Asian Option Time Notation and the Turnbull-Wakeman Approximation

# Different notations for times variable in Haug's book


# Different notations for times variable in Haug's book












I am reading the book by Haug, 2007 on the pages 186-188 one can find the Turnbull and Wakeman approximation for arithmetic avarage rate option.

> The approximation adjusts the mean and variance so that they are consistent with the exact moments of the arithmetic average. The adjusted mean, $b_a$ and variance, $\sigma_a$, are then used as input in the generalized BSM formula: $$ c \approx S e^{(b_a - r)T}N(d_1)-K e^{-rT}N(d_2), $$ $$ p \approx K e^{-rT}N(d_2) - S e^{(b_a - r)T}N(d_1), $$ $$ d_1 = \frac{\ln(S/K)+(b_a+\sigma^2_a/2)T}{\sigma_a\sqrt{T}}, \quad d_2 = d_1 - \sigma_a\sqrt{T}. $$ The volatility and cost-of-carry of the average are given: $$ \sigma_a = \sqrt{\frac{\ln(M_2)}{T}-2b_a}, \quad b_a = \frac{\ln(M_1)}{T},$$ where $$ M_1 = \frac{e^{bT}-e^{bt_1}}{b(T-t_1)}, $$ $$ M_2 = \frac{2e^{(2b+\sigma^2)T}}{(b+\sigma^2)(2b+\sigma^2)(T-t_1)^2} + \frac{2e^{(2b+\sigma^2)t_1}}{b(T-t_1)^2}\left(\frac{1}{2b+\sigma^2} - \frac{e^{b(T-t_1)}}{b+\sigma^2}\right), $$ if $b = 0$, then $$M_1=1$$ and $$ M_2 = \frac{2e^{\sigma^2 T}-2e^{\sigma^2 t_1}(1+\sigma^2(T-t_1))}{\sigma^4(T-t_1)^2}, $$ where $t_1$ - is the time to the beginning of the average period. Extension to hold for options on futures. If the option is into the average period, the strike price must be replaced by $\hat{K}$, and option value must be multiplied by $T_2/T$, where $$ \hat{K} = \frac{T_2}{T} K - \frac{\tau}{T} S_{ave}, $$ where $\tau=T_2-T$ - is the reminding time in the average period, $T_2$ -- original time in average period in years, constant over life of options. If we are into the average period, $\tau>0$ and $\frac{T_2}{T} K - \frac{\tau}{T} S_{ave}<0$, then a call option will for certain be exercised and is equal to the expected value of the average at maturity minus the strike price $$ e^{-rT}(\mathbb{E}(S_{ave})-K),$$ where the expected average at maturity is equal to $$\mathbb{E}(S_{ave})=S_{ave}\frac{T_2-T}{T_2} + S \cdot M_1 \frac{T}{T_2}. $$

In the quote above one can see four different notations for times: $t_ 1$, $\tau$, $T_ 2$ and $T$. $T$ is standard notation, while the $t_ 1$, $\tau$, and $T_ 2$ are not.

Question. Could some one explain what does mean $\tau$, $T_ 2$ and $t_1$ with a visualisation for these notationds on a timeline?

$t_1$ - the time to the beginning of the average period.

$\tau$ - the reminding time in the average period.

$T_2$ - original time in average period in years, constant over life of options.

$T$ - time to maturity in years of options.

My attempt in days is:

Ref. Espen G. Haug. The Complete Guide To Option Pricing Formulas. McGraw Hill, 2007

## Answer by Jayfish (score 1)

https://quant.stackexchange.com/a/79095

Just reading these section for Asian option too. Also found those notation annoying. Especially, discriptions below cannot make any sense.

The adjusted strike comes from the idea:

CallOption(at maturity)

= max{ $S^{fixing}_{avg} * \frac{T_2-T}{T_2}$ +$S^{nonfixing}_{avg} * \frac{T}{T_2}$ -K , 0}

= $\frac{T}{T_2}$ max{ $S^{nonfixing}_{avg}$ -(K * $\frac{T_2}{T}$-$S^{fixing}_{avg} * \frac{T_2 -T}{T}$) , 0}

Here,

$T_2$ : times between option start date and option maturity date.

$T$ : times between pricing date to opiton maturity date.

$T_2-T$ : times from start date to pricing date, which already past. Same as $\tau$ in Haug's book.

under these notation, "the option value must multiplied by $\frac{T_2}{T}$" is incorrect obviously.

Would suggest read John Hull's book Ch.26 (26.13 Asian option in 9th edition) and its technical Note. May find it much more friendly in notation. (Especially, the moment $M_1 M_2$ in Hull's book really means the moment for underlying asset, which have a big differece compared to Haug's notation.)

Also wish there will be next version of Haug's book to correct the whole asian option chapter lol.