Understanding the Payoff Structure

Oct 13, 2025 by windailys.com 12 views

Introduction

In the realm of options pricing, the Black-Scholes model is a widely used framework for valuing European call and put options. However, there are instances where the standard Black-Scholes formula may not be applicable, and alternative valuation methods are required. One such scenario is the valuation of an "Asymptotic" call option, which has a unique payoff structure. In this article, we will delve into the derivation of a valuation formula for an "Asymptotic" call option.

Understanding the Payoff Structure

The payoff of an "Asymptotic" call option is given by the formula:

K \times \max\left[1 - \frac{K}{S_T}, 0\right]

where:

$K$ is the strike price of the option
$S_T$ is the price of the underlying asset at maturity
$\max\left[1 - \frac{K}{S_T}, 0\right]$ is the payoff function, which returns the maximum value between 1 minus the ratio of the strike price to the underlying asset's price at maturity, and 0.

Derivation of the Valuation Formula

To derive the valuation formula for an "Asymptotic" call option, we can start by considering the following:

When the underlying asset's price at maturity is greater than the strike price, the payoff function is 0, and the option expires worthless.
When the underlying asset's price at maturity is less than or equal to the strike price, the payoff function is 1 minus the ratio of the strike price to the underlying asset's price at maturity.

Using this insight, we can rewrite the payoff function as:

K \times \max\left[1 - \frac{K}{S_T}, 0\right] = K \times \left(1 - \frac{K}{S_T}\right) \times \mathbb{I}_{\left\{S_T \leq K\right\}}

where $\mathbb{I}_{\left\{S_T \leq K\right\}}$ is an indicator function that takes the value 1 when the underlying asset's price at maturity is less than or equal to the strike price, and 0 otherwise.

Applying the Risk-Neutral Valuation Framework

To derive the valuation formula, we can apply the risk-neutral valuation framework, which states that the value of a derivative security is equal to the discounted expected value of its payoff under the risk-neutral measure.

Using this framework, we can write the valuation formula for the "Asymptotic" call option as:

V(S_t, t) = e^{-r(T-t)} \times \mathbb{E}^Q \left[ K \times \left(1 - \frac{K}{S_T}\right) \times \mathbb{I}_{\left\{S_T \leq K\right\}} \right]

where:

$V(S_t, t)$ is the value of the "Asymptotic" call option at time $t$ and underlying asset price $S_t$
$r$ is the risk-free interest rate
$T$ is the maturity date of the option
$\mathbb{E}^Q$ is the expected value operator under the risk-neutral measure
$\mathbb{I}_{\left\{S_T \leq K\right\}}$ is the indicator function

Simplifying the Valuation Formula

To simplify the valuation formula, we can use the fact that the underlying asset's price at maturity is lognormally distributed under the risk-neutral measure.

Using this fact, we can write the valuation formula as:

V(S_t, t) = e^{-r(T-t)} \times \int_0^K \left(1 - \frac{K}{S_T}\right) \times \frac{1}{S_T} \times \frac{1}{\sigma \sqrt{T-t}} \times \exp\left(-\frac{(\ln(S_T) - \ln(S_t) - (r - \frac{1}{2}\sigma^2)(T-t))^2}{2\sigma^2(T-t)}\right) dS_T

where:

$\sigma$ is the volatility of the underlying asset

Numerical Computation of the Valuation Formula

To compute the valuation formula numerically, we can use a variety of methods, including Monte Carlo simulation and finite difference methods.

Using a Monte Carlo simulation approach, we can generate a large number of random paths for the underlying asset's price at maturity, and then compute the average payoff of the "Asymptotic" call option over these paths.

Conclusion

In this article, we have derived a valuation formula for an "Asymptotic" call option, which has a unique payoff structure. We have applied the risk-neutral valuation framework and used the fact that the underlying asset's price at maturity is lognormally distributed under the risk-neutral measure to simplify the valuation formula. We have also discussed the numerical computation of the valuation formula using Monte Carlo simulation and finite difference methods.

References

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
Hull, J. C. (2009). Options, futures, and other derivatives. Prentice Hall.

Code

import numpy as np
from scipy.stats import norm
def asymptotic_call_option_valuation(S_t, t, K, r, T, sigma):
"""
Valuation formula for an "Asymptotic" call option.
Parameters:
S_t (float): Underlying asset price at time t
t (float): Time
K (float): Strike price
r (float): Risk-free interest rate
T (float): Maturity date
sigma (float): Volatility of the underlying asset

Returns:
float: Value of the &quot;Asymptotic&quot; call option
&quot;&quot;&quot;
# Compute the risk-neutral expectation
expectation = 0
for S_T in np.linspace(0, K, 1000):
    expectation += (1 - K/S_T) * (1/S_T) * norm.pdf(np.log(S_T) - np.log(S_t) - (r - 0.5*sigma**2)*(T-t), 0, sigma*np.sqrt(T-t))

# Discount the expectation
value = np.exp(-r*(T-t)) * expectation

return value

S_t = 100
t = 0
K = 100
r = 0.05
T = 1
sigma = 0.2
value = asymptotic_call_option_valuation(S_t, t, K, r, T, sigma)
print(value)

Note: The code provided is a simplified example and may not be suitable for production use.

Introduction

In our previous article, we derived a valuation formula for an "Asymptotic" call option, which has a unique payoff structure. In this article, we will answer some frequently asked questions about the valuation of "Asymptotic" call options.

Q: What is an "Asymptotic" call option?

A: An "Asymptotic" call option is a type of option that has a unique payoff structure. The payoff of an "Asymptotic" call option is given by the formula:

K \times \max\left[1 - \frac{K}{S_T}, 0\right]

where:

$K$ is the strike price of the option
$S_T$ is the price of the underlying asset at maturity
$\max\left[1 - \frac{K}{S_T}, 0\right]$ is the payoff function, which returns the maximum value between 1 minus the ratio of the strike price to the underlying asset's price at maturity, and 0.

Q: How is the valuation of an "Asymptotic" call option different from the valuation of a standard call option?

A: The valuation of an "Asymptotic" call option is different from the valuation of a standard call option because of its unique payoff structure. The payoff of an "Asymptotic" call option is not simply the maximum value between the strike price and the underlying asset's price at maturity, but rather a more complex function that depends on the ratio of the strike price to the underlying asset's price at maturity.

Q: Can the valuation of an "Asymptotic" call option be approximated using a standard call option valuation model?

A: No, the valuation of an "Asymptotic" call option cannot be approximated using a standard call option valuation model. The unique payoff structure of an "Asymptotic" call option requires a more complex valuation model that takes into account the specific characteristics of the option.

Q: What are some common applications of "Asymptotic" call options?

A: "Asymptotic" call options have a variety of applications in finance, including:

Hedging against extreme market movements
Speculating on the price of an underlying asset
Creating complex investment products

Q: How can the valuation of an "Asymptotic" call option be computed in practice?

A: The valuation of an "Asymptotic" call option can be computed using a variety of methods, including:

Monte Carlo simulation
Finite difference methods
Analytical solutions

Q: What are some common challenges associated with the valuation of "Asymptotic" call options?

A: Some common challenges associated with the valuation of "Asymptotic" call options include:

Complexity of the payoff structure
Sensitivity to model parameters
Limited availability of data

Q: Can the valuation of an "Asymptotic" call option be affected by market conditions?

A: Yes, the valuation of an "Asymptotic" call option can be affected by market conditions, including:

Volatility of the underlying asset
Interest rates
Market sentiment

Q: How can the valuation of an "Asymptotic" call option be used in practice?

A: The valuation of an "Asymptotic" call option can be used in a variety of ways in practice, including:

Pricing and hedging complex investment products
Managing risk in a portfolio
Speculating on the price of an underlying asset

Conclusion

In this article, we have answered some frequently asked questions about the valuation of "Asymptotic" call options. We hope that this information has been helpful in understanding the valuation of these complex options.

References

Black, F., & Scholes, M. (1973). The pricing of options and corporate liabilities. Journal of Political Economy, 81(3), 637-654.
Merton, R. C. (1973). Theory of rational option pricing. Bell Journal of Economics and Management Science, 4(1), 141-183.
Hull, J. C. (2009). Options, futures, and other derivatives. Prentice Hall.

Code

import numpy as np
from scipy.stats import norm
def asymptotic_call_option_valuation(S_t, t, K, r, T, sigma):
"""
Valuation formula for an "Asymptotic" call option.
Parameters:
S_t (float): Underlying asset price at time t
t (float): Time
K (float): Strike price
r (float): Risk-free interest rate
T (float): Maturity date
sigma (float): Volatility of the underlying asset

Returns:
float: Value of the &quot;Asymptotic&quot; call option
&quot;&quot;&quot;
# Compute the risk-neutral expectation
expectation = 0
for S_T in np.linspace(0, K, 1000):
    expectation += (1 - K/S_T) * (1/S_T) * norm.pdf(np.log(S_T) - np.log(S_t) - (r - 0.5*sigma**2)*(T-t), 0, sigma*np.sqrt(T-t))

# Discount the expectation
value = np.exp(-r*(T-t)) * expectation

return value

S_t = 100
t = 0
K = 100
r = 0.05
T = 1
sigma = 0.2
value = asymptotic_call_option_valuation(S_t, t, K, r, T, sigma)
print(value)

Note: The code provided is a simplified example and may not be suitable for production use.