Question

An asset that follows geometric Brownian motion will have a European call option with value c given by the risk-neutral expectation: c = e^{-rT}E^*left(max(Se^{(r-frac{1}{2}sigma^2)T+sigma W_T}-X,0) ight), (1) where S is the current spot price of the asset, X is the strike price for the option, r is the annual continuously compounded interest rate, T is the time to expiry, ? is the asset volatility, and W_T is a Wiener process. (a) Evaluate the integral above and show that this gives the Black-Scholes formula, c = SN(d_1) - Xe^{-rT}N(d_2), where N(z) indicates the (cumulative) standard normal probability distribution, and the parameters d_1 and d_2 are given by d_1 = frac{ln(S/X)+(r+frac{1}{2}sigma^2)T}{?sqrt{T}}, and d_2 = frac{ln(S/X)+(r-frac{1}{2}sigma^2)T}{?sqrt{T}}. (b) The same integral approach can be used to derive the expression for the ? of the call option: Delta_{coll} = frac{partial c}{partial S} = N(d_1). i. What range of values can ? have? ii. Explain why this range of values is to be expected, based on the pay-off behaviour of a call option with respect to asset price. iii. Determine an expression for the ? of the call option. (c) The vega indicates the dependence of the call option with respect to volatility: v_{call} = S_0sqrt{frac{T}{2pi}}e^{-frac{d_1^2}{2}}. i. How does the vega for a European call option compare to the vega of a European put option with the same strike price X and the same time to maturity T? Explain your answer. ii. Explain how and why options can be used to estimate the market perception of volatility.

          An asset that follows geometric Brownian motion will have a European call option with
value c given by the risk-neutral expectation:
c = e^{-rT}E^*left(max(Se^{(r-frac{1}{2}sigma^2)T+sigma W_T}-X,0)
ight),
(1)
where S is the current spot price of the asset, X is the strike price for the option, r
is the annual continuously compounded interest rate, T is the time to expiry, ? is the
asset volatility, and W_T is a Wiener process.
(a) Evaluate the integral above and show that this gives the Black-Scholes formula,
c = SN(d_1) - Xe^{-rT}N(d_2),
where N(z) indicates the (cumulative) standard normal probability distribution,
and the parameters d_1 and d_2 are given by
d_1 = frac{ln(S/X)+(r+frac{1}{2}sigma^2)T}{?sqrt{T}},
and
d_2 = frac{ln(S/X)+(r-frac{1}{2}sigma^2)T}{?sqrt{T}}.
(b) The same integral approach can be used to derive the expression for the ? of the
call option:
Delta_{coll} = frac{partial c}{partial S} = N(d_1).
i. What range of values can ? have?
ii. Explain why this range of values is to be expected, based on the pay-off
behaviour of a call option with respect to asset price.
iii. Determine an expression for the ? of the call option.
(c) The vega indicates the dependence of the call option with respect to volatility:
v_{call} = S_0sqrt{frac{T}{2pi}}e^{-frac{d_1^2}{2}}.
i. How does the vega for a European call option compare to the vega of a
European put option with the same strike price X and the same time to
maturity T? Explain your answer.
ii. Explain how and why options can be used to estimate the market perception
of volatility.

$An asset that follows geometric Brownian motion will have a European call option with value c given by the risk-neutral expectation: c = e^-rTE^*left(max(Se^(r-frac12sigma^2)T+sigma WT-X,0) ight), (1) where S is the current spot price of the asset, X is the strike price for the option, r is the annual continuously compounded interest rate, T is the time to expiry, ? is the asset volatility, and WT is a Wiener process. (a) Evaluate the integral above and show that this gives the Black-Scholes formula, c = SN(d1) - Xe^-rTN(d2), where N(z) indicates the (cumulative) standard normal probability distribution, and the parameters d1 and d2 are given by d1 = fracln(S/X)+(r+frac12sigma^2)T?sqrtT, and d2 = fracln(S/X)+(r-frac12sigma^2)T?sqrtT. (b) The same integral approach can be used to derive the expression for the ? of the call option: Deltacoll = fracpartial cpartial S = N(d1). i. What range of values can ? have? ii. Explain why this range of values is to be expected, based on the pay-off behaviour of a call option with respect to asset price. iii. Determine an expression for the ? of the call option. (c) The vega indicates the dependence of the call option with respect to volatility: vcall = S0sqrtfracT2pie^-fracd1^22. i. How does the vega for a European call option compare to the vega of a European put option with the same strike price X and the same time to maturity T? Explain your answer. ii. Explain how and why options can be used to estimate the market perception of volatility.$

Added by April R.

Question

Please give Ace some feedback