Option theory S2008 Q7 P(ii)

hello

please may I ask for some help with S2008 Q7 P(ii) a - valuation using a risk neutral portfolio :

Q7
"Consider a one-period Binomial model of a stock whose current price is 0 S = 40 . Suppose that:
• over a single period, the stock price can either move up to 60 or down to 30
• the continuously compounded risk-free rate is r = 5% per period


(ii) Calculate the price of a European call option with maturity date in one period and strike price K = 45 using each of the following methods:
(a) by constructing a risk-neutral portfolio"



Specifically the published solution with respect to valuing the option using a risk neutral portfolio states:
"First method: we construct a risk-neutral portfolio with 1 underlying asset and m call options. We choose the value of m such that this portfolio is risk neutral (its value in the upper state and in the lower state at time 1 should coincide). In this case, m = −2"

Can someone please help me understand what "value in the upper state and in the lower state at time 1 should coincide" means in the context of risk neutral

Any help would be appreciated. Pardon any lapse in decorum this is my first post.

 

It means that the value of the portfolio in the next period is known with certainty; i.e. it doesn't matter if the stock price increases or decreases.

 

hi Mugono,

Thank for your reply however Im not sure how that implies that 2 calls must be sold in order for the portfolio to be risk neutral. i.e. im not sure how the -2 is calculated.

Any guidance for me?

 

My advice would be to focus on the underlying concept set out in the core reading here. Correct me if this is outdated but this is based on the argument that holding a fraction (+delta units) of shares and a negative holding of the derivative eliminates delta risk and is therefore equivalent to investing the portfolio in a risk-free asset over the next instant.

I had a quick look at the answer; and saw that the explanation for why m = -2 was (conveniently) not explained .

The delta of a share is 1; and m is set such that the delta of the portfolio is 0. The delta of an ATM option is 0.5 [technically, this is true where the strike price equals the forward price (i.e. S*exp(r(T-t)) assuming no dividends). It is approximately true when the strike price equals the current share price. I include this here for completeness to possibly enhance your understanding - I suspect it is beyond the syllabus].

Setting m = -2 would suggest that the solution has assumed that the call option used to delta hedge the share is ATM (0.5 x 2 = 1). Stating this as an assumption would, in my view, have made the solution clearer

Aside:
The interest rate used in the original question looks to be too low. At a strike price = 45, the one period instantaneous interest rate should be around 12%. I would have been thrown too .


It was a tricky question !


Hope the above is helpful.

 

many thanks Mugono. your response was very helpful