Pricing a Piecewise European Claim Under Black–Scholes
Summary
The document derives a risk-neutral price for a European payoff that receives the asset value when the terminal price is at or below a strike and pays the negative strike when the price is above it. With zero interest rates and a lognormal terminal asset price, it splits the expectation at the strike, evaluates the truncated lognormal expectation, and expresses the result using the normal cumulative distribution function. It then seeks an equivalent representation in terms of a Black–Scholes call price with an altered volatility parameter.
The proposed algebra identifies a second volatility by matching the call formula’s normal arguments. This is a formal parameter-matching exercise, not a general hedging or trading method. The document offers no numerical validation and does not discuss whether the resulting parameter is positive or meaningful for all inputs; in particular, its expression depends on the relationship between strike and spot. The assumptions are restrictive: European maturity payoff, lognormal dynamics, and zero rates.
Key ideas
- The claim price is written as a risk-neutral expectation split across the strike threshold.
- Under the stated lognormal model, each region of the payoff can be evaluated using normal distribution probabilities.
- The derivation attempts to match the resulting expression to a Black–Scholes call with a modified volatility.
- The proposed volatility depends on strike, spot, original volatility, and maturity, so its applicability needs checking for the chosen inputs.
- The setup assumes zero interest rates and does not establish a broader pricing or hedging result.
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Full text
# Initial price of contingent claim in terms of Black-Scholes call option initial price with new volatility parameter
# Initial price of contingent claim in terms of Black-Scholes call option initial price with new volatility parameter
In this problem we assume the interest rate $r=0$. I have a European contingent claim with maturity $T$ and payout $$Y = S_T 1_{\{S_T \leq K\}} - K1_{\{S_T > K\}}$$
I need to show that the initial price of this claim can be written as the Black-Scholes call option initial price $$EC(S_0, K, \sigma, T) = S_0 \Phi(d_1) - K\Phi(d_2), \quad d_1 = -\frac{\log(K/S_0)}{\sigma\sqrt{T}} + \frac{\sigma\sqrt{T}}{2}, \quad d_2 = d_1 - \sigma\sqrt{T}$$ with an altered volatility parameter.
I am quite new to these sorts of calculations so I wanted to verify my solution here and check if it is correct!
The initial price is equal to $$\pi_0 = \mathbb{E}^{\mathbb{Q}}[Y | \mathcal{F}_0] = \mathbb{E}^{\mathbb{Q}}[S_T1_{\{S_T \leq K\}} |\mathcal{F}_0] - K \mathbb{E}^{\mathbb{Q}}[1_{\{S_T > K\}}|\mathcal{F}_0]$$
Under $\mathbb{Q}$ we can write $$S_T = S_0e^{-\sigma^2 T /2 + \sigma\sqrt{T}Z}$$ where $Z\sim N(0,1)$.
Note $S_T \leq K \iff Z \leq -d_2$ and $S_T > K \iff Z > -d_2$.
With information $\mathcal{F}_0$, we know the value of $S_0$ and so $$\pi_0 = S_0 \mathbb{E}^{\mathbb{Q}}[e^{-\sigma^2 T / 2 + \sigma \sqrt{T} Z}1_{\{Z \leq -d_2\}}] - K\mathbb{E}^{\mathbb{Q}}[1_{\{Z > -d_2\}}]$$.
Writing the first term as an integral and then using a change of variables, and using the symmetry of $\Phi$ on the second term, we have $$\pi_0 = S_0 \Phi(-d_2 - \sigma\sqrt{T}) - K\Phi(d_2)$$
So we want to find a volatility parameter $\sigma'$ such that $$-d_2(S_0, K, \sigma', T) -\sigma'\sqrt{T} = d_1(S_0, K, \sigma, T)\quad \text{ and }\quad d_2(S_0, K, \sigma', T) = d_2(S_0, K, \sigma, T)$$
Considering the second equality, one solution is $\sigma' = \sigma$, but then this does not allow for the first equality to hold. The other solution to this equation is $$\sigma' = \frac{2\log(K/S_0)}{\sigma T}$$
This choice of $\sigma'$ also allows the first equality to hold, so we have our required volatility parameter.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.