为什么risk neutral probability 要转化成forward probability

上derivate product课，对于risk neutral probability 要转化成forward probability这个问题一直没有搞明白。

有种说法是forward price在risk neutral probability  mesure 下信息不够不能计算，或者是计算不够简单。但是觉得解释有点单薄。

不知道哪位大侠知道这个问题。

如果可以很好解释这个问题，本人给10个币吧。

To fully understand this, you have to understand the concept of Numeraire. The so-called risk neutral measure is really saying the risk neutral with respect to using the continuously compounded money market account as Numeraire. The T forward measure is risk neutral with respect to using the T-maturity zero coupon bond as Numeraire. In principle, the derivative price is the same no matter what Numeraire you use because they are using expression the same price with different "unit". In practice, using T forward measure will simplify some calculation especially for pricing some of the European type options, such as cap/floor. Another popular measure is using the so-called annuity as Numeraire, for pricing swaption, for example.

为了简化计算，特别是针对利率衍生品。
1.一般而言，risk-neutral probabilities是建立在利率是常数的分析框架之上的；而绝大多数利率敏感衍生品的分析框架必须放松利率为常数的假设，即采用随机利率。
2.关于risk-neutral probabilites在利率衍生品上定价的缺陷。在neftci的巨著Introduction to the Mathematics of Financial Derivatives的p399有精彩的说明：“In contrst, the risk-neutral measure fist applies a random discount factor to a random payoff,and then does the averaging. Note that in processing this way, the risk-neutral mesure missing the opportunity of using the discount factor implied by the markets during the pricing process.Instead,the risk-neutral measure is trying to recalculate the discount factor from scrath, as if it is the pricing problem, leading to the complicated bivaiate dynamics.” 另一方面forward probabilies中。“This implies that one can replace the future value of a spot rate by the corresponding forward rates to find the current arbitrage-free price of various interest rate dependent securieties. ” “This is an important result because it eliminated the need to calclulate complex correalations between spot rates and future values of interest rate dependent prices.”
3.关于equivalent martingale measure，必须要了解Girsanov Theorme和Radon-Nikodyn Derivatives 或者Girsanov Factor。这方面的应用在陈松男所著之《金融工程学—金融商品创新选择权理论》中会有非常精彩的体现，虽然这本书通常会有一些不经意的小错误。

因为forward比spot更加liquid, 而且使用forward还可以少了交易spot的麻烦（特别是对于原材料）

这是一个非常好的问题，涉及到了equivalent martingale measure。

使用forward probability，discount变得很简单，即为满期时点的无利息债券价格P(0,T)；然后只关注在该测度下，payoff的概率分布，求其平均Except(PayOff at T under the forward measure)即可。价格等于P(0,T)乘以Except(PayOff)。

使用risk neutral probability，比如利用Monte Carlo,需要产生金利的path,以及payoff的path。然后求discount乘以PayOff的平均。

另外，Hull White的三分支树，是建立在risk neutral measure上的。从叶结点回溯到根节点的过程，就体现了求(discount乘PayOff)的平均的过程。

仕事に使っているので、体得出来ました。
本を見るとか、講義を聴くなら、実感がなかなか湧かなくても不思議ではありません。
勉強の段階で、ある程度の認識を持てば済んで、
実際に応用した時、また両方の使い分けを深く考えても構いません。

neftci的巨著Introduction to the Mathematics of Financial Derivatives的p399有精彩的说明

2楼正解，Numeriae。Neftci不算是巨著吧，只是入门的入门。