The relation between cumulative simple periodic returns and the pricing process

K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 357

6. The relation between cumulative simple periodic returns and the pricing process

Here we first derive an analogue of the standard relation between the price process and cumulative simple periodic returns. To this end, define the cumulative return process ˜ Rt by ˜ Rt = Z t 1 ˜Su− d ˜ Su, ˜ R0 = 0 34 with corresponding spot price process ˜ S = ˜S t ≤t≤T given by d ˜ St = ˜St− d ˜ Rt . 35 The solution to 34 and 35 is given by the semimartingale exponential E ˜ R see e.g., Doléans-Dade, 1970, or ˜St = ˜S0E ˜Rt = ˜S0 exp ˜R t − 1 2 [ ˜ R, ˜ R] t Y 0s ≤t 1 + 1 ˜ R s exp −1 ˜ R s + 1 2 1 ˜ R s 2 , 36 where 1 ˜ R t = ˜ Rt − ˜ Rt − and the square brackets signify the quadratic variation process of ˜ R. From represen- tation 14, we know that ˜ R is of bounded variation, sometimes called a quadratic pure jump process, and thus the continuous part of the quadratic variation process [ ˜ R, ˜ R] c t = 0 for all t, implying that [ ˜ R, ˜ R] t = P 0s ≤t 1 ˜ R s 2 . Accordingly ˜St = ˜S0 exp   ˜ R t − X 0s ≤t 1 ˜ R s   Y 0s ≤t 1 + 1 ˜ R s . 37 Since ˜ R is a pure jump process where ˜ R0 = 0, clearly ˜ Rt = X 0s ≤t 1 ˜ R s . So from representation 37, we simply get ˜St = ˜S0 Y 0s ≤t 1 + 1 ˜ R s , 38 which is indeed the present analogue of the standard relation between prices and simple periodic returns in finance. Products are sometimes cumbersome to work with, so we may try to get rid of it. Below we demonstrate two ways of doing that. From the representation of the Lévy measure 15 and the integral representation of K 1 y given by K 1 y = √ 2 Z ∞ e −y √ 1 +2u 2 du, 39 it follows that K 1 y behaves as y −1 as y → 0, νdy behaves as δπ −1 y −2 dy as y → 0, implying that the small jumps are dominating the behavior of ˜ R = ˜ R t ≤t≤T . However, if 1 ˜ R t ≤ −1 for some t, then ˜S t ≤ 0. At such a point in time the company is bankrupt and the stock is nil worth. Therefore, we technically kill the process ˜ R at the first time this happens. This we do as follows: let τ = inf{t 0 : ˜ R t , ˜ R t − ∈ C }, where C = {x, y : x ≤ y − 1, x ∈ −∞, ∞}. Such a τ is known to be an F t -stopping time. Now define Rt = ˜ R t for t τ, † for t ≥ τ, 40 358 K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 where † is the cemetery state, a point not in −∞; ∞. Here we may choose † = −∞. Our stopped process R = R t ≤t≤T is a strong Markov process with transition probabilities P t x, A = P ˜ R t ∈ A, τ t| ˜ R = x = P x ˜ R t ∈ A, τ t for x ∈ −∞, ∞, P t x, † = P x τ t , P t †, † = 1. From the representation 14 of ˜ R we notice that the last term is a compound Poisson process, and the term R −∞,−1 N t ; dy is a Poisson process with parameter λ = λα, β, δ = ν−∞, −1 = R −∞,−1 νdy, where λ can be numerically computed using 15. From this, it follows that P t x, † = P x [τ ≤ t] = 1 − e −λt , t ≥ 0. 41 Returning to the version of ˜ S given in 37, the corresponding price process ˜ S = ˜S t ≤t≤T killed at τ is ˜St = ˜S exp { ˜ Rt − R −1y yNt ; dy + R −1y ln1 + yNt; dy}, 0 ≤ t τ ∧ T , 0, τ ≤ t ≤ T . Thus, if ˜ R hits † before time T , ˜ S is set to zero, or R τ = † ⇔ ˜S τ = 0 for τ ≤ T . We simplify by noticing that since the small jumps dominate the behavior of the process R, the term R ln1 + yNt; dy − R yNt; dy will be small for most values of t , so we choose to ignore it in the remainder. This approximation seems reasonable when the process is estimated to fit stock market data, in which case it amounts to a common approximation for asset returns. Doing this, we get the following pricing process: St = S e ˜ R t , ≤ t τ ∧ T , 0, τ ≤ t ≤ T , 42 which, in view of the definition of R = R t ≤t≤T given in 40 can simply be written as St = S0 exp{Rt}, ≤ t ≤ T . 43 The relationship 43 almost gives the usual connection between prices and accumulated logarithmic returns as in Eq. 17, but takes account of bankruptcy. It may seem a little cryptic what we have done above, so let us present a second and perhaps more natural way to get rid of the product in 38. Consider the version of ˜ S given in Eq. 38, as soon as some 1 ˜ R s ≤ −1 the associated price ˜ Ss becomes zero, or negative, so we stop the process the first time = τ this happens. Taking natural logarithms in Eq. 38 and denoting the associated stopped price process by ˆ S, we then get ˆSt = ˆS exp { P 0s ≤t ln1 + 1 ˜ R s }, 0 ≤ t τ ∧ T , 0, τ ≤ t ≤ T . 44 Define ˆ Rt = P 0s ≤t ln1 + 1 ˜ R s , ≤ t τ ∧ T , −∞, τ ≤ t ≤ T . 45 Then it follows that ˆSt = ˆS0 e ˆ Rt , ≤ t ≤ T . If the intermediate period returns 1 ˜ Rs are small enough, typically the case for intra-day returns on common stocks, then ln1 + 1 ˜ Rs ≈ 1 ˜ Rs. Since the increments of the process ˜ R are independent, and if the errors are not systematic to one side or the other, it is likely that the following approximation holds with reasonable accuracy: X 0s ≤t ln1 + 1 ˜ Rs ≈ ˜ Rt , t τ, K.K. Aase Insurance: Mathematics and Economics 27 2000 345–363 359 ˆ Rt ≈ ˜ R t , ≤ t τ ∧ T , −∞, τ ≤ t ≤ T , 46 which means that ˆ Rt ≈ Rt as given in Eq. 40 in which case ˆSt ≈ St = S0 e Rt , ≤ t ≤ T , where St is the same as given in Eq. 43. We now make the following final approximation. Assumption 3. P t x, A = P x ˜ R t ∈ A, τ t = Q t x, A e −λt 47 for any t ≥ 0, and Borel set A of real. This is only correct if τ is P x independent of ˜ R for all x. In other words, we construct a new ˜τ, by possibly enlarging the probability space, having the same probability distribution as τ and being P x independent of ˜ R for all x, and consider the process ˜ R killed at ˜τ. The Feller probability transition functions of this constructed process is given by the right-hand side of 47. Assumption 3 may be reasonable in the present model if the default probability of the company is independent of the return level of its stock. The underlying process may not strictly satisfy this property, but we shall nevertheless work with the constructed process having transition probabilities given by the right-hand side of Eq. 47 in the sequel. We can now alter the analysis of Sections 3 and 4 using the above approximations. Conditioning on the event {ω : τ ω T } of no failure by time T in Eq. 20, then in Eq. 21 we get µ − λ instead of µ in the exponent. In Eq. 22, we get λ − µ instead of −µ in the exponent as well, and the equilibrium restriction between the parameters in Eq. 23 now becomes ρ − µ + λ = δ p ψ − p ψ + 2η − 2β − 1. 48 However, Eq. 24 for the equilibrium interest rate r remains unchanged. Eq. 27 gets a µ − λ instead of µ in the expression for a, while the equations for the price of the European call option are changed slightly. Eq. 28 gets two corrections in both the exponents of the exponential multiplication factors, where the term ρ changes to ρ + λ both places, and finally, in Eq. 29 the exponent of the exponential in front of the integral sign is changed from ρ to ρ + λ as well. All these changes are of course now approximations.

7. The survival hypothesis