Option Pricing Theory (first paper) Flashcards
(16 cards)
What is the primary goal of the Black-Scholes model?
The primary goal of the Black-Scholes model is to find the fair price (theoretical value) of a European call option.
What is the key assumption behind the Black-Scholes model regarding arbitrage?
The key assumption is no arbitrage, meaning there are no opportunities for risk-free profit in the market.
How does the Black-Scholes model ensure no arbitrage?
The model uses dynamic hedging—creating a portfolio of the underlying asset and the call option that is adjusted to ensure it is risk-free, earning the risk-free return.
What is meant by “delta-hedging” in the Black-Scholes model?
Delta-hedging involves adjusting the portfolio’s position in the stock and call option to neutralize the risk (i.e., making the portfolio risk-free) and ensure it earns a return equal to the risk-free rate.
What role does the risk-free rate play in the Black-Scholes model?
The risk-free rate is the return of an asset with no risk, and the Black-Scholes model assumes that the hedged portfolio earns a return equal to the risk-free rate, ensuring no arbitrage.
What is the significance of the assumption of continuous trading in the Black-Scholes model?
Continuous trading allows for the constant rebalancing of the hedged portfolio, which helps maintain a no-arbitrage condition and the risk-free rate return.
What is the Black-Scholes option pricing formula for a call option?
The formula for a European call option price is:
C=S⋅N(d1)−K⋅e −rTN(d2)
Where:
C = Call option price
S = Stock price
K = Strike price
T = Time to expiration
r = Risk-free rate
N(d1) and N(d2) = Cumulative standard normal distribution functions
What does the term no-arbitrage condition mean in the context of the Black-Scholes model?
The no-arbitrage condition means that the price of the option is set such that there are no opportunities to make a risk-free profit by exploiting price differences between the option and the underlying asset.
How is a firm’s equity interpreted in the Merton model?
Equity is modeled as a European call option on the firm’s assets with the strike price equal to the face value of debt and maturity equal to the debt’s due date.
How is the firm’s total asset value modeled in Merton’s framework?
The asset value follows a geometric Brownian motion:
dV=μVdt+σVdW
where
σ represents asset volatility and
dW is a Wiener process (random shock).
When does default occur in the Merton model?
Default happens if the firm’s asset value at debt maturity is less than the face value of the debt. This leads to debt holders seizing the firm’s assets and equity holders getting nothing.
Why does Merton model equity as an option?
Because shareholders get paid only if the asset value exceeds the debt—just like a call option pays off only if the asset exceeds the strike price. This option-like payoff justifies using option pricing to value equity.
What inputs are needed to apply the Merton model?
Current asset value
𝑉
V
Debt face value
𝐷
D
Time to maturity
𝑇
T
Asset volatility
𝜎
σ
Risk-free rate
𝑟
r
How does the Merton model handle uncertainty in the future value of assets?
It doesn’t predict a specific outcome, but models a distribution of future values based on asset volatility, allowing for a probabilistic valuation of equity and debt.
What practical insights does the Merton model provide for capital structure?
It helps measure default probability, value risky debt, and assess how changes in leverage, volatility, or time to maturity affect the market value of equity and debt.
Does the Merton model predict future asset values?
No—it models a probabilistic distribution of future asset values using volatility. It’s not about predicting one outcome, but about valuing all possible outcomes weighted by their likelihood.
