Closed form spread option aluation Petter Bjerksund and Gunnar Stensland Department of Finance, NHH Helleeien 30, N-5045 Bergen, Norway e-mail: petter.bjerksund@nhh.no Phone: +47 55 959548, Fax: +47 55 959650 This ersion: October 31, 006 Abstract This paper considers the aluation of a spread call when asset prices arelognormal. TheimplicitstrategyoftheKirkformulaistoexerciseif the price of the long asset exceeds a gien power function of the price of the short asset. We derie a formula for the spread call alue, conditional on following this feasible but non-optimal exercise strategy. Numerical inestigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much higher than the Kirk formula. Moreoer, optimizing with respect to the strategy parameters (which corresponds to the Carmona-Durrleman procedure) yields only a marginal improement of accuracy (if any). Keywords: asset prices. Spread option, closed form, aluation formula, lognormal JEL code: G1, G13, D81, C63. 1
Closed form spread option aluation Abstract This paper considers the aluation of a spread when asset prices are lognormal. The implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a gien power function of the price of the short asset. We derie a formula for the spread call alue, conditional on following this feasible but non-optimal exercise strategy. Numerical inestigations indicate that the lower bound produced by our formula is extremely accurate. The precision is much higher than the Kirk formula. Moreoer, optimizing with respect to the strategy parameters (which corresponds to the Carmona-Durrleman procedure) yields only a marginal improement of accuracy (if any). Keywords: asset prices. Spread option, closed form, aluation formula, lognormal JEL code: G1, G13, D81, C63. 1 Introduction This paper considers the aluation of a spread call when the asset prices are lognormal. The starting point is the obseration that the implicit strategy of the Kirk formula is to exercise if the price of the long asset exceeds a gien power function of the price of the short asset. We derie a formula that ealuates the spread call, conditional on following this exercise strategy. The formula consists of three terms, one for each of the two assets, and one for the strike. A standard normal cumulatie probability enter into each term, and each argument is a function of the forward prices, time to exercise, olatilities, and correlation. The formula fits well in to the tradition of Black-Scholes, Black76, and Margrabe. Numerical inestigations indicate that our formula is extremely accurate. The precision is much higher than the Kirk formula. Furthermore, the accuracy of our formula is comparable with the precision of the lower bound procedure suggested by Carmona and Durrleman, which requires a two-dimensional optimization scheme. Assumptions Consider a frictionless market with no arbitrage opportunities and with a constant riskless interest rate r. Assume two assets where the prices at the future date T are
o S 1 (T ) = F 1 exp n 1 σ 1T + σ 1 T ε1 o S (T ) = F exp n 1 σ T + σ T ε (1) () with respect to the Equialent Martingale Measure (EMM), where F 1 and F are the current forward prices for deliery at the future date T, σ 1 and σ are olatilities, and ε 1 and ε are standard normal random ariables with correlation ρ. It follows from aboe that the two asset prices are lognormal, and that the expected future price for each asset (wrt. the EMM) coincides with the current forward price. 3 The spread call ConsideraEuropeancalloptiononthepricespreadS 1 (T ) S (T ) with strike K 0 and time to exercise T. The call option pay-off at time T is C(T )=(S 1 (T ) S (T ) K) + (3) where () + denotes the positie part. The call alue at time 0 can be represented by 1 C = e rt E 0 h(s 1 (T ) S (T ) K) +i (4) where the expectation is taken with respect to the EMM, and r is the riskless interest rate. It follows from the put-call parity that the alue of a European put option on the price spread S 1 (T ) S (T ) with strike K 0 and time to exercise T is gien by P = C e rt (F 1 F K). With both S 1 (T ) and S (T ) being lognormal, there is no known general formula for the spread call alue. Howeer, closed form solutions are aailable for the following limiting cases: Firstly, if F =0, the call spread collapses into a standard call on S 1 (T ), and the alue is gien by the Black76 formula (c.f. Black (1976)). And secondly, if K =0, the call spread collapses into an option to exchange one asset for another. The option alue in this case is gien by the Margrabe formula (see Margrabe (1978)). 4 The Kirk formula In the general case, howeer, we hae to rely on either approximation formulas or extensie numerical methods. Approximation formulas allow quick calculations and facilitate analytical tractability, whereas numerical methods typically 1 See, e.g., Cox and Ross (1976), Harrison and Kreps (1979), and Harrison and Pliska (1981). 3
produces more accurate results. Practitioners are ery focused on simple calculations and real time solutions, hence a closed form approximation formula is typically the preferred alternatie. Kirk (1995) suggests the following approximation to the spread call c K = e rt {F 1 N (d K,1 ) (F + K) N (d K, )} (5) where N () denotes the standard normal cumulatie probability function, and d K,1 and d K, are gien by d K,1 = ln (F 1/ (F + K)) + 1 σ K T (6) σ K T d K, = d 1 σ K T (7) σ K = s σ 1 F F + K ρσ 1σ + µ F F + K 5 The Carmona-Durrleman procedure σ (8) Carmona and Durrleman (003a, 003b) represent the future spot prices by two independent state ariables and model the correlation by using trigonometric functions. In particular, Eqs.(1) and () aboe translate into o S 1 (T ) = F 1 exp n 1 σ 1T +(z 1 sin φ + z cos φ) σ 1 T (9) o S (T ) = F exp n 1 σ T + σ Tz (10) where z 1 and z are standard normal and independent random ariables, and cos φ = ρ where φ [0, π]. The authors consider the alue from exercising the spread option according to a feasible, but non-optimal strategy conditional on the two state ariables. In particular, the strategy is to exercise when Y θ z 1 sin θ z cos θ d (11) where θ [π, π] and d are found numerically by maximising the option alue. 3 By the put-call parity, the Kirk approximation of a put on the price spread S 1 (T ) S (T ) with strike K 0 and time to exercise T is p K = c K e rt (F 1 F K). 3 φ [0, π] and θ [π, π] translate into sin φ 0 and cos θ 0. To motiate this, obsere from Eqs. (9) and (11) that an increase in z 1 will increase the pay-off from asset 1, and push the call more in-the-money (less out-of-the-money). 4
The alue from following this strategy, which represents a lower bound to thetruespreadoptionalue,is 4 c CD = e rt E 0 [(S 1 (T ) S (T ) K) I (z 1 sin θ z cos θ d n )] ³ = e rt F 1 N d + σ 1 T cos (θ + φ) (1) o F N ³d + σ T cos θ KN (d ) see Appendix B. 5 Numerical inestigations in Carmona and Durrleman (003a) indicate that their lower bound optimization procedure produces ery accurate estimates to the true option alue. 6 A closed form spread option formula It can be erified (see Appendix C) that the Kirk formula follows from the expectation + c K = e rt E 0 S 1 (T ) a (S (T )) b E h(s (T )) bi (13) where a = F + K, b = F /(F +K), and E 0 h(s (T )) bi =exp 1 b (b 1) σ T ª F b Note that 0 b < 1 when K 0. Obsere from aboe that the implicit strategy is to exercise if and only if S 1 (T ) exceeds a scaled power function of S (T ). We want to use the insight aboe to obtain an alternatie spread option approximation formula. Now, consider the future spread call pay-off conditional on exercising if and only if S 1 (T ) exceeds a power function of S (T ), with exponent b and scalar a/e 0 h(s (T )) bi. We can express the future pay-off from following this strategy as c(t )=(S 1 (T ) S (T ) K) I S 1 (T ) a (S (T )) b E h(s (T )) bi (14) where I () represents the indicator function, assuming unity wheer the argument is true and zero otherwise. Compare Eq.(14) with Eq.(3), and obsere 4 There is a typo in Eq.(0) of Carmona and Durleman (003a) as well as in Eq.(6.3) of Carmona and Durrleman (003b). The trigonometric function entering the second term should read cos, and not sin. 5 By the put-call parity, the Carmona-Durrleman approximation of a put on the price spread S 1 (T ) S (T ) with strike K 0 and time to exercise T is p CD = c CD e rt (F 1 F K). 5
that the specified strategy is feasible but not optimal. Consequently, the spread option alue from following such a strategy represents a lower bound to the true spread option alue. Proposition: Approximate the spread call alue by the following formula c (a, b) = e rt E 0 (S 1 (T ) S (T ) K) I S 1 (T ) a (S (T )) b E h(s (T )) bi = e rt {F 1 N (d 1 ) F N (d ) KN (d 3 )} (15) where d 1,d,andd 3 are defined by d 1 = ln (F 1/a)+ 1 σ 1 bρσ 1 σ + 1 b σ T σ (16) T d = ln (F 1/a)+ 1 σ 1 + ρσ 1 σ + 1 b σ bσ T σ (17) T d 3 = ln (F 1/a)+ 1 σ 1 + 1 b σ T σ (18) T q σ = σ 1 bρσ 1σ + b σ (19) and where the constants a and b are a = F + K (0) b = F F + K (1) By the put-call parity, the put option on the price spread S 1 (T ) S (T ) with strike K and time to exercise T is approximated by p = c e rt (F 1 F K). Proof: See Appendix D. The Black-Scholes, the Black76, and the Margrabe formulas consist of one term for each component that enters into the future option pay-off. Eqs.(15)- (19) conforms with this tradition. This form is similar to Eq.(6.3) in Carmona and Durrleman (003b). Howeer, by comparing with our Eqs.(15)-(19), there should be no doubt that our representation of the arguments d 1,d,d 3 is more along the lines of the Black-Scholes, Black76, and Margrabe than the corresponding arguments found in Carmona and Durrleman (003b). In order to obtain a stricter lower bound, one could optimise the spread call alue c (a, b) aboe with respect to a and b. Optimal parameters a and b can be obtained by expanding first order conditions and applying the Newton-Raphson iteratie procedure, using Eqs.(0) and (1) as the initial guess. The first order conditions, as well as the second order partials needed for Newton-Raphson, are proided in Appendix E. Optimizing our formula with respect to a and b is in fact equialent to the Carmona and Durrleman procedure (see Appendix F). Extensie numerical 6
inestigations, howeer, indicate that with our initial choice a and b, thereis ery little to gain from implementing such an optimization procedure. Put differently, the formula stated in Eqs.(15)-(1) aboe represents a ery tight lower bound to the true spread option alue. 7 Numerical results In the following, we compare the accuracy of the Kirk approximation in Eqs.(5)- (8), our formula in Eqs.(15)-(1) aboe, and the optimal lower bound following from maximizing the formula with respect to the parameters a and b (which is similar to the Carmona-Durrleman optimization procedure). To approximate the true spread option alue, we apply a Monte Carlo simulation procedure using the first 100,000 pair of numbers from a two-dimensional Halton sequence. In order to reduce the simulation error, we use our representation of the spread call as control ariate. 6 We adopt the numerical example in Carmona and Durrleman (003a), where the annual riskless interest rate is r =0.05 and the time horizon is T =1 year. Their numerical case translate into forward prices F 1 = e (0.05 0.03) 1 110 11. and F = e (0.05 0.0) 1 100 103.05. The annualized olatilities are σ 1 =0.10 and σ =0.15. We consider different combinations of strike K and correlation ρ. In case of K =0, the spread option reduces to the Margrabe exchange option (c.f. Margrabe (op.cit.)). With K>0, the option corresponds to a call on the price spread S 1 (T ) S (T ). In case of K<0, the option represents a put on the opposite price spread, i.e. S (T ) S 1 (T ). The put alues are obtained by the put-call parity. 6 Rewrite Eq.(4) as E e rt C(T ) = c + E e rt (C(T ) c(t )) where the pay-offs C(T ) and c(t ) are defined in Eqs.(3) and (14), and c is our spread option formula in Eqs.(15)-(1). Clearly, the two pay-offs C(T ) and c(t ) are highly correlated. Consequently, the simulation error from ealuating the expectation on the RHS is much lower than the simulation error from ealuating the expectation on the LHS. 7
Table 1. Spread option alue approximation ρ -1-0.5 0 0.3 0.8 1 K 9.675 9.0056 8.3848 8.0709 7.7704 7.7538 9.6561 8.9948 8.3811 8.0701 7.7701 7.7538-0 9.6561 8.9948 8.3811 8.0701 7.7701 7.7538 9.6561 8.9948 8.3811 8.0701 7.7701 7.7538-10 0 5 15 5 1.8787 1.8686 1.8686 1.8686 15.133 15.133 15.133 15.133 1.45 1.441 1.441 1.441 7.5376 7.518 7.518 7.517 4.475 4.014 4.014 4.013 0.9114 0.9050 0.9050 0.9049 13.9180 13.9180 13.9180 13.9180 10.9543 10.956 10.956 10.956 6.559 6.4 6.4 6.41 3.1686 3.1300 3.1300 3.198 19.8917 19.8889 19.8889 19.8888 1.537 1.537 1.537 1.537 9.4431 9.4453 9.4453 9.4453 4.756 4.7445 4.7444 4.7443 1.993 1.961 1.960 1.9617 19.710 19.701 19.701 19.701 11.5618 11.5618 11.5618 11.5618 8.3649 8.3674 8.3674 8.3674 3.6907 3.6798 3.6797 3.6796 1.441 1.00 1.198 1.194 18.3816 18.3811 18.3811 18.3811 9.635 9.635 9.635 9.635 5.968 5.9670 5.9670 5.9670 1.3545 1.345 1.34 1.341 0.114 0.1041 0.1039 0.103 Number on top of each box: Kirk s formula. Second number (in italics) from top of each box: Simulation result (100,000 trials). Third number from the top of each box: Optimizing our formula wrt. a and b. Number on the bottom of each box: Our formula (a and b fixed)). Table 1 shows the results of the spread option alue approximations. The number on top in each box represents the result from the Kirk formula. The second number (in italics) from top in each box is the spread option alue obtained by Monte Carlo computation with 100,000 trials. We use the simulation results as the benchmark for the true spread option alue. The third number from top of each box is the result from optimizing our formula with respect to a and b, which is similar to the Carmona-Durrleman optimization procedure. The number at the bottom of each box represents the result from our formula. It is interesting to obsere from Table 1 that the Kirk formula iolates the lower bound (proided by our formula) for all correlations when the strike is K =5. 18.444 18.439 18.439 18.438 8.81 8.81 8.81 8.81 4.440 4.454 4.454 4.454 0.074 0.0488 0.0488 0.0479 0.0000 0.0000 0.0000 0.0000 8
Table. Pricing error ρ K -1-0.5 0 0.3 0.8 1-0 0.0191 0.0108 0.0037 0.0008 0.0003 0.0001 0.0000 0.0000-0.0001 0.0000 0.0000 0.0000-10 0.0101 0.0065 0.009 0.0010 0.0005 0.0006 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5-0.0016-0.0019-0.003-0.005-0.004-0.01 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 15 0.0158 0.0137 0.0118 0.0109 0.010 0.036-0.0001-0.0001-0.0001-0.000-0.0004-0.0009 5 0.0461 0.0386 0.0301 0.041 0.0083 0.0000-0.0001-0.000-0.0004-0.0006-0.0009 0.0000 Number on top of each box: Approximation error following from the Kirk formula. Number on the bottom of each box: Approximation error following from our formula. Table shows the approximation error associated with the Kirk formula (number on top of each box) and our formula (number on bottom of each box), as compared to the benchmark. Note that the Kirk formula seems to underprice the spread option when the strike is closer to zero, and to oerprice the spread option when the strike is further away from zero. Our formula represents a lower bound, hence the approximation error (if any) is negatie. Obsere that for all releant cases in Table, 7 our formula performs much better than the Kirk formula. In our iew, practitioners looking for a pricing formula are better off using our formula than the Kirk formula when ealuating spread options. Table 3. Improed accuracy obtained by optimization ρ K -1-0.5 0 0.3 0.8 1-0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000-10 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 5 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 15 0.0001 0.0001 0.0001 0.0001 0.000 0.0009 5 0.0001 0.0001 0.0003 0.0004 0.0007 0.0000 Number in each box: Improed accuracy obtained by optimization. Table 3 shows the improed accuracy by optimizing our formula with respect to a and b, as compared to using our formula with parameter alues for a and b in Eqs.(0) and (1). Recall that optimizing our formula corresponds to the Carmona-Durrleman optimization procedure. Hence, we may interpret the 7 When K =0, both formulas degenerate to the Margrabe exhange option formula, which represents the true alue in this case. Hence, the pricing errors are zero in these cases. 9
results in Table 3 as the gain form using their numerical optimization procedure as compared to our formula which is closed form. The results in the table indicate that the improed accuracy from implementing numerical optimization is either marginal or zero. For practical purposes, the benefits of a closed form solution are obious. In our iew, the numerical results indicate that the accuracy of our formula is comparable with the accuracy of using an optimization procedure. Consequently, practioners should settle for our formula rather than the Carmona-Durrleman procedure when ealuating spread options. 8 Conclusions This paper considers the aluation of a European spread option when the asset prices are lognormal. We derie a spread option formula that consists of three terms, one for each of the two assets and one for the strike. A standard normal cumulatie probability enters into each term, and each argument is a function of the forward prices, time to exercise, olatilities, and correlation. The formula fits well in to the tradition of Black-Scholes, Black76, and Margrabe. Numerical inestigations indicate that our formula is extremely accurate. The precision is much better than the Kirk formula, which is the current market standard in practice. Moreoer, the precision of our formula is comparable the lower bound procedure of Carmona and Durrleman, which requires a twodimensional optimization scheme. Black, F. (1976): "The pricing of commodity contracts," Journal of Financial Economics 3 (1976), pp. 167-179. Carmona, R. and V. Durrleman (003a): "Pricing and Hedging spread options in a log-normal model," Tech. report, Department of Operations Research and Financial Engineering, Princeton Uniersity, Princeton, NJ, 003. Carmona, R. and V. Durrleman (003b): "Pricing and hedging spread options," SIAM Reiew Vol. 45, No. 4, pp. 67-685. Cox, J. and S. Ross (1976): "The aluation of options for alternatie stochastic processes," Journal of Financial Economics, 3 (January-March), pp. 145-166. Harrison, J.M. and D. Kreps (1979): "Martingales and arbitrage in multiperiod security markets," Journal of Economic Theory 0 (July). pp. 381-408. Harrison, J.M. and S. Pliska (1981): "Martingales and stochastic integrals in the theory of continuous trading," Stochastic Processes and Their Applications, 11, pp. 313-316. 10
Kirk, E. (1995): "Correlation in the energy markets," In Managing Energy Price Risk (First Edition). London: Risk Publications and Enron, pp. 71-78. Margrabe, W. (1978): "The alue of an option to exchange one asset for another," Journal of Finance, 33 (1978), pp. 177-186. 11
A Biariate normal ariables - a useful result The standard biariate normal density function is defined by 1 m(x, y; ρ) = ½ π p 1 ρ exp x ρxy + y ¾ (1 ρ ) where ρ is correlation. The density function satisfies the identity exp (ax + by) 1 a +ρab + b ª m (x, y; ρ) =m (x (a + ρb),y (ρa + b);ρ) where E exp (ax + by) 1 a +ρab + b ª =1. Consequently, = = = E exp (ax + by) 1 a +ρab + b ª h(x, y) Z Z exp (ax + by) 1 a +ρab + b ª h(x, y)m(x, y; ρ)dydx Z Z h(x, y)m (x (a + ρb),y (ρa + b);ρ) dydx Z Z h(x +(a + ρb),y+(ρa + b))m (x, y; ρ) dydx = E [h(x +(a + ρb),y+(ρa + b))] B The Carmona-Durrleman result For notational conenience, write X 1 = F 1 exp 1 1 + 1 (z 1 sin φ + z cos φ) ª X = F exp 1 + z ª where z 1 and z are independent and standard normal. Consider the expectation c CD = E [(X 1 X K) I (z 1 sin θ z cos θ d )] = E [X 1 I (z 1 sin θ z cos θ d )] E [X I (z 1 sin θ z cos θ d )] E [KI (z 1 sin θ z cos θ d )] Obsere that due to the identity 1
(sin φ) +(cosφ) =1 both z (z 1 sin φ + z cos φ) and Y θ (z 1 sin θ z cos θ ) are standard normal. Ealuate the last term E [KI (z 1 sin θ z cos θ d )] = KE [I (z d )] = KN (d ) Ealuate the second term E [X I (z 1 sin θ z cos θ d )] = E F exp ª 1 + z I (z1 sin θ z cos θ d ) = F E [I (z 1 sin θ (z + )cosθ d )] = F E [I (z 1 sin θ z cos θ d + cos θ )] = F E [I (z d + cos θ )] = F N (d + cos θ ) using the result in Appendix A. And finally, ealuate the first term E [X 1 I (z 1 sin θ z cos θ d )] = E F 1 exp 1 1 + 1 (z 1 sin φ + z cos φ) ª I (z 1 sin θ z cos θ d ) = F 1 E [I (sin θ (z 1 + 1 sin φ) cos θ (z + 1 cos φ) d )] = F 1 E [I (z 1 sin θ z cos θ d + 1 (cos θ cos φ sin θ sin φ))] = F 1 E [I (z d + 1 cos (θ + φ))] = F 1 N (d + 1 cos (θ + φ)) whereweusetheresultinappendixaandtheidentity cos θ cos φ sin θ sin φ =cos(θ + φ) Now, collect the results, apply riskless discounting, and translate 1 = σ 1 T and = σ T, to obtain the result in Eq.(1). C The implicit Kirk strategy For notational conenience, write 13
X 1 = F 1 exp ª 1 1 + 1 ε 1 X = F exp ª 1 + ε ax b X b = a exp 1 b ª + b ε and consider the expectation Ã! + " Ã!# " Ã!# E X 1 axb X b = E X 1 I X 1 axb X b E axi b X 1 axb X b The two terms are ealuated as follows: " Ã!# E X 1 I X 1 axb X b = E F 1 exp ª 1 1 + 1 ε 1 I F1 exp ª 1 1 + 1 ε 1 a exp 1 b ª + b ε = F 1 E I F 1 exp 1 1 + 1 (ε 1 + 1 ) ª a exp 1 b + b (ε + ρ 1 ) ª = F 1 E I 1 ε 1 b ε ln (F 1 /a) 1 1 + bρ 1 1 b " à = F 1 E I ε ln (F 1/a)+ 1 1 bρ!# 1 + 1 b p 1 bρ 1 + b Ã! ln (F1 /a)+ 1 = F 1 N 1 bρ 1 + 1 b p 1 bρ 1 + b and " ax b Ã!# E I X b X 1 axb X b = E a exp 1 b ª + b ε I F1 exp ª 1 1 + 1 ε 1 a exp 1 b ª + b ε = ae exp 1 b ª + b ε I F1 exp ª 1 1 + 1 ε 1 a exp 1 b ª + b ε = ae I F 1 exp 1 1 + 1 (ε 1 + ρb ) ª a exp 1 b + b (ε + b ) ª = ae I 1 ε 1 b ε ln (F 1 /a)+ 1 1 bρ 1 + 1 b " à = ae I ε ln (F!# 1/a) 1 1 + bρ 1 1 b p 1 bρ 1 + b = an à ln (F1 /a) 1 1 + bρ 1 1 b p 1 bρ 1 + b Now, collect the results, apply riskless discounting, choose the constant a such that! 14
a = F + K and translate 1 = σ 1 T ; = σ T ; p 1 bρ 1 + b = σ T, to obtain the Kirk formula stated in Eqs.(5) - (8) aboe. D Deriation of the spread option formula Consider the expectation " Ã!# E (X 1 X K) I X 1 axb X b " Ã!# " Ã!# = E X 1 I X 1 axb X b E X I X 1 axb X b " Ã E I The first term is ealuated in Appendix C. The two remaining terms are ealuated as follows X 1 axb X b " Ã!# E X I X 1 axb X b = E F exp ª 1 + ε I F1 exp ª 1 1 + 1 ε 1 a exp 1 b ª + b ε = F E I F 1 exp 1 1 + 1 (ε 1 + ρ ) ª a exp 1 b + b (ε + ) ª = F E I 1 ε 1 b ε ln (F 1 /a)+ 1 1 ρ 1 1 b + b " Ã = F E I ε ln (F!# 1/a)+ 1 1 ρ 1 1 b + b p 1 bρ 1 + b Ã! ln (F1 /a) 1 = F N 1 + ρ 1 + 1 b b p 1 bρ 1 + b!# " Ã!# E KI X 1 axb X b = KE I F 1 exp ª 1 1 + 1 ε 1 a exp 1 b ª + b ε = KE 1 ε 1 b ε ln (F 1 /a)+ 1 1 1 b " = KE ε ln (F 1/a) 1 1 + # 1 b p 1 bρ 1 + b Ã! ln (F1 /a) 1 1 + 1 b = KN p 1 bρ 1 + b 15
Collect the results, and translate 1 = σ 1 T ; = σ T ; p 1 bρ 1 + b = p (σ 1 bρσ 1 σ + b σ ) T = σ T, to obtain the result stated as Eqs.(15)-(19) 1aboe. E Optimizing wrt. the exercise strategy Define H(a, b) = e rt {F 1 N (d 1 ) F N (d ) KN (d 3 )} d 1 = ln (F 1/a)+ 1 σ 1 bρσ 1 σ + 1 b σ T σ T d = ln (F 1/a)+ 1 σ 1 + ρσ 1 σ + 1 b σ bσ T σ T d 3 = ln (F 1/a)+ 1 σ 1 + 1 b σ T σ T q σ = σ 1 bσ 1σ + b σ First, establish d 1 = d a a = d 3 a = 1 aσ T σ b = bρσ d 1 b = C 1 d 1 bρ d b = C d bρ d 3 b = C 3 d 3 bρ where we for notational conenience define bρ = bσ ρσ 1 σ σ C 1 = ρσ 1σ T + bσ T σ T C = bσ T σ T σ T C 3 = bσ T σ T 16
Consequently, the first order partials of H are H a = e rt 1 {F 1 n (d 1 ) F n (d ) Kn(d 3 )} aσ T H b = e rt {F 1 n (d 1 ) d 1 F n (d ) d Kn(d 3 ) d 3 } bρ +e rt {F 1 n (d 1 ) C 1 F n (d ) C Kn(d 3 ) C 3 } Next, obtain the second order partials d 1 a = d a = d 3 a = 1 a σ T d 1 = d a b a b = d 3 a b = bρ aσ T d 1 b = σ d b = σ σ C 1bρ + σ C bρ + d b = σ σ C 3bρ + and the second order partials of H are µ 3bρ σ σ d 1 µ 3bρ σ d σ µ 3bρ σ σ d 3 H aa = e rt 1 {F 1 n (d 1 ) d 1 F n (d ) d Kn(d 3 ) d 3 } a σ T +e rt 1 {F 1 n (d 1 ) F n (d ) Kn(d 3 )} a σ T H ab = H ba = e rt F 1 n (d 1 ) d 1 F n (d ) d Kn(d 3 ) d ª bρ 3 aσ T +e rt 1 {F 1 n (d 1 ) C 1 d 1 F n (d ) C d Kn(d 3 ) C 3 d 3 } aσ T +e rt bρ {F 1 n (d 1 ) F n (d ) Kn(d 3 )} aσ T H bb = e rt F 1 n (d 1 ) d 3 1 F n (d ) d 3 Kn(d 3 ) d 3 ª 3 bρ +e rt F 1 n (d 1 ) C 1 d 1 F n (d ) C d Kn(d 3) C 3 d 3ª bρ e rt F 1 n (d 1 ) C 1d 1 F n (d ) C d Kn(d 3 ) C 3d 3 ª +e rt {F 1 n (d 1 ) d 1 F n (d ) d Kn(d 3 ) d 3 } e rt {F 1 n (d 1 ) C 1 F n (d ) C Kn(d 3 ) C 3 } bρ +e rt {F 1 n (d 1 ) F n (d ) Kn(d 3 )} σ σ 17 µ 3bρ σ σ
We now hae the necessary results to implement the Newton-Raphson iteratie procedure, using a = F + K and b = F /(F + K) as our initial guess. F Optimizing our formula and the Carmona- Durrleman procedure Compare Eq.(1) with Eqs.(15)-(19), assuming that a and b are optimal. Note that it is sufficient to show that the arguments of N () equal for each of the three terms. Firstly, let d = d 3. Secondly, let σ T cos θ = d d 3 which leads to cos θ =(ρσ 1 bσ ) /σ. Hence, we need to show that σ 1 T cos(θ + φ) = d 1 d 3. Recall that cos φ = ρ, andthatφ [0, π] and θ [π, π] (see footnote 3). Obtain the result as follows: cos(θ + φ) = cos(θ )cos(φ) sin(θ )sin(φ) s = ρσ 1 bσ ρ ( 1) σ 1 µ ρσ1 bσ σ r = ρ σ 1 bρσ σ + 1 ρ σ p 1 1 ρ σ σ = ρ σ 1 bρσ + σ 1 1 ρ σ σ = σ 1 bρσ σ 1 σ = 1 bρσ 1 σ T σ 1 T σ T = d 1 d 3 σ 1 T p 1 ρ Consequently, optimizing our formula with respect to a and b is equialent to the Carmona-Durrleman procedure. 18