Question

Assume that  
S=40, 0 = 30
, . Suppose you enter into a put ratio spread where you buy a 45-strike put and sell two 40-strike puts. If you delta-hedge this position, what investment is required? What is your overnight profit if the stock tomorrow is $39? What if the stock is $40.50?

Answer 1

At the outset, let me tell you that the information on the time to maturity is missing. The question is therefore incomplete.

In order to solve this, I therefore will have to make an assumption about the time to maturity. A customary assumption in this kind of question is 180 days time to maturity. hence, we proceed with this assumption.

Black Scholes formula for Put:

p=ke" N(-d,)-S, N(-d) where di In(S/K)+(r+62/2) - OST d. - In(S/K)+(r-62/2) 2, OVT

Delta of a put option = N(d₁) - 1

Let's find the Price and Delta of the 45-strike put:

21 Inputs 22 S 23 T 24 o 25 K 26 Risk free rate, 27 Output 28 di 29 d2 30 N(-d1)= 31 N(-d2)= 32 Value of put option, p 33 Delta of the put option 40.00 Current stock price 0.4932 180 days = 180 / 365 year 30.00% 45 Strike price of the put option 8.00% Values Formula Used -0.266474186 =(LN(C22/C25)+(C26+C24^2/2)*C23)/(C24*SQRT(C23)) -0.477148251 =C28-C24*SQRT(C23) 0.605062981 =NORMSDIST(-C28) 0.683371721 =NORMSDIST(-C29) 5.3596 =C25*EXP(-C26*C23) *C31-C22*C30 (0.6051) 'ENORMSDIST(C28)-1

Let's now find the price and delta of the 40-strike put:

21 Inputs 22 S 23 T 24 O 25 K 26 Risk free rate, 27 Output 28 d1 29 d2 30 N(-d1)= 31 N(-d2)= 32 Value of put option, p 33 Delta of the put option 40.00 Current stock price 0.4932 180 days = 180 / 365 year 30.00% 40 Strike price of the put option 8.00% Values Formula Used 0.292602868 =(LN(C22/C25)+(C26+C24^2/2)*C23)/(C24*SQRT(C23)) 0.081928803 =C28-C24*SQRT(C23) 0.384912861 ENORMSDIST(-C28) 0.467351665 =NORMSDIST(-C29) 2.5744 =C25*EXP(-C26*C23)*C31-C22*C30 (0.3849) '=NORMSDIST(C28)-1

Delta of Put ratio spread = Delta of 45-strike put - 2 x delta of 40-strike put =  (0.6051) - 2 x  (0.3849) =  0.1648

Hence, to delta hedge this portfolio of put options, we need to short  0.1648 nos. of the underlying, i.e. we need to short  0.1657 nos. of stocks

Since, we have to short, no investment is required.

When S = 39, price of the 45 strike Put =  $ 5.9859 and that of 40 strike put will be = $   2.9778 (these two values can be worked out using the same model as above, to avoid repetition, I am not reproducing the same)

Hence, gain / (loss) from the position = Gain from short position in 0.1648 shares + Gain from price change of 45 strike put option - 2 x gain from price change of 40 strike put option = -0.1648 x (39 - 40) + (5.9859 -  5.3596) - 2 x (2.9778 -  2.5744) =  (0.0158) i.e. a loss of $ 0.0158

When S = 40.50, price of the 45 strike Put =  $ 5.0598 and that of 40 strike put will be = $  2.3824 (these two values can be worked out using the same model as above, to avoid repetition, I am not reproducing the same)

Hence, gain / (loss) from the position = Gain from short position in 0.1648 shares + Gain from price change of 45 strike put option - 2 x gain from price change of 40 strike put option = -0.1648 x (40.50 - 40) + ( 5.0598 -  5.3596) - 2 x (2.3824
-  2.5744) =   0.0018 i.e. again of $ 0.0018