ISSUE: How does one explain Jarrow-Rudd formula?

JARROW-RUDD FORMULA: The analysis is based on moments and cumulants. The following terms are defined:

C(F) = call option price based on stock price distribution denoted by F.

C(A) = call price based on known distribution denoted by A.

kj(F) = adjustment based on cumulant for distribution F.

kj(A) = adjustment based on cumulant for distribution A.

k2(F) = mu2(F) or second central moment of F

k3(F) = mu3(F) or third central moment of F

k4(F) = mu4(F) – 3mu22(F) or fourth central moment of F

a(St) = density of A where the derivative of a(St) is evaluated at strike price K.

NOW, the full formula is given as:

(1)   C(F) = C(A) – X + Y

… where C(F) and C(A) are defined above. The terms X and Y are defined as follow;

(2)   X = exp[((-rt(k3(F) – k3(A)) / 3!](da(k) / dSt)

(3)   Y = exp[((-rt(k4(F) – k4(A)) / 4!](d2aK / dS2t) + eK

Now drop the term eK and rewrite the above equation (1) [see Jarrow and Rudd, 1982, p. 354] as:

(4)   C(F) = C(A) + λ1Q3 + λ2Q4

The characteristic of equation (4): -Q3 is skewness and Q4 is the kurtosis. The following definition of λj and Qj applied:

λ1 = g2(F) – g1(A)

λ2 = g2(F) – g2(A)

Q3 = -(S0ert)3(esig2t - 1)3/2 (e-rt / 3!) (da(K) / dSt)

Q4 = (S0ert)4(esig2t - 1)2 (e-rt / 4!) (d2a(K) / dS2t)

g1(F) = (k3(F)) / K23/2

g2(F) = (k4(F)) / K22(F)

The definition of gamma1 and gamma 2: g1 and g2 may be found in Stuart and Ord (1987), p. 107. The terms g1(F) and g1(A) are skewness, and g2(F) and g2(A) are kurtosis. They are defined in Stuart and Ord in terms of cumulants.

REFERENCES:

(1) Black, F., and Scholes, M. (1973): “The Pricing of Options and Corporate Liabilities,”

Joumal of Political Economy, 8 1:637-659.

(2) Jarrow, R., and Rudd, A. (1982): “Approximate Option Valuation for Arbitrary

Stochastic Processes,”Journal of Financial Economics, 10:347-369.

(3) Stuart, A., and Ord, J.K. (1987): Kendall’s Advanced Theory of Statistics, New

York: Oxford University Press.

Many thanks for the answer. I will study this and get back to you. Is it possible for me to take the interaction to the next level through email, perhaps?

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