Quantitative Finance & Derivatives
Discrete Geometric Asian Option and Arithmetic Forward Moment Formula
Price discrete n-fixing Geometric Asian calls exactly via Kemna-Vorst volatility scaling and compute the Turnbull-Wakeman discrete arithmetic forward mean M1.
Distinguish discrete n-fixing Asian volatility from the continuous sig / SQRT(3) limit
Averaging spot prices across n_fix discrete fixings dampens volatility relative to a vanilla European option, reducing option cost for corporate hedgers. A frequent mistake is applying the continuous-averaging factor sig / SQRT(3) = 0.5774 * sig to monthly or quarterly discrete fixings. For n_fix equally spaced fixings over [0, T], exact discrete geometric volatility is sig_G = sig * SQRT(((n_fix + 1) * (2 * n_fix + 1)) / (6 * n_fix^2)).
With spot S = 100 (B2), strike K = 100 (B3), r = 0.06 (B4), carry b = 0.06 (B5), sig = 0.30 (B6), T = 1.0 (B7), and n_fix = 12 monthly fixings (B8), the variance multiplier is (13 * 25) / (6 * 144) = 325 / 864 = 0.376157, so sig_G = 0.30 * 0.613317 = 0.1840 (higher than continuous 0.1732!). Effective geometric carry is b_G = 0.5 * ((n_fix + 1) / n_fix) * (b - 0.5 * sig^2) + 0.5 * sig_G^2 = 0.0251.
| Asian Quantity (n = 12 Monthly) | Exact Formula | Sample Output |
|---|---|---|
| Discrete Geometric Vol sig_G | sig * SQRT((n + 1)*(2*n + 1) / (6*n^2)) | 0.1840 (vs continuous 0.1732) |
| Effective Geometric Carry b_G | 0.5*((n+1)/n)*(b - 0.5*sig^2) + 0.5*sig_G^2 | 0.0251 |
| Kemna-Vorst Geometric Call C_Geo | Generalized BSM(S, K, r, b_G, sig_G, T) | 8.2498 (Strict lower bound on Arith) |
| Turnbull-Wakeman Arith Forward M1 | S * (1/n) * dt_1 * (1 - EXP(b*T)) / (1 - dt_1) | 103.3188 (where dt_1 = EXP(b*T/n)) |
Discrete Geometric Asian Volatility Formula
Verify that setting n_fix = 1 recovers full European volatility sig_G = sig * SQRT(2 * 3 / 6) = sig, while large n_fix approaches sig / SQRT(3).
=IF(OR(B6 <= 0, B7 <= 0, B8 < 1), "Invalid Asian Input", ROUND(B6 * SQRT(((B8 + 1) * (2 * B8 + 1)) / (6 * B8^2)), 4))Kemna-Vorst Discrete Geometric Asian Call Price Formula
By the arithmetic-geometric mean inequality, the exact discrete Geometric Asian call C_Geo is a strict lower bound (and Monte Carlo control variate) for the Arithmetic Asian call. Compare it alongside the exact discrete arithmetic first moment M1 = (S / n) * EXP(b*dt) * (1 - EXP(b*T)) / (1 - EXP(b*dt)).
=LET(s_px, B2, k_str, B3, r_rf, B4, b_car, B5, sig_v, B6, t_yr, B7, n_fx, B8, IF(OR(sig_v <= 0, t_yr <= 0, n_fx < 1), "Invalid Asian Input", LET(var_g, (sig_v^2) * ((n_fx + 1) * (2 * n_fx + 1)) / (6 * n_fx^2), sig_g, SQRT(var_g), b_geo, 0.5 * ((n_fx + 1) / n_fx) * (b_car - 0.5 * sig_v^2) + 0.5 * var_g, sig_t, sig_g * SQRT(t_yr), d_1, (LN(s_px / k_str) + (b_geo + 0.5 * var_g) * t_yr) / sig_t, d_2, d_1 - sig_t, t_1, 1 / (1 + 0.2316419 * ABS(d_1)), p_1, 0.3989422804014327 * EXP(-0.5 * d_1^2) * t_1 * (0.31938153 + t_1 * (-0.356563782 + t_1 * (1.781477937 + t_1 * (-1.821255978 + 1.330274429 * t_1)))), nd_1, IF(d_1 >= 0, 1 - p_1, p_1), t_2, 1 / (1 + 0.2316419 * ABS(d_2)), p_2, 0.3989422804014327 * EXP(-0.5 * d_2^2) * t_2 * (0.31938153 + t_2 * (-0.356563782 + t_2 * (1.781477937 + t_2 * (-1.821255978 + 1.330274429 * t_2)))), nd_2, IF(d_2 >= 0, 1 - p_2, p_2), ROUND(s_px * EXP((b_geo - r_rf) * t_yr) * nd_1 - k_str * EXP(-r_rf * t_yr) * nd_2, 4))))Try the example
Use the original values or change them to test your understanding.