One of the primary goals of this website is to help you learn the building blocks of quantitative finance by avoiding repetition and by connecting the dots so to speak. This topic will deploy this learning mechanism and underscore why I gave this website its tagline quick concepts.
If you have not already studied and developed familiarity with the mechanism for pricing a European Call you should do so before proceeding.
We now propose a general model for stochastic interest rates.
$$dr_t = \alpha(r_t , t) dt + \sigma (r_t, t)dW_t$$
Observe how this general model bears some semblance to our auxiliary stochastic process $d\widehat{S}_t = r \widehat{S}_t dt + \sigma \widehat{S}_t dW_t$ for stock prices used to value (under the assumption of a zero drift martingale) a European Call $V(S,t)$. There are however, two major differences compared to stock price model.
- The functions $\alpha$ and $\sigma$ depend on time $t$
- The functions $\alpha$ and $\sigma$ come in different “flavours”. The ultimate choice depends on the asset, commodity or interest rate product being modelled and information available from empirical data which indicates which models yield prices most consistent with market prices (where available).
With the general model we can price a zero coupon bond $P_{t,T}$ with maturity $T$. Denote $P_{t,T} = p(r,t;T) \equiv p(r_t , t)$ for simplicity.
As with the option pricing model, we use Itô’s lemma to compute $dP_{t,T}$, the change in the zero coupon bond value over the interval $t, t+dt$
$$dP_{t,T} = a_{t,T} P_{t,T} dt + b_{t,T} P_{t,T} dW_t$$ where $$a_{t,T} P_{t,T} = \partial_t P + \alpha \partial_r P + \dfrac{1}{2} \sigma^2 \partial^2_r P\quad , \quad b_{t,T} P_{t,T} = \sigma \partial_r P$$
This is a good point to pause and compare the properties of a European Option and a Zero Coupon Bond
$V(S, t)$ | $P(t;T)$ |
|
|---|---|---|
| Underlying/Primitive | ||
| Underlying increment | (general form of Vasicek etc) |
|
| Change over $t,t+dt$ - by Itô's lemma | $dV = \partial_t V dt + \partial_s V ds + \frac{1}{2}\sigma^2 \partial^2_s Vdt $ | ⇓ $dP_{t,T} = a_t P_{t,T} dt + b_{t,T}P_{t,T}dW_t$ |
| Hedged Portfolio $\Pi_t$ | ||
| Riskless $d\Pi_t$ (No $dW_t$) | $a_{t,T} = \frac{1}{p}( \partial_t P + \alpha \partial_r P + \frac{1}{2} \sigma^2 \partial^2_r P )$ |
Construct a portfolio of two zero coupon bonds with maturities $T_1, T_2$.
We then construct a hedged portfolio $\Pi_t$
$$\Pi_t = \Delta_{1,t} P_{t,T1} \;-\; \Delta_{2,t} P_{t,T2}$$and equate this to the risk free growth of the portfolio $r \Pi dt$ over the same time interval and thereby obtain a PDE for the bond price.
Group the $dt$ and the $dW_t$ terms.
In the option pricing model, we obtained expression with separate time dependent and risky $dS_t (f(W_t))$ components. We chose $\Delta_t$ to be $\dfrac{\partial V}{\partial S}$ in order to make the risky component $(\partial_s V – \Delta_t ) dS_t$ disappear. Essentially this was simply taking the grouped term that multiplies $dS_t$ and equating it to zero.
$$(\partial_s V – \Delta_t ) = 0$$ Similarly for our bond pricing portfolio we take the entire term that multiplies $dW_t$ and equate it to zero $$ (\Delta_{1t} b_{t,T1}P_{t,T1} \;-\; \Delta_{2t}b_{t,T2} P_{t,T2}) = 0$$