Pricing a European Bond Option in the Ho-Lee Interest Rate Model

# Bond price in Ho-Lee Model


# Bond price in Ho-Lee Model












I know Ho-Lee model and want to extract the price at $t$, of a European call option with strike price $K$ and exercise date $T$, on an underlying $S$-bond, but I don't know what way should I choose:

- PDE approach

- Risk Neutral Valuation

- Forward Measure

Please Help me.So Thanks.

## Answer by user16891 (score 3)

https://quant.stackexchange.com/a/19358

We assume $$dr_t=\alpha(t)dt+\beta dW_{t}^{\mathbb{Q}},$$ and $$dP(t,T)=r(t)P(t,T)dt+\Sigma(t,T)P(t,T)dW_{t}^{\mathbb{Q}},$$ in the Ho-Lee model, bond prices are given by $$P(t,T)=e^{A(t\,,\,T)-B(t\,,\,T)\,r_t}$$ where \begin{align} &B(t,T)=T-t\\ &A(t,T)=\frac{\beta^2}{2}\times\frac{(T-t)^3}{3}+\int_{t}^{T}\alpha(u)(u-T)du\,. \end{align} By application of Ito's lemma, we have $$\Sigma(t,T)=-\beta\times(T-t)\,.$$ Price at $t$ of a European call option with strike $K$ and exercise date $T$, on an underlying zero coupon $S-$bond is given by $$C(t,T,K;S)=\mathbb{E^Q}\left[e^{-\int_{t}^{T}\,r_u du}\,\mathbb{max}\{P(T,S)-K\,,\,0\}\,|\,\mathcal{F}_t\right],$$ now we use change of measure,then $$C(t,T,K;S)=P(t,T)\mathbb{E}^{\mathbb{Q}_T}\left[\mathbb{max}\{P(T,S)-K\,,\,0\}\,|\,\mathcal{F}_t\right].$$ let $Z(t)=\frac{P(t,S)}{P(t,T)}$.We assume that $Z(t)$ follows the Ito process as described by the following stochastic differential equation $$dZ(t)=\{...\}dt+\sigma(t)Z(t)dW_{t}^{\mathbb{Q}}$$ We know $Z(t)$ is a martingale under forward measur $\mathbb{Q}_T$, thus we have $$dZ(t)=\sigma(t)Z(t)dW_{t}^{\mathbb{Q}_T}.$$ We can also be written as $$Z(t)=e^{A(t\,,\,S)-A(t\,,\,T)-[B(t\,,\,T)-B(t\,,\,T)]\,r_t}$$ By application of Ito's lemma, we have $$\sigma(t)=-[B(t\,,\,T)-B(t\,,\,T)]=-\beta\times(S-T),$$ then $$\ln Z(T)\sim N\left[\ln Z(t) - \frac{1}{2}\beta^2\,(S-T)^2(T-t)\,\, ,\beta^2\,(S-T)^2(T-t)\right]$$ on the other hand \begin{align} &P(t,T)\mathbb{E}^{\mathbb{Q}_T}\left[\mathbb{max}\{P(T,S)-K\,\,0\}\,|\,\mathcal{F}_t\right]=P(t,T)\mathbb{E}^{\mathbb{Q}_T}\left[ \mathbb{max}\left\{\frac{P(T,S)}{\underbrace{P(T,T)}_{1}}-K\,,\,0\right\}\,|\,\mathcal{F}_t\right]\\ &\\ &\hspace{7.8cm}=P(t,T)\,\mathbb{E}^{\mathbb{Q}_T}\left[max\{Z(T)-K\,,\,0\}|\,\mathcal{F}_t\right]\\ &\\ &\hspace{7.8cm}=P(t,T)[Z(t)\,N(d_1)-K\,N(d_2)]\\ &\\ &\hspace{7.8cm}=P(t,S)\,N(d_1)-K\,P(t,T)\,N(d_2)\\ \end{align}

where $$d_1=\frac{\ln \frac{P(t,S)}{K\,P(t,T)}+\frac{1}{2}\beta^2\,(S-T)^2(T-t)}{\beta\,(S-T)\sqrt{T-t}}$$ and $$d_2=d_1-\beta\,(S-T)\sqrt{T-t}$$