Calculate options prices based on given options and spread prices

Question

Suppose you know the following information:

1. Futures price on a stock is 66
2. 70 strike straddle is trading at 27
3. 50-60 put spread is trading at 2.5
4. 50-60-70 put butterfly is trading at 0.2
5. Assume volatility is constant across strikes; interest rate is 0

Questions:

1. What are the fair values for the 80-strike call, 60-strike straddle, and 40-strike put
2. Now assume we have a volatility smile among the curve, how would this change your markets differently

My try:

Using put-call parity and direct definitions of the spreads, I have below equations

Call(K=70) - Put(K=70) = (Futures - K) = (66-70)

Call(K=70) + Put(K=70) = 27

Put(K=60) - Put(K=50) = 2.5

Put(K=50) + Put(K=70) - 2Put(K=60) = 0.2

Solving the above equations, I got:

Call(K=70) = 11.5

Put(K=70) = 15.5

Put(K=50) = 10.7

Put(K=60) = 13.2

Given the assumption of constant volatility, I am not sure how I should go from here to calculate values for:

Call(K=80)

Call(K=60) + Put(K=60)

Put(K=40)

Any help or hint is highly appreciated!

Answer 1

What you are given is a linear combination in instruments and corresponding (benchmark) prices, what you need are

1. invert the linear combinations to arrive at the benchmark prices
2. back out implied vols from benchmark prices
3. apply the vols to your new products.

For the first step, I'd go as

$$ \begin{align} Ax&=b\\ \begin{pmatrix}1&0&0&0&0\\ 0 & 0 & 0 & 1 & 1\\ 1 & 0 & 0 & 1 & -1\\ 0&1&-1&0&0\\ 0 & 1 & -2 & 1 & 0 \end{pmatrix}\begin{pmatrix}F\\P(50)\\P(60)\\P(70)\\C(70)\end{pmatrix}&=\begin{pmatrix}66\\27\\70\\2.5\\0.2\end{pmatrix} \\ \Rightarrow \quad x&=A^{-1}b\\ &=\begin{pmatrix}66\\20.3\\17.8\\15.5\\11.5\end{pmatrix} \end{align} $$

We can now back out the implied total variance (assuming zero interest rate) using the Black-Scholes-Model:

$$ \begin{align} C(\sigma_X^2T)&\equiv F\mathrm{N}\left(\frac{\ln(F)-ln(X)+\frac{1}{2}\sigma_X^2T}{\sigma_X\sqrt{T}} \right)-X\mathrm{N}\left(\frac{\ln(F)-ln(X)-\frac{1}{2}\sigma_X^2T}{\sigma_X\sqrt{T}} \right)\\ &\stackrel{!}{=}O(X) \end{align} $$ (accordingly for puts) where $\sigma_X^2T$, the total variance, is unknown and $O(X)$, the observed price, is given.

Using some root search method, you can now calculate the implied vols across all given strikes and option prices and obtain $\sigma_{50}^2T\approx 1.64192027$, $\sigma_{60}^2T\approx 0.71805172$, $\sigma_{70}^2T\approx 0.2494285$. You can then use these vols to price your other products.

NB: Question 1 may be ill-defined under the assumption of constant vols; probably you just have to 'pick' one vol? Question 2 then needs to be answered using the vols we just backed out.

HTH?