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Barrier Options

上海子午线About 4 min

Barrier Options

Visit the Mathema Option Pricing System, supporting FX options and structured product pricing and valuation!
Visit the Mathema Option Pricing System, supporting FX options and structured product pricing and valuation!

Stock barrier options have been traded in the over-the-counter (OTC) market since 1967. Barrier options have become extremely popular and are undoubtedly the most favored category of exotic options. The Chicago Board Options Exchange and the American Options Exchange now list up-and-out calls and down-and-out puts on stock indices. In the OTC market, a variety of barrier options are actively traded, including those on currencies, interest rates, and commodities.

European Barrier Options

Merton (1973) and Reiner and Rubinstein (1991a) proposed formulas for pricing standard barrier options (Rich, 1994). These formulas use a common set of factors:

A=ϕSe(b−r)TN(ϕx1)−ϕXe−rTN(ϕx1−ϕσT)B=ϕSe(b−r)TN(ϕx2)−ϕXe−rTN(ϕx2−ϕσT)C=ϕSe(b−r)T(H/S)2(μ+1)N(ηY1)−ϕXe−rT(H/S)2μN(ηY1−ησT)D=ϕSe(b−r)T(H/S)2(μ+1)N(ηy2)−ϕXe−rT(H/S)2μN(ηy2−ησT)E=Ke−rT[N(ηx2−ησT)−(H/S)2μN(ηy2−ησT)]F=K[(H/S)μ+λN(ηz)+(H/S)μ−λN(ηz−2ηλσT)]\begin{aligned} &A =\phi Se^{(b-r)T}N(\phi x_{1})-\phi Xe^{-rT}N(\phi x_{1}-\phi\sigma\sqrt{T}) \\ & B =\phi Se^{(\boldsymbol{b}-\boldsymbol{r})\boldsymbol{T}}N(\boldsymbol{\phi}x_{\boldsymbol{2}})-\boldsymbol{\phi}Xe^{-\boldsymbol{r}\boldsymbol{T}}N(\boldsymbol{\phi}x_{\boldsymbol{2}}-\boldsymbol{\phi}\sigma\sqrt{T}) \\ & C=\phi Se^{(b-r)T}(H/S)^{2(\mu+1)}N(\eta_{Y1})-\phi Xe^{-rT}(H/S)^{2\mu}N(\eta_{Y1}-\eta\sigma\sqrt{T}) \\ & D=\phi Se^{(b-r)T}(H/S)^{2(\mu+1)}N(\eta y_{2})-\phi Xe^{-rT}(H/S)^{2\mu}N(\eta y_{2}-\eta\sigma\sqrt{T}) \\ &E =Ke^{-rT}[N(\eta x_2-\eta\sigma\sqrt{T})-(H/S)^{2\mu}N(\eta y_2-\eta\sigma\sqrt{T})] \\ &F=K[(H/S)^{\mu+\lambda}N(\eta z)+(H/S)^{\mu-\lambda}N(\eta z-2\eta\lambda\sigma\sqrt{T})] \end{aligned}

Here:

x1=ln⁡(S/X)σT+(1+μ)σTx2=ln⁡(S/H)σT+(1+μ)σTy1=ln⁡(H2/(SX))σT+(1+μ)σTy2=ln⁡(H/S)σT+(1+μ)σTz=ln⁡(H/S)σT+λσTμ=b−σ2/2σ2λ=μ2+2rσ2\begin{aligned} x_1&=\frac{\ln(S/X)}{\sigma\sqrt T}+(1+\mu)\sigma\sqrt T & x_2&=\frac{\ln(S/H)}{\sigma\sqrt T}+(1+\mu)\sigma\sqrt T \\ y_1&=\frac{\ln(H^2/(SX))}{\sigma\sqrt T}+(1+\mu)\sigma\sqrt T & y_2&=\frac{\ln(H/S)}{\sigma\sqrt T}+(1+\mu)\sigma\sqrt T \\ z&=\frac{\ln(H/S)}{\sigma\sqrt T}+\lambda\sigma\sqrt T & \mu&=\frac{b-\sigma^2/2}{\sigma^2} & \lambda&=\sqrt{\mu^2+\frac{2r}{\sigma^2}} \end{aligned}

"Knock-in" Barrier Options

A "knock-in" barrier option is paid for today but only comes into existence if the asset price SS reaches the barrier level HH before expiration. It may also include a pre-specified cash rebate KK, which is paid if the option is not knocked in during its lifetime.

Down-and-in Call (S>HS > H)
Payoff: If S≤HS \leq H before expiration, the payoff is max⁡(S−X,0)\max(S - X, 0); otherwise, KK is paid at expiration.

cdi(X>H)=C+Eη=1,ϕ=1cdi(X<H)=A−B+D+Eη=1,ϕ=1\begin{aligned} c_{di(X>H)} &= C + E & \eta=1, \phi=1 \\ c_{di(X<H)} &= A - B + D + E & \eta=1, \phi=1 \end{aligned}

Up-and-in Call (S<HS < H)
Payoff: If S≥HS \geq H before expiration, the payoff is max⁡(S−X,0)\max(S - X, 0); otherwise, KK is paid at expiration.

cdi(X>H)=A+Eη=−1,ϕ=1cdi(X<H)=B−C+D+Eη=−1,ϕ=1\begin{aligned} c_{di(X>H)} &= A + E & \eta=-1, \phi=1 \\ c_{di(X<H)} &= B - C + D + E & \eta=-1, \phi=1 \end{aligned}

Down-and-in Put (S>HS > H)
Payoff: If S≤HS \leq H before expiration, the payoff is max⁡(X−S,0)\max(X - S, 0); otherwise, KK is paid at expiration.

pdi(X>H)=B−C+D+Eη=1,ϕ=−1pdi(X<H)=A+Eη=1,ϕ=−1\begin{aligned} p_{di(X>H)} &= B - C + D + E & \eta=1, \phi=-1 \\ p_{di(X<H)} &= A + E & \eta=1, \phi=-1 \end{aligned}

Up-and-in Put (S<HS < H)
Payoff: If S≥HS \geq H before expiration, the payoff is max⁡(X−S,0)\max(X - S, 0); otherwise, KK is paid at expiration.

pui(X>H)=A−B+D+Eη=−1,ϕ=−1pui(X<H)=C+Eη=−1,ϕ=−1\begin{aligned} p_{ui(X>H)} &= A - B + D + E & \eta=-1, \phi=-1 \\ p_{ui(X<H)} &= C + E & \eta=-1, \phi=-1 \end{aligned}

"Knock-out" Barrier Options

A "knock-out" barrier option is similar to a standard option, except that it becomes worthless if the asset price SS reaches the barrier level before expiration. It may include a pre-specified cash rebate KK, which is paid if the option is knocked out before expiration.

Down-and-out Call (S>HS > H)
Payoff: If S>HS > H before expiration, the payoff is max⁡(S−X,0)\max(S - X, 0); otherwise, KK is paid when the barrier is hit.

cdo(X>H)=A−C+Fη=1,ϕ=1cdo(X<H)=B−D+Fη=1,ϕ=1\begin{aligned} c_{do(X>H)} &= A - C + F & \eta=1, \phi=1 \\ c_{do(X<H)} &= B - D + F & \eta=1, \phi=1 \end{aligned}

Up-and-out Call (S<HS < H)
Payoff: If S<HS < H before expiration, the payoff is max⁡(S−X,0)\max(S - X, 0); otherwise, KK is paid when the barrier is hit.

cuo(X>H)=Fη=−1,ϕ=1cuo(X<H)=A−B+C−D+Fη=−1,ϕ=1\begin{aligned} c_{uo(X>H)} &= F & \eta=-1, \phi=1 \\ c_{uo(X<H)} &= A - B + C - D + F & \eta=-1, \phi=1 \end{aligned}

Down-and-out Put (S>HS > H)
Payoff: If S>HS > H before expiration, the payoff is max⁡(X−S,0)\max(X - S, 0); otherwise, KK is paid when the barrier is hit.

pdo(X>H)=A−B+C−D+Fη=1,ϕ=−1pdo(X<H)=Fη=1,ϕ=−1\begin{aligned} p_{do(X>H)} &= A - B + C - D + F & \eta=1, \phi=-1 \\ p_{do(X<H)} &= F & \eta=1, \phi=-1 \end{aligned}

Up-and-out Put (S<HS < H)
Payoff: If S<HS < H before expiration, the payoff is max⁡(X−S,0)\max(X - S, 0); otherwise, KK is paid when the barrier is hit.

puo(X>H)=B−D+Fη=−1,ϕ=−1puo(X<H)=A−C+Fη=−1,ϕ=−1\begin{aligned} p_{uo(X>H)} &= B - D + F & \eta=-1, \phi=-1 \\ p_{uo(X<H)} &= A - C + F & \eta=-1, \phi=-1 \end{aligned}

Calculation Example

import math
from scipy.stats import norm

def european_call_option(S, X, T, r, sigma):
    d1 = (math.log(S / X) + (r + 0.5 * sigma ** 2) * T) / (sigma * math.sqrt(T))
    d2 = d1 - sigma * math.sqrt(T)
    return S * norm.cdf(d1) - X * math.exp(-r * T) * norm.cdf(d2)

def down_and_in_call(S, X, T, r, sigma, H, K):
    mu = (r - 0.5 * sigma ** 2) / sigma ** 2
    lambda_ = math.sqrt(mu ** 2 + 2 * r / sigma ** 2)

    x1 = (math.log(S / X) + (1 + mu) * sigma * math.sqrt(T)) / (sigma * math.sqrt(T))
    x2 = (math.log(S / H) + (1 + mu) * sigma * math.sqrt(T)) / (sigma * math.sqrt(T))
    y1 = (math.log(H ** 2 / (S * X)) + (1 + mu) * sigma * math.sqrt(T)) / (sigma * math.sqrt(T))
    y2 = (math.log(H / S) + (1 + mu) * sigma * math.sqrt(T)) / (sigma * math.sqrt(T))
    z = (math.log(H / S) + lambda_ * sigma * math.sqrt(T)) / (sigma * math.sqrt(T))

    A = S * math.exp((r - sigma ** 2 / 2) * T) * norm.cdf(x1) - X * math.exp(-r * T) * norm.cdf(x1 - sigma * math.sqrt(T))
    B = S * math.exp((r - sigma ** 2 / 2) * T) * norm.cdf(x2) - X * math.exp(-r * T) * norm.cdf(x2 - sigma * math.sqrt(T))
    C = S * math.exp((r - sigma ** 2 / 2) * T) * (H / S) ** (2 * (mu + 1)) * norm.cdf(y1) - X * math.exp(-r * T) * (H / S) ** (2 * mu) * norm.cdf(y1 - sigma * math.sqrt(T))
    D = S * math.exp((r - sigma ** 2 / 2) * T) * (H / S) ** (2 * (mu + 1)) * norm.cdf(y2) - X * math.exp(-r * T) * (H / S) ** (2 * mu) * norm.cdf(y2 - sigma * math.sqrt(T))
    E = K * math.exp(-r * T) * (norm.cdf(-z) - (H / S) ** (2 * mu) * norm.cdf(y2 - sigma * math.sqrt(T)))
    F = K * ((H / S) ** (mu + lambda_) * norm.cdf(z) + (H / S) ** (mu - lambda_) * norm.cdf(z - 2 * lambda_ * sigma * math.sqrt(T)))

    c_di_X_H = A - B + D + E
    c_di_X_H_eta1_phi1 = c_di_X_H * 1 * 1

    return c_di_X_H_eta1_phi1

# Set parameters
S = 100  # Stock price
X = 95  # Strike price
T = 1  # Time to maturity (years)
r = 0.05  # Risk-free rate
sigma = 0.2  # Volatility
H = 90  # Barrier level
K = 5  # Cash rebate

# Calculate Down-and-in call option payoff
result = down_and_in_call(S, X, T, r, sigma, H, K)
print("Down-and-in call option payoff:", result)
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