Chapter 8 Utility Indifference ricing and Dynamic rogramming Algorithm In the Black-Scholes framework, we can perfectly replicate an option s payoff. However, it may not be true beyond the Black-Scholes world. In this chapter, we will provide a general pricing framework in terms of the concept of utility indifference. 8.1 Continuous-time Model Let S t ) t be the underlying asset price process. We illustrate the utility indifference pricing from the viewpoint of an option writer. After receiving the premium, the writer has to hedge to reduce his risk exposure. Typically, the entire investment operation to the writer consists of three steps: 1. selling the option at time for the price v for delivery at T); 2. paying out to the buyer the payoff V S T ) at time T; 3. accumulating the profits and losses arising from the self-financing hedging strategy { t } with the underlying asset. That is, at time t, the writer holds t number of underlying asset. Let Z t be the wealth process excluding the initial option premium v and the terminal payoff of the option V S T ). Then, dz t = r Z t S t ) dt + t ds t or dz t rz t dt = t dst rs t dt ), where the initial endowment Z = z. It follows d e rt t) Z t ) = t d e rt t) S t ). 75
76 CHATER 8. UTILITY INDIFFERENCE RICING So, In what follows, we make use of forward prices to simplify the notations. For example, Z t = e rt t) Z t, S t = e rt t) S t, v = e rt v, z = e rt z. dz t = t ds t. The terminal wealth of the investment realized at time T is v + Z T V S T ) = v + z + T t ds t V S T ). The purpose of the writer is to choose t so as to maximize the expected utility of the terminal wealth: T sup E [Uv + z + t ds t V S T ), where Ux) is a utility function monotonically increasing and concave function), and E is a real world expectation. We will always assume that the writer is of constant absolute risk aversion, that is, exponential utility) Ux) = e αx, α >, where parameter α is the measure of risk aversion. Bigger α corresponds to higher degree of risk aversion. In particular, α = + indicates absolute risk aversion while α = corresponds to risk neutrality. Remark 37 Note that v + z + T tds t V S T ) is likely negative. So, the power or log utility might not work in that case. The indifference price of the option, v, is defined as: T αv + z + sup E [ exp = sup E [ exp T αz + t ds t ) t ds t V S T )), which means that the writer is indifferent to his expected utilities between writing the option for the premium and writing no option. Due to the desirable separability of the exponential function, we have the following explicit expression for the indifference price vα, V ) = 1 α = 1 α 1 α 1 α ln inf E ln inf E ln inf E { t} ln inf E { t} T α [ exp α T T T t ds t V T ) t ds t t ds t + αv T t ds t. 8.1)
8.2. RICING IN DISCRETE TIME: DYNAMIC ROGRAMMING 77 In general, for a given underlying price process of S t, the two terms in 8.1) can at least be evaluated by numerical methods. Remark 38 It is worthwhile pointing out that the indifference price is independent of the initial endowment z for the exponential utility function. Remark 39 If we are in the Black-Scholes market, the utility indifference price reduces to the Black-Scholes price. Remark 4 Utility indifference prices are non-linear in the number of options. A question: the utility indifference price presented above is from the point of view of an option s writer. How do we define the utility indifference price from the point of view of an option s buyer? 8.2 ricing in Discrete Time: Dynamic rogramming arallel to the continuous case, we now consider the writing and subsequent hedging of an option in the discrete setting. Without loss of generality, we assume that the initial endowment is. Suppose that a writer writes an option of payoff V T for premium v and follows a hedging strategy { i } at moments {t i }. Let S i be the stock price at time t i. Then at the option s maturity he will end up with a profit or loss in the amount v + N 1 i= i S i+1 S i ) V S N ). Here, we stick to the forward prices for all securities. The writer with exponential risk aversion will seek for a strategy to maximize the utility: which amounts to Define max { i } E [ exp α e αv max E [ exp { i } J ) S i, t i ) = min Et i J 1) S i, t i ) = min Et i v + j=i j=i j= N 1 j= j S j+1 S j ) V S N ) j S j+1 S j ) + αv S N ) j S j+1 S j ) j S j+1 S j ) + αv S N ) ), without option). with option)
78 CHATER 8. UTILITY INDIFFERENCE RICING Then the utility indifference option price is defined by v = 1 α ln J 1) S, ) ln J ) S, ) ). The valuation of J l) S, t ), l =, 1 can be achieved with a dynamical programming procedure: J l) S i, t i ), l =, 1 satisfy with J l) S i, t i ) = min i E t i i S i+1 S i ))J l) S i+1, t i+1, 8.2) J ) S N, t N ) = 1, and J 1) S N, t N ) = expαv S N )). Eq 8.2) can be derived using the following argument. Let us only restrict attention to the case l = 1 as the case l = is just a special case such that V T =. Using the statistical independence we have J 1) S i, t i ) ) = min Et i i S i+1 S i ) [ = min Et i i ) exp i S i+1 S i ) [ = min Et i exp i S i+1 S i ) i E t i+1 min E t i+1 j=i+1 ) ] J 1) S i+1, t i+1 ), j S j+1 S j ) + αv S N ) j=i+1 ] j S j+1 S j ) + αv S N ) which is desired. The dynamic programming procedure 8.2) has a single control variable i and is easily implementable. Differentiating with respect to i we obtain the first-order condition ] = E t i [S i+1 S i )exp i S i+1 S i ))J l) S i+1, t i+1. 8.3) The above equation implicitly defines i, which, for known function JS i+1, t i+1 ), can be solved numerically. In fact the right hand side of 8.3) is a monotonic function of i and the solution is thus unique. To see this we denote the right-hand side by Apparently we have f i ) = E t i [S i+1 S i )exp i S i+1 S i ))J l) S i+1, t i+1. df i ) d i = E t i [S i+1 S i ) 2 exp{ i S i+1 S i )}J l) S i+1, t i+1 > for J l) S i+1, t i+1 ) >. Using the Newton-Raphson method we can calculate i in just a few steps of iterations. roceeding backwardly with 8.2) we will eventually obtain J l) S, ), l =, 1. For later reference we denote the optimal control for J ) and J 1) by { i } and {,1 i }, respectively.
8.3. AN EXAMLE: RICING WITH JUM RISK 79 8.3 An Example: ricing with Jump Risk Let us turn to option pricing in a market, specifically, driven by a jump-diffusion process: ds = µdt + σdb + Y 1)dq, 8.4) S where W t is a Brownian motion, and q is a oisson process defined as {, with probability 1 λdt dq = 1, with probability λdt, and Y 1 is the jump size. There is therefore a probability λdt of a jump in q in the timestep dt. The parameter λ is called the intensity of the oisson process. There is no correlation between the Brownian motion and the oisson process. If there is a jump dq = 1) then S immediately goes to the value Y S. We can model a sudden 1% fall in the asset price by Y =.9. We can generalize further by allowing Y to be a random quantity. In the following, we will restrict attention to a bivariate jump-size distribution defined as { β, with probability b lny ) = β, with probability 1 b for some constant b, 1). The β corresponds to, say for example, a 1% jump is The corresponding mean percentage jump is β = ln1 + 1%).1. k =: E Y [Y 1] = b e β + 1 b )e β 1. As a result, we can compute b by use of k and β : b = 1 + k e β e β e β. Over a small time interval δt, the jump-diffusion process can be approximated by a one-period quadrinomial tree, as is shown in Figure 1, where J > 1 is a positive integer to be determined). p J SJ S p 1 p -1 S1 S-1 p -J S-J Figure 1 The quadrinomial tree.
8 CHATER 8. UTILITY INDIFFERENCE RICING The ending nodes relate to the root S by S e iσ δt, i = J, 1, 1, J. Intuitively, the two branches in the middle correspond to diffusion, while the other two branches correspond to jump. Here we have used the approximation e β =: e Jσ δt, where [ ] β J = σ, δt and [x] means the largest integer smaller or equal to x alternatively, the nearest integer to x). Consequently, we will use b = 1 + k δt e Jσ e Jσ δt e. Jσ δt are The objective real world) probabilities to reach those nodes, matching to the subindices, p J = λδt1 b ), p J = λδt b, p 1 = 1 λδt)1 p d ), p 1 = 1 λδt)p d, where p d = eµδt e σ δt e σ δt e. σ δt In summary, the parameters we need are µ, σ, β, k and λ, as well as α. The implementation of scheme on the multi-period tree is rather straightforward. Below is the algorithm Assume the jump size J = 3). /* Algorithm for utility valuation */ /* Compute terminal values */ For i = 3N 1,...,3N + 1 compute J 1) i,n = expαv TS i,n )) J ) i,n = 1 end /* Valuate J l), l =, 1 */ For j = N 1, N 2,..., For i = 3j 1,..., 3j + 1 Solve = E ti [S,j+1 S ij )exp i S,j+1 S ij ))J l),1,j+1] for Evaluate J 1) ij = E ti,1 i S,j+1 S ij ))J 1),j+1 ], end end /* Compute the indifference price */ v = 1 ln J1) α lnj ) ). i ;
8.3. AN EXAMLE: RICING WITH JUM RISK 81 /* The end of the algorithm */ In the complete market driven by Brownian diffusion, the drift term µ t does not enter pricing. It is no longer the case in an incomplete market. We use the following parameter values: spot asset price S = 1; annualized volatility for diffusion σ = 25%; jump intensity λ = 12. maturity T = 1 months or 1/12 year; strikes X: from 85% to 115% of S, mean jump size k =.5...44 r=, σ=.25, T=1mth, k=.5, β=.1, λ=12, α=1.43 µ=.1 implied volatility.42.41 µ=.5.4 µ=.39.38.8.85.9.95 1 1.5 1.1 1.15 1.2 1.25 strike price over asset price Figure 2. rice sensitivity to the rate of stock return