# A European call option is a financial contract the gives...

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MATH 472: Homework 5 Fall 2017 Due 13:00 on Thursday, November 16. Problems 1. A European call option is a financial contract the gives its holder that right to buy one share of stock at strike price K at maturity T . The famous Black Scholes formula gives the price of such a contract under the Black Scholes model. In this model, the price process for the stock is assumed to solve a specific stochastic differential equations starting from S at time 0 . Using this differential equation one can show that one needs to know the following parameters to compute the fair price of the European call option: The maturity of the contract T in number of years The current price of the underlying stock S . The strike price of the option K . The interest rate in the market r . The volatility of the underlying stock σ (annualized). Suppose the continuously compounded interest rate r is a constant. The Black-Scholes price for a European call option with maturity T is C BS ( T, S, K, r, σ ) = SN ( d 1 ) - Ke - rT N ( d 2 ) , (1) where d 1 = ln( S/K ) + ( r + σ 2 2 ) T σ T , (2) d 2 = d 1 - σ T, (3) where S is the stock price at time 0, N is the cumulative distribution function of standard normal. (a) (20 Points) Write a matlab function C BS computing the price of an option for given ( T, S, K, r, σ ) . (b) (20 Points) Note that ( T, S, K, r ) are observable in the market and σ is not. Hence one needs to obtain the volatility of the stock. However, the price of a European call option can be directly observed in the market C market ( K ) . This data is quoted in the market for some European call options. Thus, for a given option one can compute the volatility σ implied to be plugged in the formula (1), to have C BS ( T, S, K, r, σ implied ) = C market ( K ) . This σ implied that depends on the option and is called the implied volatility of the Eurpoean call option and will be denote σ implied ( K ) . You are given the following table of strikes K and C market ( K ) . 500 550 600 650 700 750 800 850 900 950 210 . 3400 166 . 3140 126 . 5249 89 . 0857 59 . 5878 43 . 5040 31 . 3392 25 . 2330 20 . 1734 15 . 7494 Assume that S = \$700 , T = 1 / 4 (in years) and r = 0 . 03 . Compute the implied volatility σ implied ( K ) correct to six decimal places with for all 10 European call options given using: (i) Bisection method on the interval [0.0001,1], (ii) Secant method with with initial guesses 0.3, 0.35. (c) (10 Points) Plot the graph of implied volatility as a function of the strike K . 1
%Problem 1 of Exercise Sheet 5 (Part a) %Black-Scholes formula for a European call option %T is the maturity of the contract (in years) %S is the current (t=0) price of the underlying stock %K is the strike price of the option %r is the interest rate in the market %sigma is tha (annualized) volatility of the underlying stock function Europ_Call_BS=C_BS(T,S,K,r,sigma) d_1=(log(S./K)+(r+0.5*sigma.^2).*T)./(sigma.*sqrt(T)); d_2=d_1-sigma.*sqrt(T); Europ_Call_BS=S*normcdf(d_1)-K.*exp(-r.*T).*normcdf(d_2); %Problem 1 of Exercise Sheet 5 (Part b and c) %Calculate the Implied Volatility by means of