Pricing Formula for 3-Period Discrete Barrier Options Chun-Yuan Chiu Down-and-Out Call Options We first give the pricing formula as an integral and then simplify the integral to obtain a formula similar to the Black-Scholes formula he Integral Given the geometric Brownian motion S t = K r S > K +W > K K S 0 r W > S t > H H S 0 r W > S t > H H S 0 r W t > S > H H S 0 r W > H H r t+w t rewrite the important events: t K H H he fair value of a down-and-out barrier call option: B = e r E S + {St >H & S t >H & S >H} = e r r E +W {W >K & W >H & W t >H & W > } = e r r +z nx y z dxdydz where nx y z is the joint density function of W W t W a normal random vector with mean 0 0 0 and covariance matrix t t t t o Simplify In the following argument we assume K > H or equivalently K > If this is not the case by replacing all K by the argument still works fine First normalize the normal random vector Plug in x = x y = y z = z t t
to Eq to get B = e r K = e r K H t H t Ke r K H t H t H t r r + z nx y z dx dy dz + z nx y z dx dy dz nx y z dx dy dz 3 H t Now n refers to the normal joint density function n x = Σ π 3 e x Σ x with covariance matrix Σ = t t t t t t Denote by Nx y z; 3 3 the standard trivariate normal distribution function Eq 3 is clearly Ke r N H H K t t ; t t t t As for Eq we need the following lemma Lemma Completing the Square in Matrix Form If A is symmetric and invertible then x A x + b x = x + A b A x + A b 4 b A b Proof : Expand x µ A x µ + c and compare to x Ax + b x hus Eq is = K K H t H t H t e z nx y z dx dy dz Σ π 3 e x Σ x z dx dy dz 4 H t where x Σ x z can be simplified by the above lemma as follows: x = x 0 y A = Σ b = 0 z A b = t 4 b A b = b A b = x Σ x z = x A x + b x = x + A b A x + A b 4 b A b = x y t z x Σ y t z
Plug this into Eq 4 to get K H t Σ π 3 e x Σ x z dx dy dz H t where = S 0 K H t t H t nx y z dx dy dz x x y = y t z z and n stands for the joint density function of x y and z Finally we simplify Eq as S 0 N H + H + t K + t t ; t t t t Note that H = H S 0 H = H S 0 = H S 0 K = K S 0 r r r r t H t = H t = = H K + r r t K = r r t H t + = H t + t = + = + = K + r+ t r+ r+ r+ t t Put everything together to get B = S 0 N r + Ke r H N + r Up-and-Out Put Options r + t t r t t K r + K t t t t ; r t t ; t t 5 In a similar manner we first list the pricing formulae in integral form for all knock-out type options in the following table Option ype Important Events Value in Integral Form r +z nx y z dxdydz r +z Down-and-Out Call S > K S t > H S t > H S > H e r K H H Down-and-Out Put S < K S t > H S t > H S > H e r K H H Up-and-Out Call S > K S t < H S t < H S < H e r K H H Up-and-Out Put S < K S t < H S t < H S < H e r H H K r K nx y z dxdydz +z r nx y z dxdydz +z nx y z dxdydz {K>H} {H>K} 3
Simplify the last integral to get the pricing formula for an up-and-out put H S 0 N + r + r + t K r + t ; +Ke r H N + r r t K r t 3 Down-and-Out Put Options he value of a down-and-out put is zero if K H If K > H our goal is to simplify the integral e r H H K For the first term note tha r e r = e r +z nx y z dxdydz = e r H H K e r H H H H H H K r r r H H +z nx y z dxdydz +z nx y z dxdydz t t t t H3 +z nx y z dxdydz t t ; t t H H 6 7 where the first term of the RHS happens to be a down-and-out call For the second term we also use the identity H3 e r r K +z nx y z dxdydz = e r e r H H H H r r +z nx y z dxdydz +z nx y z dxdydz Subtracting the above two identities the integral that we want to simplify in Eq 7 is e r = e r H H H H K H3 = Down-and-Out Call e r r +z nx y z dxdydz H H H H r +z nx y z dxdydz where the last term of the RHS is also similar to a down-and-out call see Eq and hence has a simplified formula similar to Eq 5 hus Eq 7 can be written as Down-and-Out Call S 0 N r + r + t r + t t t ; t t e r H N + r r t r t t t ; t t his identity is similar to the derivation of the put-call parity 4
4 Up-and-Out Call Options he idea is very similar to the above section If H K the option value is zero Otherwise we use the decomposition e r H3 K = e r H3 = e r H3 H H H H H H r H r +z nx y z dxdydz H +z nx y z dxdydz + Up-and-Out Put where the first term of the RHS is also similar to an up-and-out put and has a simplified formula like Eq 6 hus the pricing formula for an up-and-out call option is H Up-and-Out Put S 0 N + r + r + t r + t t t ; t t + Ke r H N + r r t r t t t ; t t 5 Knock-In Options As an example assume we are to derive a down-and-in call he event we need is actually not S > K S t < H S t < H S < H but S > K S t < H S t < H S < H So the value in integral form is not just a single one term like e r H3 K H H r +z nx y z dxdydz If we write down the fair price as an integral there will be 7= 3 terms much more complicated than knockout type options In this situation pricing by first deriving the integral is no longer effective One way to derivative down-and-in options is through the in-out parity down-and-in call + down-and-out call = Black-Scholes call; down-and-in put + down-and-out put = Black-Scholes put; up-and-in call + up-and-out call = Black-Scholes call; up-and-in put + up-and-out put = Black-Scholes put 5