We’re rewarding the question askers & reputations are being recalculated! Read more.

# Tag Info

15

Short Version : Two main uses I'm doing an arbitrage/statarb strategy (volatility for instance) which should not be dependant on the Delta (I'm an arbitragist). I HAVE to keep a product in my portfolio, but I don't want to be EXPOSED to it (I'm a market maker). Long Version : The goal of Dynamic Hedging is not down the line to earn risk free rate of ...

12

Actually there are more than just ideas and hints concerning this topic. There is an intuitive model and solution to your question already using machinery of option theory. But don't worry, it's not a surprise that you didn't find any useful literature in your search because the proposed solution actually comes from a very different topic. In addition to ...

12

A digital call option (cash-or-nothing) can be replicated with two call options with different Strike. When we make the delta infinitely small and assume we have arbitrary strike prices. We get:

10

Regarding your 1st question, jumps are indeed unhedgeable. From a theoretical point of view, you might want to look at Merton's "Option pricing when underlying stock returns are discontinuous", the original paper that adapted Black-Scholes framework to include jumps. If you look at page 7, just after equation $(9)$: Unfortunately, in the presence of the ...

6

While another user touched on the hedging argument in order to reconcile your intuition with the correct value of the option he went off track (imho). I like to focus entirely on the hedging issue because it is key in understanding the differences in intuition and the fair price of such option. Unfortunately I have hardly ever found a simple 1-2 paragraph ...

6

Specifically, we have a generic conditional claim, $C$, that is a function of the diffusion process for the underlying, $S(t)$, and time $t$ so $C = C(S(t), t)$. As you pointed out, $C$ is an Ito process becuase it is a function of a stochastic process so we use Ito's Lemma to determine how the contingent claim varies as a function of the diffusion process $... 6 I think there is an error implicit in your question. Dynamic delta hedging, even assuming the underlying process is a continuous martingale and trading entails zero transaction costs, only eliminates the directional risk. A number of residual risks remain, most notably volatility risk, embodied in both the gamma and vega. A dynamically hedged portfolio of ... 6 A general hedging strategy Let assume that$S_1(t)$and$S_2(t)$are the price processes of your 2 stocks and that they follow a Geometric Brownian Motion (GBM): $$\forall \, i \in \{1,2\}, dS_i(t) =\mu_iS_i(t)dt + \sigma_iS_i(t)dW_i(t)$$ We assume both stocks have an instant correlation of$\rho$: $$dW_1(t)dW_2(t)=\rho dt$$ Let also$V(t)$be the value ... 6 Any non path dependent European type payoff$f(S_T)$can be replicated in a model independent way with vanilla calls and puts provided$f$is twice differentiable (in the distribution sense). This is a consequence of the Carr-Madan formula. 5 If I understand well, you have a market with 3 states: up, flat or down. You have 3 instruments: The stock The risk-free rate (50%) The option If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of ... 5 You have to differentiate here between the risk-taking and the market-making side. As a risk-taker, like e.g. a hedge-fund, you are right, you could just buy the bond! But as a market-maker you sell these options but don't want to bear the risk, so you have to counterbalance it. You could of course counterbalance it with another option which would be the ... 5 To see the exposure to FX risk and the difficulty for hedging, we assume constant interest rates and constant volatilities. Let$r_d$and$r_f$denote respectively the interest rates for USD and EUR. Moreover, let$X_t$be the exchange rate at time$t$from one unit USD to units of EUR. Finally, let$S_t$be the price level of DAX at time$t$. We assume that,... 5 Consider the case where we are interested in decomposing a continuous and piece-wise linear European payoff function$V \left( S_T \right)$over$n$intervals with$n + 1$node points$S_i$for$i = 0, 1, \ldots, n$. Without loss of generality, we assume that$S_0 = 0$and write$V_i$as short-hand for$V \left( S_i \right). We assume that the slope of the ... 4 You are treading controversial waters. It's hard to summarize, but at the risk of oversimplifying, there are three broad schools of thought: "Linear Models": Classic Examples are a string of papers from Jasmina Hasanhodzic and Andy Lo at MIT (scholar.google.com should give you plenty). For similar work related to Mutual Funds that you may be able to ... 4 A good place to start is Hagan's paper Convexity Conundrum ...available on the web. 4 I don't have much experience in the matter, but I've been doing some related literature research recently and I think these links can be helpful: A rather recent study from CME A (possible a bit biased) report by BlackRock A report by Lyxor (asset manager affialiated to Societe Generale) 4 Note that \begin{align*} S_T^2 = 2\int_0^{S_T} k dk. \end{align*} Then \begin{align*} S_T^2 &= 2S_T^2-2\int_0^{S_T} k dk\\ &=2S_T\int_0^{S_T}dk-2\int_0^{S_T} k dk\\ &=2\int_0^{S_T} (S_T-k)dk\\ &=2\int_0^{\infty} (S_T-k)^+dk. \end{align*} For the partition0=k_0 < k_1 < \cdots < k_n < \infty, \begin{align*} S_T^2 &=2\int_0^{\... 4 I assume your tradeV(S,K,t,T)is European. Its payoff is: \begin{align} V(S,K,T,T)&=C^2(S,K,T,T) \\[3pt] &=\max(S_T-K,0)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}(S_T-K)^2 \\[3pt] &=\boldsymbol{1}_{\{S_T\geq K\}}f(S_T) \end{align} wheref(x)=(x-K)^2$. By Carr-Madan's static replication formula (see this question or this paper), we ... 3 Regarding the dividends: In order to avoid jumps on ex-dividend date, you can make the simplifying assumption that dividends are paid continuously and adjust the returns of the assets. The size of dividends could be estimated from historical data or can be set proportionally to the asset price. 3 While not really an answer, here are my thoughts on the problem. For starters, I would approach the problem as one of whether a portfolio which is constantly rebalanced over some horizon can be replicated using vanilla options. Obviously the portfolio will have to be rolled over when the options reach expiration. Then the rest, such as payoff diagram and ... 3 That's impossible. Since neither the vanilla options nor the underlyings have any exposure to the correlation, no portfolio of these instruments can either. 3 It is my understanding that a replicating portfolio for a put involves short selling stock and lending money. You cannot statically replicate an option. So this is not true in general, you'll need to re-balance your replicating portfolio (underlying + cash) dynamically if you want to replicate the option. This will imply sometimes buying stock and borrowing ... 3 Let me know whether this helps, but the author mentions a paper from Fujii and Takahashi; I have been looking for it on the internet and I have found what seems to be a version of it: Collateral Posting and Choice of Collateral Currency. I think they give a relatively transparent explanation$-$in terms of funding costs$-$of why the discount rate for ... 3 Gap risk contracts. These are daily-restriking putspreads that pay & cancel only if the underlying drops more than (say) 20% as measured vs yesterday's closing level. Contracts can range from as short as 6 months to 10 years. Cannot replicate that using Europeans. 3 In a cash settled swaption the payoff is settled using the cash annuity contractually computed using the swap rate. Thus is you work out the replication procedure you will find that CMS replication is exact when you replicate on cash settled swaption (at least when$\delta=0$, that is for CMS with fixing in arrears), because Hagan's "street approximation" is ... 3 The pricing of options is married with the concept of a hedging strategy that replicates the effect of the option. If you can only long or short a stock that will not replicate the greeks, it only creates delta. It is the commitment to the strategy that achieves it. For example if the price goes up and you are committed to buying more to increase your delta ... 3 The key point here is that the portfolio must be self-financing, namely the initial option premium$V_0$should be enough to allow you to hedge it throughout its life. If not, the option price$V_0$is either too low or too high. Because the option is written on the asset$S$, buying or selling$S$is how you neutralize the changes in value of the option: ... 3 There is no terminal$\mathcal{F}_T$mesurable payoff$g$such that$e^{-r(T-t)} E_t[g] = C(S_t, t, T, K)^2$, simply because$E_t[g]$must be a martingale and$e^{r(T-t)} C(S_t, t, T, K)^2$is not. So any deal that has npv$C(S_t, t, T, K)^2$must involve a stream of intermediary payoffs$ h(S_t,t) dt$, which you can solve for by plugging$V(S,t) = C(S, t, ...

3

We should be able to replicate the payoff exactly in each of the two regions $S_{T}\leq k_{1}$ and $S_{T}\geq k_{2}$. From the first, $$a_{0}+a_{1}S_{T}+a_{3}(k_{2}-S_{T}) =S_{T}$$ so, matching coefficients, $a_{0}+a_{3}k_{2}=0$ and $a_{1}-a_{3}=1$. From the second, $$a_{0}+a_{1}S_{T}+a_{2}(S_{T}-k_{1})=P_{0}$$ so, matching coefficients, \$a_{0}-a_{2}k_{1}=P_{...

2

When performing a tracking error optimization, you will obtain the same result by using the tracking error squared, which is just the variance of the relative portfolio weights. This would be just finding the minimum variance portfolio, but with conditions on the weights. For instance, it would be equivalent to instead set up the variance minimization ...

Only top voted, non community-wiki answers of a minimum length are eligible