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Jamshidian’s Trick for Pricing Swaptions as Bond Options

Article Quant Q&A · Author: reteip
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Summary

Jamshidian’s trick expresses a payer swaption price as a sum of options on zero-coupon bonds. The approach first finds a critical short rate at which the discounted value of the swap’s cash flows reaches the exercise threshold. In the Vasicek example, analytical bond prices reduce this step to solving an equation involving the model’s bond price factors and the critical rate.

The document asks how to find that rate when a short-rate model has no analytical zero-coupon bond formula, using Exponential Vasicek as an example. It does not answer the question or provide a numerical demonstration, so it leaves the numerical method and any assumptions about solving the equation unexplained. The cited references are offered as further reading, but the document itself gives no detail about numerical techniques or the conditions under which the decomposition applies.

Key ideas

  • Jamshidian’s trick decomposes a swaption into a sum of zero-coupon bond options.
  • The decomposition requires finding a critical short rate that makes the swap cash-flow value equal the exercise threshold.
  • For Vasicek, analytical zero-coupon bond prices provide an equation for that critical rate.
  • The document leaves open how to find the rate when bond prices are available only numerically.

Tags

Fixed incomeOptionsDerivatives pricing
Full text
# Jamshidian's trick for Swaptions


# Jamshidian's trick for Swaptions












Following Brigo$^1$ p.77, we can decompose the price of a swaption as a sum of Zero-Coupon bond options (Jamshidian's Trick).

To do so, the authors suggest to find $r^*$ the value of the spot rate at $t$ for which $ \sum_{i=1}^n c_i P(t,T_i, r*)= 1$

As an example they show this for the Vasicek Model, where $A()$ and $B()$ give the analytical ZCB Prices.

$ \sum_{i=1}^n c_i A(t,T_i) e^{-B(t,T_i)r*} = 1$

Assume we have a short-rate model for which we do not have an analytical expression for the ZCB prices (e.g. Exponential Vasicek), how can we still find the value of $r^*$?

See another $^2$ step-by-step explanation on the problem

1) http://link.springer.com/book/10.1007%2F978-3-540-34604-3

2) https://papers.ssrn.com/sol3/papers2.cfm?abstract_id=2246054

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.