Advanced Option
Pricing Models

Part II · The BSM engine

4.Risk-neutral measure and Girsanov

Pricing by discounted expectation under $\mathbb{P}$ would make $V$ depend on the equity risk premium $\mu - r$, which the no-arbitrage hedging argument shows it cannot. The resolution is the risk-neutral measure $\mathbb{Q}$, equivalent to $\mathbb{P}$, under which the discounted stock is a martingale. Girsanov's theorem [9] supplies the explicit Radon-Nikodym derivative

and the change of variable $dX^{\mathbb{P}} = dX^{\mathbb{Q}} - \theta\,dt$ converts $dS = \mu S\,dt + \sigma S\,dX^{\mathbb{P}}$ into $dS = rS\,dt + \sigma S\,dX^{\mathbb{Q}}$. The risk premium is absorbed into the measure; the option price comes out the same.

5.The Black-Scholes PDE

Form the hedged portfolio $\Pi = V(S,t) - \Delta\,S$. Applying (3) to $V$ and choosing $\Delta = \partial V/\partial S$ kills the $dS$ term and forces the residual $d\Pi$ to be deterministic. By no arbitrage $d\Pi = r\Pi\,dt$, giving the Black-Scholes PDE [1, 2]

with terminal condition $V(S,T) = \text{Payoff}(S)$.

6.Closed-form European and binary calls

The standard substitution $V = e^{-r(T-t)} U$, $\tau = T - t$, $\xi = \log S$, $x = \xi + (r - \tfrac{1}{2}\sigma^2)\tau$, reduces (7) to the heat equation. Direct evaluation of the risk-neutral expectation against the lognormal density and a complete-the-square step (see e.g. Hull [8] Sec. 15) gives

with

$N(d_2)$ is $\mathbb{Q}(S_T > E)$; $N(d_1)$is the same probability under the share-numéraire measure $\bar{\mathbb{Q}}$. For $S_0 = E = 100$, $T = 1$, $\sigma = 0.20$, $r = 0.05$, one obtains $d_1 = 0.35$, $d_2 = 0.15$, $C = 10.4506$, $C_{\text{bin}} = 0.5323$, the numbers the studio below reproduces.

7.The Greeks

Differentiating (8) and using the identity $S\,n(d_1) = E e^{-r(T-t)} n(d_2)$ that drops out of the parity of $d_1, d_2$:

• $\Delta = N(d_1)$, sensitivity to spot.
• $\Gamma = n(d_1)/(S\sigma\sqrt{T-t})$, peaks at-the-money; spikes for binaries.
• $\nu = S\sqrt{T-t}\,n(d_1)$, exposure to vol level.
• $\Theta = -\frac{S\sigma n(d_1)}{2\sqrt{T-t}} - rE e^{-r(T-t)}N(d_2)$, time decay.
• $\rho = E(T-t)e^{-r(T-t)}N(d_2)$, rate sensitivity.

Part III · The Heston engine

8.The Heston SDE

BSM assumes a constant volatility $\sigma$. Heston (1993) [3] lets the variance itself be stochastic. Under $\mathbb{Q}$,

The variance process is a Cox-Ingersoll-Ross / square-root mean-reverting diffusion. Parameter roles:

• $\kappa$, speed of mean reversion of variance.
• $\theta$, long-run variance.
• $\sigma$, vol-of-vol; controls smile width.
• $\rho$, correlation between asset and variance shocks; controls smile asymmetry.
• $v_0$, initial variance; controls short-maturity ATM level.

The Feller condition $2\kappa\theta > \sigma^2$ keeps $v_t > 0$ pathwise. Heston remains well-posed if it is violated, but the variance can touch zero.

9.The characteristic function of log S_T

Heston's key analytic result: although the density of $\log S_T$ has no closed form, its characteristic function $\varphi(u) = \mathbb{E}^{\mathbb{Q}}[e^{i u \log S_T}]$ does. With $x_0 = \log S_0$,

where, in the Albrecher et al. (2007) “little trap” form [5], algebraically equivalent to Heston's original but free of the branch-cut discontinuity at long maturities,

Derivation: substitute $f(x,v,t) = e^{C(u,T-t) + D(u,T-t) v + i u x}$ into the Feynman-Kac PDE for $\varphi$ implied by (10); matching coefficients in $v$ gives a Riccati ODE for $D$ with closed-form solution, and integrating yields $C$. The full derivation runs to several pages and is reproduced in Heston [3] and rederived more carefully in Albrecher et al. [5]. The studio below implements (12) directly in TypeScript with a hand-rolled complex-number kernel.

10.The Carr-Madan Fourier inversion

Even given (11) and (12), a Fourier inversion of $\varphi(u)$ does not give the call price directly because the call payoff is not integrable in the strike. Carr and Madan [4] resolve this by introducing a damping parameter $\alpha > 0$. Let $k = \log E$, $c(k)$ the call price as a function of log-strike, and define

For $\alpha$ large enough that $\mathbb{E}^{\mathbb{Q}}[S_T^{1+\alpha}] < \infty$, $c_\alpha \in L^1$ and its Fourier transform exists. Direct computation gives

Fourier inversion of (14) and using (13) yields

This is the Carr-Madan formula. With $\varphi$given by (12), the call price is now a single one-dimensional real integral. The original paper recommends evaluating it with the FFT to price a whole strip of strikes at once, but for the on-page studio Simpson's rule on a truncated grid $[0, u_{\max}]$ is faster to implement and adequate. The studio uses $u_{\max} = 120$, $N = 1024$ nodes, $\alpha = 1.5$.

11.Numerical integration and the “little trap”

Two subtleties matter in implementation:

1. Branch cut of log. Heston's original $g$ uses $(\kappa - \rho\sigma i u + d)/(\kappa - \rho\sigma i u - d)$; at long maturities $e^{dT}$ blows up and the principal-branch log wraps around the cut, producing visible kinks in $\varphi$. Albrecher et al. [5] show that swapping numerator and denominator in $g$, with a matching sign change inside $C$, gives a formulation in which $\Re(d) > 0$ keeps $g e^{-dT}$ bounded, eliminating the wrap. The studio implements (12) in this safe form throughout.
2. Damping choice. Too small an $\alpha$ violates the integrability condition; too large amplifies the truncation error at large $v$. Carr and Madan recommend $\alpha \approx 1.5$ for equity-index parameters; the studio uses this.

12.The implied volatility smile

Given a Heston call price $c(E)$, inverting the BSM formula for the volatility $\sigma_{\text{imp}}(E)$ that reproduces it gives the implied volatility curve. Key empirical patterns the studio below lets you reproduce:

• $\rho < 0$ tilts the smile into the equity skew: low-strike puts are richer than high-strike calls.
• $\sigma$ (vol-of-vol) widens the smile symmetrically.
• $\kappa$ controls the term-structure flattening: as $T \to \infty$ the smile flattens toward the long-run level $\sqrt{\theta}$ at rate $\sim 1/\kappa$.
• $v_0$ versus $\theta$ controls whether the term structure is upward or downward sloping at the short end.

Part IV · The Monte Carlo engine

13.Discretisation: Euler, Milstein, exact log-Euler

For an Itô SDE $dG = a\,dt + b\,dX$ over a grid of $M$ steps $\delta t = T/M$, three schemes the studio considers:

Euler-Maruyama. Freezing the integrands at the left endpoint of each subinterval gives

Milstein.Keeping one more term of the Itô-Taylor expansion (a double Itô integral whose evaluation uses $\int_{t_n}^{t_{n+1}}\!\!\int_{t_n}^{s} dX_u\,dX_s = \tfrac{1}{2}[(\Delta X_n)^2 - \delta t]$) [7] adds the correction

For GBM ($a = rS$, $b = \sigma S$): $b\,\partial_S b = \sigma^2 S$ and the correction becomes $\tfrac{1}{2}\sigma^2 S_n(\phi_n^2 - 1)\,\delta t$.

Exact log-Euler. For constant-coefficient SDEs like GBM, equation (5) is exact at any $\delta t$. The studio's Monte Carlo engine uses (5) with $\delta t = T$, eliminating discretisation bias entirely.

14.Strong and weak convergence

Two notions of “the scheme approximates the true solution well”:

• Strong order $\gamma$: $\sqrt{\mathbb{E}|G_T^{\delta t} - G_T^{\text{exact}}|^2} \le C\,\delta t^\gamma$. Path-wise accuracy. Euler: $\gamma = 1/2$; Milstein: $\gamma = 1$.
• Weak order $\beta$: $|\mathbb{E}[g(G_T^{\delta t})] - \mathbb{E}[g(G_T^{\text{exact}})]| \le C_g\,\delta t^\beta$. Distributional accuracy; what matters for pricing. Euler: $\beta = 1$; Milstein: $\beta = 1$.

So Milstein gives no weak-order advantage for plain-vanilla pricing, but it pays off for path-dependent payoffs where the path itself must be accurate (barriers, Asian, lookback). For European or binary on GBM, the exact step uniformly dominates.

15.The Monte Carlo estimator

With $N$ i.i.d. draws $S_T^{(i)}$ from (5),

By linearity, $\mathbb{E}[\widehat V_N] = V$ (unbiased). By the strong law, $\widehat V_N \to V$ a.s. (consistent). With $\sigma_P^2 = \operatorname{Var}[\text{Payoff}(S_T)]$,

The CLT-implied 95% confidence interval is $\widehat V_N \pm 1.96\,\widehat{\mathrm{SE}}$. The $1/\sqrt N$ rate is the central cost of plain Monte Carlo: a tenfold reduction in standard error costs a hundredfold $N$.

16.Antithetic variates

Pair each draw $\phi_i$ with its negation $-\phi_i$ (both $\mathcal{N}(0,1)$ by symmetry) and define $X_i = \text{Payoff}(S_T(\phi_i))$, $Y_i = \text{Payoff}(S_T(-\phi_i))$. The antithetic estimator

is unbiased. For its variance, the within-pair calculation gives

Variance reduction proof. If $\text{Payoff}\circ S_T$ is monotone in $\phi$, then $X_i = f(\phi_i)$ and $Y_i = f(-\phi_i)$ are images of $\phi_i$ under an increasing and a decreasing transformation respectively. The FKG / Chebyshev sum inequality gives $\operatorname{Cov}(g_1(Z), g_2(Z)) \le 0$ when $g_1$ increases and $g_2$ decreases [6]. Strict inequality holds whenever both functions are non-constant on a set of positive measure (true for any non-trivial payoff). So (21) is strictly less than $\sigma_P^2/2$, and the antithetic estimator strictly beats plain Monte Carlo on $2N$ independent draws.

17.The strike-near-median amplification

The studio routinely measures a variance-reduction factor

for the base case. The binary's outsized gain has a clean geometric explanation. At ATM, the strike $E$ sits close to the median of $S_T$ (Sec. 3). For an antithetic pair $(\phi, -\phi)$, the two terminal stocks $S_T(\phi)$ and $S_T(-\phi)$ almost always land on opposite sides of the median, and hence opposite sides of $E$. The two indicators are then exactly $\{0, 1\}$ and the pair mean is deterministically $1/2$. Within-pair variance collapses to (almost) zero, and the estimator's variance is reduced by an order well beyond the $\sim 2\times$ a smooth monotone payoff would predict.

Part V · Conclusion

18.Conclusion

What stitches the three engines together is a single object: the risk-neutral expectation of the discounted payoff. The Black-Scholes-Merton closed form is the one place in derivative pricing where that expectation integrates by hand, and the resulting $d_1, d_2$ identities are worth knowing in the bone, both as benchmarks and as the limit every other engine should reduce to. Heston is what the same expectation looks like when the variance refuses to be constant, and the Carr-Madan formula is the trick that lets a single one-dimensional integral do the work of a two-dimensional joint density. Monte Carlo is what is left when the model has nothing closed-form to offer, and antithetic variates is one of the cheapest improvements you can layer on top, free for any monotone payoff and surprisingly powerful when the strike sits near the median of the terminal distribution.

In practice the three engines belong in a chain rather than in competition. The closed form is the truth that any simulation engine should reproduce before it is trusted on anything else. The exact log-Euler step is the right Monte Carlo discretisation whenever the model is constant-coefficient geometric Brownian motion, because it carries zero discretisation bias at any step size. Carr-Madan inversion takes over the moment the volatility becomes stochastic, provided the characteristic function exists and is tractable, and the Albrecher “little trap” form is what keeps it numerically honest at long maturities. Beyond that, when neither a closed form nor a tractable transform is available, the burden shifts back to Monte Carlo with as much variance reduction as the payoff structure allows, and every reported price still travels with a standard error. Built and validated together, the three engines cover most of what an equity-derivatives desk needs from a pricing library, and the three studios above are the working proof.

19.References

1. [1]Black, F. & Scholes, M. (1973). The Pricing of Options and Corporate Liabilities. Journal of Political Economy, 81(3), 637-654.
2. [2]Merton, R. C. (1973). Theory of Rational Option Pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
3. [3]Heston, S. L. (1993). A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options. The Review of Financial Studies, 6(2), 327-343.
4. [4]Carr, P. & Madan, D. B. (1999). Option Valuation Using the Fast Fourier Transform. Journal of Computational Finance, 2(4), 61-73.
5. [5]Albrecher, H., Mayer, P., Schoutens, W. & Tistaert, J. (2007). The Little Heston Trap. Wilmott Magazine, January 2007, 83-92.
6. [6]Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering. Springer, Stochastic Modelling and Applied Probability 53.
7. [7]Kloeden, P. E. & Platen, E. (1992). Numerical Solution of Stochastic Differential Equations. Springer, Applications of Mathematics 23.
8. [8]Hull, J. C. (2018). Options, Futures, and Other Derivatives (10th ed.). Pearson.
9. [9]Wilmott, P. (2006). Paul Wilmott on Quantitative Finance (2nd ed., Vols. 1-3). Wiley.
10. [10]Ahmad, R. (2026). CQF Module 3 lecture series (JA263.1 through JA263.12). Certificate in Quantitative Finance.
11. [11]Andersen, L. (2008). Simple and Efficient Simulation of the Heston Stochastic Volatility Model. Journal of Computational Finance, 11(3), 1-42.