Demonstrate the arbitrage opportunity can be exploited. Assume we traded 100 call options with option options selling for $6.5 (Additional information, if we use binomial tree to calculate the call option price: the call option should be $5.14. Spot price of asset is $30 and chance of going up to $40 in next period is 55% while chance of going down to $22.5 is 45% )
The solution first calculates the hedge ratio (ie. delta which is equal to ($10-0)/($40-$22.5)=0.5714 shares per option. Thus, we need to purchase 57.14 shares.
and then it calculates the portfolio value in up-move. The value of portfolio = 57.14*40- (100*10)=1,286. i dunno understand why the value of short options is 100*10 instead of 100*0.5714*10.
Is that 1 option only covers 0.5714 shares?
I found another example in the book supporting what i think.
Example 2:
A owns 60,000 shares of Co. B Stock with spot price is $50. A call option with strike price of $50 is selling for $4 and has a delta of 0.6.
Calculate no. of option needed to create a delta hedge
Calculate the effect on portfolio value of $1 increase in the price of Co. B stock.
For part (1): no of option needed =60,000/0.6=100,000 options
For part (2): the total value of increase in the portfolio= 60,000*1 (total value increase of stock) - 0.6*100,000 (the total value decrease in short option)
In example 1, the short call option value = change in option value @ t=1 (i.e.$10) * no. of option (i.e.100) while in example 2, the value of short options = delta (i.e. 0.6) * number of options (i.e.100,000)*change in option value @t=1 (i.e.$1)
Thanks but could you pls elaborate more why in example 1, the short call option value = change in option value @ t=1 (i.e.$10) * no. of option (i.e.100) while in example 2, the value of short options = delta (i.e. 0.6) * number of options (i.e.100,000)*change in option value @t=1 (i.e.$1)
Should i multiply delta when i calculate the value of short options? Thanks!
The value of short option is - (100*10) because you shorted 100 options, and each option costs you -10$
And the value of the portfolio which containing 57.14 shares & 100 short options is 57.14*40- (100*10)=1,286 ($)
In the example 2, the portfolio contains 60,000 shares and 100,000 short call option. When the price of share increases 1$, each call option increase: Variation(Call) with Variation (Call) = Delta * Variation(Share) = 0.6 * 1$ = 0.6$
So, The value of portfolio is 60,000*1$ - 100,000*0.6$
And I change what I said in the post above.
Delta hedge is a strategy of keeping ( delta * shares and the rest in cash) in your portfolio from inception until the expiration of the option. At the maturity, the value of your portfolio will be exactly the the value of (Price (T) - K) at the maturity.
what you thought, a call can hedge a share, now I don’t think understand you.