain on the Option Ito's Lemma: 1 dC = Ct dt + CS dS + 2 CSS S2 S2 dt dS 1 = Ct dt + CS S S + 2 CSS S2 S2 dt Thus: expected gain on the option is 1 (B.14) E(dC) = Ct dt + CS S S dt + 2 CSS S2 S2 dt 1 (B.15) C C = Ct + CS S S + 2 CSS S2 S2 Black-Scholes: 1 (B.13) C r = Ct + CS S (r–r*) + 2 CSS S2 S2 • (r*– r) replaces the true drift of S, S in equation (B.15). • the domestic risk-free rate replaces the true expected return on the contingent claim, C in equation (B.15). Thus: if we replace the true drift of S, s, by r-r*—that is, if we replace Et(St+dt) by Ft,t+dt—, we can discount the future option value at r rather than at c. P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431 International Financial Markets and the FirmLognormal Option Pricing Model page 8-44 P. Sercu and R. Uppal Version January 1994 Printout Rabi' Awwal 12, 1431