Jump Diffusionモデルの由来

最近マートン教授の論文を読んでいます。ジャンプ・デイフュージョン・モデルはブラックショールズと似ているところがあるけどポアソン・プロセスも含まれている。裏付けの株の変動性とリスクがない利子の変数の数式がわかりづらいです。
Merton's use of Ito's Lemma to derive the instantaneous expected return on an option and instantaneous volatility is unclear.
The formula is c = sigma(i=1 to infinity)pi*ci, where pi is the Poisson probability of i jumps occurring, namely
pi = exp(-lambda*i)(lambda^i)/i!
and ci is the Black-Scholes vanilla call when exactly i jumps occur, which is the Black-Scholes formula, with substitutions for r and sigma (rj and sigmaj).
rj = r-lambda*k-i*gamma/t, where k is the percentage jump size and gamma = ln(1+k)
sigmaj^2 = sigma^2+i*delta^2/t, where delta^2 is the variance of ln(Y), where Y is the value such that k = expectation(Y-1).

もう一つな不明店が発生した。
Yp-1 = w2(F(SY,t)-F(S,t)-(dF/dS)(S,t)(SY-S))/F(s,t) is not trivially derived. It seems to assume that w1 = -w2(sigmaw/sigma).
These statements are both true, as
w1*sigma+w2*sigmaw = 0
Yp-1 = w1*(Y-1)+w2*(F(SY,t)-F(S,t))/F(S,t)
and sigmaw = (dF/dS)(S,t)*sigma*S/F(s,t),
so one can substitute the first and third formulas into the second to derive the expression above.
次の問題は株価の微分計算です。ds/s は変更率を表す。the partial differential equation for dS/S was solved to yield S/S0.

dS/S= (alpha-lambda*k)dt+sigma*dZ+(Y-1)
S/S0 = exp(((alpha-lambda*k-sigma^2/2)t+sigma*Z)*Y(n))
where Y(n)=pi(i=1 to n)(Yi)
The partial derivatives dS/dt, dS/dZ, and dS/dq do not initially seem to validate this pair.
However, upon further review, if the stochastic derivative dZ/dt=sigma/2, the dS/dt term of S*(alpha-lambda*k) would be correct. The other derivative, dS/dz=sigma, is easily derived, and the (Y-1) term in dS/S results from the dS/dq computation, where lambda*k*t counts as q.
Another problem involves ambiguity about the use of "n" in the Poisson PDF. ｎがジャンプ数かジャンプの平均値かが曖昧です。In fact, the answer is that the intensity, lambda(1+k) describes the instantaneous expected change due to jumps. Multiplying by the time to maturity t induces the expected jump change. If n jumps each with Y-value 1/lambda occurred over t, the change due to jumps would be n. Whether this is the correct formula for the probability of n jumps (as opposed to using the S jump intensity value lambda alone) is unclear. In probability terms, it involves a substitution for the intensity into the original weighted sum formula. However, changing lambda, the intensity of the stock price jump process, to lambda' (intensity for the option) will change the weights for each term i in the summation from (exp(-lambda))(lambda^i)/i! to (exp(-lambda'))(lambda'^i)/i!, which could result in different weights for the conditional if prices and a different final option value.
もう一つの問題点は正式と実際の変動率です。 The formal instantaneous variance in the option pricing fn expression of vn^2 = sigma^2+n*delta^2/t can be derived by considering that, over the lifetime t, the variance from the non-jump component is sigma^2 and the variance from the jump component is n*delta^2 (if n jumps occur).
The differential equation used to derive the jump-enhanced option price is as follows.
rF=(1/2)*sigma^2S^2(d^2F/dS^2)+(r-lambda*k)S(dF/dS)+dF/dt+lambda*Expectation(F(SY,t)-F(S,t))
The derivation of this formula is based on the values of alphaw, sigmaw, and the formula (alpha-r)/sigma=(alphaw-r)/sigmaw.
The derivation is unclear, but apparently the solution to this is F(S,t)=sigma(n=0 to infinity) (((exp(-lambda*t))*(lambda*t)^n/n!)(Expectation(W(SXn*exp(-lambda*k*t),t;E,sigma^2,r)))
Two other comments made about the solution to the PDE are relevant. Firstly, the author claimed that the derivation involved assuming that jump risk can be diversified away. This is apparently due to the assumption that the expected return on the option as well as the stock be r. One further comment is that the appendix derivation computes the time derivative with a sigma(n=0 to infinity)Pn*expectation(W2). It is unclear what is meant by W1 and W2 in this passge, although it is expected that they are the N(d1) and N(d2) terms of Black-Scholes (later, it was found that this could not be the case, either). W11 is also unclear. The author does not explain dN(d1)/dt or dN(d1)/ds, although the latter just the Gaussian PDF, it is expected. The former would be the integral with respect to S of dd1/dt, it is expected. Lastly, the Poisson term in dF/dt seems to be wrong, as the summation(0 to n)*1, and d^2F/dS^2=sigma(n=0 to infinity)(Pn*expectation(d^2W/dS^2)). By the chain rule, dW/dS = (dW/dV)(dV/dS). Since dV/dS=(1/S)V, dW/dS = (V/S)(W1), and Fs = sigma(n=0 to infinity)*2. d^2W/dS^2 = d/dV(dV/dS)*3=(d^2W/dV^2)(dV/dS)^2=W11(V/S)^2. Thus, d^2F/dS^2=sigma(n=0 to infinity)(Pn*expectation(W11(V/S)^2)).
dF/dt was also verified, as, in the derivative of sigma(P)'s (d/dt)(lambda*t)^n the summation's 0 term is 0, so can be dropped and, when n>=1 the n in the numerator and n factor in the n! denominator cancel. In addition, dW/dt = (dV/dt)(dW/dV) + dW/dt(where V is treated as a constant) = (dV/dt)(W1) + W2. As of yet, no clear mathematical error in the article has been found, although ambiguities persist. 曖昧なところはない訳ではありません
One such ambiguity is the probability, over interval [t, t+h] of more than one jump. The number of jumps follows a Poisson distribution with intensity lambda. Thus, the probability of 1 jump in h time interval is lambda*h. However, an additional O(h) term is added to this probability as well as to the probability of 0 jumps (1-lambda*h+O(h)). Lastly, the probability of more than one jump would it is expected, be sigma(i=2 to infinity)((lambda)^i*h)
However, the author provides a value of O(h), suggesting that the probabilities of 0 jumps, 1 jump, and more than 1 jump are mutually exclusive, which it would seem they are not.

*1:lambda*exp(-lambda*t)(lambda*t)^(n-1)/n!)(n-lambda*t) is the expected result. The points about W1, W11, and W2 were clarified to be the partial derivatives dW/dV, d^2W/dV^2, and dW/dt, as W is a function of V (1st parameter) and t(2nd parameter). Next, the derivatives dF/dS and d^2F/dS^2 were clarified, as dF/dS is sigma(n=0 to infinity)(Pn*expectation(dW/dS

*2:V/S)(W1

*3:dW/dV)(dV/dS