Problem 3 Consider again the exchange opti... [FREE SOLUTION]

Chapter 9: Problem 3

Consider again the exchange option in Example 9.5. Now assume that $W_{1}$ and $W_{2}$ are no longer independent, but that the local correlation is given by $d W_{1} \cdot d W_{2}=e d t$. (We still assume that both Wiener processes have unit variance parameter, i.e. that $\left.d \bar{W}_{1}^{2}=d \bar{W}_{2}^{2}-d t .\right)$ How will this affect the Black-Scholes-type formula given in the example?

Short Answer

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Answer: The local correlation between the two Wiener processes affects the expressions of $d_1$ and $d_2$ in the Black-Scholes formula for the exchange option. Instead of using the correlation coefficient between the two assets ($\rho$), the term $2\rho \sigma_1\sigma_2$ in the $d_1$ and $d_2$ expressions is replaced with the term $2e$ due to the given local correlation. This results in a modified Black-Scholes formula that considers the given local correlation.

Step by step solution

- Review the Original Black-Scholes Formula for the Exchange Option

In Example 9.5, the Black-Scholes formula for the exchange option (Option to exchange one asset for another) is given by: $$C(S_{1}, S_{2}, t) = S_{1}N(d_1) - S_{2}N(d_2),$$ where $d_1 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})+\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2\rho \sigma_{1}\sigma_{2})(T-t)}}$ and $d_2 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})-\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2\rho \sigma_{1}\sigma_{2})(T-t)}}$. Here, $C(S_1, S_2, t)$ is the option price, $S_1$ and $S_2$ are the first and second asset prices, $T-t$ is the time until expiration, $\sigma_{1}$ and $\sigma_{2}$ represent the volatilities of the two assets, and $\rho$ represents the correlation coefficient between the two assets. In this exercise, we are given that the local correlation between the two Wiener processes is expressed as $dW_1 \cdot dW_2 = edt$. We will use this to modify the original formula.

- Determine the Effect of Local Correlation on Option Pricing

The given correlation between the Wiener processes, $dW_1 \cdot dW_2 = edt$, will have an impact on the formula for $d_1$ and $d_2$, as their expressions depend on the correlation between the two assets. Instead of using the correlation coefficient between the two assets ($\rho$) in the $d_1$ and $d_2$ expressions, we will now use the given local correlation.

- Modify the Black-Scholes Formula Based on Local Correlation

The modified Black-Scholes formula will involve replacing the term $2\rho \sigma_1\sigma_2$ in $d_1$ and $d_2$ with the term $2e$ due to the given local correlation $dW_1 \cdot dW_2 = edt$. The modified formulas for $d_1$ and $d_2$ are as follows: $d_1 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})+\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2e)(T-t)}}$ and $d_2 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})-\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2e)(T-t)}}$. Now, we can rewrite the modified Black-Scholes formula for the exchange option as: $$C(S_{1}, S_{2}, t) = S_{1}N(d_1) - S_{2}N(d_2),$$ Now, with $d_1 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})+\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2e)(T-t)}}$ and $d_2 = \dfrac{\ln(\dfrac{S_{1}}{S_{2}})-\left(\sigma_{1}^2-\sigma_{2}^2\right)(T-t)}{\sqrt{(\sigma_{1}^2+\sigma_{2}^2-2e)(T-t)}}$. By following these steps, we have obtained the modified Black-Scholes formula for the exchange option considering the local correlation between the Wiener processes given by $dW_1 \cdot dW_2 = edt$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Exchange Option Pricing

The concept of exchange option pricing is a fascinating area in financial mathematics. It refers to the valuation of options that give the holder the right to exchange one asset for another. Unlike standard options, which provide the right to buy or sell an asset, exchange options involve two underlying assets.
To understand how we price such options, the Black-Scholes model is frequently employed. In the given exercise, the exchange option's price is determined using a formula similar to the standard Black-Scholes, but adapted to consider both assets.
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Short Answer

Step by step solution

- Review the Original Black-Scholes Formula for the Exchange Option

- Determine the Effect of Local Correlation on Option Pricing

- Modify the Black-Scholes Formula Based on Local Correlation

Key Concepts

Exchange Option Pricing

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