1.Arbitrage-free valuation of fixed-rate, option-free bonds entails discounting each of the bond’s future cash flows at its corresponding spot rate. （理想状态，通常债券会在spot rate 上加多 z - spred 估值）
2. 二叉树利率路径路线为 ： 2的（n-1） 次方，n 为到期年限减去1。 For a n-period binomial tree, there are 2(n-1) possible paths.
3. 债券按照路径估值模式：先计算每条利率路径的pv，再对各个路径的pv值求平均。In the pathwise valuation approach, the value of the bond is simply the average ofthe values of the bond at each path.
4. Mortgage-backed securities have pathdependent cash flows on account of the embedded prepayment option. The Monte Carlo simulation method should be used for valuing MBS as the binomial tree backward induction process is inappropriate for securities with path-dependent cash flows. MBS 是含权债券，不适用二叉树。
5.波动率 利率 和 利率曲线形状对含权债券价格的影响
When interest rate volatility increases, the value of a callable bond (where the investor is short the call option) decreases and the value of a putable bond (where the investor is long the put option) increases. （波动率 --期权价值 ---含权债券价值，波动率不需要分析对价格影响，就可以直接影响到期权价值，所以波动率的分析主要看 买入 还是卖出期权）。波动率大对买入方有利，callable 是卖出期权， putable 是买入期权。）
As interest rates decline, the short call in a callable bond limits the bond’s upside, so the value of a callable bond rises less rapidly than the value of an otherwise equivalent straight bond. （买入一个callable bond 等于卖出一个call 的期权， 利率下跌，债券价格上涨，short call 的亏钱。）
As interest rates increase, the long put in a putable bond hedges against the loss in value; the value of a putable bond falls less rapidly than the value of an otherwise equivalent straight bond. (买入一个putable 等于买入一个put，利率上升，价格下跌，买入put 方赚钱。）
Call option value is inversely related to the level of interest rates, while put option value varies directly with the level of interest rates. （利率上涨，债券下跌，call 没有价值。）
Because a higher interest rate scenario limits the probability of the call option being in the money, the value of a call option will be lower for an upward sloping yield curve. As an upward-sloping yield curve becomes flatter, the call option value increases.
The value of a put option increases with interest rates. When the yield curve is
upward sloping, the probability of the put option going in the money is higher. Put option value therefore declines as an upward-sloping yield curve flattens.
6. OAS 的计算： 债券的先按照无风险利率折现计算理论价格1，再把折现利率改成 rf +oas 后 ，理论价格1会变成市场价格。
7. how interest rate volatility affects option-adjusted spreads.
when the estimated (or assumed) volatility (of benchmark rates) used in a binomial tree is higher, the computed value of a callable bond will be lower—and therefore closer to its true market price. The constant spread that needs to be added to the benchmark rates to correctly price the bond is therefore lower.
To summarize, as the assumed level of volatility used in an interest rate tree increases, the computed OAS (for a given market price) for a callable bond decreases. Similarly, the computed OAS of a putable bond increases as the assumed level of volatility in the binomial tree increases.
假设的波动率越大，理论的callable 越小，折算回市场价格所需的oas 越小。
假设的波动率越大，理论的putable 越大，折算回市场价格所需的oas 越大。
8. Effective duration of floater ≈ time (years) to next reset.
an increase (decrease) in rates would decrease the effective duration of a putable (callable) bond.
callable bond 临界点左边 convexity 小于0，右边大于0，两点连线后函数在连线下方叫做convexity 大于0，convexity大于0债券价格意味上升快，下跌慢。
When rates are high, callable bonds are unlikely to be called and will exhibit positive convexity. When the underlying call option is near the money, its effective convexity turns negative; the upside potential of the bond’s price is limited due to the call (while the downside is not protected). Putable bonds exhibit positive convexity throughout.
9. callable bonds will have lower one-sided down-duration than one-sided up-duration（callable ：down 和up 指的是利率下跌和上涨，临界点左边斜率小于右边）; the price change of a callable when rates fall is smaller than the price change for an equal increase in rates. Conversely, a near-the-money putable bond will have larger one-sided down duration than one-sided up-duration.
（puatable ：down 和up 指的是利率下跌和上涨，临界点左边边斜率大于右边）
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