Internal Design of HWA Engine

Implementation

Zero Coupon Bond Price

The zero coupon bond price for Hull White One Factor Model is calculated using the following:

$$P(t,T) = A(t,T) e ^ {-B(t,T) r(t)}$$

$$B(t,T) = 1 - \frac {e ^ {-(T-t)} } {a}$$

$$lnA(t,T) = ln \frac{P(0,T)}{P(0,t)} - B(t,T) -\frac{\delta ln P(0,t)}{\delta t} - \frac {\sigma ^ {2}}{4a^{2}} (e ^ {-aT} - e ^ {-at}) ^ {2} (e ^ {2at} - 1)$$

These input parameters are:

$a$ - the mean reversion

$\sigma$ - the volatility

$t$ - the current time

$T$- the maturity

$r(t)$ - the short rate at time t

Equity Option Pricing

The price at time t of a European Call option with strike X, maturity T on a discount bond maturing at time S is given by:

$$ZBC(t,T,S,X) = P(t,S)\theta(h) - XP(t,T)\theta(h-\sigma^{p})$$

The price at time t of a European Put option with strike X, maturity T on a discount bond maturing at time S is given by:

$$ZBP(t,T,S,X) = XP(t,T)\theta(-h+\sigma^{p}) - P(t,T)\theta(-h)$$

The terms are derived from:

$${\sigma^{p}} = \sigma \sqrt{1 - \frac{e ^ {-2a(T-t)}}{2a}} B(T,S)$$

$$h = \frac{1}{\sigma^{p}} ln(\frac{P(t,S)}{P(t,T)X}) + \frac{\sigma ^ {p}}{2}$$

Cap/Floor

ZBC & ZBP can be used to price caps & floors since they can be viewed as portfolios of zero-bond options:

$$Cap(t,T,N,X) = N \sum_{i=1}^{n}[P(t,t_{i-1})\theta(-h_i + \sigma_p^i) - (1+X_{Ti})P(t,t_i)\theta(-h_i)]$$

$$Flr(t,T,N,X) = N \sum_{i=1}^{n}[(1+X_{Ti})P(t,t_i)\theta(h_i)-P(t,t_{i-1})\theta(h_i-\sigma_p^i)]$$

The terms are derived from:

$${\sigma_p^i} = \sigma{\sqrt{\frac{1-e^{2a(t_{i-1}-t)}}{2a}}}B(t_{i-1},t)$$

$${h_i} = {\frac{1}{\sigma_p^i}} ln (\frac{P(t,t_i)(1+X_{Ti})}{P(t,t_{i-1})}) + \frac{\sigma_p^i}{2}$$

Implemention

The framework is split into host and kernel code.

Kernel

The kernel directory contains the 3 kernels based on the above formula:

• HWA_K0.cpp contains the bond pricing engine
• HWA_k1.cpp contains the option pricing engine
• HWA_k2.cpp contains the cap/floor engine

Host

The host code (main.cpp) contains the OpenCL calls to invoke each of the kernels and test for accuracy compared to the CPU model (cpu.cpp).