Question

You are attempting to formulate an investment strategy. In particular, you short a put option with strike price X₁ equals to $95 and you long another put option with strike price X₂ equals to $115. Both put options have the same underlying stock and will expire at time T.

(a) Plot the payoff structure of this investment strategy as a function of S_T, which stands for the price of the underlying stock at maturity. The X-axis for S_t and the Y-axis is the total payoff. [hint: you can first calculate the total payoff in the three scenarios: S_T < 100, 100 ≤  S_T ≤ 115 and S_T > 115.]

(b) given the following information:

Stock price right now S₀ = $100, X₁ = $95, X₂ = $115; the annualised risk free rate at 2%; volatility of the stock σ = 0.18; T equals to 5 years. The stock has a policy of not paying out any dividends. BAsed on these information, calculate the cost of establishing the above investment strategy based on the Black-Sholes model.

Answer 1

Part (a)

Payoff from a long put position = max (X - S, 0)

Payoff from a short put position = - max (X - S, 0)

Hence, total payoff from the investment strategy, Payoff = short a put option with strike price X1 equals to $95 and you long another put option with strike price X2 equals to $115 = - max (X1 - ST, 0) + max (X2 - ST, 0) = - max (95 - ST, 0) + max (115 - ST, 0)

Scenario 1:  ST < 95 (the question has stated it to be $ 100 which is wrong, the correct limit should be $ 95), Since T < 95, hence both the put options will be exercised. Hence, Payoff = - (95 - ST) + 115 - ST = $ 20

Scenario 2: 95 ≤ ST ≤ 115 : The short position will not be exercised, the long position will be exercised. Hence, Payoff = -0 + (115 - ST); Maximum value of this will be when ST = 95 and the max value = 115 - 95 = 20 and the minimum vale will be when ST = 115 and the minm value = 115 - ST = 115 - 115 = 0

Scenario 3: ST > 115.None of the two options will be exercised. Hence payoff will be 0

Hence, the plot of the payoff structure will be:

Total payoff ($) oor ܝ ܕܘ .orܘܝܝ 250 300 Stock Price, ST ($)

Part (b)

Let's find the price of the put option using Black Scholes model.

Price of the 95 Put option:

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

Please see the snapshot from my excel model:

21 Inputs 22 5 23 T 24 25 K 26 Risk free rate, 27 Output 28 d1 29 d2 30 N(-d1)= 31 N(-d2)= 100.000 5.000 0.18 95.000 Strike price of the put option 2% Values Formula Used 0.57713733=(LN(C22/C25)+(C26+C24^2/2)*C23)/(C24*SQRT(C23)) 0.174645094 =C28-C24*SQRT(C23) 0.281923344 =NORMSDIST(-C28) 0.430679257 =NORMSDIST(-C29) 8.82866 =C25*EXP(-C26*C23)*C31-C22*C30

115 Put option

22 s Bc. 21 Inputs 100.000 23 T 5.000 24 o 0.18 25 K 115.000 Strike price of the put option 26 Risk free rate, 2% 27 Output Formula Used 28 d1 0.102456778 =(LN(C22/C25)+(C26+C24^2/2)*C23)/(C24*SQRT(C23)) 29 d2 -0.300035458 =C28-C24*SQRT(C23) 30 N(-d1)= 0.459197059 =NORMSDIST(-C28) 31 N(-d2)= 0.617924945 =NORMSDIST(-C29) 32 Value of put option, p 18.37928 =C25*EXP(-C26*C23) *C31-C22*C30

the cost of establishing the above investment strategy based on the Black-Sholes model = 18.37928 -  8.82866 = $  9.5506