Valuing a Down-and-In Claim from Down-and-Out Option Prices
Summary
The document works through an arbitrage-free valuation problem for a down-and-in claim paying the underlying stock value at maturity if a barrier is touched. The barrier equals the strike of a down-and-out call. The given market prices are for the down-and-out call and a down-and-out zero-coupon bond, while the question asks whether a down-and-out put price is also needed.
The solution decomposes the claim’s payoff into the stock payoff minus the down-and-out call payoff and the strike multiplied by the down-and-out bond payoff. This payoff identity avoids needing a separate down-and-out put price. Applying the stated prices and assuming zero interest rates gives a value of 61. The result depends on the barrier coinciding with the strike and on the zero-rate assumption; the document does not establish that the same calculation applies unchanged with different barrier or financing terms.
Key ideas
- A barrier claim can be valued by matching its maturity payoff with a portfolio of simpler claims.
- When the barrier equals the call strike, the down-and-in stock claim decomposes into stock minus a down-and-out call and strike times a down-and-out bond.
- This payoff identity eliminates the need to infer a down-and-out put price.
- The stated valuation assumes zero interest rates and the specified barrier relationship.
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# Pricing 'Down and In' claims
# Pricing 'Down and In' claims
I came across this question in a sheet of practice problems which has me a bit stumped.
> A down-and-out call option with maturity T, strike K = 100 and barrier L = K coinciding with the strike, trades at the price 5 SEK. The underlying stock is dividend-free and trades at 120 SEK, and the down-and-out version of a zero-coupon T-bond with face value 50 SEK and barrier L trades at 27 SEK. What is the arbitrage-free value of the down-and-in version of the T-claim X = S(T) with barrier L?
I have tried to combine the In/Out parity inequality with the (barrier) put/call parity, except that I can't (as far as I can see) calculate the down and out put value required:
Letting $\Pi(t)$ denote the price of our claim $X$, with $\Pi_{LO}, \Pi_{LI}$ denoting the corresponding down and out at $L$ contract, and down and in respectively. Then
$$\Pi_{LI} = \Pi_{t} - \Pi_{LO}$$
where $\Pi_t$ is just the stock price $S(t)$, and using the put/call parity we have
$$P_{LO} = KB_{LO} + C_{LO} - S_{LO}$$ where $P,B,C,S$ denoted the put price, zero coupon bond price, EU call price and the stock price. In the question $KB_{LO}$ and $ C_{LO}$ are given, but not $P_{LO}$ so I would need to find it, which I am not sure is possible only given the above information.
Any ideas on how to approach this?
## Answer by Gordon (score 6, accepted)
https://quant.stackexchange.com/a/37572
Let \begin{align*} \tau = \inf\{t: t \ge 0, S_t \le L \}. \end{align*} Then the down-out-call option has payoff \begin{align*} (S_T-K, 0)^+\pmb{1}_{\tau >T}, \end{align*} and the down-out version zero-coupon $T$-maturity bond has payoff \begin{align*} \pmb{1}_{\tau >T}. \end{align*} Moreover, for the down-in payoff $X$, since $L=K$, \begin{align*} X &= S_T \pmb{1}_{\tau \le T} \\ &=S_T - S_T \pmb{1}_{\tau >T}\\ &=S_T - (S_T-K) \pmb{1}_{\tau >T} - K \pmb{1}_{\tau >T}\\ &=S_T - (S_T-K)^+ \pmb{1}_{\tau >T} - K \pmb{1}_{\tau >T}. \end{align*} Assuming zero interest rate, then the value of $X$ is given by \begin{align*} 120-5 - 2 \times 27 = 61. \end{align*}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.