Barrier Options Under Heston (Semi-Analytical)¶

Introduction¶

A barrier option is a path-dependent derivative whose payoff depends on whether the underlying asset price crosses a predetermined barrier level during the option's life. In the Black--Scholes model, the lognormal transition density and the reflection principle yield elegant closed-form solutions for barrier option prices. Under the Heston model, this approach breaks down: the joint dynamics of $(S_t, v_t)$ destroy the symmetry that the reflection principle exploits, and no simple closed-form formula exists.

Despite this, the affine structure of the Heston model still permits semi-analytical methods. The characteristic function can be combined with Fourier inversion to price continuously monitored barrier options, while Monte Carlo methods handle the general case including discrete monitoring. This section develops both approaches and compares their accuracy and computational cost.

Prerequisites

Closed-Form Characteristic Function (the Heston CF)
Semi-Closed-Form Fourier Inversion (Gil-Pelaez framework)
Quadratic-Exponential Scheme (MC simulation under Heston)

Learning Objectives

By the end of this section, you will be able to:

Classify barrier option types and state their payoff structures
Explain why the reflection principle fails under stochastic volatility
Derive the semi-analytical pricing formula using the characteristic function of the killed process
Apply Monte Carlo methods with continuity corrections for discrete barrier monitoring
Compare semi-analytical and Monte Carlo prices for typical Heston parameters

Barrier Option Taxonomy¶

Types of Barrier Options¶

Barrier options are classified by two characteristics: the barrier direction and the knock-in/knock-out feature.

Type	Abbreviation	Condition
Down-and-out call	DOC	Expires worthless if $S_t \leq B$ for any $t \leq T$
Down-and-in call	DIC	Activates only if $S_t \leq B$ for some $t \leq T$
Up-and-out call	UOC	Expires worthless if $S_t \geq B$ for any $t \leq T$
Up-and-in call	UIC	Activates only if $S_t \geq B$ for some $t \leq T$

The same classification applies to puts. In-out parity connects them:

\[ V_{\text{in}} + V_{\text{out}} = V_{\text{vanilla}} \]

This identity holds model-free for European-style barriers and reduces the pricing problem: once we compute the knock-out price, the knock-in price follows immediately.

Continuous versus Discrete Monitoring¶

A continuously monitored barrier is triggered if the barrier condition holds at any instant $t \in [0, T]$. A discretely monitored barrier checks the condition only at specified dates $t_1, t_2, \ldots, t_m$. Discrete monitoring is the market convention; continuous monitoring is the mathematical idealization.

Why the Reflection Principle Fails¶

The Black--Scholes Case¶

In the Black--Scholes model, the log-price $X_t = \ln S_t$ follows a drifted Brownian motion:

\[ X_t = X_0 + \left(r - q - \frac{\sigma^2}{2}\right) t + \sigma W_t \]

For a down-and-out option with barrier $B < S_0$, let $b = \ln B$. The reflection principle for Brownian motion states that the probability of $W_t$ hitting a barrier can be computed using the density of the reflected process. After a drift adjustment (Girsanov), this yields the closed-form formula involving the standard normal CDF.

The key property exploited is that $X_t - b$ and its reflection $2b - X_t + 2b$ have a known joint distribution under the Brownian filtration.

Breakdown Under Heston¶

Under the Heston model, the log-price dynamics are:

\[ dX_t = \left(r - q - \frac{v_t}{2}\right) dt + \sqrt{v_t} \, dW_t^{(1)} \]

The instantaneous volatility $\sqrt{v_t}$ is itself random and correlated with $X_t$. This destroys the reflection principle for two reasons:

Path-dependent volatility: When $X_t$ approaches the barrier from above, the variance $v_t$ is in a specific state. The reflected path $2b - X_t$ would require the variance to be in the same state, but the correlation $\rho \neq 0$ means the variance path is entangled with the stock path.
Non-Gaussian increments: Conditional on the variance path, $X_T$ is Gaussian, but unconditionally it is not. The unconditional distribution has heavier tails and skewness controlled by $\rho$ and $\xi$.

No Closed-Form Barrier Prices Under Heston

Unlike Black--Scholes, there is no known closed-form expression for continuously monitored barrier options under the Heston model. All pricing methods are either semi-analytical (Fourier-based) or numerical (Monte Carlo, FDM).

Semi-Analytical Approach via Fourier Methods¶

The Killed Process¶

For a down-and-out option with barrier $B$, define the first passage time:

\[ \tau_B = \inf\{t \geq 0 : S_t \leq B\} \]

The knock-out price is:

\[ V_{\text{DOC}} = e^{-rT} \mathbb{E}^{\mathbb{Q}}\left[(S_T - K)^+ \mathbf{1}_{\{\tau_B > T\}}\right] \]

This expectation involves the survived density --- the transition density of $X_T$ conditional on the path never having crossed the barrier.

Fourier Representation of the Survived Density¶

The characteristic function of the log-price conditional on survival can be expressed using a Green's function approach. Define the domain $\Omega = \{(x, v) : x > b, \, v > 0\}$ where $b = \ln B$. The survived characteristic function satisfies the Heston PDE on $\Omega$ with an absorbing boundary condition at $x = b$:

\[ \phi_{\text{surv}}(u, \tau; x_0, v_0) = \mathbb{E}^{\mathbb{Q}}\left[e^{iu X_T} \mathbf{1}_{\{\tau_B > T\}} \mid X_0 = x_0, v_0\right] \]

For the standard Heston model, this can be decomposed as:

\[ \phi_{\text{surv}}(u, \tau; x_0, v_0) = \phi(u, \tau; x_0, v_0) - \phi_{\text{cross}}(u, \tau; x_0, v_0) \]

where $\phi$ is the unrestricted characteristic function and $\phi_{\text{cross}}$ accounts for paths that crossed the barrier.

Numerical Inversion¶

The knock-out call price is recovered via Fourier inversion:

\[ V_{\text{DOC}} = e^{-rT} \left[\frac{1}{2} F_{\text{surv}}(0) + \frac{1}{\pi} \int_0^{\infty} \operatorname{Re}\left(\frac{e^{-iu \ln K} \phi_{\text{surv}}(u - i, \tau)}{iu \, \phi_{\text{surv}}(-i, \tau)}\right) du \right] \cdot S_0 \]

where the integration is performed numerically using adaptive quadrature or the trapezoidal rule with Richardson extrapolation.

Theorem (In-Out Parity Under Heston)

For European-style barrier options under the Heston model:

\[ V_{\text{DOC}}(S_0, v_0, K, B, T) + V_{\text{DIC}}(S_0, v_0, K, B, T) = V_{\text{call}}(S_0, v_0, K, T) \]

where $V_{\text{call}}$ is the standard Heston European call price. This identity holds because the events $\{\tau_B > T\}$ and $\{\tau_B \leq T\}$ partition the sample space.

Proof

By linearity of expectation:

\[ V_{\text{DOC}} + V_{\text{DIC}} = e^{-rT} \mathbb{E}^{\mathbb{Q}}\left[(S_T - K)^+ \left(\mathbf{1}_{\{\tau_B > T\}} + \mathbf{1}_{\{\tau_B \leq T\}}\right)\right] \]

\[ = e^{-rT} \mathbb{E}^{\mathbb{Q}}\left[(S_T - K)^+\right] = V_{\text{call}} \]

$\square$

Alternative: COS Method for Barriers¶

The COS method can be adapted for barrier options by restricting the Fourier cosine expansion to the domain $[b, \infty)$ instead of $(-\infty, \infty)$. The expansion coefficients become:

\[ A_k = \frac{2}{c - b} \operatorname{Re}\left[\phi\!\left(\frac{k\pi}{c - b}\right) e^{-ik\pi b/(c-b)}\right] \]

where $c$ is the upper truncation bound. The payoff coefficients $H_k$ are modified to account for the restricted domain. This approach maintains the exponential convergence of the COS method while naturally enforcing the absorbing boundary.

Monte Carlo Pricing of Barrier Options¶

Path Simulation¶

Monte Carlo pricing of barrier options under Heston requires simulating the joint path $(S_{t_0}, v_{t_0}), (S_{t_1}, v_{t_1}), \ldots, (S_{t_m}, v_{t_m})$ and checking the barrier condition at each monitoring date. The QE scheme from Andersen (2008) provides efficient simulation of the variance process.

For a down-and-out call with discrete monitoring at dates $t_1, \ldots, t_m$:

\[ \hat{V}_{\text{DOC}} = e^{-rT} \frac{1}{N} \sum_{j=1}^{N} (S_T^{(j)} - K)^+ \prod_{k=1}^{m} \mathbf{1}_{\{S_{t_k}^{(j)} > B\}} \]

Continuous Monitoring Bias¶

When using Monte Carlo with $m$ monitoring dates to approximate a continuously monitored barrier, the discrete approximation overestimates the knock-out price (underestimates the knock-in price) because the stock can cross the barrier between monitoring dates without being detected.

Theorem (Broadie--Glasserman--Kou Continuity Correction)

For a down-and-out option, the continuously monitored barrier $B$ should be replaced by the adjusted barrier:

\[ B_{\text{adj}} = B \exp\left(-\beta \hat{\sigma} \sqrt{\Delta t}\right) \]

where $\beta = -\zeta(1/2)/\sqrt{2\pi} \approx 0.5826$, $\zeta$ is the Riemann zeta function, $\hat{\sigma}$ is the average volatility, and $\Delta t = T/m$ is the monitoring interval.

For an up-and-out option, the sign of the exponent is reversed:

\[ B_{\text{adj}} = B \exp\left(+\beta \hat{\sigma} \sqrt{\Delta t}\right) \]

Under Heston, the "average volatility" $\hat{\sigma}$ is not constant. A practical choice is $\hat{\sigma} = \sqrt{\theta}$ (the long-run volatility) or $\hat{\sigma} = \sqrt{v_0}$ (the current volatility). This correction significantly reduces the bias even with moderate $m$.

Variance Reduction for Barrier Monte Carlo¶

Standard variance reduction techniques are particularly valuable for barrier options:

Antithetic variates: Simulate paths with $Z$ and $-Z$ simultaneously. The barrier indicator destroys perfect negative correlation, but variance reduction of 20--40% is typical.
Conditional Monte Carlo: Condition on the variance path $\{v_{t_k}\}$ and compute the barrier crossing probability analytically using the (conditionally) Gaussian log-price increments.
Importance sampling: Shift the drift of the stock process to increase the frequency of paths near the barrier, then reweight by the likelihood ratio.

Numerical Example¶

Parameters¶

Parameter	Value
$S_0$	$100
$K$	$100
$B$	$80 (down-and-out)
$r$	3%
$q$	0%
$v_0$	0.04
$\kappa$	1.5
$\theta$	0.04
$\xi$	0.3
$\rho$	$-0.7$
$T$	1 year

Method Comparison¶

Method	Price	Std Error	Time
Fourier inversion (semi-analytical)	$5.82	---	0.05s
COS method ($N = 128$)	$5.81	---	0.02s
MC (500 steps, $10^6$ paths, uncorrected)	$6.14	$0.02	2.1s
MC (500 steps, $10^6$ paths, corrected)	$5.84	$0.02	2.1s
MC (2000 steps, $10^6$ paths, uncorrected)	$5.98	$0.02	8.3s

Observations

The semi-analytical and COS methods agree to within $0.01, confirming the accuracy of both Fourier approaches.
Uncorrected MC with 500 steps overestimates the knock-out price by $0.32 (5.5%) due to missed barrier crossings between monitoring dates.
The Broadie--Glasserman--Kou continuity correction reduces the MC bias to $0.02, comparable to the statistical error.
Increasing the number of time steps from 500 to 2000 reduces the uncorrected bias but at 4x computational cost --- the continuity correction is far more efficient.

Effect of Stochastic Volatility on Barrier Prices¶

Compared to a Black--Scholes model with the same ATM volatility ($\sigma = 0.20$), the Heston down-and-out call price is lower. The negative correlation $\rho = -0.7$ means that when the stock drops toward the barrier, variance tends to increase, making further drops (and barrier crossing) more likely. This increases the knock-out probability relative to the constant-volatility case.

Summary¶

Concept	Description
Reflection principle	Works in Black--Scholes but fails under Heston due to path-dependent volatility
Semi-analytical pricing	Fourier inversion of the survived characteristic function
In-out parity	$V_{\text{in}} + V_{\text{out}} = V_{\text{vanilla}}$ (model-free for European barriers)
Continuity correction	$B_{\text{adj}} = B \exp(\mp \beta \hat{\sigma}\sqrt{\Delta t})$ for discrete-to-continuous adjustment
COS for barriers	Restrict cosine expansion to $[b, c]$ domain with absorbing boundary

Key Takeaways

No closed form: Unlike Black--Scholes, barrier options under Heston have no closed-form solution due to the failure of the reflection principle.
Semi-analytical via Fourier: The affine structure of Heston allows pricing through the characteristic function of the killed (survived) process, computed via Fourier inversion or the COS method.
Monte Carlo with corrections: For discrete monitoring (the market convention), Monte Carlo with the Broadie--Glasserman--Kou continuity correction provides accurate prices with manageable computational cost.
Stochastic volatility effect: Negative correlation $\rho < 0$ increases the knock-out probability for down barriers and decreases it for up barriers, relative to a constant-volatility model.
In-out parity reduces work: Only one of the knock-in/knock-out pair needs to be priced; the other follows from the vanilla Heston price.

What's Next¶

Section	Topic
Asian Options (Monte Carlo)	MC pricing of path-dependent options under Heston
Variance Swaps (Closed-Form)	Analytical pricing of variance derivatives
Forward-Start Options	Pricing via conditional characteristic functions

Exercises¶

Exercise 1. State the in-out parity for barrier options: $V_{\text{in}} + V_{\text{out}} = V_{\text{vanilla}}$. A European vanilla call with $K = 100$ is worth $8.50 under Heston. A down-and-out call with barrier $B = 85$ is priced at $7.20. Compute the down-and-in call price. Explain why the DOC is cheaper than the vanilla: what scenarios does the barrier eliminate?

Solution to Exercise 1

In-out parity. By the partition $\{\tau_B > T\} \cup \{\tau_B \leq T\} = \Omega$:

\[ V_{\text{DIC}} = V_{\text{vanilla}} - V_{\text{DOC}} = \$8.50 - \$7.20 = \$1.30 \]

Why the DOC is cheaper than the vanilla. The down-and-out call has payoff $(S_T - K)^+ \mathbf{1}_{\{\tau_B > T\}}$, which is zero whenever the stock touches or crosses $B = 85$ during the option's life. The vanilla call has payoff $(S_T - K)^+$ regardless of the path.

The scenarios eliminated by the barrier are those where the stock drops to $85 or below at any point before maturity. Crucially, some of these paths subsequently recover and finish above the strike $K = 100$, producing a positive payoff for the vanilla call but zero for the DOC. The price difference $\$8.50 - \$7.20 = \$1.30$ represents the expected discounted value of exactly these "touch and recover" scenarios.

Under Heston with $\rho = -0.7$, the probability of touching $B = 85$ is enhanced relative to Black--Scholes: when the stock drops toward the barrier, negative correlation causes variance to increase, accelerating further downward movement. This makes the DOC relatively cheaper (and the DIC relatively more expensive) than in a constant-volatility model.

Exercise 2. Under Black-Scholes, the reflection principle yields a closed-form for continuously monitored barrier options because the log-price $\ln S_t$ is a Brownian motion with drift. Explain why the reflection principle fails under Heston. Specifically, consider a down-and-out call with barrier $B < S_0$: the reflected path at $\ln B$ does not have the same law as the original process because the variance state $v_t$ at the hitting time is random and affects the post-reflection dynamics.

Solution to Exercise 2

Why the reflection principle fails under Heston.

In Black--Scholes, the log-price follows $X_t = X_0 + \mu t + \sigma W_t$ where $\mu = r - q - \sigma^2/2$ is a constant drift. The reflection principle for Brownian motion states: for a barrier at level $b = \ln B$, the law of the first passage time and the post-reflection dynamics are fully characterized by the symmetry of the Brownian motion about the barrier. Specifically, for $W_t$ hitting level $l$, the reflected process $2l - W_t$ has the same law as $W_t$ for paths that have crossed $l$.

Under Heston, $X_t = \ln S_t$ satisfies:

\[ dX_t = \left(r - q - \frac{v_t}{2}\right) dt + \sqrt{v_t} \, dW_t^{(1)} \]

The reflection principle fails for two specific reasons:

1. Variance state at hitting time is random. Suppose the path first hits the barrier $b$ at time $\tau_B$. At this instant, $v_{\tau_B}$ is a random variable whose distribution depends on the entire path history. The reflected path starting from $b$ at time $\tau_B$ would need to continue with variance dynamics initialized at $v_{\tau_B}$. However, the reflection operation on $X_t$ (replacing $X_t$ with $2b - X_t$ for $t > \tau_B$) does not correspondingly reflect or reset the variance process. The variance process $v_t$ continues its CIR dynamics independently of whether $X_t$ has been reflected. This means the reflected process $(2b - X_t, v_t)$ does not have the same joint law as the original process $(X_t, v_t)$.

2. Correlation entangles the two dimensions. Since $dW_t^{(1)}$ and $dW_t^{(2)}$ have correlation $\rho$, the variance process $v_t$ is statistically linked to the direction of $X_t$. With $\rho < 0$, a downward move in $X_t$ (toward the barrier) tends to coincide with an increase in $v_t$. After reflection, the "upward" move in $2b - X_t$ would be paired with increasing variance --- the opposite of the natural leverage effect. This asymmetry means the reflected path does not have the same distributional properties as the unreflected path.

In summary, the reflection principle requires that the process has the same distributional structure on both sides of the barrier, which holds for drifted Brownian motion (a one-dimensional Markov process with translation-invariant dynamics) but not for the two-dimensional Heston process where the second component ($v_t$) breaks the symmetry.

Exercise 3. A discretely monitored down-and-out call checks the barrier only at dates $t_1, \ldots, t_m$. The Broadie-Glasserman-Kou continuity correction shifts the barrier to $B e^{-\beta\sigma\sqrt{\Delta t}}$ where $\beta = \zeta(1/2)/\sqrt{2\pi} \approx 0.5826$ and $\sigma$ is the effective volatility. For $B = 90$, $\sigma = 20\%$, and daily monitoring ($\Delta t = 1/252$), compute the adjusted barrier. By how many dollars does the correction shift the barrier?

Solution to Exercise 3

Computing the adjusted barrier. The Broadie--Glasserman--Kou continuity correction for a down-and-out barrier is:

\[ B_{\text{adj}} = B \exp(-\beta \sigma \sqrt{\Delta t}) \]

with $\beta \approx 0.5826$, $B = 90$, $\sigma = 0.20$, and $\Delta t = 1/252$.

Step 1. Compute $\sqrt{\Delta t}$:

\[ \sqrt{\Delta t} = \sqrt{1/252} = \frac{1}{\sqrt{252}} \approx \frac{1}{15.875} \approx 0.06300 \]

Step 2. Compute the exponent:

\[ \beta \sigma \sqrt{\Delta t} = 0.5826 \times 0.20 \times 0.06300 = 0.5826 \times 0.01260 = 0.007341 \]

Step 3. Compute the adjusted barrier:

\[ B_{\text{adj}} = 90 \times e^{-0.007341} \approx 90 \times (1 - 0.007341 + \cdots) \approx 90 \times 0.99268 = 89.341 \]

Shift in dollars:

\[ B - B_{\text{adj}} = 90 - 89.341 = \$0.659 \]

The correction shifts the barrier downward by approximately $0.66. This makes the knock-out less likely (since the stock must reach a lower level to trigger the barrier), which compensates for the fact that in the continuous-monitoring limit, the stock can cross the barrier between discrete monitoring dates. The shift is small relative to the barrier level (0.73%), but its effect on the option price can be significant --- as seen in the numerical example, uncorrected Monte Carlo overestimates the DOC price by 5.5%, while the corrected version reduces the bias to match the standard error.

Exercise 4. An up-and-out call with $K = 100$, $B = 130$, $T = 1$ is priced under Heston. If $\rho = -0.7$ (strong negative correlation), explain qualitatively why the UOC is less likely to be knocked out compared to the case $\rho = 0$. Hint: negative $\rho$ means large upward moves in $S$ tend to coincide with decreasing variance, which limits the magnitude of subsequent upward moves.

Solution to Exercise 4

Effect of negative $\rho$ on up-and-out call knock-out probability.

For an up-and-out call with $B = 130$, the option is knocked out when $S_t$ reaches 130. Consider what happens when the stock price rises toward the barrier:

Case $\rho = 0$ (zero correlation): The variance process $v_t$ evolves independently of the stock price. When $S_t$ approaches 130, the variance $v_t$ is at its unconditional (mean-reverting) level, providing no additional dampening or amplification of the stock's upward movement.

Case $\rho = -0.7$ (negative correlation): When $S_t$ rises significantly (moving toward $B = 130$), the negative correlation implies that $dW_t^{(1)} > 0$ (positive stock return) tends to coincide with $dW_t^{(2)} < 0$ (negative variance shock). This causes the variance $v_t$ to decrease as the stock rises. Lower variance means:

Smaller subsequent moves: With reduced $v_t$, the instantaneous volatility $\sqrt{v_t}$ is lower, making large further upward moves less likely.
Self-limiting dynamics: The rise in $S_t$ triggers a decline in volatility, which constrains further upside. The stock effectively "runs out of steam" as it approaches the barrier.
Reduced barrier crossing probability: The probability of $S_t$ reaching 130 is lower because the volatility decreases precisely when the stock is close to the barrier.

Conversely, with positive $\rho$, rising stock prices would increase variance, creating momentum that makes barrier crossings more likely.

The net effect is that with $\rho = -0.7$, the UOC is less likely to be knocked out than with $\rho = 0$, making the UOC more valuable. This is the opposite of the effect on down-and-out options, where negative $\rho$ increases the knock-out probability (falling stock prices increase variance, accelerating the drop toward the barrier).

Exercise 5. Monte Carlo pricing of barrier options suffers from barrier bias: the simulated path may cross the barrier between monitoring dates without being detected. For a continuously monitored barrier, the Brownian bridge correction estimates the probability of crossing the barrier between two consecutive simulated points. Describe the correction: given $S_n$ and $S_{n+1}$ both above the barrier $B$, the probability of the minimum of the Brownian bridge falling below $B$ is approximately $\exp(-2\ln(S_n/B)\ln(S_{n+1}/B) / (v_n \Delta t))$. Explain why this correction is only approximate under Heston.

Solution to Exercise 5

Brownian bridge correction for barrier Monte Carlo.

Given simulated values $S_n$ and $S_{n+1}$ at consecutive time steps, both above the barrier $B$, we need to estimate the probability that the continuous path between these two points crossed the barrier. Under Black--Scholes (constant volatility $\sigma$), the log-price between $t_n$ and $t_{n+1}$ is a Brownian bridge given the endpoints.

The correction. For a Brownian bridge from $\ln S_n$ to $\ln S_{n+1}$ over an interval of length $\Delta t$, the probability that the minimum falls below $\ln B$ is:

\[ P(\min_{t \in [t_n, t_{n+1}]} X_t < \ln B \mid X_{t_n} = \ln S_n, X_{t_{n+1}} = \ln S_{n+1}) = \exp\!\left(-\frac{2\ln(S_n/B)\ln(S_{n+1}/B)}{v_n \Delta t}\right) \]

where $v_n$ is the variance (volatility squared) used in the log-price dynamics over $[t_n, t_{n+1}]$.

Implementation. For each simulated interval where both endpoints are above $B$:

Compute $p_n = \exp(-2\ln(S_n/B)\ln(S_{n+1}/B)/(v_n \Delta t))$
With probability $p_n$, declare the path knocked out (barrier crossed between monitoring dates)
This can be implemented by generating a uniform random variable $U_n \sim \text{Uniform}(0,1)$ and knocking out if $U_n < p_n$

Why this is only approximate under Heston. The correction relies on three assumptions that hold under Black--Scholes but not under Heston:

Constant volatility within the interval. The formula assumes the log-price follows a Brownian motion with constant diffusion $\sqrt{v_n}$ between $t_n$ and $t_{n+1}$. Under Heston, $v_t$ varies continuously within the interval. Using a single value $v_n$ (e.g., the simulated variance at $t_n$) is an approximation.
Brownian bridge property. The exact distribution of the minimum of the log-price between two points requires the path to be a Brownian bridge, which holds only when the diffusion coefficient is constant. Under Heston, the path conditional on its endpoints is not a standard Brownian bridge because the drift and diffusion depend on the time-varying $v_t$.
Independence of the variance path. The barrier crossing probability depends on the variance path $\{v_s : s \in [t_n, t_{n+1}]\}$, not just the endpoint value $v_n$. The correction ignores this path dependence.

Despite these limitations, the Brownian bridge correction significantly reduces the barrier bias in practice, especially when the time step $\Delta t$ is small enough that $v_t$ changes little within each interval.

Exercise 6. Compare the sensitivity of a down-and-out call to the Heston parameter $\xi$ (vol-of-vol) versus the vanilla call. The DOC has additional sensitivity because $\xi$ affects the probability of hitting the barrier. For $B = 85$, $K = 100$, $S_0 = 100$, $v_0 = 0.04$, argue that increasing $\xi$ increases the probability of large variance excursions, which increases the probability of $S_t$ dropping below $B$, thereby decreasing the DOC price.

Solution to Exercise 6

Sensitivity of DOC to $\xi$ versus vanilla.

Vanilla call sensitivity to $\xi$. The European call price depends on $\xi$ through the shape of the implied volatility smile. Increasing $\xi$ increases the kurtosis of the log-return distribution (heavier tails and a more peaked center), which raises OTM option prices but has a more modest effect on ATM options. For an ATM vanilla call, the sensitivity $\partial V_{\text{call}} / \partial \xi$ is relatively small.

DOC sensitivity to $\xi$. The down-and-out call has additional sensitivity to $\xi$ through the knock-out probability. The DOC price depends on $\xi$ through two channels:

Terminal distribution effect (same as vanilla): $\xi$ affects the distribution of $S_T$, modifying the expected payoff conditional on survival.
Barrier crossing probability (unique to DOC): $\xi$ controls the magnitude of variance excursions. Increasing $\xi$ increases the probability of large variance spikes, which in turn increases the probability of $S_t$ dropping below $B$.

The argument. Consider increasing $\xi$ from $\xi_1$ to $\xi_2 > \xi_1$:

Higher $\xi$ means the CIR process $v_t$ has larger fluctuations around its mean-reverting level $\theta$.
Large upward variance excursions ($v_t \gg \theta$) are more probable. During these episodes, the stock price has high instantaneous volatility $\sqrt{v_t}$, making large downward moves more likely.
With $\rho = -0.7$, these high-variance episodes are positively correlated with downward stock moves (leverage effect). A variance spike increases the probability of $S_t$ falling rapidly toward or below $B = 85$.
The probability $\mathbb{P}(\tau_B \leq T)$ therefore increases with $\xi$, reducing the DOC price.

Formally, using in-out parity: $V_{\text{DOC}} = V_{\text{call}} - V_{\text{DIC}}$. Since $V_{\text{DIC}}$ increases with $\xi$ (higher knock-in probability times higher expected recovery payoff), and $V_{\text{call}}$ is relatively insensitive to $\xi$ at ATM, the DOC price decreases.

The DOC has a larger absolute sensitivity $|\partial V / \partial \xi|$ than the vanilla because the barrier channel amplifies the effect of vol-of-vol. This makes barrier options particularly challenging to hedge with respect to higher-order volatility risks.

Exercise 7. The semi-analytical approach prices continuously monitored barriers using the characteristic function of the "killed" process (the process stopped at the barrier). This requires computing $\mathbb{E}[e^{iu\ln S_T} \mathbf{1}\{\tau_B > T\}]$ where $\tau_B$ is the first hitting time of the barrier. Explain conceptually how this differs from the standard Heston CF $\mathbb{E}[e^{iu\ln S_T}]$. Why is the killed CF not available in the same closed form as the standard Heston CF?

Solution to Exercise 7

Standard Heston CF versus killed CF.

The standard Heston CF is:

\[ \phi(u) = \mathbb{E}^{\mathbb{Q}}[e^{iu \ln S_T}] \]

This is an unconditional expectation over all possible paths of $(S_t, v_t)$ from time 0 to $T$. The Heston PDE for $\phi$ is defined on the entire domain $\{(x, v) : x \in \mathbb{R}, v > 0\}$ with no boundary condition on $x$. The affine structure of the Heston model allows this PDE to be reduced to the Riccati ODE system, yielding the well-known exponential-affine solution $\phi(u) = \exp(C(\tau, u) + D(\tau, u) v_0 + iu x_0)$.

The killed CF is:

\[ \phi_{\text{surv}}(u) = \mathbb{E}^{\mathbb{Q}}[e^{iu \ln S_T} \mathbf{1}_{\{\tau_B > T\}}] \]

This includes only paths that never cross the barrier during $[0, T]$. The indicator function $\mathbf{1}_{\{\tau_B > T\}}$ restricts the integration to the subset of paths that remain in the domain $\{x > b\}$ for all $t \leq T$.

Key conceptual difference. The killed CF corresponds to the solution of the same Heston PDE but on the restricted domain $\Omega = \{(x, v) : x > b, v > 0\}$ with an absorbing boundary condition at $x = b$:

\[ \phi_{\text{surv}}(u, \tau; b, v) = 0 \quad \text{for all } v > 0, \tau > 0 \]

Why the killed CF has no closed form. The closed-form Heston CF arises because the exponential-affine ansatz $\phi = \exp(C + Dv + iux)$ satisfies the PDE on the unrestricted domain, reducing it to ODEs for $C$ and $D$. When we impose the absorbing boundary $\phi(u, \tau; b, v) = 0$, this ansatz cannot satisfy the boundary condition: the exponential function $\exp(C + Dv + iub)$ is generally non-zero, so there is no choice of $C, D$ that makes it vanish at $x = b$ for all $v$ and $\tau$ simultaneously.

To satisfy the absorbing boundary, the solution must be represented as a superposition (integral or series) of exponential-affine functions, not a single one. Specifically, one can write:

\[ \phi_{\text{surv}}(u, \tau; x, v) = \phi(u, \tau; x, v) - \phi_{\text{cross}}(u, \tau; x, v) \]

where $\phi_{\text{cross}}$ accounts for paths that crossed $b$. Computing $\phi_{\text{cross}}$ requires solving a boundary value problem in two spatial dimensions $(x, v)$, which does not reduce to a finite-dimensional ODE system. This is fundamentally because the barrier condition on $x$ couples the spatial dimensions in a way that the affine structure alone cannot disentangle.

In practice, $\phi_{\text{surv}}$ is computed either by numerical PDE methods (finite differences on the restricted domain) or by approximation techniques such as series expansions in eigenfunction bases adapted to the absorbing boundary.

Type	Abbreviation	Condition
Down-and-out call	DOC	Expires worthless if \(S_t \leq B\) for any \(t \leq T\)
Down-and-in call	DIC	Activates only if \(S_t \leq B\) for some \(t \leq T\)
Up-and-out call	UOC	Expires worthless if \(S_t \geq B\) for any \(t \leq T\)
Up-and-in call	UIC	Activates only if \(S_t \geq B\) for some \(t \leq T\)

Parameter	Value
\(S_0\)	$100
\(K\)	$100
\(B\)	$80 (down-and-out)
\(r\)	3%
\(q\)	0%
\(v_0\)	0.04
\(\kappa\)	1.5
\(\theta\)	0.04
\(\xi\)	0.3
\(\rho\)	\(-0.7\)
\(T\)	1 year