Centre for Risk & Insurane Studies enhaning the understanding of risk and insurane OPTIONS ON NORMAL UNDERLYINGS Paul Dawson, David Blake, Andrew J G Cairns, Kevin Dowd CRIS Disussion Paer Series 007.VII
OPTIONS ON NORMAL UNDERLYINGS Abstrat The seminal otion riing work of Blak and Sholes [1973] and Merton [1973] was rediated on the rie of the underlying asset being lognormally distributed. Ever sine it beame lear that a geometri Brownian motion roess rovides a more lausible model of asset ries than its arithmeti equivalent, it has been assumed that an otion riing model for a normally distributed underlying asset was redundant. Nevertheless, 34 years after Blak and Sholes [1973] and Merton [1973], we identify a ontemorary need for suh a model: namely when we wish to rie an otion on a survivor swa. In this ase, an otion-riing model based on a normal underlying is not some flawed relative of Blak-Sholes, as it is usually onsidered to be, but is instead the key to riing this tye of swation orretly and hene, a very useful tool in the raidly emerging universe of mortality derivatives. Aordingly, this aer derives the all and ut valuation models for otions on normal underlying assets, and derives their Greeks. It then shows how this otion riing model an be used to rie swations on survivor swas. Paul Dawson David Blake Andrew J G Cairns Kevin Dowd This version:19 November 007 PRELIMINARY VERSION COMMENTS WELCOME. NOT TO BE QUOTED WITHOUT PERMISSION FROM AUTHORS Corresonding author: Paul Dawson, Kent State University, e-mail:dawson1@kent.edu 1
OPTIONS ON NORMAL UNDERLYINGS 1. Introdution The seminal otion riing work of Blak and Sholes [1973] and Merton [1973] was rediated on the assumtion that the geometri (or ontinuously omounded) returns of the asset under otion are normally distributed, or equivalently, that ries of that asset are lognormally distributed. Subsequent researh (e.g. Cox and Ross [1976]) has onsidered other distributions and, eseially sine Rubinstein [1994], researh has analyzed the distributions imlied in market ries of otions aross a range of strike ries. One ase whih has not been given muh attention is that in whih the rie of an asset, rather than its returns, is normally distributed. This ase was famously onsidered by Bahelier s model of arithmeti Brownian motion (Bahelier [1900]). However, suh a distribution would allow the underlying asset rie to beome negative, and this unomfortable imliation an be avoided by using a geometri Brownian motion (GBM) instead. Consequently, the Bahelier model ame to be was regarded as an instrutive dead end. The lak of interest in an otion-riing model with a normally distributed underlying was therefore hardly surrising. Nonetheless, we suggest here that it is remature to onlude that an otion riing model with a normal underlying is of no use. An examle of suh a requirement arises from some reent work on survivor derivatives. Dowd, Blake, Cairns and Dawson [006] identify a remium, π, in the riing of survivor swas, whih must be ermitted to beome negative. Dawson, Blake, Cairns and Dowd [007] then go on to establish that π is the essential stohasti variable in the riing of survivor swations, and further show that the distribution
of π is aroximately normal. Thus, riing a survivor swation requires an otion riing model with a normal underlying. The rinial urose of the resent aer is to rovide suh a model. Aordingly, setion derives the formulae for the all and ut otions for a Euroean otion with a normal underlying and resents their Greeks. Setion 3 disusses how the model an be alied to rie swations on survivor swas. Setion 4 tests the model and setion 5 onludes. The derivation of the Greeks is resented in the aendix.. Model Derivation For the remainder of the aer, we onsider an asset with forward rie F, with - < F <. We do not onsider the ase of an otion on a normally distributed sot rie, as this is an obvious seial ase of an otion on a forward rie. We denote the value of Euroean all and ut otions by and resetively. The strike rie and maturity of the otions are denoted by X and τ resetively. The annual risk-free interest rate is denoted by r and the annual volatility rate (or the annual standard deviation of the rie of the asset) is denoted by σ. We first establish the ut-all arity ondition. The ut-all arity ondition for the otions under onsideration in the resent aer is the same as that aliable in Blak [1976] for otions on forward ontrats with lognormally distributed ries, i.e. = e F-X 1 Proof of this ondition follows the same reasoning as Stoll [1969]. Consider an investor who holds a all otion in tandem with a short osition in an otherwise idential ut. At maturity, either the investor will hoose to exerise the all otion or the ut otion will be exerised against him/her. Either way, the investor will aquire the forward ontrat at the strike rie, 3
X. The investor has thus reliated a forward ontrat at a rie of X. A zero value forward ontrat has a rie of F. The forward value of the ortfolio of long all and short ut is thus F X. Its resent value is then e -rτ (F X) and it follows that = e F X = rτ e F X 3 QED The Blak-Sholes-Merton dynami hedging strategy an be imlemented if there is assumed to be a liquid market in the underlying asset. In suh irumstanes, a risk-free ortfolio of asset and otion an be onstruted and the value of an otion is simly the resent value of its exeted ayoff. The values of all and ut otions an then be resented as ( τ ) ( ( τ τ ) ) ( 4) ( τ ) ( ( τ τ )) ( 5) = e P F > X E F F > X X = e P F < X X E F F < X in whih F τ reresents the forward rie at otion exiry, and F τ ~ N(F,σ τ). If N(z) is the standard normal umulative density funtion of z, with z ~ N(0,1), the orresonding robability density funtion, N' ( z), is: z 1 N'( z) = e 6 π and it follows that ( X F) στ ( X F) στ 1 P( Fτ > X) = e df X π = 1 X 1 e π df ( 8) ( 7) Defining d = F X σ t then gives: 4
X F P F > X = N σ τ F X = N ( 10) σ τ = N d 1 ( 9) ( 11) We next onsider the onditional exeted value of F τ, i.e. the exeted value of F at exiry given that the all otion has exired in the money: ( ) EF F X τ τ > = X X ( X F) F τ e στ df π 1 e π ( 1 ) ( X F) στ df shows that: A well-known result from exeted shortfall theory see, e.g. Dowd [005,. 154] - X Fτ e π ( X F) στ df = F + σ τ ( 13) ( X F) 1 στ 1 e df X π N = F + σ τ 1 N = F + σ τ N d Substituting (11) and (15) into (4) gives: '( d) N( d) ( 14) '( d) ( 15) X F N ' σ τ X F N σ τ ' = e N d F+ σ τ X N d N d ( 16 ) and so gives us the all otion riing formula we are seeking in (17) below. Substituting (17) into (3) then gives ( σ τ ) = e F X N d + N' d 17 5
( σ τ ') ( 18) (( 1 ) σ τ ') ( 19) (( 1 ) σ τ ') ( 0) = e F X N d + N d F+ Xe = + e F X N d N d = + e X F N d N d Thus, the orresonding ut otion formula is given by (1) below. ( σ τ ) = e X F N d + N' d 1 Table 1 resents the Greeks. Their derivation an be found in the aendix. Insert Table 1 about here 3. A ratial aliation As noted earlier, a ratial illustration of the usefulness of this otion riing model an be found in the riing of survivor swations or otions on survivor swas. Dowd et al. [006] roose a survivor swa ontrat in whih the reeive-fixed arty ommits to making a ayment stream based on the atual survivorshi rate of a seified ohort and reeives in return a fixed ayment stream, based on the exeted survivorshi rate of that ohort exeted at the time of the swa ontrat formation multilied by (1+π). The term π is a risk remium refleting the otential errors in the exetation and π an be ositive or negative, deending on whether greater longevity (π > 0) or lesser longevity (π < 0) is ereived to be the greater risk. It an also be zero, when the risks of greater longevity and lesser longevity exatly balane. Tyially, however, we would exet π to be in the region lose to zero, and in this region, and Dawson et al. [007] go on to show that the distribution of π is aroximately normal when underlying aggregate mortality shoks obey the beta roess set out in Dowd et al. [006,. 5-7]. They then roose a Euroean survivor swation ontrat, in whih the otion holder has the right, but not the obligation, to enter into a survivor swa ontrat on re-seified terms at some time in the future. These otions an take one of two forms: a ayer swation, equivalent to 6
our earlier all, in whih the holder has the right but not the obligation to enter into a ayfixed swa at the seified future time; and a reeiver swation, equivalent to our earlier ut, in whih the holder has the right but not the obligation to enter into a reeive-fixed swa at the seified future time. In order to rie the swation using the usual dynami hedging strategies assumed for riing uroses, we are also imliitly assuming that there is a liquid market in the underlying asset, the forward survivor swa. Naturally, we reognize that this assumtion is not yet emirially valid, but we would defend it as a natural starting oint, not least sine survivor swations annot exist without survivor swas. We now onsider an examle alibrated on swations that mature in 5 years time and are based on a ohort of US males who will be 70 when the swations mature. The strike rie of the swation is a seified value of π and for this examle, we shall use an at-the-money forward otion, i.e. X is set at the revailing level of π for the forward ontrat used to hedge the swation. Setting the otion at the money forward means that the ayer swation remium and the reeiver swation remium are idential at all times. Using the same mortality table as Dowd et al. [006], and assuming, as they did, longevity shoks, ε i, drawn from a beta distribution 1 with arameters 1000 and 1000 and a yield urve flat at 6%, Monte Carlo analysis with 10,000 trials shows the distribution of π for the forward swa to have the following values: Mean 0.001156 Annual variane 0.000119 Skewness 0.008458 Kurtosis 3.0941 1 See Dowd et al [006] for the signifiane of the longevity shoks, ε i, and the beta distribution.. 7
The distribution is shown in Figure 1 below. Figure 1 Distribution of the values of the π for a 45 year survivor swa starting in 5 years time, from a Monte Carlo simulation of 10,000 trials with ε i values drawn from a beta distribution with arameters (1000, 1000). A normal distribution lot is suerimosed. 450 400 350 Frequeny 300 50 00 150 100 50 0-0.1 0.0 0.1 π A Jarque-Bera test on these data gives a value of 0.476. Given that the test statisti has a χ distribution with degrees of freedom, this test result is onsistent with a normal distribution. Figure below shows the otions remia for both ayer and reeiver swations aross π values sanning + 3 standard deviations from the mean. The gamma, or onvexity, familiar in more onventional otion riing models is also seen here. 8
Figure Premia for the seified survivor swations 3.000%.500%.000% Premium 1.500% 1.000% 0.500% 0.000% 3.38%.95%.51%.08% 1.64% 1.0% 0.77% 0.33% -0.10% -0.54% -0.97% -1.41% -1.84% -.8% -.7% -3.15% π Payer Reeiver As with onventional interest rate swations, the remia are exressed in erentage terms. However, whereas with interest rate swations, the remia are onverted into urreny amounts by multilying by the notional rinial, with survivor swations, the urreny amount is determined by multilying the erentage remium by 50 t= 1 N A PS() t exiry, t t in whih N is the ohort size, A is the disount fator alying from otion exiry until time t, exiry, t is the ayment er survivor due at time t, S(t) is the roortion of the original ohort exeted to survive until time t, suh exetation being observed at the time of the otion ontrat, and Pt 9
where all members of the ohort are assumed to be dead after 50 years. N A PS() t is known with ertainty at the time of otion riing. Figure 3 below shows the hanging value of these at the money forward ayer and reeiver swations as time asses. The raid rie deay as exiry aroahes, again familiar in more onventional otions, is also seen here. 50 t= 1 exiry,t t Figure 3 Otions remia against time 0.8000% 0.7000% 0.6000% 0.5000% Premium 0.4000% 0.3000% 0.000% 0.1000% 0.0000% 5 4.5 4 3.5 3.5 1.5 1 0.5 0 Years to exiry This aroah is equivalent to that used in the riing of an amortizing interest rate swa, in whih the notional rinial is redued by re-seified amounts over the life of the swa ontrat. 10
4. Testing the model The derivation of the model is rediated on the assumtion that imlementation of a dynami hedging strategy will eliminate the risk of holding long or short ositions in suh otions. We test the effetiveness of this strategy by simulating the returns to dealers with (searate) short 3 ositions in ayer and reeiver swations, and who undertake daily rehedging over the 5 year (1,50 trading days) life of the swations. We use Monte Carlo simulation to model the evolution of the underlying forward swa rie, assuming a normal distribution. We assume a dealer starting off with zero ash and borrowing or deositing at the risk-free rate in resonse to the ashflows generated by the dynami hedging strategy. As Merton [1973, 165] states, Sine the ortfolio requires zero investment, it must be that to avoid arbitrage rofits, the exeted (and realized) return on the ortfolio with this strategy is zero. Merton s model was rediated on rehedging in ontinuous time, whih would lead to exeted and realized returns being idential. In ratie, traders are fored to use disrete time rehedging, whih is modeled here. One onsequene of this is that on any individual simulation, the realized return may differ from zero, but that over a large number of simulations, the exeted return will be zero. This is atually a joint test of three onditions: i. The otion riing model is orretly seified equations (17) and (1) above, ii. iii. The hedging strategy is orretly formulated equations (A1) and (A) below, and The realized volatility of the underlying asset rie mathes the volatility imlied in the rie of the otion trade. We an isolate this ondition by foreasting results when this ondition does not hold and omaring observation with foreast. The dealer who has sold an otion at too low an imlied volatility will exet a loss, whereas the dealer sells 3 The returns to long ositions will be the negative of returns to short ositions. 11
at too high an imlied volatility an exet a rofit. This exeted rofit or loss of the dealer s ortfolio, E[V ], at otion exiry is: ( σ σ ) ( 5) rτ = e imlied atual imlied E V ( d) ( 6) = τn' σ σ imlied in whih σ imlied and σ atual reresent, resetively the volatilities imlied in the otion rie and atually realized over the life of the otion. We have onduted simulations aross a wide set of senarios, using different values of π, σ imlied and σ atual and different degrees of moneyness. In all ases, we ran 50,000 trials and in all ases, the results were as foreast. By way of examle 4, we illustrate in Table the results of the trials of the otion illustrated in Figures 1 and. The t-statistis relate to the differenes between the observed and the foreast mean outomes. atual Insert Table about here The reader will note that the differenes between the observed and exeted means are very low and statistially very insignifiant. This reinfores our assertion that the model rovides aurate swation ries. 5. Conlusion Ever sine it beame lear that a GBM roess rovides a more lausible model of asset ries than an arithmeti Brownian motion roess, it has been taken for granted that there was no 4 Results of the full range of Monte Carlo simulations are available on request from the orresonding author. 1
oint develoing an otion riing model for a normally distributed underlying. Nevertheless, 34 years after Blak and Sholes [1973] and Merton [1973], we suggest that there are ossible irumstanes in whih we might need suh a model, and a ontemorary examle is when we wish to rie a swation on a survivor swa. In this ase, an otion-riing model based on a normal underlying is not some flawed relative of Blak Sholes, as it is usually onsidered to be, but is instead the key to orretly riing this tye of swation and hene, a very useful tool in the raidly emerging universe of mortality derivatives. 13
Aendix Derivation of the Greeks Delta (, ) The otion s deltas follow immediately from (17) and (1) : Gamma( Γ, Γ ) = = e N d F = = e N d F ( A1) rτ ( A) e N d Γ = = = F F F d N( d) = e F d ( A4) e = N' ( d) σ τ ( A5) (A3) ( d) ( A9) e N d Γ = = = F F F ( d) N( d) e F d e = N' ( d) ( A8) σ τ e = N' σ τ ( A7) ( A6) Rho ( Ρ, Ρ ) Ρ = = τ r Ρ = = τ r ( A10) ( A11) 14
Theta ( Θ, Θ ) τ e τ F X N d tn d A1 σ ' = + By the rodut rule = + + τ τ τ + Let A = σ τ F X N d + N' d e τ ( A14) Let B= e σ τ τ F X N d + N' d ( A15) = A+ B τ ( A16) A = ( ) + σ τ = ( ) + σ τ F X N d N' d e re τ F X N d N' d = r A18 ( F X) N( d) σ τn' ( d) e e ( F X) N( d) σ τn' ( d) ( A13) = r + B τ Alying the sum rule ( A19) = + = + τ τ τ Let C = e ( F X) N( d) τ ( A1) Let D= e σ τn' ( d) τ ( A) = r + C + D τ ( A3) ( A17) σ τ ' σ τ ' ( A0) B e F X N d N d e F X N d e N d The hain rule then imlies ( A4) C = e F X N d = e F X N d = ( ) ' e d F X N d τ d τ d τ ( A5) ( ) ' e d F X N d = r + D τ τ ( A6) 15
By the rodut rule By hain rule D= e σ τn' d e N' d σ τ e σ τ N' d τ τ τ Let E = e N' ( d) σ τ τ ( A8) Let F= e σ τ N' ( d) τ ( A9) e σ N' ( d) E = e N' ( d) σ τ = τ τ ( A30) = + ( A7) ( ) ' σ ' e d F X N d e N d = r + + F τ τ τ ( A31) d σ τ ' ' τ d σ τ ( A3) F = e N d = e N d = e d N d σ τ τ ' ( A33) ( ) ' σ ' σ τ ' e d F X N d e N d e d N d = r + + τ τ τ τ ( A34) Tidying u ( X) τ σ σ τ = + r d F d r e N '( d) + + τ τ τ τ ( σ τ σ τ ) ( A36) e N' d = r + d F X + + d τ e N' d = r + d + + d τ = r + = r + σ τ ( σ τ σ τ σ τ ) ( A37) e N d τ σ ' e N d τ ' ( A38) ( A39) ( A35) Sine it is onventional for ratitioners to quote theta as the hange in an otion s value as one day asses 16
Θ = τ ' r e σn d 730 τ ( A40 ) The equivalent value for a ut otion an be obtained quite easily from ut-all arity and equation (A40). = e F X ( 1) e F X τ τ τ σ ' r re F X τ = ( A41) e N d = + ( A4) σ ' r( e ( F X) ) re F X τ e N d = + + σ ' r re F X τ e N d = + τ σ τ ( A44) r e N ' d + 4 re F X Θ = 730 τ ( A43 ( A45) ) Vega, e F X N d N d A46 σ τ ' = + = e F X N d + e N d Let G = e ( F X) N( d) ( A48) Let H = e σ τn' ( d) ( A49) = G + H ( A50) σ τ ' ( A47) 17
By the hain rule ( A5) G = e F X N d = e F X N d = = ( ) ' e d F X N d σ e d N' d By the rodut and hain rules τ d d ( A54) ( A53) d e N d e N d e N d d d e N' ( d) τ e σ τn' ( d) ( A56) σ = + ( A55) H = σ τ ' ' σ τ σ τ ' = + = + ( 1 ) ' ( A57) e τ d N d ( 1 ) ' τ ' ( A58) = + = τ e N d e τ d N d e d N d ' ( A59) Sine ratitioners generally resent vega in terms of a one erentage oint hange in volatility, we resent vega here as = e τ N d 100 ' ( A60 ) Put-all arity shows that the vega of a ut otion equals the vega of a all otion. = e F X ( 1) e F X = = ( A61) QED 18
Referenes Bahelier, Louis. (1900) Théorie de la Séulation. Paris: Gauthier-Villars. Blak, Fisher, [1976]. The Priing of Commodity Otions. Journal of Finanial Eonomis, 3: 167 179. Blak, Fisher and Myron Sholes, [1973]. The Priing of Otions and Cororate Liabilities. Journal of Politial Eonomy. 81: 637-654. Cox, John C. and Stehen A. Ross, [1976]. The Valuation of Otions for Alternative Stohasti Proesses. Journal of Finanial Eonomis. 3: 145 166. Dawson, Paul, David Blake, Andrew J G Cairns and Kevin Dowd, [007]. Comleting the Market for Survivor Derivatives. Working Paer. Dowd, Kevin, [005]. Measuring Market Risk. Seond edition. John Wiley. Dowd, Kevin, Andrew J G Cairns, David Blake and Paul Dawson, [006]. Survivor Swations. Journal of Risk and Insurane. 73: 1 17. Merton, Robert C., [1973]. Theory of Rational Otion Priing. Bell Journal of Eonomis and Management Siene. 4: 141 183. Rubinstein, Mark, [1994]. Imlied Binomial Trees. Journal of Finane. 49: 771 818. Stoll, Hans, [1969]. The Relationshi between Put and Call Otion Pries. Journal of Finane. 14: 319 33. 19
Otion value Table 1 Summary of the model and its Greeks Calls Delta Gamma Rho (er erentage oint rise in rates) Puts rτ e ( F X) N( d) σ τn' ( d ) e ( X F) N( d) + σ τn' ( d ) e rτ e N d e N d σ τ τ 100 r τ τ σ ' N' ( d ) N' ( d ) e σ τ τ 100 r τ τr e σn ' d + 4 τre F X Theta (for 1 day assage of time) r e N ( d ) Vega (er erentage oint rise in volatility) 730 τ 730 e τ N' ( d ) e τ N' ( d) 100 100 τ σ imlied σ atual Exeted 1.088998% Table Results of Monte Carlo simulations of delta hedging strategy Payer Reeiver Mean Standard t-stat Exeted value Mean Standard t-stat value deviation deviation 1.088998% 0.0000% 0.000% 0.09% 0.40 0.0000% 0.000% 0.036% 0.4 1.088998% 1.088998% 0.988998% 0.089% 0.0894% 0.1897% 0.45 0.089% -0.0894% 0.1853% 0.46 1.188998% -0.089% -0.0891% 0.87% 0.4-0.089% -0.0891% 0.1% 0.4 Simulations arried out using @Risk, with 50,000 trials. 0