債券オプションの値段計算式

最近ヴァシチェックの利子模型を学びながらオプションの値を計算する方法に目を通した。ブラック・ショールズに似ているところもあるし違うところもあります。Interestingly, the formula was derived by Jamshidian. He begins with a term-structure equation which, being general,includes a continuous payout rate h. However, the solution contains some ambiguous terms like "g" (learned to be the terminal value for a given rate, i.e. g(r). The solution to this, U(r,t) or the asset value, is, intuitively, the expected present value of the cash flows. E(g*exp(-Y)+integral(h*exp(-Y))),where Y is the integral of r, so exp(-Y) is the value of a zero-coupon bond with notional of 1. Expectation is just an integral over values and their probabilities, so the E expression can be rewritten as the sum of an integral on g and a double integral on h.
The expression for the Vasicek rate, r, as a solution to dr = a(b-r)dt+sigma*dz is somewhat unclear, as Jamshidian cites Arnold's stochastic differential equations text. This is interestingly derived through a change of variable in the Ornstein-Uhlenbeck equation for dr. Replacing r with f=r*exp(a*t), taking the derivative df/dt, and rewriting dr as df, substituting for dr with the original form, enables the correct form to be derived via algebra and some change of integral bound manipulations. The final value is r=r*exp(-a(s-t))+ravg(1-exp(-a(s-t)))+sigma*integral(s to t)(exp(-a(s-u))dw(u)). This is apparently used to determine the bivariate distribution of r and Y (convenience for the bond price formula), and the r,y distribution is used to obtain p and G. r's movements are normally distributed, as given by the original dr expression (the random component is Brownian motion). In addition, since Y is an integral of r values, it is essentially a summation. As a sum of normal variables is also normally distributed, Y is normal. Since a distribution for an ordered pair of normal variables would also be normal, the (r,Y) pair is bivariate normal. 正規分配の平均と変動率も算出出来ます。Given G in terms of known variables, the the zero-coupon bond price can be extrapolated from the U formula by substituting h=0 and g=1. The expression for P(r,t,T), which is not a function of the integrand, can be moved outside of the integral, and the remaining multiple of g is the probability of the forward rate, thus, the first term of U, P(r,t,T)E(g(R)) is obtained. 瞬間の利率と債券値の分配を由来する方法をまだわからない。 The second term of U, integral(P(rts)E(h(R(rts),s)), is derived similarly. The expression for the discounted continuous payment integral(h*exp(-Y)ds) is converted to a double integral, integral(integral(G(r,r',t,s)h(r',s))dr')ds, which again contains the G term above. This G can be factored into P(r,t,s) and p(r,t,s,Y), where P is the bond price and p is the probability density function of a stochastic variable with mean equal to the forward rate for s observed at t. The second term double integral thus equates to integral(P(r,t,s)E(h(R(r,t,s),s))ds. Thus, the first part of Jamshidian's thesis (about the value of an asset that is the function of a dynamic interest rate (e.g. a bond or interest rate derivative). The ancillary portion defining P(r,t,s) was derived. f (forward rate) was not yet derived.
The second part, namely C=L*P(r,t,s)N(h)-KP(r,t,T)N(h-sigmap) is essentially derived using the Black-Scholes formula with substitution. Black-Scholes for a European call with no dividend is derived using the present value of the expectation of max(S-K,0) at expiry. The derivation is carried out using the assumption of lognormality for S, integration for the expectation, and Ito's Lemma. The formula is: C=S0N(d1)-Kexp(-r*(T-t))N(d2), where d2=d1-sigma*sqrt(T-t), and d1= (ln(S0exp(rT)/K))/(sigma*sqrt(T-t))+sigma*sqrt(T-t). The substitution to carry out is: h for d1, L*P(r,t,s) for S0, and sigmap for sigma*sqrt(T-t). S0 is simple, as r is the current short rate, and P(r,t,s) moves the notional L back to the present. sigmap is derived by writing out the exponential for P in terms of r. The logarithm of P is then linear in r, the only stochastic variable. P is thus lognormal, with volatility equal to the absolute value of the multiplier of r, which is (1-exp(-a(s-T))/a. h is derived by substituting for S0, sigmap for sigma*sqrt(T-t) and P(r,t,T) for exp(rT) into the d1 expression. Substituting h-sigmap for d2 yields the Jamshidian pricing formula for a zero coupon bond option from Black-Scholes.　　将来の利率、計算式の詳細とプォトフォリオのオプションの値はまだ書いていませんがほとんどの定理の証拠が出来ました。