Encyclopedia of Quantitative Finance, IV Volume Set

ICAPM see Capital Asset Pricing Model

Implied Volatility: Large Strike Asymptotics

Let S_t be the price of a risky asset at time t [0, T] and B_t = B (t, T) the time-t value of one monetary unit received at time T. Assuming suitable no-arbitrage conditions, there exists a probability measure , called the (T -forward) pricing measure, under which the B_t-discounted asset price

is a martingale and so are B_t-discounted time-t option prices, such as C_t/B (t, T), where C_t denotes the time-t value of a European call option with maturity T and payoff (S_T − K)⁺. With focus on t = 0 and writing C instead of C₀, we have

Let us remark that in the case of deterministic interest rates r (·), one can rewrite this as^a

If we now make the assumption that there exists σ > 0, the Black–Scholes volatility