Advanced Topics in Derivative Pricing Models Topic 2 - Lookback style derivatives 2.1 Product nature of lookback options 2.2 Pricing formulas of European lookback options • Floating strike lookback options • Fixed strike lookback options 2.3 Rollover strategy and strike bonus premium 2.4 Differential equation formulation 2.5 Multistate lookback options 2.6 Dynamic fund protection • Fixed number of resets 1 2.1 Product nature of lookback options The payoff of a lookback option depends on the minimum or maxi- mum price of the underlying asset attained during certain period of the life of the option. Let T denote the time of expiration of the option and [T0;T ] be the lookback period. We denote the minimum value and maximum value of the asset price realized from T0 to the current time t (T0 ≤ t ≤ T ) by mt = min S T0 ξ T0≤ξ≤t and Mt = max S T0 ξ T0≤ξ≤t 2 • A floating strike lookback call gives the holder the right to buy at the lowest realized price while a floating strike lookback put allows the holder to sell at the highest realized price over the lookback period. • Since S ≥ mT and MT ≥ S so that the holder of a floating T T0 T0 T strike lookback option always exercise the option. • Hence, the respective terminal payoff of the lookback call and put are given by S − mT and MT − S . T T0 T0 T • A fixed strike lookback call (put) is a call (put) option on the maximum (minimum) realized price. The respective terminal payoff of the fixed strike lookback call and put are max(MT − T0 X; 0) and max(X − mT ; 0), where X is the strike price. T0 3 • An interesting example is the Russian option, which is in fact a perpetual American lookback option. The owner of a Russian option on a stock receives the historical maximum value of the asset price when the option is exercised and the option has no pre-set expiration date. Under the risk neutral measure, the stochastic price process of the underlying asset is governed by ! ! 2 dSt St σ = r dt + σ dZt or d ln = dUt = r − dt + σ dZt; St S0 2 2 St σ where Ut = ln and µ = r − . S0 2 4 2.2 Pricing formulas of European lookback options We define the following stochastic variables mT y = ln t = minfU ; ξ 2 [t; T ]g T S ξ MT Y = ln t = maxfU ; ξ 2 [t; T ]g; T S ξ and write τ = T − t. Here, S is the asset price at the current time t (dropping the subscript t for brevity). Downstream barrier For y ≤ 0 and y ≤ u, we can deduce the following joint distribution function of UT and yT from the transition density function of the Brownian motion with the presence of a downstream barrier ! ! −u + µτ 2µy −u + 2y + µτ P [U ≥ u; y ≥ y] = N p − e σ2 N p : T T σ τ σ τ 5 Illustration of [UT ≥ u; YT ≥ y] Sξ Uξ = ln is visualized as the restricted Brownian motion with S0 constant drift rate µ and downstream absorbing barrier y. ξ downstream ≥ yT y T ≥ barrier UT u y y T u U ξ 6 Upstream barrier For y ≥ 0 and y ≥ u, the corresponding joint distribution function of UT and YT is given by ! ! u − µτ 2µy u − 2y − µτ P [U ≤ u; Y ≤ y] = N p − e σ2 N p : T T σ τ σ τ By taking y = u in the above two joint distribution functions, we obtain the respective distribution function for yT and YT ! ! − + 2µy + y µτ 2 y µτ P (yT ≥ y) = N p − e σ N p ; y ≤ 0; σ τ ! σ τ ! y − µτ 2µy −y − µτ P (Y ≤ y) = N p − e σ2 N p ; y ≥ 0: T σ τ σ τ The density function of yT and YT can be obtained by differentiating the above distribution functions. 7 Illustration of [UT ≤ u; YT ≤ y] ξ ≤ Upstream UT u ≤ T YT y barrier u y U Y T ξ Time frame S t S T T 0 t T For convenience, we write t t t T S = St, M = M and m = m . M T0 M T0 T0 T0 T t m T0 m T0 8 European fixed strike lookback options Consider a European fixed strike lookback call option whose terminal payoff is max(MT − X; 0). The value of this lookback call option at T0 the current time t is given by h i −rτ T cfix(S; M; t) = e E max(max(M; Mt ) − X; 0) ; where S = S; Mt = M and τ = T − t, and the expectation is t T0 taken under the risk neutral measure. The payoff function can be simplified into the following forms, depending on M ≤ X or M > X: (i) M ≤ X T T max(max(M; Mt ) − X; 0) = max(Mt − X; 0) (ii) M > X T T max(max(M; Mt ) − X; 0) = (M − X) + max(Mt − M; 0): 9 Define the function H by −rτ T H(S; τ; K) = e E[max(Mt − K; 0)]; where K is a positive constant. Once H(S; τ; K) is determined, then ( H(S; τ; X) if M ≤ X c (S; M; τ) = fix e−rτ (M − X) + H(S; τ; M) if M > X − = e rτ max(M − X; 0) + H(S; τ; max(M; X)): • cfix(S; M; τ) is independent of M when M ≤ X because the terminal payoff is independent of M when M ≤ X. • When M > X, the terminal payoff is guaranteed to have the floor value M − X. If we subtract the present value of this guaranteed floor value, then the remaining value of the fixed strike call option is equal to a new fixed strike call but with the strike being increased from X to M. 10 Recall that when X is a non-negative random variable, we have Z 1 E[X] = [1 − FX(t)] dt; if X is continuous: 0 T − Since max(Mt K; 0) is a non-negative random variable, its ex- pected value is given by the integral of the tail probabilities where H(S; τ; K) − = e rτ E[max(MT − K; 0)] Z 1 t −rτ T = e P [Mt − K ≥ x] dx 0 " # Z 1 T − M z = e rτ P ln t ≥ ln dz; z = x + K K S S Z 1 [ ] −rτ y z 1 y = e Se P [YT ≥ y] dy; y = ln dy = dz; z = Se ln K S z Z S " ! !# 1 − 2µy − − −rτ y y + µτ y µτ = e Se N p + e σ2 N p dy K ln S σ τ σ τ 11 p −rτ = SN(d) − e 2 KN(d − σ τ) 3 2 ( )− 2r ( p ) − σ S σ2 2r + e rτ S 4erτ N(d) − N d − τ 5 ; 2r K σ where ( ) S σ2 ln K + r + 2 τ d = p : σ τ The European fixed strike lookback put option with terminal payoff max(X − mT ; 0) can be priced in a similar manner. Write m = mt T0 T0 and define the function −rτ T h(S; τ; K) = e E[max(K − mt ; 0)]: 12 The value of this lookback put can be expressed as −rτ pfix(S; m; τ) = e max(X − m; 0) + h(S; τ; min(m; X)); where Z 1 −rτ T h(S; τ; K) = e P [max(K − mt ; 0) ≥ x] dx Z0 K −rτ T T = e P [K − mt ≥ x] dx 0 ≤ max(K − mt ; 0) ≤ K Z0 K −rτ T = e P [mt ≤ z] dz; z = K − x 0 Z ln K −rτ S y z = e Se P [yT ≤ y] dy; y = ln 0 Z " ! S !# ln K − 2µy −rτ S y y µτ y + µτ = e Se N p + e σ2 N p dy 0 σ τ σ τ 2 − p − σ = e rτ KN(−d + σ τ) − SN(−d) + e rτ S 2 2r 3 ( ) 2 ( ) S −2r/σ 2rp 4 N −d + τ − erτ N(−d)5 : K σ 13 European floating strike lookback options By exploring the pricing relations between the fixed and floating lookback options, we can deduce the price functions of floating strike lookback options from those of fixed strike options. Consider a European floating strike lookback call option whose terminal payoff is S − mT , the present value of this call option is given by T T0 −rτ T cf`(S; m; τ) = e E[ST − min(m; mt )] −rτ T = e E[(ST − m) + max(m − mt ; 0)] − = S − me rτ + h(S; τ; m) p 2 −rτ −rτ σ = SN(dm) − e mN(dm − σ τ) + e S 2 2r 3 ( )− 2r ( p ) 4 S σ2 2r rτ 5 N −dm + τ − e N(−dm) ; m σ where ( ) S σ2 ln m + r + 2 τ dm = p : σ τ 14 Consider a European floating strike lookback put option whose ter- minal payoff is MT −S , the present value of this put option is given T0 T by −rτ T pf`(S; M; τ) = e E[max(M; Mt ) − ST ] −rτ T = e E[max(Mt − M; 0) − (ST − M)] − = H(S; τ; M) − (S − Me rτ ) p 2 −rτ −rτ σ = e MN(−dM + σ τ) − SN(−dM ) + e S 2 3 2r ( )− 2r ( ) S 2 2rp 4erτ N(d ) − σ N d − τ 5 ; M M M σ where ( ) S σ2 ln M + r + 2 τ d = p : M σ τ Remark Through H(S; τ; M) and H(S; τ; X), we can deduce the fixed-floating relation between lookback call and put; the form of which is depen- dent on either M ≤ X or M > X.