For the case of European call options, we shall prove that The call options price is...

 

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For the case of European call options, we shall prove that ⚫The call options price is increasing with respect to the asset price and •The call options price is convex with respect to the asset price. We let c(St, T;X) be the current price of an European call option on a stock with maturity date T and strike price X. (a) Using no arbitrage pricing principle, show that for any positive constant k. c(kS,T; kX)=kc(S₁, T; X) (Hint: c(ks,,T; kX) can be viewed as the price of call option which the holder has the right to buy k units of the underlying asset at price kX at maturity date T.) (b) Using the result of (a) and the fact that the call options price is decreasing with respect to the strike price (i.e. property 2), show that the call price is increasing with respect to asset price. That is, for any S₁ <S₂, we have c(S₁,T;X) <c(S₂, T; X) ...... (1) (Hint: Let S₁ = h₁X and S₂ = h₂X (where h₁, h₂ are some constants) and rewrite the inequality (1) using the result of (a).) (c) Using similar method as in (b), show that the call price is a convex function of the asset price. That is, for any S₁ <S₂ and 0 ≤≤ 1, we have c(So, T; X) Ac(S₁, T; X) + (1-λ)c(S₂, T;X), where S₁ = AS₁ + (1-2)52. For the case of European call options, we shall prove that ⚫The call options price is increasing with respect to the asset price and •The call options price is convex with respect to the asset price. We let c(St, T;X) be the current price of an European call option on a stock with maturity date T and strike price X. (a) Using no arbitrage pricing principle, show that for any positive constant k. c(kS,T; kX)=kc(S₁, T; X) (Hint: c(ks,,T; kX) can be viewed as the price of call option which the holder has the right to buy k units of the underlying asset at price kX at maturity date T.) (b) Using the result of (a) and the fact that the call options price is decreasing with respect to the strike price (i.e. property 2), show that the call price is increasing with respect to asset price. That is, for any S₁ <S₂, we have c(S₁,T;X) <c(S₂, T; X) ...... (1) (Hint: Let S₁ = h₁X and S₂ = h₂X (where h₁, h₂ are some constants) and rewrite the inequality (1) using the result of (a).) (c) Using similar method as in (b), show that the call price is a convex function of the asset price. That is, for any S₁ <S₂ and 0 ≤≤ 1, we have c(So, T; X) Ac(S₁, T; X) + (1-λ)c(S₂, T;X), where S₁ = AS₁ + (1-2)52.

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To prove the properties of European call options lets go through each step a Using the no arbitrage pricing principle we can show that ckS T kX kcS T ... View the full answer