Digital barrier option contract with exponential random time

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1 IMA Journal of Applie Mathematics Avance Access publishe June 9, IMA Journal of Applie Mathematics ) Page of 9 oi:.93/imamat/hxs3 Digital barrier option contract with exponential ranom time Doobae Jun Department of Mathematics an Statistics, York University, oronto, ON, Canaa an Hyejin Ku Department of Mathematics an Statistics, York University, oronto, ON, Canaa Corresponing author: [Receive on 3 March ; revise on January ] In this paper, we erive a close-form valuation formula for a igital barrier option which is initiate at ranom time τ with exponential istribution. We also provie a simple example an graphs to illustrate our result. Keywors: igital barrier option; exponential time.. Introuction Barrier options are a wiely use class of path-epenent erivative securities. hese options either cease to exist or come into existence when some pre-specifie asset price barrier is hit uring the option s life. Merton 973) has erive a own-an-out call price by solving the corresponing partial ifferential equation with some bounary conitions. Reiner & Rubinstein 99) publishe close-form pricing formulas for various types of single barrier options. Rich 997) also provie a mathematical framework to value barrier options. Moreover, Kunitomo & Ikea 99) erive a pricing formula for ouble barrier options with curve bounaries as the sum of an infinite series. Geman & Yor 996) followe a probabilistic approach to erive the Laplace transform of the ouble barrier option price. Pelsser ) use contour integration for Laplace transform inversion to price new types of ouble barrier options. In these papers, the unerlying asset price is monitore for barrier hits or crossings uring the entire life of the option. Heynen & Kat 994) stuie partial barrier options where the unerlying price is monitore uring only part of the option s lifetime. Partial barrier options have two classes. One is forwar starting barrier options where the barrier appears at a fixe ate strictly after the option s initial starting ate. he other is early ening barrier options where the barrier isappears at a specifie ate strictly before the expiry ate. hey can be applie for various types of options accoring to the clients nees as controlling the starting or ening time of the monitoring perio. Also, they can be use as components to synthetically create other types of exotic options. Digital barrier options cash-or-nothing barrier options) of knock-in type, a part of stanar barrier options, pay out a fixe amount of money if the unerlying asset price crosses a given barrier an arrives above a certain value at expiry ate. For igital barrier options of knock-out type, the payoff is a fixe amount if the unerlying asset price never hits a given barrier an is above a certain level at expiry. c he authors. Publishe by Oxfor University Press on behalf of the Institute of Mathematics an its Applications. All rights reserve.

2 of9 D. JUN AND H. KU his paper stuies a igital barrier option where the monitoring perio for barrier hits commences at the exponentially istribute ranom time, an lasts for the perio of length. We note that in the relevant literature, the starting time of monitoring perio has been given by a preetermine time. his paper investigates the case that the monitoring perio starts at ranom, which is triggere by an event. he exponential istribution has been wiely use in many areas, incluing life testing, reliability an operations research ue to its nice mathematical form an the characterizing memoryless property. See for example Epstein & Sobel 954), Marshall & Olkin 967) an Embrechts & Schmili 994). It is well known that an exponential ranom variable has two important interpretations; one is the waiting time until the first arrival an the other is the lifetime of an item that oes not age. he use of barrier options, igital options an other path-epenent options has increase ramatically by financial institutions. hese instruments are use for a wie variety of heging, risk management an investment purposes. See Golman et al. 979), Hull 6, Chapter ), Rich 997) an Reiner & Rubinstein 99), an the references therein for etails. Moreover, in recent years, many insurance companies have introuce more innovative esigns for their savings proucts. hese proucts usually have barrier option-like features which allow the policyholer to participate in favourable investment performance while maintaining a floor guarantee on the benefit level. Suppose that the barrier for the unerlying asset price appears by an event, such as efault of a bon. We assume the istribution of the time of this event is exponential with mean /, an its value oes not epen on the path of the unerlying asset price of the option. In this paper, we erive a closeform valuation formula for a igital barrier option of knock-out type which initiates at exponential time. Numerical results show the option price as a function of parameter with ifferent own barriers an strike prices. his paper is organize as follows. Section presents the explicit price formula for igital knock-out options where monitoring of the barrier starts at time that is exponentially istribute. Section 3 shows an example an graphs to illustrate our result.. Digital barrier options Let r be the risk-free interest rate an σ> be a constant. We assume the price of the unerlying asset S t follows a geometric Brownian motion S t = S expr σ /)t + σ W t ) where W t is a stanar Brownian motion uner the risk-neutral probability P. Let X t = /σ ) lns t /S ) an μ = r/σ σ/. hen X t = μt + W t. We efine the minimum for X t to be m b a = inf X t) t [a,b] an enote by E P the expectation operator uner the P-measure an by {F t } a filtration for Brownian motion W t. Suppose that τ is an exponential ranom variable with parameter an a barrier option with the lifetime of length is initiate at time τ, i.e. the monitoring perio for asset-price barrier is [τ, τ + ]. We fix a own barrier D< S ), a strike price K an assume K > D. We efine = /σ ) lnd/s ) an k = /σ ) lnk/s ). Consier a igital barrier option of knock-out type which pays out the amount of A when the unerlying asset price oes not cross the barrier D in [τ, τ + ] an is greater than the strike price K at

3 DIGIAL BARRIER OPION CONRAC WIH EXPONENIAL RANDOM IME 3of9 expiration time τ +. he payoff of this option is zero if the unerlying asset price crosses the barrier D or falls below the strike K at expiration. We erive a close-form formula for the price of this option. heorem. he value V of a igital knock-out option with the lifetime of length which initiates at exponential ranom time τ with parameter is where [ V = A e r + r N ) f+ N + ) e +r) f k + N ) + e +r) f+ k N + ) e +r) f k N + k )] ) D μ/σ [ A e r S + r N e +r) f k + k N + e +r) f k + N f m ± = = + )] + ) + f N + ) + e +r) f k N ) ) + k + μ ± ) eμ± +r)+μ )m, k + μ, = k an N ) is the cumulative stanar normal istribution function. Proof. he igital knock-out option value V at time is given by V = E P [e rτ+) A {m τ+ τ >,S τ+ >K} ] = AE P [e rτ+) {m τ+ = A = A τ >,X τ+ >k} ] e t e rt+) PX t z)pm t+ t >, X t+ > k z) t e r e +r)t πt e z μt) /t tpm > z, X > k z) z. ) he inner integral in the last equation of ) is the form of Laplace transform see Fusai & Roncoroni, 8, p. 5). Writing in more etail, e r e +r)t πt e z μt) /t t = e μz r = e +r+μ /)t πt e z) /4t t eμz r z +r)+μ.

4 4of9 D. JUN AND H. KU hus V = A e r = A e r e μ z) N eμz z +r)+μ Pm > z, X > k z) z [ ) eμz z +r)+μ k + z + μ N k z + μ )] z where the last equation is from the Proposition A.8.3 of Musiela & Rutkowski 5, p. 656) an N ) is the cumulative stanar normal istribution function. We have the following equation with two integrals for V: V = A e r [ S ) +r)+μ k + z + μ eμz z N z ) D μ/σ ) ] +r)+μ k z + μ e μz z N z. For the first integral ) +r)+μ k + z + μ I := eμz z N z, since <, we can write I = + ) eμ+ +r)+μ k + z + μ )z N z = I + I. ) +r)+μ k + z + μ eμ )z N z We now calculate I an I by the integration by parts. Let a ± μ) = μ ± ).

5 DIGIAL BARRIER OPION CONRAC WIH EXPONENIAL RANDOM IME 5of9 hen I = [a + μ) e μ+ +r)+μ k + z + μ )z N = a + μ)n )] [ a + μ) e μ+ +r)+μ )z exp ) ] k + z + μ z π ) k + μ a + μ) e μ+ ) +r)+μ k + + μ ) N [ a + μ) exp μ + )z ) ] k + z + μ z π ) k + μ = a + μ)n a + μ) e μ+ ) +r)+μ k + + μ ) N a + μ) e +r)+μ+ +r)+μ )k = a + μ)n exp π ) k + μ +r)+μ a + μ) e μ+ ) N a + μ) e +r)+μ+ +r)+μ )k N + a + μ) e +r)+μ+ +r)+μ )k N z k ) z ) k + μ k ) k ) an I = a μ)n ) k + μ +r)+μ a μ) e +r)+μ )k k ) N. Since a + μ) a μ) = / + r), I = ) ) k + μ +r)+μ k + μ + r N a + μ) e μ+ ) N +r)+μ a + μ) e +r)+μ+ )k k ) N + a + μ) e +r)+μ+ +r)+μ )k N a μ) e +r)+μ +r)+μ )k N k ) k ).

6 6of9 D. JUN AND H. KU Similarly, we compute the secon integral for V ) e μz z +r)+μ k z + μ N z = ) ) k + μ +r)+μ k + μ + r N + a μ) e μ ) ) N +r)+μ a μ) e +r)+μ )k ) + k ) N + a μ) e +r)+μ +r)+μ )k ) N a + μ) e +r)+μ+ +r)+μ )k ) N + k ) k ). Remark. Consier a igital knock-out barrier option which is initiate at the occurrence of an event. Suppose that the event has not yet occurre at time t >. hen the istribution of its payoff function conitione on F t ) is the same as the one at time, which is implie by the memoryless property of exponential istribution an the Markov property of S t. hus the value V in heorem. remains the same, except S t in replacement of S,attimet. Remark.3 In this section, we treat a own-an-out igital option. he methoology we evelop also works well for the prices of up-an-in, up-an-out an own-an-in options. 3. Example an graphs 3. Example In this subsection, we present an example to explain how heorem. is applie to value a financial contract. Suppose Companies A an B make a eal: Company B makes a payment of $ 6 to Company A in the event of Company A s efault. his contract contains clauses requiring that the stock price of Company B oes not fall below the barrier level D for the perio of length starting from the time of Company A s efault, an at the en of that perio, the stock price is above the strike level K. his conition is interprete as Company B is not in financial ifficulties for a certain perio in the event of Company A s efault. Assume that efault time of Company A has the exponential istribution with parameter. We note that might be etermine by historical efault ata accoring to Company A s bon rating. It might also be given through the negotiations between the companies. We assume that the stock price of Company B follows a geometric Brownian motion with initial price S = an volatility =.3. Also, in this example, D = 8, K =, =.5, r = an =.. hen how much is the premium that Company A shoul pay to Company B for this contract? By the price formula in heorem., we have the premium V = $,4 to the nearest ollar.

7 DIGIAL BARRIER OPION CONRAC WIH EXPONENIAL RANDOM IME 7of9 4.5 x Value D=7 D=8 D=9 D= Fig.. Digital barrier option value with varying an own barrier D option parameters: S =, K =, A = 6, σ =.3, r =.5 an =.5). Value x 5 K=9 K= K= Fig.. Digital barrier option value with varying an strike price K option parameters: S =, D = 8, A = 6, σ =.3, r =.5 an =.5). 3. Graphs We illustrate the properties of our solution obtaine in heorem.. Figure shows how the igital knock-out option value V changes when the parameter varies from. to.5 an own barrier D takes the values of 7, 8 9 or 95. he other parameters are given by S =, K =, A = 6, σ =.3, r =.5 an =.5. Figure shows how the igital barrier option value V changes when the parameter varies from. to.5 an strike price K takes the values of 9, or. he other parameters are given by S =, D = 8, A = 6, σ =.3, r =.5 an =.5.

8 8of9 D. JUN AND H. KU 6 x 5 Value D K 95 9 Fig. 3. Digital barrier option values with varying own barrier D an strike price K option parameters: S =, =, A = 6, σ =.3, r =.5 an =.5). In Figure, we observe that the igital knock-out option value V increases as increases for D = 7, 8 or 9. An interpretation of this is as follows: if the exponential parameter increases, τ is more likely to be small. hen, the probability of the unerlying asset price crossing the own barrier D in the time perio of length gets smaller, which results in the increase of V. However, if the own barrier gets closer to initial stock price S, this argument oes not work an V may not increase see the graph for D = 95). In Fig., we observe if the strike price K takes the larger value, the igital knock-out option price becomes smaller. his property of V is the same as that of the regular barrier option price. Also, one can expect for a fixe, V increases when D or K ecreases. Figure 3 emonstrates this esiring property of V. Funing his research has been partially supporte by Natural Sciences an Engineering Research Council of Canaa, Discovery grant References Embrechts, P. & Schmili, H. 994) Moelling of extremal events in insurance an finance. Math. Methos Oper. Res., 39, 34. Epstein, B. & Sobel, M. 954) Some theorems relevant to life testing from an exponential istribution. Ann. Math. Stat., 5, Fusai, G. & Roncoroni, A. 8) Implementing Moels in Quantitative Finance: Methos an Cases. Springer- Verlag, Berlin: Springer. Geman, H. & Yor, M. 996) Pricing an heging ouble-barrier options: a probabilistic approach. Math. Finance, 6, Golman,M.B.,Sosin,H.B.&Gatto,M.A.979) Path epenent options: buy at the low, sell at the high. J. Finance, 34, 7. Heynen, R. C. & Kat, H. M. 994) Partial barrier options. J. Financ. Eng., 3, Hull, J. 6) Options, Futures, an Other Derivatives, 6th en. Englewoo Cliffs, NJ: Prentice-Hall. Kunitomo, N. & Ikea, M. 99) Pricing options with curve bounaries. Math. Finance,,

9 DIGIAL BARRIER OPION CONRAC WIH EXPONENIAL RANDOM IME 9of9 Marshall, A. W. & Olkin, I. 967) A multivariate exponential istribution. J. Amer. Statist. Assoc., 6, Merton, R. C. 973) heory of rational option pricing. Bell J. Econ. Manage. Sci., 4, Musiela,M.&Rutkowski,M.5) Martingale Methos in Financial Moelling, n en. Springer-Verlag, Berlin: Springer. Pelsser, A. ) Pricing ouble barrier options using Laplace transforms. Finance Stoch., 4, Reiner, E. & Rubinstein, M. 99) Breaking own the barriers. Risk, 4, Rich, D. 997) he mathematical founations of barrier option pricing theory. Av. Futures Options Res., 7, 67 3.

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